diff --git a/.github/workflows/verify.yml b/.github/workflows/verify.yml
index e85afe14dd..6197f36b50 100644
--- a/.github/workflows/verify.yml
+++ b/.github/workflows/verify.yml
@@ -16,7 +16,7 @@ jobs:
       fail-fast: false
       matrix:
         python-version: ['3.11', '3.13', '3.14']
-        group: [community, producers, certificates, ternary, research, tests]
+        group: [community, skip, producers, certificates, ternary, research, tests]
     steps:
       - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
         with:
diff --git a/Makefile b/Makefile
index a733e4b348..63262aeb87 100644
--- a/Makefile
+++ b/Makefile
@@ -9,17 +9,22 @@ community-audit-check:
 community-followup-check:
 	python3 scripts/audit_followup_candidate.py --candidate-root . --check docs/research/community-followup-arithmetic.json
 
-.PHONY: verify-community verify-producers verify-certificates verify-ternary verify-tests
+.PHONY: verify-community verify-skip verify-producers verify-certificates verify-ternary verify-tests
 # CI runs these in separate checkouts. Keep local verification sequential:
 # different groups regenerate certificates that another group may read.
 verify:
 	$(MAKE) verify-community
+	$(MAKE) verify-skip
 	$(MAKE) verify-producers
 	$(MAKE) verify-certificates
 	$(MAKE) verify-ternary
 	$(MAKE) verify-research
 	$(MAKE) verify-tests
 
+verify-skip:
+	$(MAKE) skip-strips-verify
+	$(MAKE) skip-clones-verify
+
 verify-community: community-audit-check community-followup-check copied-reversed-producer copied-reversed-check copied-fixed-reversed-producer copied-fixed-reversed-check
 	$(MAKE) copied-fixed-verify
 	$(MAKE) climbed-48-verify
@@ -269,6 +274,26 @@ copied-fixed-check:
 
 copied-fixed-verify: copied-fixed-producer copied-fixed-check
 
+.PHONY: skip-strips-verify skip-strips-producer skip-strips-check
+skip-strips-producer:
+	python3 research/skip-strips/producer.py
+
+skip-strips-check:
+	python3 research/skip-strips/audit.py
+	python3 research/skip-strips/verify.py --output research/skip-strips/certificate.json
+	cd research/skip-strips && python3 -m unittest discover -s ../../tests -p 'test_skip_strips.py' -v
+
+skip-strips-verify: skip-strips-producer skip-strips-check
+
+.PHONY: skip-clones-verify skip-clones-producer skip-clones-check
+skip-clones-producer:
+	python3 research/skip-clones/producer.py
+
+skip-clones-check:
+	python3 research/skip-clones/witness.py --output research/skip-clones/certificate.json
+	python3 -m unittest discover -s tests -p 'test_skip_clones.py' -v
+
+skip-clones-verify: skip-clones-producer skip-clones-check
 .PHONY: climbed-48-verify climbed-48-producer climbed-48-check
 climbed-48-producer:
 	python3 research/climbed-48/producer.py
diff --git a/NOTICE b/NOTICE
index 3769520e39..3466388d0d 100644
--- a/NOTICE
+++ b/NOTICE
@@ -203,3 +203,24 @@ Ryan S / princezuda contributes historical Lean certificate arithmetic, algebrai
 frame/movement/guard lemmas and independent paired-circuit checks (#26). Original
 source notices and historical snapshots are preserved. Neither formal package
 claims a full formal proof of the multiplication algorithm or a new kappa.
+
+Skip-prefix strip producer, 8 October 2026
+By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5)
+assistance. The skip-prefix leave-one-out strip layout in
+research/skip-strips/skip_graph.py is new. Everything else reuses Chafik
+Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification
+code, exact data recovery and support-order sorting, together with:
+- Rohan Arun's PR #44 weighted carrier matching optimizer;
+- RaD (hipotures) PR #41 alternating point order;
+- every predecessor credited above.
+Complete attribution and the conditional proof boundary are recorded in
+research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain
+applicable.
+
+Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones):
+Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain
+source-partition clone operation is adapted from RaD / hipotures, PR #51,
+with original envelopes retained here. The scalar base is Avi Eisenberg's
+PR #53 skip-prefix layout, prepared with Anthropic Claude assistance.
+All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain
+in force. This extension is provided under the repository's Apache-2.0 terms.
diff --git a/README.md b/README.md
index 9e01040cbb..c2cc519bdf 100644
--- a/README.md
+++ b/README.md
@@ -28,6 +28,33 @@ AI-assistance disclosures remain credited in the source notices.
 [Maintainer review](docs/research/community-followup-review.md) ·
 [Integration record](docs/research/community-followup-integration.md)
 
+## Additional conditional circuit result: PR #54
+
+The [paid-clone construction](research/skip-clones/PROOF.md) gives
+**κ = 141532521/3125000000000 = 4.529040672e-5 > 2^-15**, conditional on
+OpenAI's base theorem and the retained analytic and fixed-tape interfaces.
+This branch result is additional to the maintainer-reviewed PR #49 witness
+above; it does not extend that review's scope or claim global optimality.
+
+Starting from Avi Eisenberg's [PR #53 skip-prefix strips](research/skip-strips/PROOF.md),
+606 explicitly paid whole-chain clones preserve the original core/cover
+envelopes. The pinned weighted matchings give R23=32693, R25=43056 and
+W=159592676, with every added operation charged and both complete dirty-basis
+orientations replayed. The conditional saving is 0.686876% above PR #53.
+The source-partition/whole-chain operation is adapted from RaD / hipotures
+(PR #51); this original-envelope specialization is by Chafik Boukhalfa with
+OpenAI Codex assistance. All predecessor notices remain in force.
+
+PR #53 itself reassociates each leave-one-out strip as
+`s_j = A_(j-2) + (v_(j-1) + B_(j+1))`, preserving every scalar term while
+creating nested consumer envelopes. Its matched roles are R23=32946 and
+R25=43409, giving W=160799739 and conditional κ=4498144/10^11.
+
+Run `make verify-skip` for both complete producer replays and their exact
+arithmetic and adversarial checks, or `make skip-clones-verify` for PR #54.
+Local `make verify` and every Python-version CI matrix include these checks
+alongside all reviewed community and historical checks.
+
 ## What changed
 
 The community work combines recursive batching and partial-swap frames with
diff --git a/docs/ci-verification.md b/docs/ci-verification.md
index d35768e22e..639378308f 100644
--- a/docs/ci-verification.md
+++ b/docs/ci-verification.md
@@ -12,17 +12,18 @@ The Python 3.13 log shows these major sequential costs:
 | Prime-field network certificate | 83 seconds |
 | Endpoint-gauge producer regeneration | 54 seconds |
 
-The workflow now distributes each Python version across six independent
+The workflow now distributes each Python version across seven independent
 checkouts:
 
 | Make target | Coverage |
 |---|---|
 | `verify-community` | PR #39 and follow-up arithmetic, reversed/fixed producers, PR #48 and #49 complete replays |
+| `verify-skip` | PR #53 skip-prefix and PR #54 paid-clone complete producers, exact arithmetic, and focused adversarial tests |
 | `verify-producers` | Copied-center, structured-bulk, endpoint-gauge and partial-swap producers and certificates |
 | `verify-certificates` | Remaining exact arithmetic, historical constructions and research audits |
 | `verify-ternary` | Complex-compression and ternary checkpoint audits and patches |
 | `verify-research` | 28 recovered historical audits and their regenerated certificates |
-| `verify-tests` | All 419 current and recovered tests and all 20 historical patch-application checks |
+| `verify-tests` | All current and recovered tests and all 20 historical patch-application checks |
 
 Every group runs on **all three** Python versions (3.11, 3.13 and 3.14).
 Both pinned Lean packages remain separate jobs. No producer replay, negative
@@ -32,13 +33,15 @@ regenerated artifact cannot silently diverge from its recorded certificate.
 
 At the original five-group split, `make -n verify` expands to the same multiset of
 **94 leaf commands**, including identical multiplicities for duplicate checks.
-That split changed scheduling rather than mathematics or certificate inputs.
-The subsequent research reconciliation adds 28 audit commands (122 total leaf
-commands) and additional tests; it does not remove any existing check.
+The original split changed scheduling rather than mathematics or certificate inputs.
+The additional `verify-skip` group preserves those commands and adds the
+PR #53/#54 producer, exact checker and focused adversarial tests. The research
+reconciliation adds all 28 recovered audit commands and additional tests.
+Both additions remain in the combined workflow; no existing check is removed.
 The complete old run remains useful as a comparison; the new workflow must
 pass independently in clean checkouts.
 
-Local `make verify` invokes the six groups sequentially. Do not launch them
+Local `make verify` invokes the seven groups sequentially. Do not launch them
 concurrently in the same worktree: one group can read certificates another
 group regenerates. CI avoids that race by using a separate runner and checkout
 for each group. Directly invoking a group works from a fresh checkout; the
diff --git a/research/skip-clones/PROOF.md b/research/skip-clones/PROOF.md
new file mode 100644
index 0000000000..975bf774f4
--- /dev/null
+++ b/research/skip-clones/PROOF.md
@@ -0,0 +1,270 @@
+# Proof and review boundary for paid clones on PR53 skip-prefix graphs
+
+The pinned finite construction targets
+T(n)=O(n*(log n)^(1-kappa)) with **kappa=141532521/3125000000000** and bit saving
+**a_b=1132311451/25000000000000**. This is a **0.686876%** improvement over PR53. Fresh finite replay has
+passed for both dimensions; the complete verification and source freeze
+remain reproducible through the commands in the README. It inherits the mathematical
+interfaces in [the copied/fixed proof](../copied-fixed/PROOF.md), pinned by
+`SOURCE.json`; those interfaces remain conditional hypotheses, not results
+proved merely by executing this directory's tests.
+
+## The finite construction and its semantic invariant
+
+Take a=23, b=25, m=575. Source lines are indexed by triples, with
+v_h=binomial(h,3), rational address metric G_h=I-J/9, and bit payloads
+over F2. Address-field and payload-field arithmetic remain distinct.
+
+The base is the unchanged PR53 graph at commit
+`3ffd4021995c959ac02d12920e0279ae97dd03c7`, by **Avi Eisenberg with
+Anthropic Claude Opus 5.5 assistance**. `cloned_graph.py` imports its
+pinned `research/skip-strips/skip_graph.py` and then replays the new paid
+clone jobs. Its paired exclusion construction uses a skip-prefix layout
+for every leave-one-out strip vector. If A_i and B_i are the usual prefix
+and suffix sums, the ordinary side output omitting v_j is
+A_(j-1)+B_(j+1). PR53 computes it as
+A_(j-2)+(v_(j-1)+B_(j+1)), with the explicit boundary cases in its generator.
+Associativity and the disjoint index ranges show this is exactly the same
+indexed leave-one-out sum. The total and every omitted-point output retain
+their exact scalar meanings.
+
+The new placement makes successive prefix and side consumers have nested
+covers, and provides further nested suffix continuations. These explain
+why more carrier matches become available; all selected matches are still
+checked individually. At the top level PR53 reverses the value list;
+at deeper levels it uses PR48's support-based order and restores original
+output indices. Its totals retain PR48's rules, and global common-point
+orders retain RaD's alternating construction. This increment adds no new
+schedule bits, point-order choices, source dimensions, or tensor basis.
+The complete inherited skip-prefix argument is retained in
+[PR53's proof](../skip-strips/PROOF.md), with its original attribution.
+
+For a common core C and cover M, retain the original rational envelope
+E(C,M)={x: supp(x) subset M, x_i=t for i in C, sum_i x_i=3t}. It has rank
+|M|-|C| for an addition, with |C| in {1,2}, and lies in a nondegenerate
+common-point hyperplane. Side outputs are orthogonal to their target
+triple line; center envelopes have rank h-1. The inherited envelope
+formula and exact bounded-minor certificate apply to every such core/cover
+configuration; they do not depend on a particular summand order.
+
+## Paid source-partition clones with original envelopes
+
+The extra operation is a specialization of RaD / hipotures's PR51 paid
+whole-chain cloning construction, developed with substantial OpenAI Codex
+assistance. It is applied after the unchanged PR53 skip-prefix construction.
+Every existing gate keeps its literal original envelope E(C,M), and each
+clone receives exactly the original envelope of its parent gate.
+
+Choose an addition x with at least two independent result-carrier chains.
+Let f be the first use of one selected chain. Find two distinct previously
+unused provider gates g1,g2, both earlier than f in the actual rank/node
+chronology, and operands y,z consumed by these providers. Require exact
+disjoint scalar supports supp(y) union supp(z)=supp(x), with empty
+intersection, and require their core intersection and cover union to equal
+x's original core and cover. Require E(g1) and E(g2) to lie in E(x).
+Insert the paid addition x'=y+z immediately before f and redirect every
+use of that complete continuation chain from x to x'. The original x
+retains at least one other chain. No partial continuation chain is moved.
+
+The new sum has exactly x's scalar form and original envelope. Its operand
+frames fit E(x) because they fit their provider frames, which fit E(x).
+All moved consumers previously contained E(x), so they still contain the
+new producer's frame. Providers are earlier than the insertion point;
+source operands retain their original producer identity in this conservative
+specialization. The replay explicitly rejects a provider operand that was
+itself redirected by another selected clone. Thus no circular source
+substitution or unavailable earlier value is assumed. Disjoint job chains,
+distinct unused provider capacities, and exact topological reconstruction
+are checked before every resulting output is re-expanded in global source
+coordinates.
+
+Retain provider g1 at the first actual input use of x', and g2 at its
+second. These are new, distinct uses; each provider capacity was unused.
+Transport every old selected continuation together with its complete
+chain. The two added continuations preserve their actual source operand,
+are causal, and have nested original envelopes. Hence for each clone the
+addition count rises by one, q is unchanged, and the legal retained-link
+count rises by two: R=c+q-mu falls by exactly one. This is a charged new
+scalar gate, not a free identification of two dirty carriers. The final
+selected matching may be recomputed, but every selected use and the full
+physical word must then pass the independent recount and compiler again.
+The complete dirty-basis check supplies the semantic guarantee for arbitrary
+old auxiliary contents; equal scalar supports alone would not supply it.
+
+At h=23 the two rounds add 241+12=253 gates; at h=25 they add
+340+13=353 gates. All **606 new gates** and **1,212 added retained links**
+are accounted for. The final finite counts are:
+
+| h | Additions c | Designated uses q | Matched links mu | Roles R |
+|---|---:|---:|---:|---:|
+| 23 | 40,582 | 5,336 | 13,225 | 32,693 |
+| 25 | 53,828 | 6,925 | 17,697 | 43,056 |
+
+The complete fixed I+J profiles are recomputed on these actual new DAGs and
+selected matching uses. Their unmodified rank masses are 752,951 and
+1,077,600; copied centers reduce them by 506 and 600 respectively. Every
+source triple, center endpoint, common tensor basis, data-pair geometry,
+paid endpoint correction, complex phase, and assembly interface remains
+unchanged. The original-envelope projector and bounded-minor argument below
+therefore apply directly. PR51's enlarged signed positive frames and negative
+basis are not inputs to this construction.
+
+## Actual role allocation, physical word, and finite gates
+
+Let c_h be additions, q_h designated uses, and mu_h selected continuations.
+The number of physical roles is R_h=c_h+q_h-mu_h and retained-center loss is
+ell_h=h(h-1). A continuation is allowed only along the profiler's original
+causal-order and nested-envelope relation, with no repeated donor or
+recipient use. Selected matching edges are finite inputs. Search scores
+and approximate matching objectives are not proof inputs.
+
+For each h, `producer.py` must pass all of the following independent gates:
+
+1. Rebuild the exact PR53 graph and replay the pinned integer clone-job files,
+   verify every scalar output, and export the actual binary DAG.
+2. Expand additions into dense global triple bitsets; check disjointness,
+   exact common core and cover, nested operand envelopes, every designated
+   output, and exactly h retained centers.
+3. Validate the matching binary structure and injectivity in Python,
+   independently recount every selected use, and then pass it to the C++
+   profiler. C++ independently requires every edge to be admissible.
+4. Reconstruct every actual transition's fixed I+J rational profile,
+   compare modular ranks, and check the inherited exact minor bounds.
+5. Compile every physical role trajectory from the exported DAG and selected
+   uses; check terminal uniqueness, source and output coefficients, every
+   frame containment, reverse-complement incidence, and copied rank mass.
+6. Apply the full chronological XOR word to the complete formal basis of
+   inputs, targets, and all dirty auxiliary roles. Both forward and reverse
+   complementary orientations must produce the required shear and restore
+   every auxiliary exactly. These are full finite linear checks, not random
+   dirty-state tests.
+
+Writing files with `--write` does not by itself certify them. After they
+are frozen, ordinary replay compares every generated scalar/profile/timeline
+record with its pin. Verification checks the source manifest before replay
+and never regenerates it. The separate exact witness also checks that
+all counts are nonnegative integers, original and fixed rank sums agree,
+zero-width blocks are absent, and exactly h full center cleanup blocks
+exist before the copied-center modification.
+
+The scalar injection V, mixer L, and scatter J satisfy JLV=I. For an
+arbitrary dirty auxiliary vector z, the word L,J,L^-1,V,L,J,L^-1,V
+scatters JLz+JL(z+Vx)=x and restores z. The independently compiled
+complete-basis test checks this finite word and its dual. The inherited
+ordered-affine physical realization is still a separate mathematical
+interface requiring review.
+
+## Copied centers, fixed profiles, and complete cost
+
+Retain the copied-center replacement in both orientations. For each center,
+replace its original full h-block by a singleton while retaining the
+transformed copy's charged profile. The rank-mass reduction is h-1 per
+center, not twice h-1. Copy, read, erase, and parking passes remain paid.
+The N separate rank-one endpoint corrections remain paid as well. These
+are the same complete role streams used by the inherited fixed-tape
+construction; this increment adds no independent row index or gather.
+
+The explicit envelope projector formula and its rank-two/three/four
+corrections are unchanged. The inherited exact CRT argument bounds every
+minor numerator strictly below the selected modulus product and proves
+denominator invertibility. A nonzero modular minor proves nonzero rational
+rank; a supposedly nonzero minor vanishing modulo all selected primes
+would contradict the strict numerator bound. This certifies ordered zeros
+as well as nonzeros. Only consecutive increasing pivots form a contiguous
+recursive child. No separated pivots are gathered.
+
+The compatible tensor basis, all-weight geometry, physical offsets, and
+data corner proof are inherited unchanged. The primary modular sweep
+certifies 4,073,290 triple pairs; direct rational elimination recovers the
+other ten with all 47 prescribed pivots and 346 required ordered zeros
+per pair. Hence all N=4,073,300 data pairs have the profile
+9 singletons +21+17+481. No pair is dropped. `base.data_corners()` replays
+the exact finite data recovery; source hashes pin its proof interfaces.
+
+Write B_a=v_b R_a, B_b=v_a R_b,
+W=2N+B_a+B_b, and L=v_b ell_a+v_a ell_b=2,226,400. The complete child list is
+
+| Class | Multiplicity | Recursive widths |
+|---|---:|---|
+| a auxiliary exterior | B_a | 23 and 529 |
+| b auxiliary exterior | B_b | 25 and 525 |
+| Every data macro | 2N | 9 singletons, 21, 17, 481 |
+| a local internal | v_b | Complete copied fixed profile at h=23 |
+| b local internal | v_a | Complete copied fixed profile at h=25 |
+| a data growth | 2N | 1 and 21 |
+| b data growth | 2N | 1 and 23 |
+| Paid endpoint correction | N | 1 |
+
+The weighted rank is s=Wm-N+L, so the rank deficit N-L=1,846,900 is
+independent of the graph and clone choices. Every child width is at most 529<m.
+For the pinned candidate R_23=32,693 and R_25=43,056, yielding
+W=159,592,676 and s=91,763,941,800. The independent complete-dirty-basis
+checks cover 36,235 basis vectors and 310,810 elementary operations at h=23,
+and 47,656 basis vectors and 410,536 elementary operations at h=25, in both
+orientations. Every selected carrier and fixed-profile transition is replayed.
+Changing the graph and selected matching changes the role counts and full child histogram;
+neither fewer roles alone nor a favorable floating-point score proves an
+improved exponent.
+
+## Exact objective and conditional assembly
+
+For the complete multiplicities n_t, the bit recurrence's characteristic
+is M(a_b)=sum_t n_t*t/(Wm)*exp(a_b*log(m/t)). The exact acceptance test is
+M(a_b)<1. The integer-outward logarithm series and rational exponential
+bounds are the pinned inherited arithmetic routines. The next bit grid
+point must have a rigorous lower moment greater than one. The pinned PR53
+child list, and any additional contemporaneous comparison list, must
+also have lower moment greater than one at the newly claimed a_b. Thus
+the improvement is a changed finite construction, not a relabeling of
+the previous saving. These comparisons are scoped to the stored networks.
+
+The complex network and its saving a_c=717/10^7 are unchanged, including
+all copied-center inverse phases and paid corrections. The finite bridge
+retains the full scalar charge G, semantic envelope 64(W_c+m_c+G+1)^3,
+completed-child guard, halving degrees, and product-row stock. The balanced
+assembly is the inherited physical layout, not an algebraic deletion of
+prefix work. Its extra top-axis bits, common low-coordinate groups,
+arbitrary-coordinate permutations and inverses, whole active/control/row/
+spectator fields, FFT twiddle order, and bulk Gaussian passes remain paid.
+
+The witness uses the same exact assembly parameter formulas and requires
+all 47 strict inequalities and seven positive cost margins. It checks the
+eventual cutoff conditions, including complete-row padding intervals,
+prime setup and exact recovery, and rejects the next kappa grid point.
+Only under the inherited analytic and fixed finite-alphabet multitape
+interfaces does this yield T(n)=O(n*(log n)^(1-kappa)). Strict margins absorb
+the fixed polylogarithms; the inherited fixed-machine extension handles
+the bounded range below the eventual cutoff.
+
+## Remaining mathematical review and credit
+
+The ordered-affine partial-swap compiler, fixed-tape cost, copied-stream
+parking, complete spectators, common ambient-basis transfer, balanced
+positional layout, and semantic/bulk Gaussian/recovery interfaces remain
+inherited dependencies. Finite replay supplies evidence for their concrete
+inputs but does not prove the all-size machine theorem. No Lean build,
+global search optimum, exact weighted-matching optimum, or measured
+practical multiplication speedup is claimed.
+
+This increment is by **Chafik Boukhalfa with OpenAI Codex assistance**.
+The immediate skip-prefix graph is **Avi Eisenberg with Anthropic Claude
+Opus 5.5 assistance, PR53**, building on **Chafik Boukhalfa with OpenAI
+Codex assistance, PR43/46/48**. PR44's matching discovery and profiler,
+and PR47's summand climbing, are also Rohan Arun with Anthropic Claude
+assistance. Point-order and independent role-compiler work are **RaD /
+hipotures, PR41**. The copied compiler retains **icekylinx** and **Dominik
+Scholz** copyright and Apache-2.0 notices verbatim. The full predecessor
+and analytic attribution chain, including James Chang, Zhihao Chen,
+Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt, OpenAI,
+David Harvey and Joris van der Hoeven, is retained in the linked inherited
+proof and source files. Nothing in this new note supersedes those notices.
+The contemporaneous PR50 benchmark is **Rohan Gupta / gupt1156 with
+Antigravity assistance**, commit `5581d15c`. Its pinned child list is used
+only for exact comparison, not as a construction component.
+
+The paid source-partition / whole-chain clone construction is credited to
+**RaD / hipotures, PR51**, with substantial **OpenAI Codex assistance**,
+PR head `ee552125fb82c1664740fcc6c5eb97cad74d6bbd` and primary research
+commit `6d74d1181a985b5b82debf3eb3e1a7fdbc869671`. This increment supplies
+its conservative original-envelope specialization, integration with the
+PR53 skip-prefix graphs, exact replay, and the resulting new complete child list.
diff --git a/research/skip-clones/README.md b/research/skip-clones/README.md
new file mode 100644
index 0000000000..fedbeefe61
--- /dev/null
+++ b/research/skip-clones/README.md
@@ -0,0 +1,79 @@
+# Conditional saving 4.529040672e-5 from paid clones on skip-prefix graphs
+
+**κ=141532521/3125000000000=4.529040672×10^-5**, conditional on the inherited
+analytic and fixed finite-alphabet multitape interfaces. This is **0.686876%
+above PR53**. The exact bit saving is **1132311451/25000000000000**.
+
+The unchanged base is Avi Eisenberg's PR53 skip-prefix exclusion graph,
+with Anthropic Claude Opus 5.5 assistance. We apply RaD / hipotures's PR51
+paid whole-chain source-partition construction, specialized to the original
+core/cover envelopes, and recompute the selected matching and every fixed
+I+J transition profile. This increment was prepared by Chafik Boukhalfa
+with OpenAI Codex assistance.
+
+| Quantity | PR53 | This construction |
+|---|---:|---:|
+| h=23 roles | 32,946 | 32,693 |
+| h=25 roles | 43,409 | 43,056 |
+| Width W | 160,799,739 | 159,592,676 |
+
+Two rounds add 253 paid gates at h=23 and 353 at h=25. Every gate recreates
+an exact disjoint source partition and moves one complete continuation
+chain. Two unused compatible provider capacities retain its two inputs.
+All **606 gates and 1,212 added retained links are charged**, saving one
+role per clone. Every new gate keeps its parent's original envelope.
+Source dimensions, fixed I+J geometry, exact recovery of all data pairs,
+copied centers, paid endpoint corrections and complex assembly remain.
+
+The [local proof](PROOF.md) gives the scalar and dirty-state argument,
+complete cost, inherited hypotheses, and attribution. Finite checks do not
+replace expert mathematical review of the inherited all-size interfaces.
+No exact matching optimum, global graph optimum or practical speedup is claimed.
+
+## Reproduction
+
+```sh
+make skip-clones-verify
+```
+
+`producer.py` rebuilds PR53 and the pinned clone jobs, verifies every exact
+scalar output, expands all global supports, checks every actual matched
+use before C++ profiling, recomputes the fixed profiles under exact CRT
+bounds, and checks the complete physical word on every dirty basis vector
+in both orientations. Saved finite records must exactly match fresh replay.
+
+`witness.py` verifies source pins, strict integer counts, the complete child
+list and exact moment, all 47 strict assembly conditions and seven margins,
+eventual cutoffs, next-grid rejection, and lower moments excluding prior
+networks including PR53. `make verify` retains historical PR53 and copied/fixed
+checks as well as this new target. Integrated run receipts are recorded
+separately; copied discovery output is not a completed repository replay.
+
+`clone-jobs-h.json` pins each preceding carrier matching, every whole-chain
+edit, and the source/output graph hashes. The strict parser rejects floating
+JSON numbers, duplicate keys, booleans used as indices, and malformed fields.
+Replay rejects incomplete chains, colliding or used provider capacities,
+wrong chronology, changed core/cover envelopes, and false source partitions.
+`parameters.json` stores exact numerator/denominator pairs plus the grid.
+Verification never refreshes the source manifest. Maintainers explicitly
+freeze reviewed files with `pin_sources.py`; certificates and run receipts
+are derived outputs and excluded from that manifest.
+
+## Attribution
+
+**Avi Eisenberg / ikeboy with Anthropic Claude Opus 5.5 assistance, PR53**:
+skip-prefix base graph. **RaD / hipotures with substantial OpenAI Codex
+assistance, PR51**: paid whole-chain and alternative source-partition
+construction. **Chafik Boukhalfa with OpenAI Codex assistance, PR43/46/48**:
+original-envelope composition, independent replay checks and exact data
+recovery; this increment adds the conservative clone specialization and
+certificate integration. **Rohan Arun with Anthropic Claude assistance,
+PR44**: weighted matching discovery and replay profiler; PR37/39/40 and
+other inherited contributions retain their notices. **RaD / hipotures,
+PR41**, supplies alternating points and the independent physical compiler.
+
+Original copyright, Apache-2.0 and assistance disclosures for icekylinx,
+Dominik Scholz, James Chang, Zhihao Chen, Aurel Prosz / Paureel, Swapnil
+Jain, eumemic, Douglas Colkitt, OpenAI, David Harvey, Joris van der Hoeven,
+and all other credited predecessors remain in the pinned sources and
+inherited proofs. The historical directories are unchanged.
diff --git a/research/skip-clones/SOURCE.json b/research/skip-clones/SOURCE.json
new file mode 100644
index 0000000000..d018679d33
--- /dev/null
+++ b/research/skip-clones/SOURCE.json
@@ -0,0 +1,133 @@
+{
+  "base": {
+    "author": "Avi Eisenberg with Anthropic Claude Opus 5.5 assistance",
+    "commit": "3ffd4021995c959ac02d12920e0279ae97dd03c7",
+    "pull_request": 53,
+    "repository": "CrocSwap/integer-mult-bounds"
+  },
+  "comparison_PR50": {
+    "author": "Rohan Gupta (gupt1156) with Antigravity assistance",
+    "commit": "5581d15c",
+    "role": "Contemporaneous benchmark only; no construction component imported"
+  },
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+    "scripts/structured_bulk_assembly.py": "e99467e14211fba699878ed036e1e2362eb8e395dc22f9ecdcecc7fbf2cc736d"
+  },
+  "increment": {
+    "author": "Chafik Boukhalfa with OpenAI Codex assistance",
+    "construction": "Paid original-envelope whole-chain clones on the PR53 skip-prefix graph family; clone idea credited to RaD / hipotures PR51 with OpenAI Codex assistance"
+  },
+  "scope": "Explicit reviewed source/input freeze. Certificates and run receipts are derived outputs."
+}
diff --git a/research/skip-clones/certificate.json b/research/skip-clones/certificate.json
new file mode 100644
index 0000000000..fb4073ed52
--- /dev/null
+++ b/research/skip-clones/certificate.json
@@ -0,0 +1,995 @@
+{
+  "assembly": {
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+    "constraints": {
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+      "K_geometry": "3125188718576207955896183/4166855385241832955896183000000000000",
+      "a_below_b": "660188549/25000000000000",
+      "a_positive": "1132311451/25000000000000",
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+      "alpha_below_one_fourth": "2083050255466417421540967332955896183/8333710770483665911792366000000000000",
+      "alpha_positive": "377437154499056407124167044103817/8333710770483665911792366000000000000",
+      "artificial_boundary": "233345411322189809743511437498867688549/33334843081934663647169464000000000000",
+      "b_below_one_over32": "311783/10000000",
+      "beta_below_one": "19/20",
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+    "research/skip-clones/witness.py": "9e0e972d957560015946b15452bf8d4b75aeecf0ce4a49999f9f913b21abbf89"
+  },
+  "next_bit_grid_lower": "191511211200000724300782590079140790945998721153348928587695561410999994227587028933456906170137047195425072938566695576810922006863068688427/191511211200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "scope": "Pinned paid whole-chain clones on the PR53 skip-prefix graph family; fixed I+J profiles, exact data recovery, copied centers and paid endpoint corrections. Fresh finite replay is required; no exact matching optimality or global search optimality claim.",
+  "status": "Conditional exact arithmetic witness; general transfer proofs are inherited dependencies"
+}
diff --git a/research/skip-clones/clone-jobs-23.json b/research/skip-clones/clone-jobs-23.json
new file mode 100644
index 0000000000..ba33adbcc2
--- /dev/null
+++ b/research/skip-clones/clone-jobs-23.json
@@ -0,0 +1,107550 @@
+{
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diff --git a/research/skip-clones/clone-jobs-25.json b/research/skip-clones/clone-jobs-25.json
new file mode 100644
index 0000000000..3dba51dff2
--- /dev/null
+++ b/research/skip-clones/clone-jobs-25.json
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+      ],
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+          ]
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+        }
+      ]
+    }
+  ]
+}
diff --git a/research/skip-clones/clone_io.py b/research/skip-clones/clone_io.py
new file mode 100644
index 0000000000..ff8a05e205
--- /dev/null
+++ b/research/skip-clones/clone_io.py
@@ -0,0 +1,67 @@
+"""Exact input boundary for the PR53 skip-prefix clones. Apache-2.0.
+
+Prepared by Chafik Boukhalfa with OpenAI Codex assistance.
+"""
+from hashlib import sha256
+from pathlib import Path
+import json
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def loads(text):
+    def reject(value):
+        raise ValueError('Noninteger JSON number forbidden: '+value)
+    def unique(pairs):
+        result = {}
+        for key, value in pairs:
+            require(key not in result, 'Duplicate JSON key: '+key)
+            result[key] = value
+        return result
+    return json.loads(text, parse_float=reject, parse_constant=reject,
+                      object_pairs_hook=unique)
+
+
+def read(path):
+    return loads(Path(path).read_text())
+
+
+def integer(value, name, minimum=0):
+    require(type(value) is int and value >= minimum, 'Invalid integer: '+name)
+    return value
+
+
+def bit_list(value, name):
+    require(type(value) is list, 'Bit schedule must be a JSON array: '+name)
+    require(all(type(bit) is int and bit in (0, 1) for bit in value),
+            'Bits must be integer 0 or 1: '+name)
+    return value[:]
+
+
+REPLAY_INPUTS = ['cloned_graph.py', 'clone-jobs-23.json', 'clone-jobs-25.json',
+                 'clone_io.py', 'producer.py', 'profiles.cpp', 'links-23.uses', 'links-25.uses']
+CERTIFICATE_INPUTS = REPLAY_INPUTS + ['witness.py', 'parameters.json', 'comparison-pr53.json',
+                                    'PROOF.md', 'README.md']
+CERTIFICATE_INPUTS += [f'{kind}-{h}.json' for h in (23, 25)
+                       for kind in ('original', 'profiles', 'scalar', 'timeline')]
+
+
+def check_sources(required=()):
+    source = read(HERE/'SOURCE.json')
+    require(type(source.get('files')) is dict, 'Missing pinned source dictionary')
+    current = [path.name for path in HERE.iterdir()
+               if path.suffix in ('.py', '.cpp', '.uses', '.json', '.md') and path.name not in
+               ('SOURCE.json', 'certificate.json', 'validation.json', 'producer-receipt.json')]
+    required = list(required)+current
+    for name in required:
+        require('research/skip-clones/'+name in source['files'], 'Missing source pin: '+name)
+    for name, digest in source['files'].items():
+        require(sha256((ROOT/name).read_bytes()).hexdigest() == digest,
+                'Pinned inherited source changed: '+name)
+    return source
diff --git a/research/skip-clones/cloned_graph.py b/research/skip-clones/cloned_graph.py
new file mode 100644
index 0000000000..f4fe2d1fb1
--- /dev/null
+++ b/research/skip-clones/cloned_graph.py
@@ -0,0 +1,270 @@
+"""Replay pinned whole-chain clones with original core/cover envelopes.
+
+Adapted from RaD/hipotures PR51 alternative_producer_clones.py, applied to Avi Eisenberg's PR53 skip-prefix graph (with Anthropic Claude assistance), retaining
+Chafik Boukhalfa/OpenAI Codex's original-envelope graph family and finite
+checks. The source-partition cloning idea and full-chain rewrite are credited
+to RaD; this exact original-envelope specialization was prepared with OpenAI
+Codex assistance. Apache-2.0. No search or optimization runs in this module.
+"""
+from hashlib import sha256
+from itertools import combinations
+from pathlib import Path
+import array
+import struct
+import sys
+from clone_io import read, require, integer
+import importlib.util
+
+ROOT = Path(__file__).resolve().parents[2]
+_spec = importlib.util.spec_from_file_location("skip_clones53_base_graph", ROOT/"research/skip-strips/skip_graph.py")
+_prior_graph = importlib.util.module_from_spec(_spec)
+_spec.loader.exec_module(_prior_graph)
+base_graph = _prior_graph.graph
+
+HERE = Path(__file__).resolve().parent
+
+
+def index(value, length, name, minimum=0):
+    integer(value, name, minimum)
+    require(value < length, 'Out-of-range '+name)
+    return value
+
+
+def pair(value, name):
+    require(type(value) is list and len(value) == 2, 'Expected pair: '+name)
+    return value
+
+
+def digest(c):
+    require(sys.byteorder == 'little', 'Little-endian binary interface required')
+    roots = list(c.outputs.values())
+    kinds = [int(len(target) == 1) for _, target in c.outputs]
+    parts = [struct.pack('<4I', c.h, len(c.inputs), len(c.args), len(roots))]
+    parts.append(array.array('I', (y for args in c.args for y in (args or (0, 0)))).tobytes())
+    for code, values in [('Q', c.core), ('Q', c.union), ('I', roots), ('I', kinds)]:
+        parts.append(array.array(code, values).tobytes())
+    parts.append(bytes(int(x in c.active) for x in range(len(c.args))))
+    return sha256(b''.join(parts)).hexdigest()
+
+
+def scalar_supports(c, verify_outputs=True):
+    h, n = c.h, len(c.args)
+    triples = list(combinations(range(h), 3))
+    require(c.inputs == triples, 'Wrong triple source enumeration')
+    require(len(c.core) == len(c.union) == n, 'Wrong envelope array length')
+    supports = [0]*n
+    additions = 0
+    for node in sorted(c.active):
+        index(node, n, 'active node', 1)
+        args = c.args[node]
+        if args is None:
+            index(node, len(triples)+1, 'source node', 1)
+            mask = sum(1 << i for i in triples[node-1])
+            require(c.core[node] == c.union[node] == mask, 'Wrong source envelope')
+            supports[node] = 1 << (node-1)
+        else:
+            require(type(args) in (list, tuple) and len(args) == 2, 'Invalid addition')
+            a, b = args
+            for child in (a, b):
+                index(child, node, 'addition operand', 1)
+                require(child in c.active, 'Inactive addition operand')
+            require(not supports[a]&supports[b], 'Overlapping scalar sources')
+            supports[node] = supports[a]|supports[b]
+            require(c.core[node] == c.core[a]&c.core[b], 'Wrong addition core')
+            require(c.union[node] == c.union[a]|c.union[b], 'Wrong addition cover')
+            require(c.core[node].bit_count() in (1, 2), 'Addition has no valid common core')
+            additions += 1
+    require(additions == c.additions, 'Wrong addition count')
+    require(all(node in c.active for node in range(1, len(triples)+1)), 'Missing source')
+    if verify_outputs:
+        expected_keys = {(common, triple) for triple in triples for common in triple}
+        expected_keys.update((common, (common,)) for common in range(h))
+        require(set(c.outputs) == expected_keys, 'Incomplete terminal output enumeration')
+        for (common, target), node in c.outputs.items():
+            require(node in c.active, 'Inactive terminal output')
+            omitted = set(target)-{common}
+            expected = sum(1 << i for i, triple in enumerate(triples)
+                           if common in triple and not omitted.intersection(triple))
+            require(supports[node] == expected, 'Wrong exact scalar output')
+            require(c.core[node] == 1 << common, 'Wrong output common point')
+            require(c.union[node] == ((1 << h)-1)-sum(1 << x for x in omitted), 'Wrong output cover')
+    return supports
+
+
+class ClonedCircuit:
+    def __init__(self, old, args, core, union, outputs, additions, clone_count):
+        self.h = old.h
+        self.inputs = old.inputs[:]
+        self.args, self.core, self.union = args, core, union
+        self.outputs, self.additions = outputs, additions
+        self.active = set(range(1, len(args)))
+        self.merged = old.merged
+        self.clone_count = clone_count
+
+    def verify(self):
+        scalar_supports(self)
+        return dict(h=self.h, inputs=len(self.inputs), partial_outputs=len(self.outputs),
+                    additions=self.additions, roles=self.additions+len(self.outputs),
+                    merged_additions=self.merged, cloned_additions=self.clone_count,
+                    all_additions_disjoint=True, all_partial_outputs_exact=True,
+                    every_node_has_common_point=True, circuit_sha256=digest(self))
+
+
+def replay_round(c, document):
+    require(type(document) is dict and set(document) == {
+        'source_graph_sha256', 'output_graph_sha256', 'matching_before', 'jobs'}, 'Unexpected clone round fields')
+    for key in ('source_graph_sha256', 'output_graph_sha256'):
+        value = document[key]
+        require(type(value) is str and len(value) == 64 and
+                all(x in '0123456789abcdef' for x in value), 'Invalid graph digest')
+    require(digest(c) == document['source_graph_sha256'], 'Clone source graph differs from pin')
+    scalar = scalar_supports(c)
+    n = len(c.args)
+    ranks = [0]*n
+    for x in c.active:
+        ranks[x] = 1 if c.args[x] is None else c.union[x].bit_count()-c.core[x].bit_count()
+    order = sorted(c.active, key=lambda x: (ranks[x], x))
+    times = {x: i for i, x in enumerate(order)}
+    roots = list(c.outputs.values())
+    users = [[] for _ in range(n)]
+    for x in order:
+        if c.args[x] is not None:
+            for pos, child in enumerate(c.args[x]):
+                users[child].append(2*x+pos)
+    for j, node in enumerate(roots):
+        users[node].append((1 << 31)|j)
+
+    def gate(node):
+        index(node, n, 'gate', 1)
+        require(node in c.active and c.args[node] is not None, 'Inactive or source gate')
+        return node
+
+    def use(e):
+        integer(e, 'use index')
+        require(e < 1 << 32, 'Use index does not fit uint32')
+        if e >> 31:
+            j = index(e & 0x7fffffff, len(roots), 'terminal use')
+            return roots[j], roots[j], len(order)+j, n+j
+        owner = gate(e//2)
+        return c.args[owner][e%2], owner, times[owner], owner
+
+    def nested(a, b):
+        return not (c.core[b]&~c.core[a] or c.union[a]&~c.union[b])
+
+    incoming, donors, next_use = set(), set(), {}
+    links = document['matching_before']
+    require(type(links) is list, 'Prior matching must be an array')
+    for entry in links:
+        donor, e = pair(entry, 'prior matching edge')
+        gate(donor)
+        value, target, _, event = use(e)
+        require(donor not in donors and e not in incoming, 'Prior matching collision')
+        require(value in c.args[donor], 'Prior carrier does not preserve actual operand')
+        require((ranks[donor], donor) < (ranks[target], event), 'Noncausal prior carrier')
+        require(nested(donor, target), 'Nonnested prior carrier')
+        pos = c.args[donor].index(value)
+        next_use[2*donor+pos] = e
+        donors.add(donor)
+        incoming.add(e)
+
+    jobs = document['jobs']
+    require(type(jobs) is list and jobs, 'Clone jobs must be a nonempty array')
+    capacities, parents, formal, move, before, terminal = set(), set(), set(), {}, {}, []
+    for i, job in enumerate(jobs):
+        require(type(job) is dict and set(job) == {
+            'parent', 'providers', 'source_children', 'first', 'chain'}, 'Unexpected clone job fields')
+        parent = gate(job['parent'])
+        providers = pair(job['providers'], 'providers')
+        children = pair(job['source_children'], 'source children')
+        first_value, _, first_time, _ = use(job['first'])
+        require(first_value == parent and job['first'] not in incoming, 'Clone must start an actual parent chain')
+        require(sum(e not in incoming for e in users[parent]) >= 2, 'Parent would lose its last chain')
+        require(parent not in parents and parent not in formal, 'Conflicting clone parent')
+        require(not set(children)&parents, 'Clone partition depends on another cloned parent')
+        require(providers[0] != providers[1], 'Providers must be distinct')
+        for provider, child in zip(providers, children):
+            gate(provider)
+            index(child, n, 'source child', 1)
+            require(child in c.active and child in c.args[provider], 'Provider lacks named source child')
+            require(provider not in donors and provider not in capacities, 'Provider capacity already used')
+            require(times[provider] < first_time, 'Provider is not earlier than clone placement')
+            require(nested(provider, parent), 'Provider does not fit original parent envelope')
+        left, right = children
+        require(not scalar[left]&scalar[right] and scalar[left]|scalar[right] == scalar[parent],
+                'Clone partition does not equal parent with disjoint sources')
+        require(c.core[left]&c.core[right] == c.core[parent] and
+                c.union[left]|c.union[right] == c.union[parent], 'Clone original envelope differs from parent')
+        chain = job['chain']
+        require(type(chain) is list and chain, 'Clone chain must be a nonempty array')
+        expected, seen, e = [], set(), job['first']
+        while True:
+            require(e not in seen, 'Prior chain contains a cycle')
+            seen.add(e)
+            require(use(e)[0] == parent, 'Chain does not preserve parent value')
+            expected.append(e)
+            if e not in next_use:
+                break
+            e = next_use[e]
+        require(all(type(e) is int for e in chain) and chain == expected, 'Clone must replace one whole continuation chain')
+        for e in chain:
+            require(e not in move, 'Clone chains collide')
+            move[e] = i
+        if job['first'] >> 31:
+            terminal.append(i)
+        else:
+            before.setdefault(job['first']//2, []).append(i)
+        capacities.update(providers)
+        parents.add(parent)
+        formal.update(children)
+
+    args, core, union, mapping, clones = [None], [0], [0], {}, {}
+
+    def clone(i):
+        job = jobs[i]
+        parent, node, children = job['parent'], len(args), []
+        for provider, old_child in zip(job['providers'], job['source_children']):
+            edge = 2*provider+c.args[provider].index(old_child)
+            require(edge not in move, 'Conservative clone source was itself replaced')
+            require(old_child in mapping, 'Clone source is not yet available')
+            children.append(mapping[old_child])
+        require(all(child < node for child in children), 'Clone is not topological')
+        clones[i] = node
+        args.append(tuple(children))
+        core.append(c.core[parent])
+        union.append(c.union[parent])
+
+    for old in order:
+        for i in before.get(old, []):
+            clone(i)
+        mapping[old] = len(args)
+        if c.args[old] is None:
+            require(mapping[old] == old <= len(c.inputs), 'Source IDs changed')
+            args.append(None)
+        else:
+            args.append(tuple(clones[move[e]] if e in move else mapping[c.args[old][e%2]]
+                              for e in (2*old, 2*old+1)))
+        core.append(c.core[old])
+        union.append(c.union[old])
+    for i in terminal:
+        clone(i)
+    outputs = {}
+    for j, (target, old) in enumerate(c.outputs.items()):
+        e = (1 << 31)|j
+        outputs[target] = clones[move[e]] if e in move else mapping[old]
+    result = ClonedCircuit(c, args, core, union, outputs, c.additions+len(jobs),
+                           getattr(c, 'clone_count', 0)+len(jobs))
+    scalar_supports(result)
+    require(digest(result) == document['output_graph_sha256'], 'Clone output graph differs from pin')
+    return result
+
+
+def graph(h, document=None):
+    require(type(h) is int and h in (23, 25), 'Uncertified clone dimension')
+    document = read(HERE/f'clone-jobs-{h}.json') if document is None else document
+    require(type(document) is dict and set(document) == {'h', 'rounds'}, 'Unexpected clone document fields')
+    require(type(document['h']) is int and document['h'] == h, 'Wrong clone dimension')
+    require(type(document['rounds']) is list and document['rounds'], 'Clone rounds must be a nonempty array')
+    c = base_graph(h)
+    for stage in document['rounds']:
+        c = replay_round(c, stage)
+    return c
diff --git a/research/skip-clones/comparison-pr49.json b/research/skip-clones/comparison-pr49.json
new file mode 100644
index 0000000000..7d4c1f1d4a
--- /dev/null
+++ b/research/skip-clones/comparison-pr49.json
@@ -0,0 +1,41 @@
+{
+  "PR": 49,
+  "W": 177284805,
+  "author": "Rohan Arun with Anthropic Claude assistance",
+  "child_multiplicities": {
+    "1": 1470464589,
+    "10": 17797998,
+    "11": 16931473,
+    "12": 11717465,
+    "13": 10011762,
+    "14": 7294220,
+    "15": 5890208,
+    "16": 3710153,
+    "17": 10812346,
+    "18": 1603698,
+    "19": 734413,
+    "2": 109578854,
+    "20": 516833,
+    "21": 16346100,
+    "22": 116886,
+    "23": 91869475,
+    "25": 85459605,
+    "3": 73855438,
+    "4": 47248693,
+    "481": 8146600,
+    "5": 42058490,
+    "525": 85459605,
+    "529": 83678600,
+    "6": 34479921,
+    "7": 35357164,
+    "8": 24947410,
+    "9": 22730739
+  },
+  "commit": "f95d2910e027495983b53cae1693cf535abf2569",
+  "kappa": [
+    4123863984,
+    100000000000000
+  ],
+  "m": 575,
+  "total_rank": 101936915975
+}
diff --git a/research/skip-clones/comparison-pr50.json b/research/skip-clones/comparison-pr50.json
new file mode 100644
index 0000000000..e06f2b360d
--- /dev/null
+++ b/research/skip-clones/comparison-pr50.json
@@ -0,0 +1,42 @@
+{
+  "PR": 50,
+  "W": 176993671,
+  "author": "Rohan Gupta (gupt1156) with Antigravity assistance",
+  "child_multiplicities": {
+    "1": 1471256203,
+    "10": 17993843,
+    "11": 16697218,
+    "12": 11829383,
+    "13": 9939289,
+    "14": 7326006,
+    "15": 5837768,
+    "16": 3671191,
+    "17": 10814324,
+    "18": 1588863,
+    "19": 660215,
+    "2": 109376224,
+    "20": 447626,
+    "21": 16347871,
+    "22": 100947,
+    "23": 91673975,
+    "25": 85363971,
+    "3": 73570261,
+    "4": 47358104,
+    "481": 8146600,
+    "5": 41854733,
+    "525": 85363971,
+    "529": 83483100,
+    "6": 34655227,
+    "7": 35121874,
+    "8": 25106892,
+    "9": 22509134
+  },
+  "commit": "5581d15c",
+  "kappa": [
+    516177337,
+    12500000000000
+  ],
+  "m": 575,
+  "source_certificate_sha256": "7e5ad25a7f3e5e71a046f001e73349b7cfb684f01ae497ced38f9816e5c604f5",
+  "total_rank": 101769513925
+}
diff --git a/research/skip-clones/comparison-pr51.json b/research/skip-clones/comparison-pr51.json
new file mode 100644
index 0000000000..6e8aa3dfaf
--- /dev/null
+++ b/research/skip-clones/comparison-pr51.json
@@ -0,0 +1,41 @@
+{
+  "PR": 51,
+  "W": 175503731,
+  "author": "RaD / hipotures with OpenAI Codex assistance",
+  "child_multiplicities": {
+    "1": 1517393402,
+    "10": 16984843,
+    "11": 17691781,
+    "12": 11801129,
+    "13": 10960915,
+    "14": 7807390,
+    "15": 6896072,
+    "16": 4447532,
+    "17": 11165105,
+    "18": 1988166,
+    "19": 1049560,
+    "2": 68144546,
+    "20": 606464,
+    "21": 16443505,
+    "23": 91494575,
+    "25": 84053431,
+    "3": 71742789,
+    "4": 39518446,
+    "481": 8146600,
+    "5": 41151236,
+    "525": 84053431,
+    "529": 83303700,
+    "6": 30348323,
+    "7": 35808325,
+    "8": 22889585,
+    "9": 23201302
+  },
+  "commit": "ee552125fb82c1664740fcc6c5eb97cad74d6bbd",
+  "kappa": [
+    4171211781,
+    100000000000000
+  ],
+  "m": 575,
+  "source_certificate_sha256": "091716ca0c7be460a859e8ece04bbd2461b6990c14888bd7ed10aea12c708be3",
+  "total_rank": 100912798425
+}
diff --git a/research/skip-clones/comparison-pr53.json b/research/skip-clones/comparison-pr53.json
new file mode 100644
index 0000000000..8a22de1069
--- /dev/null
+++ b/research/skip-clones/comparison-pr53.json
@@ -0,0 +1,42 @@
+{
+  "PR": 53,
+  "W": 160799739,
+  "author": "Avi Eisenberg (ikeboy) with Anthropic Claude Opus 5.5 assistance",
+  "child_multiplicities": {
+    "1": 1452430151,
+    "10": 14613119,
+    "11": 14333669,
+    "12": 9755496,
+    "13": 8143127,
+    "14": 6009417,
+    "15": 4660559,
+    "16": 3021073,
+    "17": 10157467,
+    "18": 1363624,
+    "19": 518903,
+    "2": 105266377,
+    "20": 431135,
+    "21": 16346100,
+    "22": 97405,
+    "23": 83966675,
+    "25": 76877339,
+    "3": 68998298,
+    "4": 42070289,
+    "481": 8146600,
+    "5": 36521953,
+    "525": 76877339,
+    "529": 75775800,
+    "6": 29567558,
+    "7": 30289390,
+    "8": 21172811,
+    "9": 18937487
+  },
+  "commit": "3ffd4021995c959ac02d12920e0279ae97dd03c7",
+  "kappa": [
+    140567,
+    3125000000
+  ],
+  "m": 575,
+  "source_certificate_sha256": "126e1f37b881334b68830c2943797eb7cd490df9509f4d8d162128a5f9993503",
+  "total_rank": 92458003025
+}
diff --git a/research/skip-clones/links-23.uses b/research/skip-clones/links-23.uses
new file mode 100644
index 0000000000..00ce373a99
Binary files /dev/null and b/research/skip-clones/links-23.uses differ
diff --git a/research/skip-clones/links-25.uses b/research/skip-clones/links-25.uses
new file mode 100644
index 0000000000..2e3f92258f
Binary files /dev/null and b/research/skip-clones/links-25.uses differ
diff --git a/research/skip-clones/original-23.json b/research/skip-clones/original-23.json
new file mode 100644
index 0000000000..7df8199d2f
--- /dev/null
+++ b/research/skip-clones/original-23.json
@@ -0,0 +1,38 @@
+{
+  "R": 32693,
+  "baseline_R": 45918,
+  "c": 40582,
+  "h": 23,
+  "histogram": [
+    5727,
+    41847,
+    36822,
+    6030,
+    16110,
+    7223,
+    4218,
+    1840,
+    3298,
+    833,
+    2865,
+    250,
+    2456,
+    289,
+    3516,
+    840,
+    2234,
+    810,
+    8594,
+    5873,
+    0,
+    165,
+    23,
+    23
+  ],
+  "loss": 506,
+  "matched": 13225,
+  "orientation_changes": 9168,
+  "q": 5336,
+  "rank_sum": 752951,
+  "v": 1771
+}
diff --git a/research/skip-clones/original-25.json b/research/skip-clones/original-25.json
new file mode 100644
index 0000000000..ca29fdfc17
--- /dev/null
+++ b/research/skip-clones/original-25.json
@@ -0,0 +1,40 @@
+{
+  "R": 43056,
+  "baseline_R": 60753,
+  "c": 53828,
+  "h": 25,
+  "histogram": [
+    7453,
+    54426,
+    48049,
+    7718,
+    22018,
+    9080,
+    4055,
+    3022,
+    5070,
+    893,
+    2724,
+    468,
+    3707,
+    443,
+    2751,
+    168,
+    4406,
+    1536,
+    2945,
+    618,
+    11276,
+    7596,
+    0,
+    221,
+    25,
+    25
+  ],
+  "loss": 600,
+  "matched": 17697,
+  "orientation_changes": 12707,
+  "q": 6925,
+  "rank_sum": 1077600,
+  "v": 2300
+}
diff --git a/research/skip-clones/parameters.json b/research/skip-clones/parameters.json
new file mode 100644
index 0000000000..34325cb9cb
--- /dev/null
+++ b/research/skip-clones/parameters.json
@@ -0,0 +1,11 @@
+{
+  "bit_saving": [
+    1132311451,
+    25000000000000
+  ],
+  "kappa": [
+    141532521,
+    3125000000000
+  ],
+  "grid_denominator": 100000000000000
+}
diff --git a/research/skip-clones/pin_sources.py b/research/skip-clones/pin_sources.py
new file mode 100644
index 0000000000..ebf3ef5165
--- /dev/null
+++ b/research/skip-clones/pin_sources.py
@@ -0,0 +1,44 @@
+#!/usr/bin/env python3
+"""Explicitly freeze reviewed sources and available finite inputs; never run by verification.
+
+Prepared by Chafik Boukhalfa with OpenAI Codex assistance. Apache-2.0.
+"""
+from hashlib import sha256
+from pathlib import Path
+import json
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+
+
+def run():
+    names = set()
+    for directory in ('copied-fixed', 'skip-strips'):
+        path = ROOT/'research'/directory
+        names.update(json.loads((path/'SOURCE.json').read_text())['files'])
+        names.add(str((path/'SOURCE.json').relative_to(ROOT)))
+        names.update(str(p.relative_to(ROOT)) for p in path.glob('*.py'))
+    # Explicit native graph dependency closure, including the binary format.
+    names.update(str(p.relative_to(ROOT)) for p in (ROOT/'scripts/partial_swap').glob('*.py'))
+    names.update(str(p.relative_to(ROOT)) for p in (ROOT/'scripts/partial_swap').glob('*.hpp'))
+    names.add('scripts/exclusion_circuit.py')
+    for path in HERE.iterdir():
+        if path.suffix in ('.py', '.cpp', '.uses', '.md', '.json') and path.name not in (
+                'SOURCE.json', 'certificate.json', 'validation.json', 'producer-receipt.json'):
+            names.add(str(path.relative_to(ROOT)))
+    source = dict(
+        base=dict(repository='CrocSwap/integer-mult-bounds', pull_request=53,
+                  commit='3ffd4021995c959ac02d12920e0279ae97dd03c7',
+                  author='Avi Eisenberg with Anthropic Claude Opus 5.5 assistance'),
+        increment=dict(author='Chafik Boukhalfa with OpenAI Codex assistance',
+                       construction='Paid original-envelope whole-chain clones on the PR53 skip-prefix graph family; clone idea credited to RaD / hipotures PR51 with OpenAI Codex assistance'),
+        comparison_PR50=dict(author='Rohan Gupta (gupt1156) with Antigravity assistance',
+                             commit='5581d15c', role='Contemporaneous benchmark only; no construction component imported'),
+        scope='Explicit reviewed source/input freeze. Certificates and run receipts are derived outputs.',
+        files={name: sha256((ROOT/name).read_bytes()).hexdigest() for name in sorted(names)})
+    (HERE/'SOURCE.json').write_text(json.dumps(source, indent=2, sort_keys=True)+'\n')
+    return len(names)
+
+
+if __name__ == '__main__':
+    print('Pinned', run(), 'sources and finite inputs')
diff --git a/research/skip-clones/producer-receipt.json b/research/skip-clones/producer-receipt.json
new file mode 100644
index 0000000000..5aca36e75f
--- /dev/null
+++ b/research/skip-clones/producer-receipt.json
@@ -0,0 +1,322 @@
+{
+  "23": {
+    "dag_sha256": "18d85ae0400f822cd365cf0a7f5fd3f5adda28d6825b449a896b45e0db249204",
+    "dense_audit": {
+      "envelope_inclusions": 81164,
+      "exact_global_additions": 40582,
+      "exact_global_outputs": 5336,
+      "retained_centers": 23,
+      "source_lines": 1771
+    },
+    "fixed_profile": {
+      "R": 32693,
+      "blocks": [
+        0,
+        291051,
+        22021,
+        15300,
+        9379,
+        7607,
+        6019,
+        6441,
+        4517,
+        3813,
+        2862,
+        2766,
+        1905,
+        1493,
+        1038,
+        750,
+        495,
+        276,
+        174,
+        0,
+        45,
+        23,
+        0,
+        23
+      ],
+      "crt_disagreements": 0,
+      "crt_matrices": 5631,
+      "distinct_matrices": 61279,
+      "field_prime": 2305843009213693951,
+      "frames": 36628,
+      "h": 23,
+      "loss": 506,
+      "rank_sum": 752951,
+      "v": 1771
+    },
+    "full_timeline": {
+      "additions": 40582,
+      "center_rank": 22,
+      "center_terminal_count": 23,
+      "complete_dirty_basis": {
+        "all_dirty_restore": true,
+        "auxiliary_basis_vectors": 32693,
+        "elementary_word_length": 310810,
+        "inputs": 1771,
+        "orientations": [
+          "forward",
+          "reverse-complement"
+        ],
+        "targets": 1771,
+        "total_basis_vectors": 36235
+      },
+      "copied_histogram": [
+        11063,
+        41870,
+        36822,
+        6030,
+        16110,
+        7223,
+        4218,
+        1840,
+        3298,
+        833,
+        2865,
+        250,
+        2456,
+        289,
+        3516,
+        840,
+        2234,
+        810,
+        8594,
+        5873,
+        0,
+        165,
+        23,
+        0
+      ],
+      "copied_rank_sum": 752445,
+      "designated_outputs": 5336,
+      "full_output_coefficients": 3136441,
+      "h": 23,
+      "native_zero_rank_bookkeeping": 5727,
+      "physical_frame_transitions": 86500,
+      "rational_frame_classes": 36626,
+      "retained_links": 13225,
+      "reverse_complement_frame_incidences": 113857,
+      "roles": 32693,
+      "selected_links_sha256": "6f0aa3c2a3da689b22862ce626b0225d07258b466f2a64682be7a2e02a60aa51",
+      "status": "independent scalar, rational frames, physical compiler and rank timeline PASS",
+      "v": 1771,
+      "zero_rank_scope": "Explicit output incidences include no-operation frame checks omitted from native histogram bookkeeping; only positive ranks enter the recurrence"
+    },
+    "independent_recount": {
+      "R": 32693,
+      "baseline_R": 45918,
+      "c": 40582,
+      "h": 23,
+      "histogram": [
+        5727,
+        41847,
+        36822,
+        6030,
+        16110,
+        7223,
+        4218,
+        1840,
+        3298,
+        833,
+        2865,
+        250,
+        2456,
+        289,
+        3516,
+        840,
+        2234,
+        810,
+        8594,
+        5873,
+        0,
+        165,
+        23,
+        23
+      ],
+      "loss": 506,
+      "matched": 13225,
+      "orientation_changes": 9168,
+      "q": 5336,
+      "rank_sum": 752951,
+      "v": 1771
+    },
+    "links_sha256": "b9d69517798c792466a68d09a0d69f21fa68eef15cb0436d6bc2fd71149a65a6",
+    "scalar": {
+      "additions": 40582,
+      "all_additions_disjoint": true,
+      "all_partial_outputs_exact": true,
+      "circuit_sha256": "18d85ae0400f822cd365cf0a7f5fd3f5adda28d6825b449a896b45e0db249204",
+      "cloned_additions": 253,
+      "every_node_has_common_point": true,
+      "h": 23,
+      "inputs": 1771,
+      "merged_additions": 5379,
+      "partial_outputs": 5336,
+      "roles": 45918
+    }
+  },
+  "25": {
+    "dag_sha256": "571d1da11d085e98da2b0ad7247d6eda9f623ca7bb0b015aaf8734e2ebb48858",
+    "dense_audit": {
+      "envelope_inclusions": 107656,
+      "exact_global_additions": 53828,
+      "exact_global_outputs": 6925,
+      "retained_centers": 25,
+      "source_lines": 2300
+    },
+    "fixed_profile": {
+      "R": 43056,
+      "blocks": [
+        0,
+        387587,
+        30674,
+        18228,
+        11305,
+        10562,
+        8695,
+        8645,
+        6001,
+        5662,
+        4434,
+        4390,
+        2922,
+        2589,
+        1976,
+        1615,
+        1054,
+        752,
+        544,
+        290,
+        185,
+        0,
+        55,
+        25,
+        0,
+        25
+      ],
+      "crt_disagreements": 0,
+      "crt_matrices": 7434,
+      "distinct_matrices": 82331,
+      "field_prime": 2305843009213693951,
+      "frames": 48677,
+      "h": 25,
+      "loss": 600,
+      "rank_sum": 1077600,
+      "v": 2300
+    },
+    "full_timeline": {
+      "additions": 53828,
+      "center_rank": 24,
+      "center_terminal_count": 25,
+      "complete_dirty_basis": {
+        "all_dirty_restore": true,
+        "auxiliary_basis_vectors": 43056,
+        "elementary_word_length": 410536,
+        "inputs": 2300,
+        "orientations": [
+          "forward",
+          "reverse-complement"
+        ],
+        "targets": 2300,
+        "total_basis_vectors": 47656
+      },
+      "copied_histogram": [
+        14378,
+        54451,
+        48049,
+        7718,
+        22018,
+        9080,
+        4055,
+        3022,
+        5070,
+        893,
+        2724,
+        468,
+        3707,
+        443,
+        2751,
+        168,
+        4406,
+        1536,
+        2945,
+        618,
+        11276,
+        7596,
+        0,
+        221,
+        25,
+        0
+      ],
+      "copied_rank_sum": 1077000,
+      "designated_outputs": 6925,
+      "full_output_coefficients": 5290000,
+      "h": 25,
+      "native_zero_rank_bookkeeping": 7453,
+      "physical_frame_transitions": 114581,
+      "rational_frame_classes": 48675,
+      "retained_links": 17697,
+      "reverse_complement_frame_incidences": 150712,
+      "roles": 43056,
+      "selected_links_sha256": "110ddaf97f2d244afd6d80a3f361b1c65dba2cdc36455c2379cf83f9e9ec7d8a",
+      "status": "independent scalar, rational frames, physical compiler and rank timeline PASS",
+      "v": 2300,
+      "zero_rank_scope": "Explicit output incidences include no-operation frame checks omitted from native histogram bookkeeping; only positive ranks enter the recurrence"
+    },
+    "independent_recount": {
+      "R": 43056,
+      "baseline_R": 60753,
+      "c": 53828,
+      "h": 25,
+      "histogram": [
+        7453,
+        54426,
+        48049,
+        7718,
+        22018,
+        9080,
+        4055,
+        3022,
+        5070,
+        893,
+        2724,
+        468,
+        3707,
+        443,
+        2751,
+        168,
+        4406,
+        1536,
+        2945,
+        618,
+        11276,
+        7596,
+        0,
+        221,
+        25,
+        25
+      ],
+      "loss": 600,
+      "matched": 17697,
+      "orientation_changes": 12707,
+      "q": 6925,
+      "rank_sum": 1077600,
+      "v": 2300
+    },
+    "links_sha256": "31e37bbfd8459f5094bf40368009f4e39c72c5820ca6f31298e7c648038bac91",
+    "scalar": {
+      "additions": 53828,
+      "all_additions_disjoint": true,
+      "all_partial_outputs_exact": true,
+      "circuit_sha256": "571d1da11d085e98da2b0ad7247d6eda9f623ca7bb0b015aaf8734e2ebb48858",
+      "cloned_additions": 353,
+      "every_node_has_common_point": true,
+      "h": 25,
+      "inputs": 2300,
+      "merged_additions": 7008,
+      "partial_outputs": 6925,
+      "roles": 60753
+    }
+  }
+}
diff --git a/research/skip-clones/producer.py b/research/skip-clones/producer.py
new file mode 100644
index 0000000000..f77953730a
--- /dev/null
+++ b/research/skip-clones/producer.py
@@ -0,0 +1,124 @@
+#!/usr/bin/env python3
+"""Full replay of the skip-prefix cloned producers with PR #43's independent checks.
+
+For h in (23,25): rebuild the scalar DAG (cloned_graph.py), verify the scalar
+identity, run PR #43's dense global-bitset audit, profile the pinned carrier
+matching with profiles.cpp (LINKS_IN mode, admissibility asserted per edge),
+recount every matched use independently (PR #43 producer.recount), and replay
+the complete physical timeline and dirty basis in both orientations
+(PR #43 original_timeline.check). All outputs must equal the pinned JSON.
+
+The independent checkers are imported unchanged from research/copied-fixed
+(Chafik Boukhalfa, PR #43; RaD/hipotures PR #41 timeline compiler).
+Adapted from Rohan Arun with Anthropic Claude assistance (PR49).
+Prepared by Chafik Boukhalfa with OpenAI Codex assistance. Apache-2.0.
+"""
+from hashlib import sha256
+from pathlib import Path
+import argparse
+import gc
+import importlib.util
+import json
+import os
+import shlex
+import struct
+import subprocess
+import sys
+import tempfile
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+PR43 = ROOT/'research/copied-fixed'
+sys.path.insert(0, str(ROOT/'scripts'))
+sys.path.insert(0, str(PR43))
+_spec = importlib.util.spec_from_file_location('skip_clones53_pr43_producer', PR43/'producer.py')
+pr43_producer = importlib.util.module_from_spec(_spec)
+_spec.loader.exec_module(pr43_producer)
+import original_timeline as timeline       # noqa: E402  (PR #43 / RaD timeline compiler)
+from verify import exactness, require     # noqa: E402  (PR #43 CRT minor bounds)
+from partial_swap.graph import export     # noqa: E402
+sys.path.insert(0, str(HERE))
+from cloned_graph import graph           # noqa: E402
+
+
+from clone_io import read, check_sources, REPLAY_INPUTS  # noqa: E402
+
+
+def links(path):
+    raw = Path(path).read_bytes()
+    require(len(raw) >= 8, 'Links file header')
+    n, count = struct.unpack_from('<2I', raw)
+    require(len(raw) == 8+8*count, 'Links file format')
+    edges = [list(struct.unpack_from('<2I', raw, 8+8*i)) for i in range(count)]
+    require(all(0 < donor < n for donor, _ in edges), 'Invalid donor node')
+    require(len({donor for donor, _ in edges}) == count, 'Duplicate donor')
+    require(len({use for _, use in edges}) == count, 'Duplicate recipient use')
+    return n, edges
+
+
+def run(work, write=False):
+    require(not sys.flags.optimize, 'Assertions must remain enabled')
+    check_sources(REPLAY_INPUTS)
+    pr43_producer.check_sources()
+    exe = work/'profiles'
+    subprocess.run([*shlex.split(os.environ.get('CXX', 'c++')), '-O3', '-std=c++17',
+                    '-include', 'algorithm', '-I', str(ROOT/'scripts/partial_swap'),
+                    str(HERE/'profiles.cpp'), '-o', str(exe)], check=True)
+    receipt = {}
+    for h in (23, 25):
+        print('Rebuilding cloned PR53 producer h='+str(h), flush=True)
+        c = graph(h)
+        scalar = c.verify()
+        degree, ranks, outputs, audit = pr43_producer.dense_audit(c)
+        dag, out_links = work/f'h{h}.bin', work/f'h{h}.uses'
+        export(c, dag)
+        pinned = HERE/f'links-{h}.uses'
+        n, selected = links(pinned)
+        require(n == len(c.args), 'Matching DAG node count differs')
+        independent = pr43_producer.recount(c, degree, ranks, outputs, pinned)
+        env = dict(os.environ, LINKS_IN=str(pinned))
+        original = json.loads(subprocess.check_output([str(exe), str(dag), str(out_links)],
+                                                      text=True, env=env))
+        exported_n, exported = links(out_links)
+        require(n == exported_n == len(c.args), 'Matching DAG node count differs')
+        require(sorted(map(tuple, selected)) == sorted(map(tuple, exported)), 'Profiler matching differs from pin')
+        require(original == independent, 'Independent physical recount failed')
+        fixed = read(str(dag)+'.round3_rankone_certified_profiles.json')
+        require(fixed['crt_disagreements'] == 0, 'CRT disagreement')
+        exactness(h)
+        row = dict(original, dag_path=str(dag))
+        timeline.basis.cache_clear(); timeline.contained.cache_clear()
+        compiled = timeline.check({'producer': row, 'selected_links': {'links': selected}}, True)
+        compiled = {k: v for k, v in compiled.items() if k not in ('containment_cache', 'dag_sha256')}
+        timeline.basis.cache_clear(); timeline.contained.cache_clear()
+        outputs_now = {f'scalar-{h}.json': scalar, f'original-{h}.json': original,
+                       f'profiles-{h}.json': fixed, f'timeline-{h}.json': compiled}
+        for name, value in outputs_now.items():
+            if write:
+                (HERE/name).write_text(json.dumps(value, indent=2, sort_keys=True)+'\n')
+            require(json.loads(json.dumps(value)) == read(HERE/name), 'Replay mismatch: '+name)
+        print('PASS h=%d scalar, dense audit, pinned-links profile, independent recount, '
+              'full timeline and dirty basis (both orientations)' % h, flush=True)
+        receipt[str(h)] = dict(scalar=scalar, dense_audit=audit, independent_recount=independent,
+                               fixed_profile=fixed, full_timeline=compiled,
+                               links_sha256=sha256(pinned.read_bytes()).hexdigest(),
+                               dag_sha256=sha256(dag.read_bytes()).hexdigest())
+        del c
+        gc.collect()
+    return receipt
+
+
+if __name__ == '__main__':
+    parser = argparse.ArgumentParser()
+    parser.add_argument('--work-dir', type=Path)
+    parser.add_argument('--output', type=Path)
+    parser.add_argument('--write', action='store_true', help='Regenerate pinned JSON (maintainers only)')
+    args = parser.parse_args()
+    if args.work_dir:
+        args.work_dir.mkdir(parents=True, exist_ok=True)
+        result = run(args.work_dir, args.write)
+    else:
+        with tempfile.TemporaryDirectory(prefix='skip-clone-producers-') as directory:
+            result = run(Path(directory), args.write)
+    if args.output:
+        args.output.write_text(json.dumps(result, indent=2, sort_keys=True)+'\n')
diff --git a/research/skip-clones/profiles-23.json b/research/skip-clones/profiles-23.json
new file mode 100644
index 0000000000..fcd358ddf6
--- /dev/null
+++ b/research/skip-clones/profiles-23.json
@@ -0,0 +1,38 @@
+{
+  "R": 32693,
+  "blocks": [
+    0,
+    291051,
+    22021,
+    15300,
+    9379,
+    7607,
+    6019,
+    6441,
+    4517,
+    3813,
+    2862,
+    2766,
+    1905,
+    1493,
+    1038,
+    750,
+    495,
+    276,
+    174,
+    0,
+    45,
+    23,
+    0,
+    23
+  ],
+  "crt_disagreements": 0,
+  "crt_matrices": 5631,
+  "distinct_matrices": 61279,
+  "field_prime": 2305843009213693951,
+  "frames": 36628,
+  "h": 23,
+  "loss": 506,
+  "rank_sum": 752951,
+  "v": 1771
+}
diff --git a/research/skip-clones/profiles-25.json b/research/skip-clones/profiles-25.json
new file mode 100644
index 0000000000..482b384f85
--- /dev/null
+++ b/research/skip-clones/profiles-25.json
@@ -0,0 +1,40 @@
+{
+  "R": 43056,
+  "blocks": [
+    0,
+    387587,
+    30674,
+    18228,
+    11305,
+    10562,
+    8695,
+    8645,
+    6001,
+    5662,
+    4434,
+    4390,
+    2922,
+    2589,
+    1976,
+    1615,
+    1054,
+    752,
+    544,
+    290,
+    185,
+    0,
+    55,
+    25,
+    0,
+    25
+  ],
+  "crt_disagreements": 0,
+  "crt_matrices": 7434,
+  "distinct_matrices": 82331,
+  "field_prime": 2305843009213693951,
+  "frames": 48677,
+  "h": 25,
+  "loss": 600,
+  "rank_sum": 1077600,
+  "v": 2300
+}
diff --git a/research/skip-clones/profiles.cpp b/research/skip-clones/profiles.cpp
new file mode 100644
index 0000000000..410e50f475
--- /dev/null
+++ b/research/skip-clones/profiles.cpp
@@ -0,0 +1,107 @@
+#include <algorithm>
+// Optimal-matching variant of research/copied-fixed/profiles.cpp (PR #43, Chafik
+// Boukhalfa), itself adapted from Dominik Scholz PR35 and icekylinx PR32.
+// Change: with LINKS_IN=<file>, the carrier matching is read from a pinned file
+// instead of Hopcroft-Karp. Every pinned edge must lie in the same admissible
+// adjacency (causal order and frame inclusion); donors and uses are distinct.
+// This replay-only adaptation omits the discovery edge dumper. The original
+// optimizer and profiler are preserved in references/copied-fixed/pr44.
+// Replay-only adaptation prepared with OpenAI Codex assistance.
+// Prepared by Rohan Arun with Anthropic Claude assistance. Apache-2.0.
+// Adapted at h=23,25; original PR35 by Dominik Scholz, PR32 by icekylinx.
+// Dimension adaptation at 45/47; extra primes for all source-growth cases.
+// Exact bounded-minor inequalities are checked by verify.py. Original credit below.
+// Copyright 2026 icekylinx. Apache-2.0; AI-assisted handoff integration.
+// Original round3_rankone_profiles.cpp. Requires GCC/Clang unsigned __int128.
+#include <algorithm>
+#include <array>
+#include <cassert>
+#include <cstdint>
+#include <fstream>
+#include <iostream>
+#include <vector>
+#include <map>
+#include <unordered_map>
+#include <cmath>
+#include <cstdlib>
+using U=uint32_t;using V=uint64_t;
+#include "binary_io.hpp"
+int main(int argc,char**argv){assert(argc==2||argc==3);std::ifstream f(argv[1],std::ios::binary);U hdr[4];read_array(f,hdr);U h=hdr[0],v=hdr[1],n=hdr[2],q=hdr[3];assert(h==23||h==25);
+std::vector<std::array<U,2>>args(n);std::vector<V>core(n),cover(n);std::vector<U>roots(q),kind(q);std::vector<uint8_t>active(n);readv(f,args);readv(f,core);readv(f,cover);readv(f,roots);readv(f,kind);readv(f,active);
+std::vector<U>ranks(n),degree(n),begin(n+1);V c=0,inputs=0,loss=0;
+for(U x=1;x<n;x++)if(active[x]){if(args[x][0]){c++;assert(args[x][0]<x&&args[x][1]<x);ranks[x]=popcount64(cover[x])-popcount64(core[x]);for(U y:args[x])degree[y]++;}else{ranks[x]=1;inputs++;}}
+assert(inputs==v);for(U x:roots)degree[x]++;for(U x=1;x<n;x++)begin[x+1]=begin[x]+degree[x];
+std::vector<U>uses(begin.back()),cursor(begin.begin(),begin.end()-1);for(U x=1;x<n;x++)if(active[x]&&args[x][0]){uses[cursor[args[x][0]]++]=2*x;uses[cursor[args[x][1]]++]=2*x+1;}for(U j=0;j<q;j++)uses[cursor[roots[j]]++]=(1U<<31)|j;
+auto nd=[&](U e)->U{return e>>31?roots[e&0x7fffffff]:e/2;};
+auto before=[&](U a,U b){U x=nd(a),y=nd(b);if(ranks[x]!=ranks[y])return ranks[x]<ranks[y];V ox=a>>31?V(n)+(a&0x7fffffff):x,oy=b>>31?V(n)+(b&0x7fffffff):y;return ox<oy;};
+auto incl=[&](U a,U b){U x=nd(a),y=nd(b);return !(core[y]&~core[x])&&!(cover[x]&~cover[y]);};
+auto adjacency=[&](U donor,auto&& action){for(U value:args[donor])for(U j=begin[value];j<begin[value+1];j++)if(before(donor*2,uses[j])&&incl(donor*2,uses[j]))if(action(j))return true;return false;};
+std::vector<U>donors;for(U x=1;x<n;x++)if(active[x]&&args[x][0]&&adjacency(x,[](U){return true;}))donors.push_back(x);
+// Descending donor order changes the retained-carrier network while preserving
+// the exhaustive admissibility checks and maximum matching cardinality.
+const char* rev=getenv("DONOR_REVERSE"); if(!rev||rev[0]!='0') std::reverse(donors.begin(),donors.end());
+std::vector<U>leftmatch(n),rightmatch(uses.size()),distance(n,UINT32_MAX),queue;V matches=0;U phase=0,inf=UINT32_MAX,shortest=inf;
+const char* lin=getenv("LINKS_IN");
+assert(lin && "A pinned carrier matching is required");
+if(lin){std::ifstream lf(lin,std::ios::binary);U hd[2];read_array(lf,hd);assert(hd[0]==n);std::vector<U>isdonor(n);for(U x:donors)isdonor[x]=1;for(U i=0;i<hd[1];i++){U ed[2];read_array(lf,ed);U x=ed[0],e=ed[1];assert(isdonor[x]&&!leftmatch[x]);U jj=UINT32_MAX;for(U value:args[x])for(U j=begin[value];j<begin[value+1];j++)if(uses[j]==e)jj=j;assert(jj!=UINT32_MAX);bool ok=false;adjacency(x,[&](U j){if(j==jj)ok=true;return ok;});assert(ok);assert(!rightmatch[jj]);leftmatch[x]=jj+1;rightmatch[jj]=x;matches++;}}
+else while(true){queue.clear();shortest=inf;for(U x:donors){if(!leftmatch[x]){distance[x]=0;queue.push_back(x);}else distance[x]=inf;}for(U at=0;at<queue.size();at++){U x=queue[at];if(distance[x]>=shortest)continue;adjacency(x,[&](U j){U y=rightmatch[j];if(!y)shortest=distance[x]+1;else if(distance[y]==inf){distance[y]=distance[x]+1;queue.push_back(y);}return false;});}if(shortest==inf)break;
+auto aug=[&](auto&&self,U x)->bool{bool ok=adjacency(x,[&](U j){U y=rightmatch[j];if((!y&&distance[x]+1==shortest)||(y&&distance[y]==distance[x]+1&&self(self,y))){leftmatch[x]=j+1;rightmatch[j]=x;return true;}return false;});if(!ok)distance[x]=inf;return ok;};V gained=0;for(U x:donors)if(!leftmatch[x]&&aug(aug,x)){matches++;gained++;}std::cerr<<"h="<<h<<" phase "<<++phase<<" length "<<shortest<<" gained "<<gained<<" total "<<matches<<"\n";assert(gained);}
+std::vector<int64_t>hist(h+1);for(U x=1;x<n;x++)if(active[x]){assert(degree[x]);U r=ranks[x];if(args[x][0]){hist[r]+=degree[x]-1;hist[h-r]++;for(U y:args[x]){assert(r>=ranks[y]);hist[r-ranks[y]]++;}}else hist[1]+=degree[x];}
+for(U j=0;j<q;j++){U r=ranks[roots[j]];if(kind[j]){hist[r]++;hist[h]++;loss+=r;}else{assert(r<=h-1);hist[h-1-r]++;hist[1]++;}}
+V changed=0;for(U donor:donors)if(leftmatch[donor]){U e=uses[leftmatch[donor]-1],target=nd(e),value=e>>31?target:args[target][e&1];assert(value==args[donor][0]||value==args[donor][1]);if(value==args[donor][0])changed++;U ru=ranks[donor],rv=ranks[value],rt=ranks[target];assert(rt>=ru&&ru>=rv);hist[h-ru]--;hist[rv]--;hist[rt-rv]--;hist[rt-ru]++;}
+V R=c+q-matches,sum=0;for(U r=0;r<=h;r++){assert(hist[r]>=0);sum+=r*hist[r];}assert(sum==h*R+2*loss);
+if(argc==3){std::ofstream out(argv[2],std::ios::binary);U header[2]={n,U(matches)};write_array(out,header);for(U donor:donors)if(leftmatch[donor]){U edge[2]={donor,uses[leftmatch[donor]-1]};write_array(out,edge);}assert(out);}
+std::cout<<"{\"h\":"<<h<<",\"v\":"<<v<<",\"c\":"<<c<<",\"q\":"<<q<<",\"baseline_R\":"<<c+q<<",\"matched\":"<<matches<<",\"R\":"<<R<<",\"orientation_changes\":"<<changed<<",\"rank_sum\":"<<sum<<",\"loss\":"<<loss<<",\"histogram\":[";for(U r=0;r<=h;r++){if(r)std::cout<<",";std::cout<<hist[r];}std::cout<<"]}"<<std::endl;
+// Reconstruct the actual transition multiset before changing the common basis.
+struct Frame{V core,cover;U rank;};std::vector<Frame>frames{{0,0,0},{0,0,h}};std::map<std::pair<V,V>,U>frame_lookup;std::vector<U>fi(n);
+for(U x=1;x<n;x++)if(active[x]){auto key=std::make_pair(core[x],cover[x]);auto z=frame_lookup.find(key);if(z==frame_lookup.end()){U id=frames.size();frames.push_back({core[x],cover[x],ranks[x]});frame_lookup[key]=id;fi[x]=id;}else fi[x]=z->second;}
+std::map<std::pair<U,U>,int64_t>transitions;int64_t singles=0;
+auto edge=[&](U a,U b,int64_t count){if(a==b)return;transitions[{a,b}]+=count;};
+for(U x=1;x<n;x++)if(active[x]){U r=ranks[x];if(args[x][0]){edge(0,fi[x],degree[x]-1);edge(fi[x],1,1);for(U y:args[x])edge(fi[y],fi[x],1);}else edge(0,fi[x],degree[x]);}
+for(U j=0;j<q;j++){U r=ranks[roots[j]];if(kind[j]){edge(0,fi[roots[j]],1);edge(0,1,1);}else singles+=h-r;}
+for(U donor:donors)if(leftmatch[donor]){U e=uses[leftmatch[donor]-1],target=nd(e),value=e>>31?target:args[target][e&1];edge(fi[donor],1,-1);edge(0,fi[value],-1);edge(fi[value],fi[target],-1);edge(fi[donor],fi[target],1);}
+V p=2305843009213693951ULL;
+auto mul=[&](V a,V b)->V{return (__uint128_t)a*b%p;};
+auto power=[&](V a,V b)->V{V r=1;for(;b;b>>=1,a=mul(a,a))if(b&1)r=mul(r,a);return r;};
+auto sub=[&](V a,V b)->V{return (a+p-b)%p;};
+std::vector<std::vector<V>>matrix_cache(frames.size());
+auto matrix=[&](U id)->const std::vector<V>&{auto&A=matrix_cache[id];if(!A.empty())return A;A.assign(h*h,0);if(id==0)return A;if(id==1){for(U i=0;i<h;i++)A[i*h+i]=1;return A;}
+ auto f=frames[id];U c=popcount64(f.core);V out=f.cover&~f.core;U nn=popcount64(out);
+ if(c==3){V factor=power(6*(h+1),p-2);for(U i=0;i<h;i++)for(U j=0;j<h;j++){V wi=3+((f.core>>i)&1);V zj=((f.core>>j)&1)?3*(h+1)-10:p-10;A[i*h+j]=mul(mul(wi,zj),factor);}return A;}
+ assert(c==1||c==2);V s=3-c,d=s*s+(c-1)*nn,den=3*(h+1)*d,inv=power(den,p-2);
+ for(U i=0;i<h;i++)for(U j=0;j<h;j++){V oi=(out>>i)&1,oj=(out>>j)&1,wi=3+((f.core>>i)&1),zj=((f.core>>j)&1)?3*(h+1)-10:p-10;
+  V num=(mul(s*oi,zj)+3*(h+1)*s*wi*oj+mul(nn*wi,zj))%p;num=sub(num,3*(h+1)*(c-1)*oi*oj);A[i*h+j]=(V(i==j&&oi)+mul(num,inv))%p;
+ }return A;};
+
+std::vector<int64_t>blocks(h+1);blocks[1]=singles;V done=0,matrices=0,crt_matrices=0,crt_disagreements=0;U longest=0;std::map<U,V>correction_hist;
+for(auto&[key,count]:transitions){if(!count)continue;assert(count>0);U aa=key.first,bb=key.second;U rr=frames[bb].rank-frames[aa].rank;assert(frames[bb].rank>=frames[aa].rank);if(!rr)continue;
+ if(rr<=2){blocks[1]+=count*rr;continue;}if(aa==0&&bb==1){blocks[h]+=count;continue;}
+ if(aa>1&&bb>1){
+  assert(!(frames[bb].core&~frames[aa].core));
+  assert(!(frames[aa].cover&~frames[bb].cover));
+  U ca=popcount64(frames[aa].core),cb=popcount64(frames[bb].core);
+  assert(ca==3||frames[aa].core==frames[bb].core||(ca==2&&cb==1));
+ }
+ const auto&A=matrix(aa);const auto&B=matrix(bb);std::vector<V>M(h*h);for(U x=0;x<h*h;x++)M[x]=sub(B[x],A[x]);std::vector<std::pair<U,U>>pivots;
+ for(U i=0;i<h;i++){int j=h-1;while(j>=0&&!M[i*h+j])j--;if(j<0)continue;pivots.push_back({i,U(j)});V inv=power(M[i*h+j],p-2);for(U k=i+1;k<h;k++){V z=mul(M[k*h+j],inv);if(z)for(U col=0;col<=U(j);col++)M[k*h+col]=sub(M[k*h+col],mul(z,M[i*h+col]));}}
+ assert(pivots.size()==rr);
+// All minors have numerator bounded independently of matrix size, since the
+// matrix is a 0/1 diagonal mask plus a correction of rank at most four.
+// Source growth and core-two -> core-one use all three primes at these dimensions.
+if(aa>1&&bb>1&&(popcount64(frames[aa].core)==3||(popcount64(frames[aa].core)==2&&popcount64(frames[bb].core)==1))){
+ assert(h==23||h==25);crt_matrices++;matrix_cache[aa].clear();matrix_cache[bb].clear();std::vector<std::vector<std::pair<U,U>>> all{pivots};
+ for(V extra:{2147483647ULL,524287ULL}){p=extra;matrix_cache[aa].clear();matrix_cache[bb].clear();const auto&C=matrix(aa);const auto&D=matrix(bb);std::vector<V>Y(h*h);for(U z=0;z<h*h;z++)Y[z]=sub(D[z],C[z]);std::vector<std::pair<U,U>> pp;
+  for(U i=0;i<h;i++){int j=h-1;while(j>=0&&!Y[i*h+j])j--;if(j<0)continue;pp.push_back({i,U(j)});V inv=power(Y[i*h+j],p-2);for(U k=i+1;k<h;k++){V z=mul(Y[k*h+j],inv);if(z)for(U col=0;col<=U(j);col++)Y[k*h+col]=sub(Y[k*h+col],mul(z,Y[i*h+col]));}}
+  if(pp!=pivots)crt_disagreements++;all.push_back(std::move(pp));
+ }
+ p=2305843009213693951ULL;matrix_cache[aa].clear();matrix_cache[bb].clear();
+ std::vector<int> corner((h+1)*(h+1));for(U i=0;i<h;i++)for(U j=0;j<h;j++){int best=0;for(auto&pp:all){int rank=0;for(auto [row,col]:pp)rank+=row<=i&&col>=j;best=std::max(best,rank);}corner[(i+1)*(h+1)+j]=best;}
+ pivots.clear();for(U i=0;i<h;i++)for(U j=0;j<h;j++){int z=corner[(i+1)*(h+1)+j]-corner[i*(h+1)+j]-corner[(i+1)*(h+1)+j+1]+corner[i*(h+1)+j+1];assert(z==0||z==1);if(z)pivots.push_back({i,j});}assert(pivots.size()==rr);
+}
+U run=0;for(U j=0;j<pivots.size();j++){if(j&&pivots[j].first==pivots[j-1].first+1&&pivots[j].second==pivots[j-1].second+1)run++;else{if(run){blocks[run]+=count;longest=std::max(longest,run);}run=1;}}if(run){blocks[run]+=count;longest=std::max(longest,run);}matrices++;matrix_cache[aa].clear();matrix_cache[bb].clear();
+ if(++done%10000==0)std::cerr<<"profiles "<<done<<" of "<<transitions.size()<<"\n";
+}
+V mass=0;for(U t=1;t<=h;t++){assert(blocks[t]>=0);mass+=t*blocks[t];}assert(mass==sum);
+std::string target=std::string(argv[1])+".round3_rankone_certified_profiles.json";std::ofstream out(target);assert(out);out<<"{\"h\":"<<h<<",\"v\":"<<v<<",\"R\":"<<R<<",\"loss\":"<<loss<<",\"rank_sum\":"<<mass<<",\"field_prime\":"<<p<<",\"frames\":"<<frames.size()<<",\"distinct_matrices\":"<<matrices<<",\"crt_matrices\":"<<crt_matrices<<",\"crt_disagreements\":"<<crt_disagreements<<",\"blocks\":[";for(U t=0;t<=h;t++){if(t)out<<",";out<<blocks[t];}out<<"]}\n";std::cerr<<"CRT matrices "<<crt_matrices<<" differing modular profiles "<<crt_disagreements<<"\n";std::cerr<<"RANKONE h "<<h<<" longest "<<longest<<" output "<<target<<"\n";
+
+}
diff --git a/research/skip-clones/scalar-23.json b/research/skip-clones/scalar-23.json
new file mode 100644
index 0000000000..7ce3cc946f
--- /dev/null
+++ b/research/skip-clones/scalar-23.json
@@ -0,0 +1,13 @@
+{
+  "additions": 40582,
+  "all_additions_disjoint": true,
+  "all_partial_outputs_exact": true,
+  "circuit_sha256": "18d85ae0400f822cd365cf0a7f5fd3f5adda28d6825b449a896b45e0db249204",
+  "cloned_additions": 253,
+  "every_node_has_common_point": true,
+  "h": 23,
+  "inputs": 1771,
+  "merged_additions": 5379,
+  "partial_outputs": 5336,
+  "roles": 45918
+}
diff --git a/research/skip-clones/scalar-25.json b/research/skip-clones/scalar-25.json
new file mode 100644
index 0000000000..4702db6a08
--- /dev/null
+++ b/research/skip-clones/scalar-25.json
@@ -0,0 +1,13 @@
+{
+  "additions": 53828,
+  "all_additions_disjoint": true,
+  "all_partial_outputs_exact": true,
+  "circuit_sha256": "571d1da11d085e98da2b0ad7247d6eda9f623ca7bb0b015aaf8734e2ebb48858",
+  "cloned_additions": 353,
+  "every_node_has_common_point": true,
+  "h": 25,
+  "inputs": 2300,
+  "merged_additions": 7008,
+  "partial_outputs": 6925,
+  "roles": 60753
+}
diff --git a/research/skip-clones/timeline-23.json b/research/skip-clones/timeline-23.json
new file mode 100644
index 0000000000..83666e6078
--- /dev/null
+++ b/research/skip-clones/timeline-23.json
@@ -0,0 +1,57 @@
+{
+  "additions": 40582,
+  "center_rank": 22,
+  "center_terminal_count": 23,
+  "complete_dirty_basis": {
+    "all_dirty_restore": true,
+    "auxiliary_basis_vectors": 32693,
+    "elementary_word_length": 310810,
+    "inputs": 1771,
+    "orientations": [
+      "forward",
+      "reverse-complement"
+    ],
+    "targets": 1771,
+    "total_basis_vectors": 36235
+  },
+  "copied_histogram": [
+    11063,
+    41870,
+    36822,
+    6030,
+    16110,
+    7223,
+    4218,
+    1840,
+    3298,
+    833,
+    2865,
+    250,
+    2456,
+    289,
+    3516,
+    840,
+    2234,
+    810,
+    8594,
+    5873,
+    0,
+    165,
+    23,
+    0
+  ],
+  "copied_rank_sum": 752445,
+  "designated_outputs": 5336,
+  "full_output_coefficients": 3136441,
+  "h": 23,
+  "native_zero_rank_bookkeeping": 5727,
+  "physical_frame_transitions": 86500,
+  "rational_frame_classes": 36626,
+  "retained_links": 13225,
+  "reverse_complement_frame_incidences": 113857,
+  "roles": 32693,
+  "selected_links_sha256": "6f0aa3c2a3da689b22862ce626b0225d07258b466f2a64682be7a2e02a60aa51",
+  "status": "independent scalar, rational frames, physical compiler and rank timeline PASS",
+  "v": 1771,
+  "zero_rank_scope": "Explicit output incidences include no-operation frame checks omitted from native histogram bookkeeping; only positive ranks enter the recurrence"
+}
diff --git a/research/skip-clones/timeline-25.json b/research/skip-clones/timeline-25.json
new file mode 100644
index 0000000000..a5f32e882b
--- /dev/null
+++ b/research/skip-clones/timeline-25.json
@@ -0,0 +1,59 @@
+{
+  "additions": 53828,
+  "center_rank": 24,
+  "center_terminal_count": 25,
+  "complete_dirty_basis": {
+    "all_dirty_restore": true,
+    "auxiliary_basis_vectors": 43056,
+    "elementary_word_length": 410536,
+    "inputs": 2300,
+    "orientations": [
+      "forward",
+      "reverse-complement"
+    ],
+    "targets": 2300,
+    "total_basis_vectors": 47656
+  },
+  "copied_histogram": [
+    14378,
+    54451,
+    48049,
+    7718,
+    22018,
+    9080,
+    4055,
+    3022,
+    5070,
+    893,
+    2724,
+    468,
+    3707,
+    443,
+    2751,
+    168,
+    4406,
+    1536,
+    2945,
+    618,
+    11276,
+    7596,
+    0,
+    221,
+    25,
+    0
+  ],
+  "copied_rank_sum": 1077000,
+  "designated_outputs": 6925,
+  "full_output_coefficients": 5290000,
+  "h": 25,
+  "native_zero_rank_bookkeeping": 7453,
+  "physical_frame_transitions": 114581,
+  "rational_frame_classes": 48675,
+  "retained_links": 17697,
+  "reverse_complement_frame_incidences": 150712,
+  "roles": 43056,
+  "selected_links_sha256": "110ddaf97f2d244afd6d80a3f361b1c65dba2cdc36455c2379cf83f9e9ec7d8a",
+  "status": "independent scalar, rational frames, physical compiler and rank timeline PASS",
+  "v": 2300,
+  "zero_rank_scope": "Explicit output incidences include no-operation frame checks omitted from native histogram bookkeeping; only positive ranks enter the recurrence"
+}
diff --git a/research/skip-clones/validation.json b/research/skip-clones/validation.json
new file mode 100644
index 0000000000..ec6c11305a
--- /dev/null
+++ b/research/skip-clones/validation.json
@@ -0,0 +1,217 @@
+{
+  "adversarial": {
+    "clone_and_json_cases": 58,
+    "counts_receipt_sha256": "6e9bfc084f3927d4becfd16ba7a64c1b6adae0a8e78ab57ed724ecaa20027979",
+    "receipt_sha256": "ccc40ae895196d4678341dd95499146cc4771930ad63403f0795012eaa61d9d2",
+    "rejected": 65,
+    "total": 65,
+    "witness_count_cases": 7
+  },
+  "bit_saving": "1132311451/25000000000000",
+  "completed_utc": "2026-10-08T16:51:09.960483+00:00",
+  "evidence_sha256": {
+    "focused.log": "fef2558e9244993e59041d304bd387499150121bf092c9880ebe3cb9c20d09b8",
+    "producer-replay.json": "0d4e0c8d7c2461d727e38b905cfeb3c0179a94528503965bc88dd8d0252cb279",
+    "producer-replay.log": "735e8d4e2fb27938e72f2325e641ad982640eb4da33ff9152008debfc55ebc9b"
+  },
+  "focused_tests": {
+    "command": "python3 -m unittest discover -s tests -p 'test_skip_clones.py' -v",
+    "passed": 11
+  },
+  "followup_changes": "This validation receipt only; source, finite fixtures and arithmetic unchanged after the complete run.",
+  "full_repository_verification": {
+    "command": "make verify",
+    "exit_code": 0,
+    "focused_tests": {
+      "copied_fixed": 17,
+      "skip_clones": 11,
+      "skip_strips": 20
+    },
+    "historical_patch_checks": 18,
+    "integrated_tests": 218,
+    "log_sha256": "7f99daad7df965b3733d7818a6c72528358fa2cac02dd5cf03cc3f6abada2dbc"
+  },
+  "kappa": "141532521/3125000000000",
+  "merge_validation": {
+    "bit_saving_unchanged": "1132311451/25000000000000",
+    "certificate_changes": {
+      "skip-clones": [
+        {
+          "current_sha256": "f80930a97396003d6bf400596b7fb781d5b01168c0964b55524afc5876deaa26",
+          "path": [
+            "local_sha256",
+            "research/skip-clones/SOURCE.json"
+          ],
+          "previous_sha256": "9908a7eeba68cc2560c2c6e8418fb25d86ffb1bb163c5dfd0e9e721ea5e52578"
+        }
+      ],
+      "skip-strips": [
+        {
+          "current_sha256": "00677e810cfc107f788853c3e9c4994272e6f0d1edd6449564d970a8d583e980",
+          "path": [
+            "local_sha256",
+            "research/skip-strips/SOURCE.json"
+          ],
+          "previous_sha256": "5cae23859745d8b7b90a2ee40cf2c3e90a6c247a0b60ad17e75c8084b9339610"
+        },
+        {
+          "current_sha256": "d0be783f61ac4b7981e7e48a8c1139b52572302ba76c191c0e9a1b16a1a05e0e",
+          "path": [
+            "sources",
+            "files",
+            "scripts/partial_swap/binary_io.hpp"
+          ],
+          "previous_sha256": "7e143eb75bb342ac8f0c82a60005e7c35c9ec827de479bf6fb773151f3af5fa1"
+        }
+      ]
+    },
+    "certificate_numeric_changes": 0,
+    "completed_utc": "2026-10-08T17:11:08.702344+00:00",
+    "conflicts_resolved": [
+      "Makefile",
+      "NOTICE",
+      "README.md"
+    ],
+    "focused_tests": {
+      "command": "python3 -m unittest discover -s tests -p 'test_skip_clones.py' -v",
+      "log_sha256": "ab8b40a864019615dd467596b6909c8f13c06863f7c6ecd62163295d2f9fb59c",
+      "passed": 11
+    },
+    "followup_changes": "This merge_validation receipt only; all tracked source, inputs, certificates and patches remained byte-identical to the tested merge commit after complete verification.",
+    "full_repository_verification": {
+      "command": "make verify",
+      "exit_code": 0,
+      "historical_patch_checks": 20,
+      "integrated_tests": 277,
+      "log_sha256": "f7149674168053105c19275344d5d1bb1fb9211049259c4177f529b85d4badb4",
+      "tested_commit": "03aff8b1981a17257ebab00effd1ec2d7db57e3f",
+      "tested_tree": "3357043c7280117456453a7e37b034203ca3a720",
+      "tracked_regeneration_byte_clean": true
+    },
+    "independent_static_audit": {
+      "receipt_sha256": "e91e37628b7ca59d80834fff075b6a1db0ac07621b43fa7fb05502d95c06809c",
+      "status": "PASS",
+      "tested_tree": "3357043c7280117456453a7e37b034203ca3a720"
+    },
+    "integration": {
+      "added_skip_leaf_commands": 7,
+      "ci_groups": [
+        "community",
+        "skip",
+        "producers",
+        "certificates",
+        "ternary",
+        "tests"
+      ],
+      "ci_python_versions_preserved": [
+        "3.11",
+        "3.13",
+        "3.14"
+      ],
+      "existing_make_verify_leaf_commands_preserved": 94,
+      "formal_ci_jobs_preserved": true,
+      "local_formal_lean_builds_performed": false,
+      "main_notices_and_reviewed_status_preserved": true
+    },
+    "kappa_unchanged": "141532521/3125000000000",
+    "merged_main": "64ed4cbf872b86b81c22f104967b88d24b8e4548",
+    "previous_published_head": "84eb0b067741dc2690da837743fda06d133da865",
+    "proof_scope": "Conditional on the inherited base theorem and mathematical interfaces; no Lean proof of the new bound or exact/global optimization claim.",
+    "python": "3.14.8",
+    "source_pin_checks": {
+      "copied-fixed": {
+        "entries": 56,
+        "mismatches": 0,
+        "source_sha256": "2432832310377091f4bef6fc3bf2a334830af34ec26447ae256cdfa24977eff3"
+      },
+      "skip-clones": {
+        "entries": 113,
+        "mismatches": 0,
+        "source_sha256": "f80930a97396003d6bf400596b7fb781d5b01168c0964b55524afc5876deaa26"
+      },
+      "skip-strips": {
+        "entries": 70,
+        "mismatches": 0,
+        "source_sha256": "00677e810cfc107f788853c3e9c4994272e6f0d1edd6449564d970a8d583e980"
+      }
+    },
+    "status": "PASS: merged current main, complete make verify, reproducible generated artifacts, unchanged exact bound",
+    "tested_commit": "03aff8b1981a17257ebab00effd1ec2d7db57e3f",
+    "tested_tree": "3357043c7280117456453a7e37b034203ca3a720"
+  },
+  "proof_scope": "Conditional on the inherited base theorem and mathematical interfaces; no Lean or exact optimization claim",
+  "scalar_dense_carrier_fixed_profile_and_complete_dirty_basis": "PASS on both axes, including weighted selected uses",
+  "status": "PASS: complete make verify and independent finite/adversarial replay",
+  "tested_commit": "7210de7d0f9ccaeb64c3f60f188419e02be08d95",
+  "upstream_research_integration": {
+    "bit_saving_unchanged": "1132311451/25000000000000",
+    "certificate_byte_identical_to_previous_head": true,
+    "completed_utc": "2026-10-08T18:02:33.827551+00:00",
+    "conflicts_resolved": [
+      ".github/workflows/verify.yml",
+      "docs/ci-verification.md"
+    ],
+    "followup_changes": "Only this validation receipt differs from the tested merged tree; numerical certificates, finite witnesses, source manifests and all upstream research artifacts are unchanged.",
+    "fresh_verification": {
+      "command": "make -j1 verify-research verify-skip verify-tests",
+      "exit_code": 0,
+      "focused_tests": {
+        "skip_clones": 11,
+        "skip_strips": 20
+      },
+      "historical_patch_checks": 20,
+      "integrated_tests": 450,
+      "log_sha256": "5cc6118ec28f372aa9fc39c98fdbc1aad052fc7382ca6b5308d3268302f0aeda",
+      "process_receipt_sha256": "32cdfa8df1a96780c9353a072f011ceddc30de6defca5bb8a599fe09279711da",
+      "recovered_research_audits": 28,
+      "tracked_regeneration_byte_clean": true
+    },
+    "integration": {
+      "all_upstream_python_and_patch_check_commands_preserved": true,
+      "ci_groups": [
+        "community",
+        "skip",
+        "producers",
+        "certificates",
+        "ternary",
+        "research",
+        "tests"
+      ],
+      "ci_python_versions": [
+        "3.11",
+        "3.13",
+        "3.14"
+      ],
+      "formal_ci_jobs_preserved": true,
+      "static_receipt_sha256": "2f3abca03b7817640b534d9fa1cf72a6c5650781c3206a2464354f9ddf39c016",
+      "upstream_added_files_preserved_byte_for_byte": 152
+    },
+    "kappa_unchanged": "141532521/3125000000000",
+    "merged_main": "d55bcd62ceb8f322bcf311a5a66e94c3ac7724f0",
+    "previous_published_head": "b1c704d7a92b60af4e50f09a0b049df09312d642",
+    "source_pin_checks": {
+      "copied-fixed": {
+        "entries": 56,
+        "manifest_unchanged": true,
+        "mismatches": 0,
+        "source_sha256": "2432832310377091f4bef6fc3bf2a334830af34ec26447ae256cdfa24977eff3"
+      },
+      "skip-clones": {
+        "entries": 113,
+        "manifest_unchanged": true,
+        "mismatches": 0,
+        "source_sha256": "f80930a97396003d6bf400596b7fb781d5b01168c0964b55524afc5876deaa26"
+      },
+      "skip-strips": {
+        "entries": 70,
+        "manifest_unchanged": true,
+        "mismatches": 0,
+        "source_sha256": "00677e810cfc107f788853c3e9c4994272e6f0d1edd6449564d970a8d583e980"
+      }
+    },
+    "status": "PASS: additive latest-main integration, recovered upstream audits, existing skip witnesses and all tests",
+    "tested_tree": "d11297c86814f7db0f8aa1d51710dba936811cae",
+    "verification_scope": "Fresh scoped integration verification. Prior complete make verify PASS records above remain tied to their original tested commits; no new full make verify or local Lean run is claimed."
+  },
+  "verified_utc": "2026-10-08T16:40:53.036795+00:00"
+}
diff --git a/research/skip-clones/witness.py b/research/skip-clones/witness.py
new file mode 100644
index 0000000000..73ea15dc32
--- /dev/null
+++ b/research/skip-clones/witness.py
@@ -0,0 +1,224 @@
+#!/usr/bin/env python3
+"""Exact arithmetic witness for paid original-envelope clones on PR53 skip-prefix graphs.
+
+Avi Eisenberg/Anthropic Claude PR53 supplies the skip-prefix base; RaD / hipotures
+with OpenAI Codex assistance PR51 supplies paid whole-chain source partitions.
+Chafik Boukhalfa/OpenAI Codex PR43/46/48 supplies the original-envelope finite
+checkers, exact data recovery and arithmetic; Rohan Arun/Anthropic Claude PR44
+supplies weighted-matching discovery. All retained notices and source pins apply.
+Prepared with OpenAI Codex assistance. Apache-2.0; general transfer proofs remain
+inherited dependencies.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from hashlib import sha256
+from math import comb
+from pathlib import Path
+import argparse
+import importlib.util
+import json
+import sys
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+PR43 = ROOT/'research/copied-fixed'
+sys.path.insert(0, str(ROOT/'scripts'))
+sys.path.insert(0, str(PR43))
+sys.path.insert(0, str(HERE))
+from clone_io import read, require, integer, check_sources, CERTIFICATE_INPUTS
+from cloned_graph import graph
+
+_spec = importlib.util.spec_from_file_location('skip_clones53_pr43_verify', PR43/'verify.py')
+base = importlib.util.module_from_spec(_spec)
+_spec.loader.exec_module(base)
+_spec = importlib.util.spec_from_file_location('skip_clones53_balanced_assembly', PR43/'balanced_assembly.py')
+balanced = importlib.util.module_from_spec(_spec)
+_spec.loader.exec_module(balanced)
+moment, js = base.moment, base.js
+assembly, cutoffs = balanced.assembly, balanced.cutoffs
+AC = base.AC
+PR53_KAPPA = Q(4498144, 10**11)
+PR48_KAPPA = Q(411862541, 10**13)
+
+
+def rational_pair(value, name):
+    require(type(value) is list and len(value) == 2, 'Expected rational pair: '+name)
+    return Q(integer(value[0], name+' numerator', 1), integer(value[1], name+' denominator', 1))
+
+
+def parameters():
+    values = read(HERE/'parameters.json')
+    require(set(values) == {'bit_saving', 'kappa', 'grid_denominator'}, 'Unexpected parameter keys')
+    ab = rational_pair(values['bit_saving'], 'bit_saving')
+    kappa = rational_pair(values['kappa'], 'kappa')
+    grid = Q(1, integer(values['grid_denominator'], 'grid_denominator', 1))
+    require(PR53_KAPPA < kappa < ab < 1, 'Savings do not improve PR53')
+    require((ab/grid).denominator == (kappa/grid).denominator == 1, 'Savings must lie on declared grid')
+    return ab, kappa, grid
+
+
+def counts(row, names):
+    for name in names:
+        integer(row[name], name)
+
+
+def histogram(value, length, name):
+    require(type(value) is list and len(value) == length, 'Wrong histogram length: '+name)
+    for item in value:
+        integer(item, name)
+    return value
+
+
+def profile():
+    a, b = 23, 25
+    m, N = a*b, comb(a, 3)*comb(b, 3)
+    rows = [read(HERE/f'original-{h}.json') for h in (a, b)]
+    for h, row in zip((a, b), rows):
+        counts(row, ('h', 'v', 'R', 'c', 'q', 'matched', 'loss', 'rank_sum', 'baseline_R', 'orientation_changes'))
+        require(row['h'] == h and row['v'] == comb(h, 3), 'Wrong local dimension or source count')
+        require(row['baseline_R'] == row['c']+row['q'], 'Wrong unmatched role count')
+        require(row['R'] == row['baseline_R']-row['matched'], 'Wrong matched role count')
+        require(row['loss'] == h*(h-1), 'Wrong retained center loss')
+        hist = histogram(row['histogram'], h+1, 'rank histogram')
+        require(sum(t*n for t, n in enumerate(hist)) == row['rank_sum'] == h*row['R']+2*row['loss'],
+                'Original rank mass')
+    W = 2*N+sum(N//row['v']*row['R'] for row in rows)
+    L = sum(N//row['v']*row['loss'] for row in rows)
+    data = base.data_corners()
+    good, bad = data['good'], data['fallback']
+    require(good+bad == N, 'Wrong data-pair denominator')
+    parts = {'data': Counter({1: 2*(9*good+47*bad), 21: 2*good, 17: 2*good, 481: 2*N}),
+             'paid_endpoint_copy': Counter({1: N})}
+    for row in rows:
+        h = row['h']
+        f = read(HERE/f'profiles-{h}.json')
+        counts(f, ('h', 'v', 'R', 'loss', 'rank_sum', 'crt_disagreements',
+                   'field_prime', 'frames', 'distinct_matrices', 'crt_matrices'))
+        for key in ('h', 'v', 'R', 'loss', 'rank_sum'):
+            require(f[key] == row[key], 'Fixed/original mismatch: '+key)
+        require(f['crt_disagreements'] == 0 and f['field_prime'] == 2**61-1, 'Invalid CRT profile')
+        blocks = histogram(f['blocks'], h+1, 'fixed blocks')[:]
+        require(blocks[0] == 0 and blocks[h] == h, 'Expected zero null blocks and h full center cleanups')
+        require(sum(t*n for t, n in enumerate(blocks)) == f['rank_sum'], 'Fixed profile mass')
+        blocks[h] -= h
+        blocks[1] += h
+        require(sum(t*n for t, n in enumerate(blocks)) == h*row['R']+row['loss'], 'Copied profile mass')
+        scalar = read(HERE/f'scalar-{h}.json')
+        counts(scalar, ('h', 'inputs', 'additions', 'merged_additions', 'partial_outputs', 'roles'))
+        require(all(scalar[key] is True for key in ('all_additions_disjoint', 'all_partial_outputs_exact',
+                                                  'every_node_has_common_point')), 'Scalar replay flags')
+        require(scalar['h'] == h and scalar['inputs'] == row['v'] and scalar['additions'] == row['c'],
+                'Scalar dimensions differ')
+        timeline = read(HERE/f'timeline-{h}.json')
+        counts(timeline, ('h', 'v', 'roles', 'retained_links', 'additions', 'center_rank',
+                          'center_terminal_count', 'copied_rank_sum', 'designated_outputs',
+                          'full_output_coefficients', 'native_zero_rank_bookkeeping',
+                          'physical_frame_transitions', 'rational_frame_classes',
+                          'reverse_complement_frame_incidences'))
+        histogram(timeline['copied_histogram'], h+1, 'timeline histogram')
+        require(timeline['copied_histogram'][1:] == [row['histogram'][t]+(h if t == 1 else -h if t == h else 0)
+                                                    for t in range(1, h+1)], 'Timeline histogram differs')
+        require(timeline['h'] == h and timeline['roles'] == row['R'] and timeline['retained_links'] == row['matched'],
+                'Physical timeline differs')
+        require(timeline['copied_rank_sum'] == h*row['R']+row['loss'], 'Physical copied mass')
+        dirty = timeline['complete_dirty_basis']
+        counts(dirty, ('auxiliary_basis_vectors', 'total_basis_vectors', 'elementary_word_length',
+                       'inputs', 'targets'))
+        require(dirty['all_dirty_restore'] is True and dirty['auxiliary_basis_vectors'] == row['R'] and
+                dirty['total_basis_vectors'] == 2*row['v']+row['R'] and
+                dirty['orientations'] == ['forward', 'reverse-complement'], 'Incomplete dirty-basis replay')
+        rep, bank = N//row['v'], N//row['v']*row['R']
+        parts[f'internal_{h}'] = Counter({t: n*rep for t, n in enumerate(blocks) if t and n})
+        parts[f'exterior_{h}'] = Counter({h: bank, m-2*h: bank})
+        parts[f'data_growth_{h}'] = Counter({1: 2*N, h-2: 2*N})
+    hist = sum(parts.values(), Counter())
+    total_rank = sum(t*n for t, n in hist.items())
+    require(total_rank == W*m-N+L, 'Complete rank mass')
+    require((m, N, L) == (575, 4073300, 2226400) and W < 160799739, 'Physical constants or PR53 width')
+    require(max(hist) == 529 and all(type(t) is int and 0 < t < m and type(n) is int and n > 0
+                                   for t, n in hist.items()), 'Improper children')
+    return dict(m=m, N=N, W=W, L=L, total_rank=total_rank, deficit=W*m-total_rank,
+                maxchild=max(hist), child_multiplicities=dict(sorted(hist.items())), parts=parts)
+
+
+def excluded(path, ab):
+    c = read(path)
+    counts(c, ('m', 'W'))
+    children = {}
+    for key, multiplicity in c['child_multiplicities'].items():
+        require(key.isdecimal() and str(int(key)) == key, 'Invalid child width key')
+        width = int(key)
+        require(0 < width < c['m'], 'Comparison child does not contract')
+        children[width] = integer(multiplicity, 'comparison multiplicity', 1)
+    rank_mass = sum(t*n for t, n in children.items())
+    require(0 < rank_mass < c['m']*c['W'], 'Comparison rank deficit')
+    # The pinned PR40 fixture predates the redundant total_rank metadata.
+    if 'total_rank' in c:
+        require(rank_mass == integer(c['total_rank'], 'comparison rank mass'), 'Comparison rank mass')
+    lower = moment(c['m'], c['W'], children, ab)['lower']
+    require(lower > 1, path.name+' network not excluded at new saving')
+    return lower
+
+
+def run():
+    require(not sys.flags.optimize, 'Assertions must remain enabled')
+    check_sources(CERTIFICATE_INPUTS)
+    base.check_sources()
+    for h in (23, 25):
+        rebuilt = graph(h).verify()
+        require(rebuilt == read(HERE/f'scalar-{h}.json'), 'Cloned scalar record differs from exact graph replay')
+    ab, kappa, grid = parameters()
+    p = profile()
+    exact = moment(p['m'], p['W'], p['child_multiplicities'], ab)
+    require(exact['upper'] < 1, 'Bit characteristic failed')
+    nxt = moment(p['m'], p['W'], p['child_multiplicities'], ab+grid)
+    require(nxt['lower'] > 1, 'Next bit grid point unexpectedly certified')
+    prior = base.baseline.certificate()
+    phase = prior['complex']['counts']
+    phase_row = read(ROOT/'certificates/copied-centers-complex-input.json')
+    bridge = base.baseline.finite_bridge(p, phase, [phase_row, phase_row])
+    final = assembly(bridge, ab, kappa, a_complex=AC)
+    require(len(final['constraints']) == 47 and len(final['margins']) == 7, 'Changed assembly interface')
+    require(all(value > 0 for value in final['constraints'].values()), 'Nonpositive assembly constraint')
+    eventual = cutoffs(bridge, final)
+    try:
+        assembly(bridge, ab, kappa+grid, a_complex=AC)
+    except (AssertionError, ValueError):
+        pass
+    else:
+        raise ValueError('Next kappa grid point unexpectedly accepted')
+    controls = {path.name: excluded(path, ab) for path in sorted(HERE.glob('comparison-pr*.json'))}
+    controls['comparison-pr48.json'] = excluded(ROOT/'research/skip-strips/comparison-pr48.json', ab)
+    controls.update({name: excluded(PR43/name, ab) for name in
+                     ('comparison-pr40.json', 'comparison-pr41.json', 'comparison-pr42.json',
+                      'comparison-pr43.json', 'comparison-pr44.json', 'comparison-pr46.json', 'comparison-pr47.json')})
+    comparisons = {'PR53': PR53_KAPPA, 'PR48': PR48_KAPPA}
+    for path in sorted(HERE.glob('comparison-pr*.json')):
+        row = read(path)
+        if 'kappa' in row:
+            previous = rational_pair(row['kappa'], path.name+' kappa')
+            require(kappa > previous, 'Kappa does not improve '+path.name)
+            comparisons['PR'+str(row['PR'])] = previous
+    require(Q(1, 2**15) < kappa < Q(1, 2**14), 'Unexpected dyadic bracket')
+    local = sorted(path for path in HERE.iterdir() if path.suffix in ('.py', '.cpp', '.uses', '.json', '.md')
+                   and path.name not in ('certificate.json', 'validation.json', 'producer-receipt.json'))
+    return dict(status='Conditional exact arithmetic witness; general transfer proofs are inherited dependencies',
+                kappa=kappa, bit=dict(counts=p, **exact), next_bit_grid_lower=nxt['lower'],
+                complex=prior['complex'], finite_bridge=bridge, assembly=final, eventual_bounds=eventual,
+                comparison=dict(prior_savings=comparisons, ratio_PR53=kappa/PR53_KAPPA,
+                                dyadic_corollary='2^-15', next_dyadic_not_reached='2^-14'),
+                exclusion_lower_moments=controls,
+                local_sha256={str(path.relative_to(ROOT)): sha256(path.read_bytes()).hexdigest() for path in local},
+                scope='Pinned paid whole-chain clones on the PR53 skip-prefix graph family; '
+                      'fixed I+J profiles, exact data recovery, copied centers and paid endpoint corrections. '
+                      'Fresh finite replay is required; no exact matching optimality or global search optimality claim.')
+
+
+if __name__ == '__main__':
+    parser = argparse.ArgumentParser()
+    parser.add_argument('--output', type=Path)
+    args = parser.parse_args()
+    result = run()
+    if args.output:
+        args.output.write_text(json.dumps(js(result), indent=2, sort_keys=True)+'\n')
+    print('PASS conditional kappa='+str(result['kappa'])+'; 47 strict inequalities and seven margins')
diff --git a/research/skip-strips/PROOF.md b/research/skip-strips/PROOF.md
new file mode 100644
index 0000000000..e41cd36689
--- /dev/null
+++ b/research/skip-strips/PROOF.md
@@ -0,0 +1,159 @@
+# Skip-prefix strips: conditional saving 4.498144 × 10^-5
+
+Under OpenAI's base theorem and the retained analytic, fixed finite-alphabet
+and fixed multitape interfaces, the construction below gives
+
+\[
+T(n)=O\bigl(n(\log n)^{1-\kappa}\bigr),\qquad
+\boxed{\kappa=\frac{4498144}{10^{11}}=4.498144\times10^{-5}>2^{-15}.}
+\]
+
+This is about **9.21% above PR #48** (411862541/10^13). The exact bit saving
+is 4498347/10^11; the next grid value fails the same exact test, and the
+next κ grid value fails the assembly. These compare asymptotic exponent
+savings, not running times. They are conditional research claims without
+independent mathematical review or formal verification.
+
+## What changes
+
+Only the scalar producer DAG changes, in `skip_graph.py`, together with its
+pinned carrier matching, `links-{23,25}.uses`. Everything else is PR #48's
+`research/copied-fixed` pipeline, with constants recomputed for the new DAG:
+
+- original envelope labels;
+- Rohan Arun's PR #44 weighted carrier matching, pinned and replayed;
+- both fixed local bases I+J, with complete CRT-checked profiles;
+- copied centers;
+- reversed data corners, with exact rational recovery of the ten
+  primary-prime failures;
+- the balanced assembly;
+- the complex side.
+
+The producer computes, in each common-point context, every leave-two-out sum
+of the pair inputs. The paired-exclusion recursion (icekylinx PR #18/#36)
+needs, for each vertex, its *strip* values: the vertex's star sums over all
+groups except one excluded group j. That is a leave-one-out family
+s_j = Σ_{i≠j} v_i over the vertex's group values v_1, …, v_k.
+
+Earlier producers form it as s_j = A_{j-1} + B_{j+1}, with prefix sums A and
+suffix sums B (PR #48 additionally reorders the values). Then the two
+consumers of every prefix A_i are A_{i+1} and s_{i+1}. Their point covers are
+incomparable, because A_{i+1} contains group i+1 and s_{i+1} does not. So the
+second use of every prefix, and likewise of every suffix, needs a fresh role.
+
+We use the **skip-prefix** layout
+
+\[
+ s_j = A_{j-2} + \bigl(v_{j-1} + B_{j+1}\bigr).
+\]
+
+The consumers of A_i become A_{i+1} and s_{i+2}, with
+cover(A_{i+1}) ⊆ cover(s_{i+2}). The new node v_{i-1} + B_{i+1} has cover
+contained in cover(B_{i-1}), the later consumer of v_{i-1}. These are the
+nested envelopes, in causal rank order, that the inherited carrier matching
+may join. Each join turns a fresh role into a continuation of a retained
+carrier. The layout costs one more addition per strip value; the role count,
+not the addition count, drives the cost.
+
+Details of the layout:
+
+- **Top level:** the layout runs on the reversed value list.
+- **Deeper levels:** it runs on the values sorted by (support size, support),
+  following PR #48's sorting idea. The support order is ascending at h=23 and
+  descending at h=25. Outputs are restored to their original indices.
+- **Totals:** these fold in reverse input order at h=23 (PR #48's rule) and in
+  descending (support size, support) order at h=25 (PR #43/#48's rule).
+- **Point order:** the global pair order is RaD / hipotures's PR #41
+  alternating order.
+- **Zero values:** these are omitted, and their leave-one-out value is the
+  total, exactly as in the inherited code.
+
+| h | v | additions c | outputs q | matched | roles R | loss |
+|---|---|---|---|---|---|---|
+| 23 | 1771 | 40329 | 5336 | 12719 | 32946 | 506 |
+| 25 | 2300 | 53475 | 6925 | 16991 | 43409 | 600 |
+
+PR #48 has R23 = 36432 and R25 = 48329.
+
+The physical constants are m = 575, N = 4,073,300, W = 160,799,739 (PR #48:
+177,530,859) and L = 2,226,400. The complete rank is Wm − N + L =
+92,458,003,025. The deficit N − L is unchanged.
+
+The weighted matching is pinned in `links-{h}.uses`. `optimize_matching.py`
+regenerates it from PR #44's optimizer with only the graph import changed;
+it maximizes Σ c_t t ln t over all maximum-cardinality matchings. Its output
+is not trusted: `profiles.cpp` replays every pinned link. It checks that each
+link lies in the admissible adjacency (causal order and envelope inclusion)
+and that donors and uses are distinct. `producer.py` then recounts every link
+independently.
+
+## Why the inherited proofs apply
+
+All of the following are checked by the inherited code on the new DAGs:
+
+- **Scalar outputs.** Every addition joins disjoint supports, every node has a
+  common point, and every output is the exact leave-two-out sum or retained
+  total. The scalar identity [|S∩T|=1] + |S∩T| ≡ [S=T] (mod 2) and JLV = I are
+  unchanged.
+- **Physical timeline.** The full physical timeline and every dirty basis
+  vector are checked in both orientations at h = 23 and 25 by
+  `original_timeline.py`.
+- **Fixed-basis profiles.** These are recomputed from the new DAGs and pinned
+  matchings, with the inherited exactness bounds:
+  - 61,461 and 82,606 distinct transition matrices;
+  - 5,642 and 7,434 of them CRT-checked;
+  - zero modular disagreements.
+
+  Exactly h full center cleanups appear, so the copied-center replacement
+  applies unchanged.
+- **Data corners.** These do not depend on the producer. All 4,073,300 pairs
+  are replayed, and the ten primary-prime failures are recovered by exact
+  rational elimination. Every pair therefore uses 9 singletons + 21 + 17 + 481.
+
+## Exact arithmetic
+
+`verify.py` uses PR #43/#48's exact logarithm and exponential enclosures:
+
+- The moment at a_b = 4498347/10^11 is below 1.
+- Each of the following is excluded at this new bit saving by an exact lower
+  bound: the complete child lists of PR #48, #47, #46, #44, #43, #42, #41 and
+  #40, and the PR #36 baseline.
+- The inherited balanced assembly passes all 47 strict conditions and seven
+  margins at κ = 4498144/10^11. Its parameters are β = 1/20, h = 10^-12,
+  a_c = 717/10^7, C1 = 1 and row stock p^2000.
+
+## Reproduce
+
+```sh
+python3 research/skip-strips/producer.py --work-dir /tmp/skip-strips
+python3 research/skip-strips/verify.py
+python3 research/skip-strips/optimize_matching.py --work-dir /tmp/skip-opt   # optional, needs scipy
+```
+
+`producer.py --record` rewrites the JSON records. Without `--record` it checks
+them against fresh regeneration.
+
+## Scope and attribution
+
+This is a conditional finite certificate. It covers the same written proof
+obligations as PR #43–#48: the base theorem, the residual compiler, fixed-tape
+recursion, the physical meaning of carrier continuations and copied centers,
+the analytic/semantic/routing/recovery interfaces and eventual thresholds.
+None of these are re-proved here, and there is no formal verification or
+claim of global optimality.
+
+The skip-prefix strip layout is new here, prepared with assistance from
+Claude (Anthropic). Everything else is inherited, with all original notices:
+
+- Chafik Boukhalfa PR #43/#46/#48: pipeline, recovery, the sorting idea and
+  verification code.
+- Rohan Arun PR #44: weighted matching; also PR #37/#39/#40/#42/#47.
+- RaD / hipotures PR #41: alternating order and changed DAGs.
+- Dominik Scholz PR #35/#38.
+- James Chang PR #34.
+- icekylinx PR #18/#24/#32/#36.
+- Zhihao Chen PR #21/#23/#29.
+- Paureel / Aurel Prosz, Swapnil Jain, Douglas Colkitt, OpenAI, and
+  Harvey–van der Hoeven.
+
+Apache-2.0.
diff --git a/research/skip-strips/SOURCE.json b/research/skip-strips/SOURCE.json
new file mode 100644
index 0000000000..7bd435a948
--- /dev/null
+++ b/research/skip-strips/SOURCE.json
@@ -0,0 +1,146 @@
+{
+  "PR34": {
+    "author": "James Chang (jamesyc)",
+    "commit": "7fecbe3651e095fb0f450756afdeefd2a9ee80a2",
+    "contribution": "reversed geometry and all-weight rank-cut identities"
+  },
+  "PR35": {
+    "author": "Dominik Scholz",
+    "commit": "9c345a2a11e5f4f3649f7c68214bf9a2a0a3fe9c"
+  },
+  "PR36": {
+    "author": "icekylinx",
+    "commit": "11817ccacb564bb7f98789c20dc11d3fece207e3"
+  },
+  "PR37": {
+    "author": "Rohan Arun",
+    "commit": "2f7578affce416ad4b6c41f3438ebb734f66a899",
+    "contribution": "dimension-23 reversed geometry checker"
+  },
+  "PR38": {
+    "author": "Dominik Scholz",
+    "commit": "605323bab4ba330de3d1785dc5b3ecb14e8ab747",
+    "contribution": "contemporaneous independent fixed-both composition, comparison and credit"
+  },
+  "PR39": {
+    "author": "Rohan Arun",
+    "commit": "70ae24129649f6d6d4ec6360962a80c3c42a38f1",
+    "contribution": "fixed-middle modular nonvanishing method adapted to exhaustive two-fixed pair check"
+  },
+  "PR40": {
+    "author": "Rohan Arun",
+    "commit": "43f59ff533598762cbc43a5e14af2bbbc76fabbd",
+    "contribution": "contemporaneous both-fixed baseline; matching order improvement is new here"
+  },
+  "PR41": {
+    "PR": 41,
+    "author": "RaD / hipotures",
+    "commit": "d9c96297d455053b795ee58ba108d5cf892e1c04",
+    "point_order_blob": "a47df953d275f86bf3fcb100cc425ca8854875ca",
+    "rad_checkpoint": "7e488e6b25dc1713c1f41baaf1cefbf677507cf3",
+    "subsequent_inspected_head": "03bde70c0c2a453f1ceae20c619b602841405467"
+  },
+  "PR42": {
+    "author": "Rohan Arun",
+    "commit": "c6fc08951c1776a9685802b1810a2f15dbbd4f64",
+    "contribution": "contemporaneous fixed/alternating composition and exact comparison; different carrier matching"
+  },
+  "PR43": {
+    "author": "Chafik Boukhalfa with OpenAI Codex assistance",
+    "commit": "c0fbfd3cc4fd6e95cef54ab307dfb50c69dd96f6",
+    "contribution": "prior fully checked original-envelope/fixed-basis alternating composition and independent replay checks"
+  },
+  "PR44": {
+    "author": "Rohan Arun with Anthropic Claude assistance",
+    "commit": "061414a529f1123668872db44d737e362fbe066f",
+    "contribution": "weighted matching discovery, pinned carrier links and replay profiler; no optimality claim is required"
+  },
+  "PR46": {
+    "author": "Chafik Boukhalfa with OpenAI Codex assistance",
+    "commit": "71b6c960c89295e952522dd44111df9cfc51ae90",
+    "contribution": "prior verified composition and exact rational recovery of all data pairs; this increment changes both scalar graphs and pinned matchings"
+  },
+  "PR47": {
+    "author": "Rohan Arun with Anthropic Claude assistance",
+    "commit": "117caf164e3b235ef20188aa2dc1b49617e9255c",
+    "contribution": "contemporaneous hill-climbed graph benchmark; no construction component reused in this increment"
+  },
+  "PR48": {
+    "author": "Chafik Boukhalfa",
+    "commit": "b7194b8586844956904b33e9fe18f0c75606c11b",
+    "contribution": "research/copied-fixed pipeline (weighted-matching replay, exact data recovery, verification) reused here",
+    "pull_request": 48
+  },
+  "files": {
+    "certificates/copied-centers-bit-axes.json": "ed307cef595091b11ba8f8e6ff11302ad8db190227a2fbf34c232fd680206254",
+    "certificates/copied-centers-complex-input.json": "91a9e54b455ecc36c3150abcf4a1c9138691c8faab464b5ce159ba8e376764e7",
+    "certificates/copied-centers-network.json": "7dbd675e42e22ac076280360f8750031851564bb7a56079e66236fee5d0a6cdf",
+    "notes/copied-centers-bit.tex": "ccc8191ddb51698251ed4461a4c26104501b9b2ee37ce3cbe82fa73e85ae40dd",
+    "notes/copied-centers-lemma.tex": "d0ce6d3d504996aadb098227885224e48df46101daec8893523a0a8b7364d8fe",
+    "notes/structured-bulk-bit.tex": "fddef1b1d902de49e2e4e71bf2c1708c73bc25cbfee107ed6ce3b6d785a5d284",
+    "references/copied-fixed/pr34/ASSEMBLY_PROOF.md": "88bdd35ac089891dcbd47fb7fb76418c785e6004902f7a987aad571e0a4eea51",
+    "references/copied-fixed/pr34/downstream-semantic-bulk-assembly.md": "ccdeaeac7c6a1c08483fea2285a7d089d782f2c18b7be867c3fc9f77457637fd",
+    "references/copied-fixed/pr34/review-arbitrary-routing.md": "f18f535a096abae473db71a4ae3f275d0116d7438fb18594d5b160571db670fd",
+    "references/copied-fixed/pr34/review-balanced-transform.md": "0c3a5b6f17188205d35cd421ef0c35c71f6111698f497e89da3ef7f870fef862",
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+    "references/copied-fixed/pr35/README.md": "7b136ff0f9a1520296e24195dbf981f55dc63e103e1c105735b1540aa640df3c",
+    "references/copied-fixed/pr35/fixed-basis-two-stage.tex": "cfd97834f3f67378492fd14fddb1f6774f6261164c33f57389b2f729720b7e1f",
+    "references/copied-fixed/pr35/producer.py": "afa4a68a415d2281f7ef7924bd58952e33a4ae7f84c6437a5e13786d7eb966a8",
+    "references/copied-fixed/pr35/rankone_profiles.cpp": "a644821881e6cf4a426a3d6908519889b2f0edeb0032ca384e5026bcf16877a6",
+    "references/copied-fixed/pr41/SOURCE.json": "836d3d588dc4abb24c375c3a4c65a3dc6fc6ccb186a070ee35241886baf48ccd",
+    "references/copied-fixed/pr41/check_compiled_witness.py": "f358ea1109537993e48d7e9fb1215d793d4d89424f9e6b14359f4bc522173e7b",
+    "references/copied-fixed/pr41/point_order_search.py": "3021113367ed8fda0b10e7f25a3bc751b7cb39bee05db9c0c912d3b4b1256ac6",
+    "references/copied-fixed/pr41/producer_search.py": "1fc67fef1401190156f75aaca785537c3b5fb9533fcaadd030d6e922af9f5292",
+    "references/copied-fixed/pr44/SOURCE.json": "eeb610bdc954586233b9667b49f11ecda211c120e3557621d08b769c5a010459",
+    "references/copied-fixed/pr44/optimal_graph.py": "8c7dab322e5143d4e760ef9086213a0a9a4e65cb5ead1f7b8a2c68d886e99fdb",
+    "references/copied-fixed/pr44/optimize_matching.py": "0c3ec1f9c6e3dcee5f5809a9ba5d248c99e26e0b7c3df89f0a5571815a05f22b",
+    "references/copied-fixed/pr44/profiles.cpp": "2c8c32fdfa0719247e24da41a66da18bd3eef4a8a78e13b5581df86a7be225da",
+    "research/copied-fixed/balanced_assembly.py": "4d27575dcad2b45a281a34054d311409ecd954f373c0c3a62c29bcc843f06a86",
+    "research/copied-fixed/comparison-pr40.json": "3cf55a63468df230b57089f66c4a97dbee25ca5a0771f53ef23c972774718060",
+    "research/copied-fixed/comparison-pr41.json": "58beeb816847707c6c866c72ce078b751f70ab65c5baa7c498f863005602e7a2",
+    "research/copied-fixed/comparison-pr42.json": "22d8c3347405a1df3bf970c199b5c58062c8fc34d5916c86c1353b97254c4461",
+    "research/copied-fixed/comparison-pr43.json": "3595db8b0eb1f139c00e6e5984522ba91ac693d26bc32bed07211846024757c6",
+    "research/copied-fixed/comparison-pr44.json": "d2681a5c0c0020410fe0a23fd4bae51bf31171d55572ba6a79f6c9928a7543de",
+    "research/copied-fixed/comparison-pr46.json": "f7a34054403075c51289e58b583c05ff42c43e72d286b9c3ff4fc5ae7f4bca44",
+    "research/copied-fixed/comparison-pr47.json": "d15ecc820d540a2bfea10bc7d72688240aea89245ff27de263b1de0e74e4273a",
+    "research/copied-fixed/data-corners.json": "82dc2d3e116fc003348d6e9ff0cce90fd70d3d6c156580b769ff3d38b3399e17",
+    "research/copied-fixed/links-23.uses": "b215350929d15711b502cdcf5878aeedb2cc7dc81dc764977a28d37ece17f86f",
+    "research/copied-fixed/links-25.uses": "80dd63ea393931211bece7a44329a9da4e2936575cdc3f48c27690bedc2ad0ae",
+    "research/copied-fixed/original-23.json": "a7976c03e9fb788d79e72efc9afeda0f2cb977f790a532dac28c8d15762613c1",
+    "research/copied-fixed/original-25.json": "eecd3bc090d9547f5e85244b74403713838005e94561f80f7b2774b36d672b0f",
+    "research/copied-fixed/profiles-23.json": "f44077350b6759199dd95bf17d4d5e27c2d5c5295c8dcae85234d478827f0ff7",
+    "research/copied-fixed/profiles-25.json": "746402326f54c01cf32eadb2bf1ce277fc8058dfd34f14182918f1f0e58a3fb7",
+    "research/copied-fixed/reversed/geometry-source.json": "de0f18e22e2fba877cb5beb4a4b0f3e181938c00b98e2f276708a63ae681ea17",
+    "research/copied-fixed/reversed/geometry.py": "6f8990c17781d5dd42176762a3f9c95de012248a1470f34654388e5adad3b93f",
+    "research/copied-fixed/reversed/pr34/certify_reversed.py": "77311474169f64a154c2435bcde9303fbc472cbd9ac4d2b39c76e7560fa9e3a3",
+    "research/copied-fixed/reversed/pr34/independent_controls.py": "f0450c9a8127b538df6cc36ed744ad707b9e72756af9ce5290023272999c5c65",
+    "research/copied-fixed/review/pr34-balanced-assembly-audit.txt": "3b3b6bb410f53d874fff51c535b72ed6e1636041a079579df6c7cca2b0ca4ca2",
+    "research/copied-fixed/scalar-23.json": "b5defb6a89253340440c398e2b1482b3ca71d9f9655bcfe855231de9a4e248c6",
+    "research/copied-fixed/scalar-25.json": "759366ff6b4293de865248a246c55d0dca538551194f0ac8a85867c64f8473dd",
+    "research/copied-fixed/search-record.json": "1e5298f543ff2bf07fb81c612da89358517eafbf4e33d337452390471ff427a9",
+    "research/copied-fixed/timeline-23.json": "540b91c324a0394a2e22d7f448a02ee7a3aa0aa8c2d9e75cded909033ebeacd3",
+    "research/copied-fixed/timeline-25.json": "3cf71771577b7cc05a2cb0574cd9e3fe7e905734a775ad082f16276f62373b7d",
+    "research/skip-strips/PROOF.md": "0fad3bfbd7062e1607fc95430170d12f9613cc6d850a18871f1cbc8c27434934",
+    "research/skip-strips/comparison-pr48.json": "36de96fd3aea4e0b335d3693434f004f05220bc33a4a165596288e038052734b",
+    "research/skip-strips/links-23.uses": "63e47bdaebb8a5c718a4d65e2d6f6bcb8d7548a32030e3b312157b4fa665b7bf",
+    "research/skip-strips/links-25.uses": "a8a3b0dd47d5d5a46aa4eaf844f7fbd9279e21547b5c01950a31706e700210d3",
+    "research/skip-strips/optimize_matching.py": "b8146942253fd34fafb92f6eaca4d145609e4c277704492635135095a03ddacf",
+    "research/skip-strips/original-23.json": "35caafcaf98170fe5260765248ed8bf43edc33d01e81138e8fa540067192fbb3",
+    "research/skip-strips/original-25.json": "8ecd5028fcfa9972ee9e1d241a0bc296ac4e1888c42e8f0c05f95e11de326109",
+    "research/skip-strips/profiles-23.json": "baac5249694959d063a4b0bae156610679682f0782d4254cd509dbf3bd26850f",
+    "research/skip-strips/profiles-25.json": "214bdd7a5a5c0618d0d5507bc2973b4bb06f88327ee33697823ed1924bcffdf0",
+    "research/skip-strips/scalar-23.json": "08fd09b33c22f609e241753aa38495fd4bef0bfbc4222d5f7e0301cb0aa9c2bc",
+    "research/skip-strips/scalar-25.json": "c54a47e688ba8d69fbe095f6775cd540505a80e5977ea53afe8f0051e115633a",
+    "research/skip-strips/skip_graph.py": "03ee019cdaf73391455aaf4f5dcb4873f4fdcb278179b296c02ea39eafd047e4",
+    "research/skip-strips/timeline-23.json": "1f7e9ca7efc96855ef64164d8c2ec1026b9e5f1ce95f0858ef6528aee81c150c",
+    "research/skip-strips/timeline-25.json": "1e7a692e2e73465ecd0138cfe3429dec5a91f8c87998a6318d689ff600213441",
+    "scripts/copied_centers/physical.py": "837345d1f8753eab0e70f105b1d1bb0db965b993da037b1825ddef3672609d7d",
+    "scripts/copied_centers_network.py": "409ebfbd8a41c4c4d089aa0949fcf5a949415dc26f57440ffd48b7bd8881e6a9",
+    "scripts/partial_swap/binary_io.hpp": "d0be783f61ac4b7981e7e48a8c1139b52572302ba76c191c0e9a1b16a1a05e0e",
+    "scripts/partial_swap/graph.py": "fd8c567759e26a10d2bad05508fd0355eb70fb116664d4e06e1858d0d8cb2ef9",
+    "scripts/partial_swap/paired.py": "ddcecace3a8e04d690914cbd64873ab8dde8c0c3477a255767b0ed6f639bb0c7",
+    "scripts/partial_swap/shared.py": "81e9af92cdf38243a540cdcff2062778e449467848d444da37a9553c12ca4c34",
+    "scripts/partial_swap_network.py": "acdd81b4dea2815f8838a0c3ea8ac4968f2bc13fd7c99173473d865779096782",
+    "scripts/structured_bulk_assembly.py": "e99467e14211fba699878ed036e1e2362eb8e395dc22f9ecdcecc7fbf2cc736d"
+  }
+}
diff --git a/research/skip-strips/audit.py b/research/skip-strips/audit.py
new file mode 100644
index 0000000000..34d4990f66
--- /dev/null
+++ b/research/skip-strips/audit.py
@@ -0,0 +1,85 @@
+#!/usr/bin/env python3
+"""Independent defining-property checks of the fixed projector formula.
+
+No imported projector/profiler implementation. Exact fractions only. The
+canonical tests cover every envelope size at both dimensions; coordinate
+permutation equivariance extends these identities to all cores/covers.
+The mathematical extension is stated and proved in PROOF.md.
+"""
+from fractions import Fraction as Q
+from pathlib import Path
+import argparse
+import json
+
+
+def matrix(h,c,n):
+    core=set(range(c)); outside=set(range(c,c+n))
+    s=3-c; d=s*s+(c-1)*n
+    w=[3+int(i in core) for i in range(h)]
+    z=[3*(h+1)*int(i in core)-10 for i in range(h)]
+    o=[int(i in outside) for i in range(h)]
+    return [[Q(int(i==j and i in outside))+
+             Q(s*o[i]*z[j]+3*(h+1)*s*w[i]*o[j]+n*w[i]*z[j]
+               -3*(h+1)*(c-1)*o[i]*o[j],3*(h+1)*d)
+             for j in range(h)] for i in range(h)]
+
+
+def check_projector(h,c,n):
+    A=matrix(h,c,n)
+    # Pull every image column back by L^{-1}=I-J/(h+1), and verify it lies in E.
+    for j in range(h):
+        total=sum(A[i][j] for i in range(h))
+        x=[A[i][j]-total/(h+1) for i in range(h)]
+        assert len(set(x[:c]))==1 and sum(x)==3*x[0]
+        assert all(x[i]==0 for i in range(c+n,h))
+    # Each transformed basis vector of E is fixed; hence rank=n and A^2=A.
+    for k in range(c,c+n):
+        y=[Q(int(i==k))+Q(3+int(i<c),3-c) for i in range(h)]
+        assert all(sum(A[i][j]*y[j] for j in range(h))==y[i] for i in range(h))
+    # G'=L^{-T}(I-J/9)L^{-1}; symmetry identifies the unique orthogonal projector.
+    coefficient=Q(9*h+19,9*(h+1)**2)
+    cols=[sum(A[i][j] for i in range(h)) for j in range(h)]
+    assert all(A[i][j]-coefficient*cols[j]==A[j][i]-coefficient*cols[i]
+               for i in range(h) for j in range(h))
+    if c==2:
+        # Identity proving rank <=2 for SAME-core differences: the constant
+        # rank-one term cancels, leaving a difference of two rank-one matrices.
+        w=[3+int(i<c) for i in range(h)]
+        z=[3*(h+1)*int(i<c)-10 for i in range(h)]
+        o=[int(c<=i<c+n) for i in range(h)]
+        assert all(A[i][j]==Q(int(i==j and o[i]))+Q(w[i]*z[j],3*(h+1))
+                   -Q((w[i]-o[i])*(z[j]-3*(h+1)*o[j]),3*(h+1)*(n+1))
+                   for i in range(h) for j in range(h))
+
+
+def run():
+    assert __debug__, 'Run without -O'
+    cases=[]
+    for h in (23,25):
+        for c in (1,2):
+            for n in range(1,h-c+1):
+                check_projector(h,c,n)
+                cases.append([h,c,n])
+        # Source t: independent derivation from t^T G t=2, followed by conjugation.
+        t=[int(i<3) for i in range(h)]
+        p=[Q(x+3) for x in t]
+        dual=[Q(x,2)-Q(5,3*(h+1)) for x in t]
+        assert sum(x*y for x,y in zip(p,dual))==1
+        assert all(x!=0 for x in p+dual)
+        assert all(p[i]*dual[j] == Q((3+t[i])*(3*(h+1)*t[j]-10),6*(h+1))
+                   for i in range(h) for j in range(h))
+    return dict(exact_envelope_cases=len(cases),cases=cases,source_line_cases=2,
+                checks=['image contained in envelope','identity on envelope basis',
+                        'self adjoint in transformed rational metric',
+                        'same-core rank-two difference identity','source line normalization'],
+                scope='Exact canonical size cases, with coordinate-equivariance proof in PROOF.md; '
+                      'full graph, matching and CRT profiles independently replayed by producer.py')
+
+
+if __name__=='__main__':
+    parser=argparse.ArgumentParser()
+    parser.add_argument('--output',type=Path)
+    args=parser.parse_args()
+    result=run()
+    if args.output:args.output.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n')
+    print('PASS independent exact projector audit:',result['exact_envelope_cases'],'envelopes')
diff --git a/research/skip-strips/balanced_assembly.py b/research/skip-strips/balanced_assembly.py
new file mode 100644
index 0000000000..c7fd42867d
--- /dev/null
+++ b/research/skip-strips/balanced_assembly.py
@@ -0,0 +1,196 @@
+"""Exact, import-safe balanced copied-center semantic assembly arithmetic.
+
+The PR36 copied-center guard uses E=64*(W+m+G+1)^3, including G.
+Its complex saving is 717/10^7. All balanced parameter formulas are unchanged.
+
+Adapted from Zhihao Chen PR23/29 semantic assembly (Apache-2.0), RaD's
+balanced transform / arbitrary routing / bulk resampling composition, and
+jamesyc PR34's check_composition.py at 7fecbe3. PR34 source is research/reversed-two-stage/check_composition.py in that
+pinned upstream commit. Original parameterized transcription by the
+research workers is in rectangular-dimensions/next/assembly_parameterized.py.
+
+This checks arithmetic, not physical corner existence, producer correctness,
+or inherited all-size tape/analytic hypotheses. No imports execute producers,
+parse argv, read files, change globals, or mutate the supplied finite bridge.
+"""
+from fractions import Fraction as Q
+
+
+class InvalidAssembly(AssertionError):
+    """A named exact input validation or strict inequality failed."""
+
+
+def require(condition, message):
+    if not condition:
+        raise InvalidAssembly(message)
+
+
+def rational(value):
+    # Fraction(float) silently preserves binary floating error; forbid it.
+    require(type(value) in (int, str, Q), 'expected integer or exact rational string/Fraction')
+    return Q(value)
+
+
+def integer(value):
+    x = rational(value)
+    require(x.denominator == 1, 'nonintegral graph constant')
+    return x.numerator
+
+
+def ceil(value):
+    return -(-value.numerator // value.denominator)
+
+
+def halving_degree(m, child):
+    require(0 < child < m, 'child must strictly contract')
+    degree = 1
+    while m**degree <= 2 * child**degree:
+        degree += 1
+    return degree
+
+
+def validate_bridge(finite_bridge):
+    """Recompute rather than trust stored semantic and product-stock totals.
+
+    scalar_group_upper remains a supplied, proof-certified local scalar upper
+    charge. Its producer provenance is external to this arithmetic interface.
+    """
+    result = {}
+    for name in ('bit', 'complex'):
+        data = finite_bridge[name]
+        m, W, child = (integer(data[k]) for k in ('m', 'W', 'maxchild'))
+        require(m > 1 and W > 0, 'positive arity and role volume required')
+        degree, bits = halving_degree(m, child), W.bit_length()
+        require(integer(data['halving_degree']) == degree, name + ' stale halving degree')
+        require(integer(data['wire_bits']) == bits, name + ' stale role logarithm')
+        result[name] = dict(m=m, W=W, maxchild=child,
+                            halving_degree=degree, wire_bits=bits)
+    phase = finite_bridge['complex']
+    s, scalar = (integer(phase[k]) for k in ('s', 'scalar_group_upper'))
+    m, W, child = (result['complex'][k] for k in ('m', 'W', 'maxchild'))
+    require(0 < s < m*W and scalar > 0, 'invalid complex rank/scalar bound')
+    E = 64*(W+m+scalar+1)**3
+    B = s+E
+    literal = 2*scalar*W**2+8*s+4*W+4+32*m
+    C0 = 32*m*B**2
+    semantic = dict(E=E, B=B, C0=C0, C1=1, literal_charge=literal,
+                    strict_literal_gap=E-literal,
+                    induction_gap=2*B*(m-child)-(s+E))
+    require(E > literal and semantic['induction_gap'] >= 0 and C0 > 2*B+18,
+            'semantic completed-child bound failed')
+    for key, value in semantic.items():
+        require(rational(finite_bridge['semantic'][key]) == value,
+                'stale semantic constant: '+key)
+    result['complex'].update(s=s, scalar_group_upper=scalar)
+    result['semantic'] = semantic
+    rows = finite_bridge['rows']
+    coefficient = sum(result[name]['halving_degree']*result[name]['wire_bits']
+                      for name in ('bit', 'complex'))
+    degree = integer(rows['degree'])
+    gap = degree-Q(51,25)*coefficient
+    require(degree > 0 and gap > 0, 'insufficient product row degree')
+    expected = dict(coefficient=coefficient, degree=degree,
+                    degree_gap=gap, suffix_slope=4*degree)
+    for key, value in expected.items():
+        require(rational(rows[key]) == value, 'stale product row constant: '+key)
+    result['rows'] = expected
+    return result
+
+
+def assembly(finite_bridge, a_bit, kappa, beta=Q(1,20), h=Q(1,10**12),
+             a_complex=Q(717,10**7), *, original_prefix=False,
+             old_guard=False, old_exposures=False):
+    """Return 47 strict slacks and seven margins, or raise InvalidAssembly.
+
+    The three keyword switches are negative controls: they impose the old
+    prefix, nonlinear guard, or old separate movement charges at these SAME
+    balanced parameters. They do not silently optimize an alternative family.
+    """
+    f = validate_bridge(finite_bridge)
+    a, b, kappa, beta, h = map(rational, (a_bit,a_complex,kappa,beta,h))
+    require(0 < h < Q(1,2) and kappa > 0, 'positive backoff and saving required')
+    tau, sigma = 1-a, 1-b
+    q = a*(1-2*h)
+    require(1+q != 0, 'undefined balanced epsilon')
+    lp, c, eps = 1-q, q+h/4, (1-h)/(1+q)
+    lam = (tau+lp)/2
+    G = eps*q
+    r, delta = (G+1-eps)/2, h/8
+    C1 = Q(19991,10000) if old_guard else Q(1)
+    margins = dict(g1=1-eps, g2=a, g3=G, g4=a,
+                   g5=min(1-eps-delta,r-delta), g6=1-eps-delta, g7=eps)
+    if original_prefix:
+        margins['g1'] = 1-eps*(1+c)
+    if old_exposures:
+        margins.update(g2=eps*c*a, g4=a*(1-eps))
+    internal = tau+(1-beta)*max(sigma-tau,Q(0))
+    leaf = sigma+beta*(1-sigma)
+    slacks = dict(a_positive=a, a_below_b=b-a, b_below_one_over32=Q(1,32)-b,
+        beta_positive=beta, beta_below_one=1-beta, phase_leaf_above_bit=(1-beta)*b-a,
+        q_positive=q, q_below_internal=1-internal-q, q_below_leaf=1-leaf-q,
+        c_positive=c, c_below_one=1-c, q_below_reservations=c-q,
+        lambda_above_tau=lam-tau, lambda_above_sigma=lam-sigma,
+        lambda_above_internal=lam-internal, lambda_prime_above_lambda=lp-lam,
+        compact_leaf=lp-leaf, compact_reservations=lp-(1-c), lambda_prime_below_one=q,
+        epsilon_positive=eps, epsilon_below_one=1-eps, guard_width=1-eps*C1,
+        K_geometry=1-eps*(1+c), K_dominates_log=eps*c,
+        record_suffix=1-eps, phase_local=1-eps-delta, phase_boundary=r-delta,
+        gamma_sublinear=1-eps-r, cell_above_band=eps-(1-r)/2,
+        prime_interval_packing=1-eps, alpha_positive=r, alpha_below_one=1-r,
+        alpha_below_one_fourth=Q(1,4)-r, delta_positive=delta,
+        delta_below_one_eighth=Q(1,8)-delta, short_record_fallback=eps-a,
+        small_field_exposure=1-eps-G, artificial_boundary=8-eps+r-delta-G,
+        literal_scalar_guard=Q(f['semantic']['strict_literal_gap']),
+        row_product_gap=f['rows']['degree_gap'])
+    slacks.update({name+'_above_kappa': value-kappa for name,value in margins.items()})
+    require(len(slacks) == 47 and len(margins) == 7, 'constraint list incomplete')
+    failed = {name:str(value) for name,value in slacks.items() if value <= 0}
+    require(not failed, str(failed))
+    require(min(margins.values()) == G, 'unexpected controlling margin')
+    require(1-eps-G == h and 1-eps-r == h/2, 'balanced identities failed')
+    require(1-eps*(1+c) == h-eps*h/4, 'geometric identity failed')
+    parameters = dict(a_bit=a, a_complex=b, tau=tau, sigma=sigma, beta=beta,h=h,
+        q=q,c=c,epsilon=eps,lambda_=lam,lambda_prime=lp,alpha_squared_power=r,
+        delta=delta,C0=f['semantic']['C0'],C1=C1,kappa=kappa)
+    return dict(parameters=parameters,constraints=slacks,margins=margins,
+        minimum_margin=G,absorption_gap=G-kappa,scoped_limit=a/(1+a),
+        recurrence=dict(internal=internal,leaf=leaf,reservations=1-c),finite_bridge=f)
+
+
+def cutoffs(finite_bridge, result):
+    """PR34 sufficient numeric comparisons, retaining its all-interval padding.
+
+    These are eventual arithmetic thresholds, not practical running times or
+    substitutes for additional native setup / prime / recovery thresholds.
+    """
+    f = validate_bridge(finite_bridge)
+    p = result['parameters']
+    e,c,r,beta = (p[k] for k in ('epsilon','c','alpha_squared_power','beta'))
+    ka,km,kb,ks = (ceil(1/x) for x in (r,e*c,1-e,e*beta))
+    C0 = f['semantic']['C0']
+    largest = max(f['bit']['m'],f['complex']['m'])
+    cuts = dict(guard=ceil(Q((2*C0).bit_length())/(1-e)),
+        normalization=ceil(7/(1-e-r)), alpha=16*ka*ka+1, compact=64*km*km+1,
+        geometry=ceil(3/(1-e*(1+c))),phase_cell=ceil(9/(e-(1-r)/2)),
+        period=128*kb*kb+1,stopped_leaf=ks*(4*largest).bit_length(),log_p=25,reservoir=14)
+    common = max(cuts.values())
+    require(common >= max(2*ka,2*km,2*kb,25), 'cutoff does not cover monotone regime')
+    checkpoints=[]
+    for j in range(6):
+        z=common*2**j
+        powers=dict(alpha=(z//ka,8*(z+ka)+64),compact=(z//km,32*(z+km)+192),
+            period=(z//kb,16*(z+kb)+56),
+            rows=(z//kb,f['rows']['suffix_slope']*(z+kb+8)),
+            leaf=(z//ks,4*largest))
+        require(all(lhs >= rhs.bit_length() for lhs,rhs in powers.values()),
+                'compressed-power threshold failed')
+        checkpoints.append(dict(log2_input=z,powers={name:dict(exponent=lhs,rhs=rhs,
+             rhs_bits=rhs.bit_length()) for name,(lhs,rhs) in powers.items()}))
+    return dict(cutoff_log2_input=cuts,common_cutoff=common,power_checkpoints=checkpoints,
+        all_interval_argument='For [Z,Z+k), bound RHS at Z+k and LHS at floor(Z/k). '
+          'Successive intervals double LHS; the positive affine RHS grows at most '
+          'linearly, so 2^j >= j+1 proves persistence.',
+        additional_eventual_conditions=['BHP prime threshold and distinct-prime packing',
+          'Fixed rational basis, eligible common prime and native table setup',
+          'Catalogue/descriptor domination, logarithm absorption, exact recovery'],
+        practical_runtime_claim=False)
diff --git a/research/skip-strips/certificate.json b/research/skip-strips/certificate.json
new file mode 100644
index 0000000000..14d5a0943d
--- /dev/null
+++ b/research/skip-strips/certificate.json
@@ -0,0 +1,17940 @@
+{
+  "assembly": {
+    "absorption_gap": "328930412841842130178027/50002249173499995501653000000000000",
+    "constraints": {
+      "K_dominates_log": "2249173512493252479487504498347/50002249173499995501653000000000000",
+      "K_geometry": "37502249173512495501653/50002249173499995501653000000000000",
+      "a_below_b": "2671653/100000000000",
+      "a_positive": "4498347/100000000000",
+      "alpha_below_one": "99999999999950002249173499995501653/100004498346999991003306000000000000",
+      "alpha_below_one_fourth": "24996626239700008996693999995501653/100004498346999991003306000000000000",
+      "alpha_positive": "4498347049988754132500004498347/100004498346999991003306000000000000",
+      "artificial_boundary": "2800143947104549718853312499986504959/400017993387999964013224000000000000",
+      "b_below_one_over32": "311783/10000000",
+      "beta_below_one": "19/20",
+      "beta_positive": "1/20",
+      "c_below_one": "49997750826487504498347/50000000000000000000000",
+      "c_positive": "2249173512495501653/50000000000000000000000",
+      "cell_above_band": "99999999999849997750826500004498347/200008996693999982006612000000000000",
+      "compact_leaf": "1156576500004498347/50000000000000000000000",
+      "compact_reservations": "1/4000000000000",
+      "delta_below_one_eighth": "999999999999/8000000000000",
+      "delta_positive": "1/8000000000000",
+      "epsilon_below_one": "2249173549995501653/50002249173499995501653",
+      "epsilon_positive": "49999999999950000000000/50002249173499995501653",
+      "g1_above_kappa": "1184164568797944142749/156257028667187485942665625000000",
+      "g2_above_kappa": "203/100000000000",
+      "g3_above_kappa": "328930412841842130178027/50002249173499995501653000000000000",
+      "g4_above_kappa": "203/100000000000",
+      "g5_above_kappa": "111258002010209481117167/16000719735519998560528960000000000",
+      "g6_above_kappa": "2981459046949237009935787/400017993387999964013224000000000000",
+      "g7_above_kappa": "156242971333684178626132319142749/156257028667187485942665625000000",
+      "gamma_sublinear": "1/2000000000000",
+      "guard_width": "2249173549995501653/50002249173499995501653",
+      "lambda_above_internal": "4498347/100000000000000000000000",
+      "lambda_above_sigma": "2671653000004498347/100000000000000000000000",
+      "lambda_above_tau": "4498347/100000000000000000000000",
+      "lambda_prime_above_lambda": "4498347/100000000000000000000000",
+      "lambda_prime_below_one": "2249173499995501653/50000000000000000000000",
+      "literal_scalar_guard": "9693793493753822316795466474748",
+      "phase_boundary": "3598677629990553471300004498347/80003598677599992802644800000000000",
+      "phase_leaf_above_bit": "2313153/100000000000",
+      "phase_local": "17993388349961764050500004498347/400017993387999964013224000000000000",
+      "prime_interval_packing": "2249173549995501653/50002249173499995501653",
+      "q_below_internal": "4498347/50000000000000000000000",
+      "q_below_leaf": "1156576500004498347/50000000000000000000000",
+      "q_below_reservations": "1/4000000000000",
+      "q_positive": "2249173499995501653/50000000000000000000000",
+      "record_suffix": "2249173549995501653/50002249173499995501653",
+      "row_product_gap": "6548/25",
+      "short_record_fallback": "4999775072527437133815735125732409/5000224917349999550165300000000000",
+      "small_field_exposure": "1/1000000000000"
+    },
+    "finite_bridge": {
+      "bit": {
+        "W": 160799739,
+        "halving_degree": 9,
+        "m": 575,
+        "maxchild": 529,
+        "wire_bits": 28
+      },
+      "complex": {
+        "W": 537696432,
+        "halving_degree": 20,
+        "m": 784,
+        "maxchild": 756,
+        "s": 421548223824,
+        "scalar_group_upper": 4793351472,
+        "wire_bits": 30
+      },
+      "rows": {
+        "coefficient": 852,
+        "degree": 2000,
+        "degree_gap": "6548/25",
+        "suffix_slope": 8000
+      },
+      "semantic": {
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+        "C0": 2358858462850846165593256368932965330169899180021667007021321420800,
+        "C1": 1,
+        "E": 9696565176891520488192506033216,
+        "induction_gap": 533311084729033626873772984137200,
+        "literal_charge": 2771683137698171397039558468,
+        "strict_literal_gap": 9693793493753822316795466474748
+      }
+    },
+    "margins": {
+      "g1": "2249173549995501653/50002249173499995501653",
+      "g2": "4498347/100000000000",
+      "g3": "2249173499993252479500004498347/50002249173499995501653000000000000",
+      "g4": "4498347/100000000000",
+      "g5": "3598677629990553471300004498347/80003598677599992802644800000000000",
+      "g6": "17993388349961764050500004498347/400017993387999964013224000000000000",
+      "g7": "49999999999950000000000/50002249173499995501653"
+    },
+    "minimum_margin": "2249173499993252479500004498347/50002249173499995501653000000000000",
+    "parameters": {
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+      "C1": "1",
+      "a_bit": "4498347/100000000000",
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+      "alpha_squared_power": "4498347049988754132500004498347/100004498346999991003306000000000000",
+      "beta": "1/20",
+      "c": "2249173512495501653/50000000000000000000000",
+      "delta": "1/8000000000000",
+      "epsilon": "49999999999950000000000/50002249173499995501653",
+      "h": "1/1000000000000",
+      "kappa": "140567/3125000000",
+      "lambda_": "99995501653000004498347/100000000000000000000000",
+      "lambda_prime": "49997750826500004498347/50000000000000000000000",
+      "q": "2249173499995501653/50000000000000000000000",
+      "sigma": "9999283/10000000",
+      "tau": "99995501653/100000000000"
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+      "reservations": "49997750826487504498347/50000000000000000000000"
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+  "scope": "Skip-prefix leave-one-out strips (skip_graph.py) with RaD alternating point order, newly pinned PR44-weighted carrier matching and fixed-basis internal blocks at (23,25); all actual data pairs checked for 9 singletons +21+17+481; ten failed modular tests are recovered by exact rational elimination. Exact scalar/label/matching/profile reconstruction is producer.py. OpenAI base theorem, written residual compiler, fixed-tape recursion, analytic/semantic/routing/recovery and eventual-threshold interfaces remain assumed.",
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+      "author": "James Chang (jamesyc)",
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+      "contribution": "reversed geometry and all-weight rank-cut identities"
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diff --git a/research/skip-strips/comparison-pr40.json b/research/skip-strips/comparison-pr40.json
new file mode 100644
index 0000000000..e0b14fb243
--- /dev/null
+++ b/research/skip-strips/comparison-pr40.json
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+{
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new file mode 100644
index 0000000000..f2968fb1dd
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+++ b/research/skip-strips/comparison-pr41.json
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diff --git a/research/skip-strips/comparison-pr42.json b/research/skip-strips/comparison-pr42.json
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index 0000000000..3f4378580e
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+++ b/research/skip-strips/comparison-pr42.json
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index 0000000000..503e4a74ce
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diff --git a/research/skip-strips/comparison-pr46.json b/research/skip-strips/comparison-pr46.json
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+}
diff --git a/research/skip-strips/comparison-pr47.json b/research/skip-strips/comparison-pr47.json
new file mode 100644
index 0000000000..bc387e8d37
--- /dev/null
+++ b/research/skip-strips/comparison-pr47.json
@@ -0,0 +1,34 @@
+{
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+  "child_multiplicities": {
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+}
diff --git a/research/skip-strips/comparison-pr48.json b/research/skip-strips/comparison-pr48.json
new file mode 100644
index 0000000000..b0ab3101d8
--- /dev/null
+++ b/research/skip-strips/comparison-pr48.json
@@ -0,0 +1,37 @@
+{
+ "L": 2226400,
+ "N": 4073300,
+ "W": 177530859,
+ "child_multiplicities": {
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+  "9": 22825959
+ },
+ "m": 575,
+ "source": "PR48 research/copied-fixed/certificate.json (commit b7194b8)",
+ "total_rank": 102078397025
+}
\ No newline at end of file
diff --git a/research/skip-strips/data-corners.json b/research/skip-strips/data-corners.json
new file mode 100644
index 0000000000..4cb7a2b26f
--- /dev/null
+++ b/research/skip-strips/data-corners.json
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+[18,21,22,2300,428310],
+[19,20,21,2300,428310],
+[19,20,22,2300,428310],
+[19,21,22,2300,428310],
+[20,21,22,2300,428310]
+],"pairs":4073300,"good":4073290,"fallback":10,"nonzero_pivots":191444880,"ordered_zero_checks":1409360440,"fallback_pairs":[[0,9,10,14,18,20,26],[0,11,12,6,8,10,25],[0,15,18,6,13,18,25],[1,15,17,12,13,18,24],[2,11,18,4,7,22,25],[4,7,22,6,11,18,25],[4,11,20,2,17,20,25],[5,7,21,6,11,12,24],[10,11,22,14,16,18,25],[12,13,22,4,6,10,26]]}
diff --git a/research/skip-strips/data_corners.cpp b/research/skip-strips/data_corners.cpp
new file mode 100644
index 0000000000..251f6c87cc
--- /dev/null
+++ b/research/skip-strips/data_corners.cpp
@@ -0,0 +1,77 @@
+// Exhaustive rational nonvanishing certificates via a proved finite prime.
+// Adapted from Rohan Arun PR39 geometry.cpp, with James Chang PR34's pivots.
+// Both factors are now fixed I+J; every one of the 4,073,300 triple pairs is
+// checked. A failed residue test gets a charged 47-singleton fallback.
+// Zero identities are proved separately by the inherited exact rank cuts.
+// Apache-2.0. Prepared with OpenAI Codex assistance.
+#include <cstdint>
+#include <iostream>
+#include <vector>
+#include <array>
+#include <cassert>
+using namespace std;
+constexpr int prime=1000003,a=23,b=25,d=47;
+int mul(int x,int y){return int(int64_t(x)*y%prime);}
+int power(int x,int n){int z=1;for(;n;n>>=1,x=mul(x,x))if(n&1)z=mul(z,x);return z;}
+int main(){
+ for(int i=2;i*i<=prime;++i)assert(prime%i);
+ vector<int> R,C,piv;
+ for(int i=0;i<a;i++)R.push_back(i),C.push_back(i);
+ R.push_back(a-1);C.push_back(0);
+ for(int i=0;i<a;i++)R.push_back(i),C.push_back(i);
+ piv.push_back(2*a);
+ for(int i=1;i<a-1;i++)piv.push_back(i+a+1);
+ for(int i=a-1;i<a+5;i++)piv.push_back(2*a-i);
+ for(int i=a+5;i<2*a-1;i++)piv.push_back(i-a-3);
+ piv.push_back(1);piv.push_back(0);
+ assert(piv.size()==d);
+ const int xin=mul(3*(a+1),power(2*(3*(a+1)-10),prime-2));
+ const int xout=prime-mul(a+1,power(5,prime-2));
+ const int yin=mul(3*(b+1),power(2*(3*(b+1)-10),prime-2));
+ const int yout=prime-mul(b+1,power(5,prime-2));
+ int count=0,good=0;int64_t checked=0,zeros=0;
+ vector<array<int,7>> rejected;
+ cout<<"{\"prime\":"<<prime<<",\"dimensions\":[23,25],\"records\":[\n";
+ int records=0;
+ for(int aa=0;aa<a;aa++)for(int bb=aa+1;bb<a;bb++)for(int cc=bb+1;cc<a;cc++){
+  int xs[a];for(int i=0;i<a;i++)xs[i]=(i==aa||i==bb||i==cc)?xin:xout;
+  int rowgood=0,product=1;
+  for(int u=0;u<b;u++)for(int v=u+1;v<b;v++)for(int w=v+1;w<b;w++){
+   int ys[b],M[d][d];bool available[d];
+   for(int i=0;i<b;i++)ys[i]=(i==u||i==v||i==w)?yin:yout;
+   for(int i=0;i<d;i++){available[i]=true;for(int j=0;j<d;j++){
+    int z=prime-1;if(R[i]==C[j])z+=xs[R[i]];
+    if(i%b==(a*b-d+j)%b)z+=ys[i%b];M[i][j]=z%prime;
+   }}
+   bool ok=true;int determinant=1;
+   for(int i=0;i<d;i++){
+    int c=piv[i],value=M[i][c];assert(available[c]);
+    if(!value){rejected.push_back({aa,bb,cc,u,v,w,i});ok=false;break;}
+    for(int j=c+1;j<d;j++)if(available[j]){assert(!M[i][j]);zeros++;}
+    determinant=mul(determinant,value);checked++;available[c]=false;
+    int inv=power(value,prime-2);
+    for(int r=i+1;r<d;r++)if(M[r][c]){
+     int ratio=mul(M[r][c],inv);
+     for(int j=0;j<d;j++)if(available[j]&&M[i][j]){
+      int z=M[r][j]-mul(ratio,M[i][j]);M[r][j]=z<0?z+prime:z;
+     }
+     M[r][c]=0;
+    }
+   }
+   count++;
+   if(ok){good++;rowgood++;product=mul(product,determinant);}
+  }
+  assert(product);
+  if(records++)cout<<",\n";
+  cout<<"["<<aa<<","<<bb<<","<<cc<<","<<rowgood<<","<<product<<"]";
+  if(records%250==0)cerr<<"checked first triples "<<records<<" of 1771\n";
+ }
+ assert(count==4073300 && records==1771);
+ cout<<"\n],\"pairs\":"<<count<<",\"good\":"<<good<<",\"fallback\":"<<rejected.size()
+     <<",\"nonzero_pivots\":"<<checked<<",\"ordered_zero_checks\":"<<zeros<<",\"fallback_pairs\":[";
+ for(size_t i=0;i<rejected.size();i++){
+  if(i)cout<<",";cout<<"[";
+  for(int j=0;j<7;j++){if(j)cout<<",";cout<<rejected[i][j];}cout<<"]";
+ }
+ cout<<"]}\n";
+}
diff --git a/research/skip-strips/data_recovery.py b/research/skip-strips/data_recovery.py
new file mode 100644
index 0000000000..38972cab7e
--- /dev/null
+++ b/research/skip-strips/data_recovery.py
@@ -0,0 +1,51 @@
+"""Exact rational recovery of the ten primary-prime data witnesses.
+
+The ordered pivots are James Chang's PR34 reversed geometry. PR40/42 already
+used a second prime to resolve these same finite-field failures. Here a
+separate Fraction elimination certifies every ordered pivot and every
+required zero, with no modular inference for the recovered pairs.
+Prepared with OpenAI Codex assistance. Apache-2.0.
+"""
+from fractions import Fraction as Q
+from functools import lru_cache
+
+
+@lru_cache(None)
+def recover(pairs):
+    a,b,d=23,25,47
+    pivots=[2*a]+[i+a+1 for i in range(1,a-1)]
+    pivots += [2*a-i for i in range(a-1,a+5)]
+    pivots += [i-a-3 for i in range(a+5,2*a-1)]+[1,0]
+    assert len(set(pivots))==d
+    records=[]
+    for pair in pairs:
+        aa,bb,cc,u,v,w,failed_row=pair
+        S,T={aa,bb,cc},{u,v,w}
+        x=[Q(3*(a+1),2*(3*(a+1)-10)) if i in S else -Q(a+1,5) for i in range(a)]
+        y=[Q(3*(b+1),2*(3*(b+1)-10)) if i in T else -Q(b+1,5) for i in range(b)]
+        # Exact PR34 controlled-permutation restrictions, including the
+        # physical offset of the last-d column bank.
+        row_a=list(range(a))+[a-1]+list(range(a))
+        col_a=list(range(a))+[0]+list(range(a))
+        matrix=[[Q(-1)+(x[row_a[i]] if row_a[i]==col_a[j] else 0)
+                 +(y[i%b] if i%b==(a*b-d+j)%b else 0)
+                 for j in range(d)] for i in range(d)]
+        available=set(range(d)); values=[]; zero_checks=0
+        for i,c in enumerate(pivots):
+            assert c in available
+            value=matrix[i][c]
+            if not value:raise ValueError('Rational data pivot is zero')
+            for j in available:
+                if j>c:
+                    if matrix[i][j]:raise ValueError('Ordered data zero failed')
+                    zero_checks+=1
+            values.append(str(value));available.remove(c)
+            for r in range(i+1,d):
+                ratio=matrix[r][c]/value
+                if ratio:
+                    for j in available:
+                        if matrix[i][j]:matrix[r][j]-=ratio*matrix[i][j]
+                    matrix[r][c]=Q(0)
+        records.append(dict(pair=list(pair[:6]),primary_failed_row=failed_row,
+                            pivots=values,ordered_zero_checks=zero_checks))
+    return records
diff --git a/research/skip-strips/links-23.uses b/research/skip-strips/links-23.uses
new file mode 100644
index 0000000000..4b807fb5a7
Binary files /dev/null and b/research/skip-strips/links-23.uses differ
diff --git a/research/skip-strips/links-25.uses b/research/skip-strips/links-25.uses
new file mode 100644
index 0000000000..effbc2e7e5
Binary files /dev/null and b/research/skip-strips/links-25.uses differ
diff --git a/research/skip-strips/optimize_matching.py b/research/skip-strips/optimize_matching.py
new file mode 100644
index 0000000000..ef5bfeb733
--- /dev/null
+++ b/research/skip-strips/optimize_matching.py
@@ -0,0 +1,60 @@
+#!/usr/bin/env python3
+"""Regenerate the pinned weighted carrier matchings for the skip-prefix DAGs.
+
+Rohan Arun's PR #44 optimizer, unchanged in substance: the PR #44 profiler
+(references/copied-fixed/pr44/profiles.cpp) in DUMP_EDGES mode lists every
+admissible (donor,use) edge with its exact signed block change; we maximize
+sum c_t*t*ln t over all maximum-cardinality matchings with scipy's
+min_weight_full_bipartite_matching (each donor gets a private dummy column).
+Only the graph import changes. Requires scipy; not needed to *check* the
+pinned links, which producer.py replays and validates exactly.
+Usage: python3 optimize_matching.py --work-dir DIR  (writes DIR/links-{h}.uses)
+"""
+import argparse, math, os, shlex, struct, subprocess, sys
+from pathlib import Path
+import numpy as np
+from scipy.sparse import csr_matrix
+from scipy.sparse.csgraph import min_weight_full_bipartite_matching, maximum_bipartite_matching
+HERE = Path(__file__).resolve().parent; ROOT = HERE.parents[1]
+sys.path.insert(0, str(ROOT/'scripts')); sys.path.insert(0, str(HERE))
+from partial_swap.graph import export
+from skip_graph import graph
+
+
+def solve(edges_path, h):
+    E = []
+    for line in open(edges_path):
+        p = line.split(); x, e, j = map(int, p[:3])
+        d = {int(k): int(v) for k, v in (q.split(':') for q in p[3:])}
+        assert sum(t*c for t, c in d.items()) == -h
+        E.append((x, j, e, sum(c*t*math.log(t) for t, c in d.items())))
+    donors = sorted({x for x, *_ in E}); uses = sorted({j for _, j, *_ in E})
+    di = {x: i for i, x in enumerate(donors)}; ui = {j: i for i, j in enumerate(uses)}
+    nd, nu = len(donors), len(uses)
+    r = [di[x] for x, *_ in E]; c = [ui[j] for _, j, *_ in E]
+    card = int((maximum_bipartite_matching(csr_matrix((np.ones(len(E)), (r, c)), shape=(nd, nu)), perm_type='column') >= 0).sum())
+    ws = np.array([w for *_, w in E]); C = ws.max()+1.0
+    M = csr_matrix((list(C-ws)+[C+1e6]*nd, (r+list(range(nd)), c+[nu+i for i in range(nd)])), shape=(nd, nu+nd))
+    ri, ci = min_weight_full_bipartite_matching(M)
+    emap = {(di[x], ui[j]): (x, e) for x, j, e, _ in E}
+    sel = sorted(emap[(i, k)] for i, k in zip(ri.tolist(), ci.tolist()) if k < nu)
+    assert len(sel) == card
+    return sel
+
+
+if __name__ == '__main__':
+    ap = argparse.ArgumentParser(); ap.add_argument('--work-dir', type=Path, required=True); a = ap.parse_args()
+    a.work_dir.mkdir(parents=True, exist_ok=True); exe = a.work_dir/'profiles-dump'
+    subprocess.run([*shlex.split(os.environ.get('CXX', 'c++')), '-O3', '-std=c++17', '-include', 'algorithm',
+                    '-I', str(ROOT/'scripts/partial_swap'), str(ROOT/'references/copied-fixed/pr44/profiles.cpp'), '-o', str(exe)], check=True)
+    for h in (23, 25):
+        dag = a.work_dir/f'h{h}.bin'; export(graph(h), dag)
+        subprocess.run([str(exe), str(dag)], env=dict(os.environ, DUMP_EDGES=str(a.work_dir/f'edges{h}.txt')), check=True,
+                       stdout=subprocess.DEVNULL, stderr=subprocess.DEVNULL)
+        sel = solve(a.work_dir/f'edges{h}.txt', h)
+        n = struct.unpack_from('<4I', dag.read_bytes()[:16])[2]
+        out = a.work_dir/f'links-{h}.uses'
+        out.write_bytes(struct.pack('<2I', n, len(sel))+b''.join(struct.pack('<2I', x, e) for x, e in sel))
+        pinned = HERE/f'links-{h}.uses'
+        same = pinned.exists() and sorted(struct.iter_unpack('<2I', out.read_bytes()[8:])) == sorted(struct.iter_unpack('<2I', pinned.read_bytes()[8:]))
+        print(f'h={h} matched {len(sel)}; equals pinned links: {same}')
diff --git a/research/skip-strips/original-23.json b/research/skip-strips/original-23.json
new file mode 100644
index 0000000000..c814cbb766
--- /dev/null
+++ b/research/skip-strips/original-23.json
@@ -0,0 +1,38 @@
+{
+  "R": 32946,
+  "baseline_R": 45665,
+  "c": 40329,
+  "h": 23,
+  "histogram": [
+    5474,
+    41847,
+    36573,
+    6049,
+    16095,
+    7215,
+    4218,
+    1886,
+    3482,
+    833,
+    2865,
+    273,
+    2456,
+    289,
+    3527,
+    829,
+    2257,
+    994,
+    8640,
+    5873,
+    0,
+    165,
+    23,
+    23
+  ],
+  "loss": 506,
+  "matched": 12719,
+  "orientation_changes": 8904,
+  "q": 5336,
+  "rank_sum": 758770,
+  "v": 1771
+}
diff --git a/research/skip-strips/original-25.json b/research/skip-strips/original-25.json
new file mode 100644
index 0000000000..f2877eb031
--- /dev/null
+++ b/research/skip-strips/original-25.json
@@ -0,0 +1,40 @@
+{
+  "R": 43409,
+  "baseline_R": 60400,
+  "c": 53475,
+  "h": 25,
+  "histogram": [
+    7100,
+    54426,
+    47721,
+    7718,
+    21976,
+    9097,
+    4072,
+    3033,
+    5370,
+    893,
+    2724,
+    468,
+    3707,
+    468,
+    2751,
+    168,
+    4431,
+    1536,
+    2945,
+    918,
+    11304,
+    7596,
+    0,
+    221,
+    25,
+    25
+  ],
+  "loss": 600,
+  "matched": 16991,
+  "orientation_changes": 12339,
+  "q": 6925,
+  "rank_sum": 1086425,
+  "v": 2300
+}
diff --git a/research/skip-strips/original_timeline.py b/research/skip-strips/original_timeline.py
new file mode 100644
index 0000000000..0fb348896e
--- /dev/null
+++ b/research/skip-strips/original_timeline.py
@@ -0,0 +1,341 @@
+# Adapted from RaD / hipotures PR41 check_compiled_witness.py.
+# Original envelope bases replace positive-label input. Full dirty basis checks
+# now run at both h=23 and h=25; timing floats are excluded from certificates.
+#!/usr/bin/env python3
+"""Independent explicit role compiler for exported positive-frame witnesses.
+
+No producer allocator, carrier matcher or rank-histogram function is imported.
+Signed descriptions are converted to integer rational bases. Logical source
+coefficients, every physical role trajectory, terminal incidence and both
+dirty orientations are recomputed directly from binary DAG and selected uses.
+"""
+from __future__ import annotations
+
+import argparse
+import array
+from collections import Counter
+from datetime import datetime, timezone
+from functools import lru_cache
+from hashlib import sha256
+from itertools import combinations
+import json
+from pathlib import Path
+import struct
+import sys
+import time
+
+
+def read_array(stream, code, count):
+    result = array.array(code)
+    result.frombytes(stream.read(count*result.itemsize))
+    assert len(result) == count and sys.byteorder == 'little'
+    return result
+
+
+def read_dag(path):
+    with Path(path).open('rb') as stream:
+        h,v,n,q = struct.unpack('<4I',stream.read(16))
+        args = read_array(stream,'I',2*n)
+        core = read_array(stream,'Q',n)
+        cover = read_array(stream,'Q',n)
+        roots = read_array(stream,'I',q)
+        kinds = read_array(stream,'I',q)
+        active = read_array(stream,'B',n)
+        assert not stream.read()
+    return h,v,n,q,args,core,cover,roots,kinds,active
+
+
+def original_labels(h,n,core,cover,active):
+    ranks=[0]*n;frames=[None]*n
+    for node in range(1,n):
+        if not active[node]:continue
+        c=core[node].bit_count();outside=cover[node]&~core[node]
+        if c==3:
+            assert core[node]==cover[node]
+            ranks[node]=1;symbols=tuple(int(core[node]>>i&1) for i in range(h))
+        else:
+            assert c in (1,2)
+            ranks[node]=outside.bit_count()
+            symbols=tuple(1 if core[node]>>i&1 else i+2 if outside>>i&1 else 0 for i in range(h))
+        frames[node]=(ranks[node],core[node],symbols)
+    return ranks,frames
+
+
+@lru_cache(maxsize=None)
+def basis(frame):
+    rank,forced,symbols = frame
+    h = len(symbols)
+    fs = [i for i in range(h) if forced >> i & 1]
+    if len(fs) == 3:
+        assert rank == 1 and all(symbols[i] == int(i in fs) for i in range(h))
+        return (tuple(int(i in fs) for i in range(h)),)
+    assert len(fs) in (1,2)
+    assert all(symbols[i] == 1 for i in fs)
+    classes = sorted({abs(s) for s in symbols if abs(s)>1})
+    assert len(classes) == rank
+    denom = 3-len(fs)
+    rows = []
+    for label in classes:
+        row = [0]*h
+        delta = sum(1 if s == label else -1 if s == -label else 0 for s in symbols)
+        for i,s in enumerate(symbols):
+            if i in fs:
+                row[i] = delta
+            elif abs(s) == label:
+                row[i] = denom*(1 if s>0 else -1)
+        assert any(row)
+        rows.append(tuple(row))
+    return tuple(rows)
+
+
+def vector_in(vector,frame):
+    rank,forced,symbols = frame
+    fs = [i for i in range(len(vector)) if forced >> i & 1]
+    if len(fs) == 3:
+        return all(vector[i] == (vector[fs[0]] if i in fs else 0) for i in range(len(vector)))
+    t = vector[fs[0]]
+    if sum(vector) != 3*t or any(vector[i] != t for i in fs):
+        return False
+    values = {}
+    for i,s in enumerate(symbols):
+        if i in fs:
+            continue
+        if not s:
+            if vector[i]:
+                return False
+        else:
+            assert abs(s)>1
+            value = vector[i]*(1 if s>0 else -1)
+            if abs(s) in values and values[abs(s)] != value:
+                return False
+            values[abs(s)] = value
+    return True
+
+
+@lru_cache(maxsize=300000)
+def contained(a,b):
+    return all(vector_in(vector,b) for vector in basis(a))
+
+
+def check(document, small_dirty=False):
+    row = document['producer'] if 'producer' in document else document
+    native = document.get('selected_links') or json.loads(Path(row['witness_path']).read_text())
+    h,v,n,q,args,core,cover,roots,kinds,active = read_dag(row['dag_path'])
+    ranks,frames = original_labels(h,n,core,cover,active)
+    triples = list(combinations(range(h),3))
+    assert len(triples) == v
+    active_nodes = [node for node in range(1,n) if active[node]]
+    additions = sum(bool(args[2*node]) for node in active_nodes)
+    expected_roles = additions+q-len(native['links'])
+    assert expected_roles == row['R']
+    scalar = [0]*n
+    for node in active_nodes:
+        if not args[2*node]:
+            assert node <= v
+            scalar[node] = 1 << (node-1)
+            vector = tuple(int(i in triples[node-1]) for i in range(h))
+            assert basis(frames[node]) == (vector,)
+        else:
+            a,b = args[2*node:2*node+2]
+            assert a<node and b<node and not scalar[a] & scalar[b]
+            scalar[node] = scalar[a] | scalar[b]
+            assert contained(frames[a],frames[node]) and contained(frames[b],frames[node])
+        for vector in basis(frames[node]):
+            common = (frames[node][1]&-frames[node][1]).bit_length()-1
+            assert sum(vector) == 3*vector[common]
+            assert sum(x*x for i,x in enumerate(vector) if i != common)>0
+    output_targets = []
+    center_nodes = []
+    for j,node in enumerate(roots):
+        common = (core[node]&-core[node]).bit_length()-1
+        if kinds[j]:
+            assert core[node].bit_count()==1 and ranks[node]==h-1
+            expected = sum(1<<i for i,t in enumerate(triples) if common in t)
+            output_targets.append([i for i,t in enumerate(triples) if common in t])
+            center_nodes.append(node)
+        else:
+            absent = ((1<<h)-1)^cover[node]
+            assert absent.bit_count()==2
+            excluded = [i for i in range(h) if absent>>i&1]
+            target = tuple(sorted([common,*excluded]))
+            expected = sum(1<<i for i,t in enumerate(triples) if common in t and not set(t)&set(excluded))
+            output_targets.append([triples.index(target)])
+            for vector in basis(frames[node]):
+                assert 3*sum(vector[i] for i in target) == sum(vector)
+        assert scalar[node] == expected
+    assert len(center_nodes)==h and len(set(center_nodes))==h
+    uses = [[] for _ in range(n)]
+    for node in active_nodes:
+        if args[2*node]:
+            for pos in (0,1):
+                uses[args[2*node+pos]].append(2*node+pos)
+    for j,node in enumerate(roots):
+        uses[node].append((1<<31)|j)
+    for node in center_nodes:
+        assert len(uses[node])==1, 'Retained center has a later middle producer incidence'
+
+    def use_node(e):
+        return roots[e&0x7fffffff] if e>>31 else e//2
+
+    def use_value(e):
+        return use_node(e) if e>>31 else args[e]
+
+    def order_key(node):
+        old = 1 if not args[2*node] else cover[node].bit_count()-core[node].bit_count()
+        return ranks[node],old,node
+
+    successor = {}
+    receivers = set()
+    retained = {}
+    for donor,receiver in native['links']:
+        assert donor not in retained and receiver not in receivers and args[2*donor]
+        value = use_value(receiver)
+        positions = [pos for pos in (0,1) if args[2*donor+pos] == value]
+        assert len(positions)==1
+        previous = 2*donor+positions[0]
+        assert previous not in successor
+        target = use_node(receiver)
+        assert contained(frames[donor],frames[target])
+        if not receiver>>31:
+            assert order_key(donor)<order_key(target)
+        successor[previous] = receiver
+        receivers.add(receiver)
+        retained[donor] = positions[0]
+
+    order = sorted(active_nodes,key=order_key)
+    place = {node:i for i,node in enumerate(order)}
+    edge_slots = {}
+    sources = {}
+    gates = []
+    slot_frames = []
+    slot_values = []
+    histogram = Counter()
+    transitions = 0
+
+    def allocate(frame,value):
+        slot = len(slot_frames)
+        slot_frames.append(frame)
+        slot_values.append(value)
+        histogram[frame[0]] += 1
+        return slot
+
+    def assign_chain(first,slot,value):
+        seen = set()
+        e = first
+        while True:
+            assert e not in seen and e not in edge_slots and use_value(e)==value
+            seen.add(e)
+            edge_slots[e] = slot
+            if e not in successor:
+                break
+            e = successor[e]
+
+    for node in order:
+        starts = [e for e in uses[node] if e not in receivers]
+        assert starts
+        if not args[2*node]:
+            first = allocate(frames[node],scalar[node])
+            outs = [first]+[allocate(frames[node],scalar[node]) for _ in starts[1:]]
+            ins = [first]
+            sources[node-1] = first
+        else:
+            a,b = args[2*node:2*node+2]
+            assert place[a]<place[node] and place[b]<place[node]
+            ins = [edge_slots[2*node],edge_slots[2*node+1]]
+            assert ins[0]!=ins[1]
+            for operand,slot in zip((a,b),ins):
+                assert slot_values[slot]==scalar[operand]
+                assert contained(slot_frames[slot],frames[node])
+                histogram[ranks[node]-slot_frames[slot][0]] += 1
+                transitions += 1
+                slot_frames[slot] = frames[node]
+            pivot_pos = 1-retained[node] if node in retained else 0
+            pivot = ins[pivot_pos]
+            controller = ins[1-pivot_pos]
+            slot_values[pivot] ^= slot_values[controller]
+            assert slot_values[pivot]==scalar[node]
+            outs = [pivot]+[allocate(frames[node],scalar[node]) for _ in starts[1:]]
+            if node not in retained:
+                histogram[h-ranks[node]] += 1
+                slot_frames[controller] = None
+            ins = [pivot,controller]
+        for e,slot in zip(starts,outs):
+            assign_chain(e,slot,node)
+        gates.append((node,tuple(ins),tuple(outs)))
+    assert len(slot_frames)==expected_roles
+    output_slots = []
+    scatter = [0]*v
+    for j,node in enumerate(roots):
+        slot = edge_slots[(1<<31)|j]
+        assert slot_values[slot]==scalar[node] and contained(slot_frames[slot],frames[node])
+        histogram[ranks[node]-slot_frames[slot][0]] += 1
+        transitions += 1
+        if kinds[j]:
+            histogram[ranks[node]] += 1
+            histogram[h-ranks[node]] += 1  # Copied retained center: paid copy and direct cleanup.
+        else:
+            histogram[h-1-ranks[node]] += 1
+            histogram[1] += 1
+        slot_frames[slot] = None
+        output_slots.append(slot)
+        for target in output_targets[j]:
+            scatter[target] ^= scalar[node]
+    assert scatter == [1<<i for i in range(v)], 'JLV must be the complete scalar identity'
+    assert all(frame is None for frame in slot_frames), 'Every physical role must terminate'
+    reverse_frames = [None]*expected_roles
+    for j,slot in enumerate(output_slots):
+        assert reverse_frames[slot] is None, 'Designated output terminals must have separate physical roles'
+        reverse_frames[slot] = frames[roots[j]]
+    reverse_checks = 0
+    for node,ins,outs in reversed(gates):
+        for slot in set(ins+outs):
+            assert reverse_frames[slot] is None or contained(frames[node],reverse_frames[slot])
+            reverse_frames[slot] = frames[node]
+            reverse_checks += 1
+    for input_id,slot in sources.items():
+        assert contained(reverse_frames[slot],frames[input_id+1]) and contained(frames[input_id+1],reverse_frames[slot])
+    actual_hist = [histogram[r] for r in range(h+1)]
+    expected_hist = list(row['histogram'])
+    expected_hist[1] += h
+    expected_hist[h] -= h
+    assert actual_hist[1:]==expected_hist[1:], (actual_hist,expected_hist)
+    assert sum(r*n for r,n in enumerate(actual_hist)) == h*expected_roles+h*(h-1)
+    result = dict(status='independent scalar, rational frames, physical compiler and rank timeline PASS',
+                  h=h,v=v,additions=additions,designated_outputs=q,retained_links=len(native['links']),
+                  roles=expected_roles,center_terminal_count=h,center_rank=h-1,
+                  full_output_coefficients=v*v,physical_frame_transitions=transitions,
+                  reverse_complement_frame_incidences=reverse_checks,
+                  copied_histogram=actual_hist,copied_rank_sum=sum(r*n for r,n in enumerate(actual_hist)),
+                  native_zero_rank_bookkeeping=expected_hist[0],
+                  zero_rank_scope='Explicit output incidences include no-operation frame checks omitted from native histogram bookkeeping; only positive ranks enter the recurrence',
+                  rational_frame_classes=len({frames[node] for node in active_nodes}),
+                  selected_links_sha256=sha256(json.dumps(native,sort_keys=True,separators=(',',':')).encode()).hexdigest(),
+                  dag_sha256=sha256(Path(row['dag_path']).read_bytes()).hexdigest(),
+                  containment_cache=contained.cache_info()._asdict())
+    if small_dirty:
+        R = expected_roles
+        initial = [1<<i for i in range(2*v+R)]
+        mixer = []
+        for _,ins,outs in gates:
+            if len(ins)==2:
+                mixer.append((2*v+ins[0],2*v+ins[1]))
+            for slot in outs[1:]:
+                mixer.append((2*v+slot,2*v+ins[0]))
+        J = [(v+out,2*v+slot) for j,slot in enumerate(output_slots) for out in output_targets[j]]
+        V = [(2*v+slot,input_id) for input_id,slot in sources.items()]
+        word = mixer+J+list(reversed(mixer))+V+mixer+J+list(reversed(mixer))+V
+        forward = initial.copy()
+        for target,source in word:
+            assert target!=source
+            forward[target] ^= forward[source]
+        assert forward[:v] == initial[:v] and forward[2*v:] == initial[2*v:]
+        assert forward[v:2*v] == [initial[v+i]^initial[i] for i in range(v)]
+        dual = initial.copy()
+        for target,source in reversed(word):
+            dual[source] ^= dual[target]
+        assert dual[v:2*v] == initial[v:2*v] and dual[2*v:] == initial[2*v:]
+        assert dual[:v] == [initial[i]^initial[v+i] for i in range(v)]
+        result['complete_dirty_basis'] = dict(auxiliary_basis_vectors=R,total_basis_vectors=2*v+R,
+                                             elementary_word_length=len(word),inputs=v,targets=v,
+                                             orientations=['forward','reverse-complement'],all_dirty_restore=True)
+    return result
diff --git a/research/skip-strips/producer-receipt.json b/research/skip-strips/producer-receipt.json
new file mode 100644
index 0000000000..76f0f03826
--- /dev/null
+++ b/research/skip-strips/producer-receipt.json
@@ -0,0 +1,948 @@
+{
+  "23": {
+    "dense_audit": {
+      "envelope_inclusions": 80658,
+      "exact_global_additions": 40329,
+      "exact_global_outputs": 5336,
+      "retained_centers": 23,
+      "source_lines": 1771
+    },
+    "fixed_profile": {
+      "R": 32946,
+      "blocks": [
+        0,
+        291605,
+        22062,
+        15648,
+        9468,
+        7684,
+        6081,
+        6478,
+        4554,
+        3847,
+        2897,
+        2814,
+        1950,
+        1510,
+        1052,
+        772,
+        495,
+        276,
+        174,
+        0,
+        45,
+        23,
+        0,
+        23
+      ],
+      "crt_disagreements": 0,
+      "crt_matrices": 5642,
+      "distinct_matrices": 61461,
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+        16,
+        18
+      ],
+      "pivots": [
+        "-1",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "18/31",
+        "18/31",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-30425585565647987/8112000278200320",
+        "-6915016711890063397419754697651/61258776880172525541811814400",
+        "58034460589224410256212/377247763587426343740975",
+        "-24460859725227921591208225587816/99865714290552512474992173653",
+        "-5616114623039263977852817637376/566223604750646333129820036755",
+        "4872067367087112192/31397326122812625191",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "39/68",
+        "-26/5",
+        "39/68",
+        "-26/5",
+        "39/68",
+        "-26/5",
+        "-26/5",
+        "-68016/1925",
+        "936/545"
+      ],
+      "primary_failed_row": 25
+    },
+    {
+      "ordered_zero_checks": 346,
+      "pair": [
+        12,
+        13,
+        22,
+        4,
+        6,
+        10
+      ],
+      "pivots": [
+        "-1",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "18/31",
+        "18/31",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "-24/5",
+        "1257473477600655419/1257360043121049600",
+        "697637898272342204364754225328501/898378192091765621530873036800",
+        "4862910615710407729752452/413821272486424062729765",
+        "-211879701118736765516260639861416/69965631010509871166615364966895",
+        "-1040817578056012397961768075264/140132077459481987775304655993",
+        "348004811934793728/2909385858910392337",
+        "39/68",
+        "-26/5",
+        "39/68",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "39/68",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-26/5",
+        "-68016/1925",
+        "936/545"
+      ],
+      "primary_failed_row": 26
+    }
+  ]
+}
diff --git a/research/skip-strips/producer.py b/research/skip-strips/producer.py
new file mode 100644
index 0000000000..1f0b0bd257
--- /dev/null
+++ b/research/skip-strips/producer.py
@@ -0,0 +1,216 @@
+#!/usr/bin/env python3
+"""Full graph/profile replay and an independent dense-support/count audit.
+
+Inherited generators: icekylinx PR18/32/36; profiler: PR32 and Dominik
+Scholz PR35. This audit uses direct global bitsets rather than compressed
+producer provenance and replays every exported matching use independently.
+"""
+from collections import Counter
+from itertools import combinations
+from pathlib import Path
+import argparse
+import gc
+import json
+import os
+import shlex
+import struct
+import subprocess
+import sys
+import tempfile
+
+from verify import ROOT, HERE, require, read, check_sources, exactness
+from partial_swap.graph import export
+from skip_graph import graph
+import original_timeline as timeline
+
+
+def dense_audit(c):
+    h = c.h
+    triples = list(combinations(range(h),3))
+    require(triples == c.inputs, 'Wrong input enumeration')
+    masks = [sum(1<<i for i in t) for t in triples]
+    supports = [0]*len(c.args)
+    degree = [0]*len(c.args)
+    ranks = [0]*len(c.args)
+    additions = 0
+    for x in sorted(c.active):
+        args = c.args[x]
+        require(c.core[x] and not c.core[x] & ~c.union[x], 'Invalid envelope')
+        if args is None:
+            require(1<=x<=len(triples) and c.core[x] == c.union[x] == masks[x-1], 'Source label')
+            supports[x] = 1<<(x-1)
+            ranks[x] = 1
+        else:
+            left,right = args
+            require(left<x and right<x and left in c.active and right in c.active, 'DAG order')
+            require(not supports[left]&supports[right], 'Overlapping exact source supports')
+            supports[x] = supports[left]|supports[right]
+            require(c.core[x] == c.core[left]&c.core[right], 'Wrong core')
+            require(c.union[x] == c.union[left]|c.union[right], 'Wrong cover')
+            require(c.core[x].bit_count() in (1,2), 'Invalid common core')
+            ranks[x] = c.union[x].bit_count()-c.core[x].bit_count()
+            for y in args:
+                require(not c.core[x]&~c.core[y] and not c.union[y]&~c.union[x], 'Envelope inclusion')
+                degree[y] += 1
+            additions += 1
+    outputs = list(c.outputs.items())
+    totals = 0
+    for ((common,target),node) in outputs:
+        omitted = set(target)-{common}
+        expected = sum(1<<j for j,t in enumerate(triples)
+                       if common in t and not omitted.intersection(t))
+        require(supports[node] == expected, 'Wrong exact global output')
+        require(c.core[node] == 1<<common, 'Output common point')
+        require(c.union[node] == ((1<<h)-1)-sum(1<<i for i in omitted), 'Output envelope')
+        totals += not omitted
+        degree[node] += 1
+    require(totals == h, 'Missing or duplicate centers')
+    return degree,ranks,outputs,dict(exact_global_additions=additions,
+        exact_global_outputs=len(outputs),envelope_inclusions=2*additions,
+        retained_centers=totals,source_lines=len(triples))
+
+
+def recount(c, degree, ranks, outputs, link_path):
+    h,n = c.h,len(c.args)
+    hist = Counter()
+    for x in c.active:
+        r = ranks[x]
+        require(degree[x]>0, 'Dead node')
+        if c.args[x] is None:
+            hist[1] += degree[x]
+        else:
+            hist[r] += degree[x]-1
+            hist[h-r] += 1
+            for y in c.args[x]:
+                hist[r-ranks[y]] += 1
+    for ((common,target),node) in outputs:
+        r = ranks[node]
+        if len(target) == 1:
+            hist[r] += 1
+            hist[h] += 1
+        else:
+            hist[h-1-r] += 1
+            hist[1] += 1
+    raw = link_path.read_bytes()
+    nn,nlinks = struct.unpack_from('<2I',raw)
+    require(nn == n and len(raw) == 8+8*nlinks, 'Matching record format')
+    donors,uses = set(),set()
+    orientation_changes = 0
+    for i in range(nlinks):
+        donor,use = struct.unpack_from('<2I',raw,8+8*i)
+        require(donor not in donors and use not in uses, 'Matching collision')
+        donors.add(donor)
+        uses.add(use)
+        require(donor in c.active and c.args[donor] is not None, 'Invalid donor')
+        if use>>31:
+            j = use&0x7fffffff
+            require(j<len(outputs), 'Invalid terminal use')
+            target = outputs[j][1]
+            value,event = target,n+j
+        else:
+            target = use//2
+            require(target in c.active and c.args[target] is not None, 'Invalid consumer')
+            value,event = c.args[target][use%2],target
+        require(value in c.args[donor], 'Carrier does not preserve operand')
+        require((ranks[donor],donor)<(ranks[target],event), 'Noncausal carrier')
+        require(not c.core[target]&~c.core[donor] and not c.union[donor]&~c.union[target],
+                'Carrier frames not nested')
+        ru,rv,rt = ranks[donor],ranks[value],ranks[target]
+        require(rv<=ru<=rt, 'Carrier rank order')
+        hist[h-ru] -= 1
+        hist[rv] -= 1
+        hist[rt-rv] -= 1
+        hist[rt-ru] += 1
+        orientation_changes += value == c.args[donor][0]
+    require(all(n>=0 for n in hist.values()), 'Negative physical multiplicity')
+    R = c.additions+len(outputs)-nlinks
+    mass = sum(t*n for t,n in hist.items())
+    require(mass == h*R+2*h*(h-1), 'Independent old rank accounting')
+    return dict(h=h,v=len(c.inputs),c=c.additions,q=len(outputs),baseline_R=c.additions+len(outputs),
+                matched=nlinks,R=R,orientation_changes=orientation_changes,
+                rank_sum=mass,loss=h*(h-1),histogram=[hist[r] for r in range(h+1)])
+
+
+RECORD = False
+
+
+def expect(name, value, message):
+    """Compare with the recorded JSON; in --record mode write it first."""
+    path = HERE/name
+    if RECORD:
+        path.write_text(json.dumps(value, indent=2, sort_keys=True)+'\n')
+    require(value == read(path), message)
+
+
+def run(work):
+    require(not sys.flags.optimize, 'Assertions must remain enabled')
+    check_sources()
+    exe = work/'profiles'
+    subprocess.run([*shlex.split(os.environ.get('CXX','c++')),'-O3','-std=c++17',
+                    '-I',str(ROOT/'scripts/partial_swap'),str(HERE/'profiles.cpp'),'-o',str(exe)],check=True)
+    receipt = {}
+    for h in (23,25):
+        print('Rebuilding scalar graph, original labels and fixed profiles, h='+str(h),flush=True)
+        c = graph(h)
+        scalar = c.verify()
+        expect(f'scalar-{h}.json', scalar, 'Scalar provenance mismatch')
+        degree,ranks,outputs,audit = dense_audit(c)
+        dag,links = work/f'h{h}.bin',work/f'h{h}.uses'
+        export(c,dag)
+        original = json.loads(subprocess.check_output([str(exe),str(dag),str(links)],text=True,
+                    env=dict(os.environ,LINKS_IN=str(HERE/f'links-{h}.uses'))))
+        require(sorted(struct.iter_unpack('<2I',links.read_bytes()[8:])) ==
+                sorted(struct.iter_unpack('<2I',(HERE/f'links-{h}.uses').read_bytes()[8:])),
+                'Pinned matching changed')
+        independent = recount(c,degree,ranks,outputs,links)
+        require(original == independent, 'Independent physical recount failed')
+        expect(f'original-{h}.json', original, 'Recorded physical recount mismatch')
+        actual = read(Path(str(dag)+'.round3_rankone_certified_profiles.json'))
+        expect(f'profiles-{h}.json', actual, 'Fixed profile rebuild mismatch')
+        exactness(h)
+        raw=links.read_bytes()
+        nodes,count=struct.unpack_from('<2I',raw)
+        selected=[list(struct.unpack_from('<2I',raw,8+8*i)) for i in range(count)]
+        row=dict(original,dag_path=str(dag))
+        timeline.basis.cache_clear();timeline.contained.cache_clear()
+        compiled=timeline.check({'producer':row,'selected_links':{'links':selected}},True)
+        expect(f'timeline-{h}.json', compiled, 'Independent full timeline mismatch')
+        timeline.basis.cache_clear();timeline.contained.cache_clear()
+        print('PASS complete physical timeline and every dirty basis vector, both orientations h='+str(h),flush=True)
+        receipt[str(h)] = dict(scalar=scalar,dense_audit=audit,independent_recount=independent,
+                          fixed_profile=actual,full_timeline=compiled)
+        c.support_in.cache_clear()
+        del c
+        gc.collect()
+        print('PASS complete h='+str(h)+' supports, all matching uses, rank count and CRT profiles',flush=True)
+    data_exe = work/'data-corners'
+    subprocess.run([*shlex.split(os.environ.get('CXX','c++')),'-O3','-std=c++17',
+                    str(HERE/'data_corners.cpp'),'-o',str(data_exe)],check=True)
+    print('Checking all 4,073,300 actual fixed-basis data pairs',flush=True)
+    actual_data=json.loads(subprocess.check_output([str(data_exe)],text=True))
+    require(actual_data==read(HERE/'data-corners.json'),'Full data-corner replay mismatch')
+    receipt['data_corners']=dict(pairs=actual_data['pairs'],good=actual_data['good'],
+                                fallback=actual_data['fallback'],nonzero_pivots=actual_data['nonzero_pivots'])
+    from verify import data_corners
+    recovered=data_corners()
+    require(recovered['good']==4073300 and recovered['fallback']==0,'Incomplete recovery')
+    receipt['exact_recovery']=recovered['exact_recovery']
+    print('PASS all data pairs; ten primary-prime failures recovered over Q',flush=True)
+    return receipt
+
+
+if __name__ == '__main__':
+    parser = argparse.ArgumentParser()
+    parser.add_argument('--work-dir',type=Path)
+    parser.add_argument('--output',type=Path)
+    parser.add_argument('--record',action='store_true',help='write the scalar/original/profile/timeline JSON records')
+    args = parser.parse_args()
+    RECORD = args.record
+    if args.work_dir:
+        args.work_dir.mkdir(parents=True,exist_ok=True)
+        result = run(args.work_dir)
+    else:
+        with tempfile.TemporaryDirectory(prefix='copied-fixed-') as directory:
+            result = run(Path(directory))
+    if args.output:
+        args.output.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n')
diff --git a/research/skip-strips/profiles-23.json b/research/skip-strips/profiles-23.json
new file mode 100644
index 0000000000..ad9ed7fff4
--- /dev/null
+++ b/research/skip-strips/profiles-23.json
@@ -0,0 +1,38 @@
+{
+  "R": 32946,
+  "blocks": [
+    0,
+    291605,
+    22062,
+    15648,
+    9468,
+    7684,
+    6081,
+    6478,
+    4554,
+    3847,
+    2897,
+    2814,
+    1950,
+    1510,
+    1052,
+    772,
+    495,
+    276,
+    174,
+    0,
+    45,
+    23,
+    0,
+    23
+  ],
+  "crt_disagreements": 0,
+  "crt_matrices": 5642,
+  "distinct_matrices": 61461,
+  "field_prime": 2305843009213693951,
+  "frames": 36628,
+  "h": 23,
+  "loss": 506,
+  "rank_sum": 758770,
+  "v": 1771
+}
diff --git a/research/skip-strips/profiles-25.json b/research/skip-strips/profiles-25.json
new file mode 100644
index 0000000000..3dde139a02
--- /dev/null
+++ b/research/skip-strips/profiles-25.json
@@ -0,0 +1,40 @@
+{
+  "R": 43409,
+  "blocks": [
+    0,
+    388456,
+    30787,
+    18638,
+    11459,
+    10643,
+    8798,
+    8690,
+    6041,
+    5697,
+    4489,
+    4439,
+    2976,
+    2637,
+    2027,
+    1629,
+    1063,
+    777,
+    544,
+    293,
+    185,
+    0,
+    55,
+    25,
+    0,
+    25
+  ],
+  "crt_disagreements": 0,
+  "crt_matrices": 7434,
+  "distinct_matrices": 82606,
+  "field_prime": 2305843009213693951,
+  "frames": 48677,
+  "h": 25,
+  "loss": 600,
+  "rank_sum": 1086425,
+  "v": 2300
+}
diff --git a/research/skip-strips/profiles.cpp b/research/skip-strips/profiles.cpp
new file mode 100644
index 0000000000..410e50f475
--- /dev/null
+++ b/research/skip-strips/profiles.cpp
@@ -0,0 +1,107 @@
+#include <algorithm>
+// Optimal-matching variant of research/copied-fixed/profiles.cpp (PR #43, Chafik
+// Boukhalfa), itself adapted from Dominik Scholz PR35 and icekylinx PR32.
+// Change: with LINKS_IN=<file>, the carrier matching is read from a pinned file
+// instead of Hopcroft-Karp. Every pinned edge must lie in the same admissible
+// adjacency (causal order and frame inclusion); donors and uses are distinct.
+// This replay-only adaptation omits the discovery edge dumper. The original
+// optimizer and profiler are preserved in references/copied-fixed/pr44.
+// Replay-only adaptation prepared with OpenAI Codex assistance.
+// Prepared by Rohan Arun with Anthropic Claude assistance. Apache-2.0.
+// Adapted at h=23,25; original PR35 by Dominik Scholz, PR32 by icekylinx.
+// Dimension adaptation at 45/47; extra primes for all source-growth cases.
+// Exact bounded-minor inequalities are checked by verify.py. Original credit below.
+// Copyright 2026 icekylinx. Apache-2.0; AI-assisted handoff integration.
+// Original round3_rankone_profiles.cpp. Requires GCC/Clang unsigned __int128.
+#include <algorithm>
+#include <array>
+#include <cassert>
+#include <cstdint>
+#include <fstream>
+#include <iostream>
+#include <vector>
+#include <map>
+#include <unordered_map>
+#include <cmath>
+#include <cstdlib>
+using U=uint32_t;using V=uint64_t;
+#include "binary_io.hpp"
+int main(int argc,char**argv){assert(argc==2||argc==3);std::ifstream f(argv[1],std::ios::binary);U hdr[4];read_array(f,hdr);U h=hdr[0],v=hdr[1],n=hdr[2],q=hdr[3];assert(h==23||h==25);
+std::vector<std::array<U,2>>args(n);std::vector<V>core(n),cover(n);std::vector<U>roots(q),kind(q);std::vector<uint8_t>active(n);readv(f,args);readv(f,core);readv(f,cover);readv(f,roots);readv(f,kind);readv(f,active);
+std::vector<U>ranks(n),degree(n),begin(n+1);V c=0,inputs=0,loss=0;
+for(U x=1;x<n;x++)if(active[x]){if(args[x][0]){c++;assert(args[x][0]<x&&args[x][1]<x);ranks[x]=popcount64(cover[x])-popcount64(core[x]);for(U y:args[x])degree[y]++;}else{ranks[x]=1;inputs++;}}
+assert(inputs==v);for(U x:roots)degree[x]++;for(U x=1;x<n;x++)begin[x+1]=begin[x]+degree[x];
+std::vector<U>uses(begin.back()),cursor(begin.begin(),begin.end()-1);for(U x=1;x<n;x++)if(active[x]&&args[x][0]){uses[cursor[args[x][0]]++]=2*x;uses[cursor[args[x][1]]++]=2*x+1;}for(U j=0;j<q;j++)uses[cursor[roots[j]]++]=(1U<<31)|j;
+auto nd=[&](U e)->U{return e>>31?roots[e&0x7fffffff]:e/2;};
+auto before=[&](U a,U b){U x=nd(a),y=nd(b);if(ranks[x]!=ranks[y])return ranks[x]<ranks[y];V ox=a>>31?V(n)+(a&0x7fffffff):x,oy=b>>31?V(n)+(b&0x7fffffff):y;return ox<oy;};
+auto incl=[&](U a,U b){U x=nd(a),y=nd(b);return !(core[y]&~core[x])&&!(cover[x]&~cover[y]);};
+auto adjacency=[&](U donor,auto&& action){for(U value:args[donor])for(U j=begin[value];j<begin[value+1];j++)if(before(donor*2,uses[j])&&incl(donor*2,uses[j]))if(action(j))return true;return false;};
+std::vector<U>donors;for(U x=1;x<n;x++)if(active[x]&&args[x][0]&&adjacency(x,[](U){return true;}))donors.push_back(x);
+// Descending donor order changes the retained-carrier network while preserving
+// the exhaustive admissibility checks and maximum matching cardinality.
+const char* rev=getenv("DONOR_REVERSE"); if(!rev||rev[0]!='0') std::reverse(donors.begin(),donors.end());
+std::vector<U>leftmatch(n),rightmatch(uses.size()),distance(n,UINT32_MAX),queue;V matches=0;U phase=0,inf=UINT32_MAX,shortest=inf;
+const char* lin=getenv("LINKS_IN");
+assert(lin && "A pinned carrier matching is required");
+if(lin){std::ifstream lf(lin,std::ios::binary);U hd[2];read_array(lf,hd);assert(hd[0]==n);std::vector<U>isdonor(n);for(U x:donors)isdonor[x]=1;for(U i=0;i<hd[1];i++){U ed[2];read_array(lf,ed);U x=ed[0],e=ed[1];assert(isdonor[x]&&!leftmatch[x]);U jj=UINT32_MAX;for(U value:args[x])for(U j=begin[value];j<begin[value+1];j++)if(uses[j]==e)jj=j;assert(jj!=UINT32_MAX);bool ok=false;adjacency(x,[&](U j){if(j==jj)ok=true;return ok;});assert(ok);assert(!rightmatch[jj]);leftmatch[x]=jj+1;rightmatch[jj]=x;matches++;}}
+else while(true){queue.clear();shortest=inf;for(U x:donors){if(!leftmatch[x]){distance[x]=0;queue.push_back(x);}else distance[x]=inf;}for(U at=0;at<queue.size();at++){U x=queue[at];if(distance[x]>=shortest)continue;adjacency(x,[&](U j){U y=rightmatch[j];if(!y)shortest=distance[x]+1;else if(distance[y]==inf){distance[y]=distance[x]+1;queue.push_back(y);}return false;});}if(shortest==inf)break;
+auto aug=[&](auto&&self,U x)->bool{bool ok=adjacency(x,[&](U j){U y=rightmatch[j];if((!y&&distance[x]+1==shortest)||(y&&distance[y]==distance[x]+1&&self(self,y))){leftmatch[x]=j+1;rightmatch[j]=x;return true;}return false;});if(!ok)distance[x]=inf;return ok;};V gained=0;for(U x:donors)if(!leftmatch[x]&&aug(aug,x)){matches++;gained++;}std::cerr<<"h="<<h<<" phase "<<++phase<<" length "<<shortest<<" gained "<<gained<<" total "<<matches<<"\n";assert(gained);}
+std::vector<int64_t>hist(h+1);for(U x=1;x<n;x++)if(active[x]){assert(degree[x]);U r=ranks[x];if(args[x][0]){hist[r]+=degree[x]-1;hist[h-r]++;for(U y:args[x]){assert(r>=ranks[y]);hist[r-ranks[y]]++;}}else hist[1]+=degree[x];}
+for(U j=0;j<q;j++){U r=ranks[roots[j]];if(kind[j]){hist[r]++;hist[h]++;loss+=r;}else{assert(r<=h-1);hist[h-1-r]++;hist[1]++;}}
+V changed=0;for(U donor:donors)if(leftmatch[donor]){U e=uses[leftmatch[donor]-1],target=nd(e),value=e>>31?target:args[target][e&1];assert(value==args[donor][0]||value==args[donor][1]);if(value==args[donor][0])changed++;U ru=ranks[donor],rv=ranks[value],rt=ranks[target];assert(rt>=ru&&ru>=rv);hist[h-ru]--;hist[rv]--;hist[rt-rv]--;hist[rt-ru]++;}
+V R=c+q-matches,sum=0;for(U r=0;r<=h;r++){assert(hist[r]>=0);sum+=r*hist[r];}assert(sum==h*R+2*loss);
+if(argc==3){std::ofstream out(argv[2],std::ios::binary);U header[2]={n,U(matches)};write_array(out,header);for(U donor:donors)if(leftmatch[donor]){U edge[2]={donor,uses[leftmatch[donor]-1]};write_array(out,edge);}assert(out);}
+std::cout<<"{\"h\":"<<h<<",\"v\":"<<v<<",\"c\":"<<c<<",\"q\":"<<q<<",\"baseline_R\":"<<c+q<<",\"matched\":"<<matches<<",\"R\":"<<R<<",\"orientation_changes\":"<<changed<<",\"rank_sum\":"<<sum<<",\"loss\":"<<loss<<",\"histogram\":[";for(U r=0;r<=h;r++){if(r)std::cout<<",";std::cout<<hist[r];}std::cout<<"]}"<<std::endl;
+// Reconstruct the actual transition multiset before changing the common basis.
+struct Frame{V core,cover;U rank;};std::vector<Frame>frames{{0,0,0},{0,0,h}};std::map<std::pair<V,V>,U>frame_lookup;std::vector<U>fi(n);
+for(U x=1;x<n;x++)if(active[x]){auto key=std::make_pair(core[x],cover[x]);auto z=frame_lookup.find(key);if(z==frame_lookup.end()){U id=frames.size();frames.push_back({core[x],cover[x],ranks[x]});frame_lookup[key]=id;fi[x]=id;}else fi[x]=z->second;}
+std::map<std::pair<U,U>,int64_t>transitions;int64_t singles=0;
+auto edge=[&](U a,U b,int64_t count){if(a==b)return;transitions[{a,b}]+=count;};
+for(U x=1;x<n;x++)if(active[x]){U r=ranks[x];if(args[x][0]){edge(0,fi[x],degree[x]-1);edge(fi[x],1,1);for(U y:args[x])edge(fi[y],fi[x],1);}else edge(0,fi[x],degree[x]);}
+for(U j=0;j<q;j++){U r=ranks[roots[j]];if(kind[j]){edge(0,fi[roots[j]],1);edge(0,1,1);}else singles+=h-r;}
+for(U donor:donors)if(leftmatch[donor]){U e=uses[leftmatch[donor]-1],target=nd(e),value=e>>31?target:args[target][e&1];edge(fi[donor],1,-1);edge(0,fi[value],-1);edge(fi[value],fi[target],-1);edge(fi[donor],fi[target],1);}
+V p=2305843009213693951ULL;
+auto mul=[&](V a,V b)->V{return (__uint128_t)a*b%p;};
+auto power=[&](V a,V b)->V{V r=1;for(;b;b>>=1,a=mul(a,a))if(b&1)r=mul(r,a);return r;};
+auto sub=[&](V a,V b)->V{return (a+p-b)%p;};
+std::vector<std::vector<V>>matrix_cache(frames.size());
+auto matrix=[&](U id)->const std::vector<V>&{auto&A=matrix_cache[id];if(!A.empty())return A;A.assign(h*h,0);if(id==0)return A;if(id==1){for(U i=0;i<h;i++)A[i*h+i]=1;return A;}
+ auto f=frames[id];U c=popcount64(f.core);V out=f.cover&~f.core;U nn=popcount64(out);
+ if(c==3){V factor=power(6*(h+1),p-2);for(U i=0;i<h;i++)for(U j=0;j<h;j++){V wi=3+((f.core>>i)&1);V zj=((f.core>>j)&1)?3*(h+1)-10:p-10;A[i*h+j]=mul(mul(wi,zj),factor);}return A;}
+ assert(c==1||c==2);V s=3-c,d=s*s+(c-1)*nn,den=3*(h+1)*d,inv=power(den,p-2);
+ for(U i=0;i<h;i++)for(U j=0;j<h;j++){V oi=(out>>i)&1,oj=(out>>j)&1,wi=3+((f.core>>i)&1),zj=((f.core>>j)&1)?3*(h+1)-10:p-10;
+  V num=(mul(s*oi,zj)+3*(h+1)*s*wi*oj+mul(nn*wi,zj))%p;num=sub(num,3*(h+1)*(c-1)*oi*oj);A[i*h+j]=(V(i==j&&oi)+mul(num,inv))%p;
+ }return A;};
+
+std::vector<int64_t>blocks(h+1);blocks[1]=singles;V done=0,matrices=0,crt_matrices=0,crt_disagreements=0;U longest=0;std::map<U,V>correction_hist;
+for(auto&[key,count]:transitions){if(!count)continue;assert(count>0);U aa=key.first,bb=key.second;U rr=frames[bb].rank-frames[aa].rank;assert(frames[bb].rank>=frames[aa].rank);if(!rr)continue;
+ if(rr<=2){blocks[1]+=count*rr;continue;}if(aa==0&&bb==1){blocks[h]+=count;continue;}
+ if(aa>1&&bb>1){
+  assert(!(frames[bb].core&~frames[aa].core));
+  assert(!(frames[aa].cover&~frames[bb].cover));
+  U ca=popcount64(frames[aa].core),cb=popcount64(frames[bb].core);
+  assert(ca==3||frames[aa].core==frames[bb].core||(ca==2&&cb==1));
+ }
+ const auto&A=matrix(aa);const auto&B=matrix(bb);std::vector<V>M(h*h);for(U x=0;x<h*h;x++)M[x]=sub(B[x],A[x]);std::vector<std::pair<U,U>>pivots;
+ for(U i=0;i<h;i++){int j=h-1;while(j>=0&&!M[i*h+j])j--;if(j<0)continue;pivots.push_back({i,U(j)});V inv=power(M[i*h+j],p-2);for(U k=i+1;k<h;k++){V z=mul(M[k*h+j],inv);if(z)for(U col=0;col<=U(j);col++)M[k*h+col]=sub(M[k*h+col],mul(z,M[i*h+col]));}}
+ assert(pivots.size()==rr);
+// All minors have numerator bounded independently of matrix size, since the
+// matrix is a 0/1 diagonal mask plus a correction of rank at most four.
+// Source growth and core-two -> core-one use all three primes at these dimensions.
+if(aa>1&&bb>1&&(popcount64(frames[aa].core)==3||(popcount64(frames[aa].core)==2&&popcount64(frames[bb].core)==1))){
+ assert(h==23||h==25);crt_matrices++;matrix_cache[aa].clear();matrix_cache[bb].clear();std::vector<std::vector<std::pair<U,U>>> all{pivots};
+ for(V extra:{2147483647ULL,524287ULL}){p=extra;matrix_cache[aa].clear();matrix_cache[bb].clear();const auto&C=matrix(aa);const auto&D=matrix(bb);std::vector<V>Y(h*h);for(U z=0;z<h*h;z++)Y[z]=sub(D[z],C[z]);std::vector<std::pair<U,U>> pp;
+  for(U i=0;i<h;i++){int j=h-1;while(j>=0&&!Y[i*h+j])j--;if(j<0)continue;pp.push_back({i,U(j)});V inv=power(Y[i*h+j],p-2);for(U k=i+1;k<h;k++){V z=mul(Y[k*h+j],inv);if(z)for(U col=0;col<=U(j);col++)Y[k*h+col]=sub(Y[k*h+col],mul(z,Y[i*h+col]));}}
+  if(pp!=pivots)crt_disagreements++;all.push_back(std::move(pp));
+ }
+ p=2305843009213693951ULL;matrix_cache[aa].clear();matrix_cache[bb].clear();
+ std::vector<int> corner((h+1)*(h+1));for(U i=0;i<h;i++)for(U j=0;j<h;j++){int best=0;for(auto&pp:all){int rank=0;for(auto [row,col]:pp)rank+=row<=i&&col>=j;best=std::max(best,rank);}corner[(i+1)*(h+1)+j]=best;}
+ pivots.clear();for(U i=0;i<h;i++)for(U j=0;j<h;j++){int z=corner[(i+1)*(h+1)+j]-corner[i*(h+1)+j]-corner[(i+1)*(h+1)+j+1]+corner[i*(h+1)+j+1];assert(z==0||z==1);if(z)pivots.push_back({i,j});}assert(pivots.size()==rr);
+}
+U run=0;for(U j=0;j<pivots.size();j++){if(j&&pivots[j].first==pivots[j-1].first+1&&pivots[j].second==pivots[j-1].second+1)run++;else{if(run){blocks[run]+=count;longest=std::max(longest,run);}run=1;}}if(run){blocks[run]+=count;longest=std::max(longest,run);}matrices++;matrix_cache[aa].clear();matrix_cache[bb].clear();
+ if(++done%10000==0)std::cerr<<"profiles "<<done<<" of "<<transitions.size()<<"\n";
+}
+V mass=0;for(U t=1;t<=h;t++){assert(blocks[t]>=0);mass+=t*blocks[t];}assert(mass==sum);
+std::string target=std::string(argv[1])+".round3_rankone_certified_profiles.json";std::ofstream out(target);assert(out);out<<"{\"h\":"<<h<<",\"v\":"<<v<<",\"R\":"<<R<<",\"loss\":"<<loss<<",\"rank_sum\":"<<mass<<",\"field_prime\":"<<p<<",\"frames\":"<<frames.size()<<",\"distinct_matrices\":"<<matrices<<",\"crt_matrices\":"<<crt_matrices<<",\"crt_disagreements\":"<<crt_disagreements<<",\"blocks\":[";for(U t=0;t<=h;t++){if(t)out<<",";out<<blocks[t];}out<<"]}\n";std::cerr<<"CRT matrices "<<crt_matrices<<" differing modular profiles "<<crt_disagreements<<"\n";std::cerr<<"RANKONE h "<<h<<" longest "<<longest<<" output "<<target<<"\n";
+
+}
diff --git a/research/skip-strips/reversed/geometry-source.json b/research/skip-strips/reversed/geometry-source.json
new file mode 100644
index 0000000000..618912ee49
--- /dev/null
+++ b/research/skip-strips/reversed/geometry-source.json
@@ -0,0 +1,15 @@
+{
+  "upstream": "https://github.com/CrocSwap/integer-mult-bounds/pull/34",
+  "commit": "7fecbe3651e095fb0f450756afdeefd2a9ee80a2",
+  "author": "James Chang (jamesyc)",
+  "license": "Apache-2.0, inherited repository license",
+  "source_paths": {
+    "pr34/independent_controls.py": "research/reversed-two-stage/independent_controls.py",
+    "pr34/certify_reversed.py": "research/reversed-two-stage/certify_reversed.py"
+  },
+  "sha256": {
+    "pr34/independent_controls.py": "f0450c9a8127b538df6cc36ed744ad707b9e72756af9ce5290023272999c5c65",
+    "pr34/certify_reversed.py": "77311474169f64a154c2435bcde9303fbc472cbd9ac4d2b39c76e7560fa9e3a3"
+  },
+  "scope": "Retained author geometry and independent exact controls copied byte-for-byte; runtime loader omits only unsupported host resource-limit setters."
+}
diff --git a/research/skip-strips/reversed/geometry.py b/research/skip-strips/reversed/geometry.py
new file mode 100644
index 0000000000..7ad5411860
--- /dev/null
+++ b/research/skip-strips/reversed/geometry.py
@@ -0,0 +1,89 @@
+"""Exact PR34 reversed23/25 geometry controls for a proposed PR36 transfer.
+Does not prove the copied-center producer/count integration by itself.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from hashlib import sha256
+import importlib.util,json,random,types
+HERE=Path(__file__).resolve().parent
+SOURCE=HERE/'pr34'
+
+
+def load(name,file):
+    m=types.ModuleType(name);m.__file__=str(file)
+    # macOS rejects the inherited RLIMIT_AS setter; omit only host resource caps.
+    source='\n'.join(line for line in file.read_text().splitlines() if not line.startswith('resource.setrlimit('))
+    exec(compile(source,str(file),'exec'),m.__dict__);return m
+
+
+def tree(edges,n):
+    par=list(range(n))
+    def find(x):
+        while x!=par[x]:x=par[x]
+        return x
+    for x,y in edges:
+        x,y=find(x),find(y);assert x!=y;par[x]=y
+    assert len(edges)==n-1 and len({find(x) for x in range(n)})==1
+
+
+def run():
+    manifest=json.loads((HERE/'geometry-source.json').read_text())
+    assert manifest['commit']=='7fecbe3651e095fb0f450756afdeefd2a9ee80a2'
+    for name,digest in manifest['sha256'].items():
+        assert sha256((HERE/name).read_bytes()).hexdigest()==digest, 'Pinned PR34 source changed: '+name
+    checked=load('pr34_independent_reversed23',SOURCE/'independent_controls.py')
+    author=load('pr34_author_reversed23',SOURCE/'certify_reversed.py')
+    h=23;b=25;m=h*b;d=h+b-1
+    rows,cols=checked.corner_labels(h);pivots=checked.expected_pivots(h)
+    perms=checked.completion(h,rows,cols)
+    cuts=checked.check_cuts(h,rows,cols,pivots)
+    independent=[checked.witness(h,rows,cols,k) for k in (1,2)]
+    authored=author.certify(h)
+    assert authored['pivots']==pivots and authored['permutation_completions']==perms
+    assert authored['profile']==[1]*9+[21,17,481]
+    tree([(r,h+beta) for r,beta in rows],h+b)
+    tree([(c,h+gamma) for c,gamma in cols],h+b)
+    rng=random.Random(23025)
+    def line(n):
+        p=[Q(rng.randrange(1,30)) for _ in range(n)];x=[Q(rng.randrange(1,30)) for _ in range(n)];z=sum(v*w for v,w in zip(p,x));return p,[w/z for w in x]
+    p,xi=line(h);v,nu=line(b)
+    P=[[x*y for y in xi] for x in p];B=[[x*y for y in nu] for x in v]
+    I=[[Q(i==j) for j in range(h)] for i in range(h)];J=[[Q(i==j) for j in range(b)] for i in range(b)]
+    A=[[Q(rng.randrange(-9,10),rng.randrange(1,8)) for j in range(h)] for i in range(h)]
+    D=[[Q(rng.randrange(-9,10),rng.randrange(1,8)) for j in range(b)] for i in range(b)]
+    def actual(X,Y,i,j):
+        alpha,beta=divmod(i,b);gamma,delta=divmod(j,b)
+        r=perms[beta][alpha];c=perms[delta][gamma]
+        return X[r][c]*Y[beta][delta]
+    for i in range(h):
+        for j in range(h):
+            assert actual(A,B,i,m-h+j)==v[i]*A[i][j]*nu[j+2]
+            assert actual(I,B,i,m-h+j)==Q(i==j)*v[i]*nu[j+2]
+    rs=[perms[i%b][i//b] for i in range(b)];cs=[perms[(m-b+j)%b][(m-b+j)//b] for j in range(b)]
+    assert rs==list(range(h))+[h-1,0] and cs==[h-1,0]+list(range(h))
+    for i in range(b):
+        for j in range(b):
+            assert actual(P,D,i,m-b+j)==p[rs[i]]*D[i][j]*xi[cs[j]]
+            assert actual(P,J,i,m-b+j)==Q(i==j)*p[rs[i]]*xi[cs[j]]
+    # The actual off-diagonal large-projector corner is minus its null corner.
+    for i in range(d):
+        for j in range(d):
+            r,beta=rows[i];c,gamma=cols[j]
+            null=actual(P,J,i,m-d+j)+actual(I,B,i,m-d+j)-actual(P,B,i,m-d+j)
+            expected=Q(beta==gamma)*p[r]*xi[c]+Q(r==c)*v[beta]*nu[gamma]-p[r]*xi[c]*v[beta]*nu[gamma]
+            assert null==expected
+    assert (m-d)%b==3 and (m-d)%b!=b-1
+    assert any(actual(I,B,i,m-h+i)!=v[i]*nu[i] for i in range(h))
+    return dict(status='PASS EXACT REVERSED23/25 GEOMETRY; PR36 PHYSICAL INTEGRATION SEPARATE',
+                dimensions=[h,b],m=m,d=d,rank=m-d,profile=[1]*9+[21,17,481],
+                permutations=perms,rows=rows,columns=cols,pivots=pivots,cuts=cuts,
+                rational_controls=independent,author_witness=authored['witness'],
+                physical_blocks=[dict(rows=[1,21],columns=[553,573]),dict(rows=[28,44],columns=[530,546]),dict(rows=[47,527],columns=[47,527])],
+                actual_local_and_auxiliary_contractions=True,actual_null_corner=True,both_trees=True,
+                negative_controls=['incorrect_unshifted_auxiliary_formula','old_25_23_data_residue_offset'],
+                source_sha256={p.name:sha256(p.read_bytes()).hexdigest() for p in [SOURCE/'independent_controls.py',SOURCE/'certify_reversed.py']})
+
+
+if __name__=='__main__':
+    result=run();(HERE/'geometry-certificate.json').write_text(json.dumps(result,indent=2)+'\n')
+    print(result['status'],result['profile'],len(result['cuts']))
diff --git a/research/skip-strips/reversed/pr34/certify_reversed.py b/research/skip-strips/reversed/pr34/certify_reversed.py
new file mode 100644
index 0000000000..e7367cb248
--- /dev/null
+++ b/research/skip-strips/reversed/pr34/certify_reversed.py
@@ -0,0 +1,116 @@
+"""Exact author certificate for the reversed (h,h+2) boundary family.
+
+Only Python's standard library is imported. No supplier code is executed.
+"""
+from fractions import Fraction as Q
+from pathlib import Path
+import hashlib,json,time,argparse
+
+ROOT=Path(__file__).resolve().parent
+
+def geometry(h):
+    a,b=h,h+2;d=a+b-1;m=a*b
+    rows=list(range(h))+[h-1]+list(range(h))
+    cols=list(range(h))+[0]+list(range(h))
+    re=list(zip(rows,[i%b for i in range(d)]))
+    ce=list(zip(cols,[(m-d+i)%b for i in range(d)]))
+    assert len(rows)==len(cols)==d and (m-d)%b==3
+    prescribed=[{} for _ in range(b)]
+    for labels,start in [(rows,0),(cols,m-d)]:
+        for i,label in enumerate(labels):
+            physical=start+i;alpha,beta=divmod(physical,b)
+            assert alpha not in prescribed[beta]
+            assert label not in prescribed[beta].values()
+            prescribed[beta][alpha]=label
+    perms=[]
+    for constraints in prescribed:
+        unused=iter(v for v in range(a) if v not in constraints.values())
+        perm=[constraints[i] if i in constraints else next(unused) for i in range(a)]
+        assert sorted(perm)==list(range(a));perms.append(perm)
+    piv=[2*h]+[i+h+1 for i in range(1,h-1)]+[2*h-i for i in range(h-1,h+5)]+[i-h-3 for i in range(h+5,2*h-1)]+[1,0]
+    assert sorted(piv)==list(range(d))
+    return a,b,d,m,rows,cols,re,ce,perms,piv
+
+def forest_rank(a,b,edges,selected):
+    parent=list(range(a+b))
+    def find(i):
+        while parent[i]!=i:i=parent[i]
+        return i
+    for u,v in edges:
+        u,v=find(u),find(a+v)
+        assert u!=v,'Cycle contradicts the tree/forest hypothesis'
+        parent[u]=v
+    comps={}
+    for i in range(a+b):comps.setdefault(find(i),set()).add(i)
+    value=0;constant_dependent=True
+    for vs in comps.values():
+        if len(vs)==1:continue
+        value+=min(len(vs&selected),len(vs)-1)
+        if any(i not in selected for i in vs if i<a) and any(i not in selected for i in vs if i>=a):constant_dependent=False
+    if a+b in selected and not constant_dependent:value+=1
+    return value
+
+def exact_witness(a,b,rows,cols,re,ce,piv):
+    sx=sum(i*i for i in range(1,a+1));sy=sum(i**3 for i in range(1,b+1))
+    xx=[Q(sx,(i+1)**2) for i in range(a)]
+    yy=[Q(sy,(j+1)**3) for j in range(b)]
+    assert sum(1/x for x in xx)==sum(1/y for y in yy)==1
+    mat=[[xx[r]*(r==c)+yy[be]*(be==ga)-1 for c,ga in ce] for r,be in re]
+    values=[]
+    for i,expected in enumerate(piv):
+        nz=[j for j,z in enumerate(mat[i]) if z]
+        assert nz and nz[-1]==expected,(i,expected,nz[-1] if nz else None)
+        value=mat[i][expected];values.append(str(value))
+        active=[j for j in range(expected+1) if mat[i][j]]
+        for k in range(i+1,len(rows)):
+            if mat[k][expected]:
+                factor=mat[k][expected]/value
+                for j in active:mat[k][j]-=factor*mat[i][j]
+                assert not mat[k][expected]
+    return dict(p='all ones',v='all ones',xi_denominator=sx,nu_denominator=sy,xi_numerators=[i*i for i in range(1,a+1)],nu_numerators=[i**3 for i in range(1,b+1)],pivot_values=values)
+
+def certify(h):
+    a,b,d,m,rows,cols,re,ce,perms,piv=geometry(h)
+    universe=set(range(a+b+1));cuts=[]
+    for i,j in enumerate(piv):
+        prior=sum(q>j for q in piv[:i])
+        if 1<=i<=h-2:
+            selected=set(range(i+1,a))|set(range(a+i+1,a+b))|{a+b}
+            expected=(1,0)
+        elif h+5<=i<=2*h-2:
+            t=i-h-5
+            selected=set(range(3+t,a))|set(range(a+5+t,a+b))
+            expected=(h-1-t,6+t)
+        else:
+            # All suffix columns were already selected; dimension is enough.
+            assert prior==d-1-j
+            cuts.append(dict(row=i,column=j,prior=prior,method='column_count'));continue
+        r1=forest_rank(a,b,re[:i+1],selected)
+        r2=forest_rank(a,b,ce[j+1:],universe-selected)
+        assert (r1,r2)==expected and r1+r2==prior
+        cuts.append(dict(row=i,column=j,prior=prior,method='incidence_cut',selected=sorted(selected),ranks=[r1,r2]))
+    witness=exact_witness(a,b,rows,cols,re,ce,piv)
+    profile=[1]*9+[h-2,h-6,m-2*d]
+    assert sum(profile)==(a-1)*(b-1) and max(profile)<m
+    runs=[]
+    for i,j in enumerate(piv):
+        if i and j==piv[i-1]+1:runs[-1]+=1
+        else:runs.append(1)
+    assert sorted(runs)==sorted([1]*9+[h-2,h-6])
+    # Opposite h+2/h boundary residue does not describe this actual corner.
+    assert (m-d)%b!=b-1
+    # Check complete basis constraints in both directions from the saved permutations.
+    for i,label in enumerate(rows):
+        alpha,beta=divmod(i,b);assert perms[beta][alpha]==label
+    for i,label in enumerate(cols):
+        alpha,beta=divmod(m-d+i,b);assert perms[beta].index(label)==alpha
+    return dict(h=h,a=a,b=b,m=m,d=d,row_labels=rows,column_labels=cols,row_edges=re,column_edges=ce,permutation_completions=perms,pivots=piv,runs=runs,profile=profile,rank_cuts=cuts,witness=witness)
+
+def main():
+    p=argparse.ArgumentParser();p.add_argument('--h',type=int,default=45);p.add_argument('--output',type=Path,default=ROOT/'REVERSED_45_CERTIFICATE.json');args=p.parse_args()
+    started=time.monotonic();result=certify(args.h)
+    result['checker_sha256']=hashlib.sha256(Path(__file__).read_bytes()).hexdigest()
+    args.output.write_text(json.dumps(result,indent=2)+'\n')
+    print(json.dumps(dict(h=args.h,profile=result['profile'],exact_nonzero_pivots=len(result['witness']['pivot_values']),rank_cuts=len(result['rank_cuts']),seconds=time.monotonic()-started)))
+
+if __name__=='__main__':main()
diff --git a/research/skip-strips/reversed/pr34/independent_controls.py b/research/skip-strips/reversed/pr34/independent_controls.py
new file mode 100644
index 0000000000..d5578d5c51
--- /dev/null
+++ b/research/skip-strips/reversed/pr34/independent_controls.py
@@ -0,0 +1,122 @@
+from pathlib import Path
+from fractions import Fraction
+from math import gcd
+import hashlib,json,resource,time
+resource.setrlimit(resource.RLIMIT_AS,(100*1024*1024,100*1024*1024))
+resource.setrlimit(resource.RLIMIT_CPU,(110,110))
+HERE=Path(__file__).resolve().parent
+
+def exact_rank(rows):
+    pivots={}
+    for row in rows:
+        row=list(row)
+        for j,p in pivots.items():
+            if row[j]:
+                a,b=p[j],row[j]
+                row=[a*x-b*y for x,y in zip(row,p)]
+                factor=0
+                for x in row:factor=gcd(factor,x)
+                if factor:row=[x//factor for x in row]
+        j=next((j for j,x in enumerate(row) if x),None)
+        if j is not None:
+            factor=0
+            for x in row:factor=gcd(factor,x)
+            if row[j]<0:factor=-factor
+            pivots[j]=[x//factor for x in row]
+    return len(pivots)
+
+def corner_labels(h):
+    d=2*h+1;b=h+2
+    r=list(range(h))+[h-1]+list(range(h))
+    c=list(range(h))+[0]+list(range(h))
+    return [(r[i],i%b) for i in range(d)],[(c[j],(h*b-d+j)%b) for j in range(d)]
+
+def expected_pivots(h):
+    return [2*h]+[i+h+1 for i in range(1,h-1)]+[2*h-i for i in range(h-1,h+5)]+[i-h-3 for i in range(h+5,2*h-1)]+[1,0]
+
+def completion(h,rows,cols):
+    b=h+2;m=h*b;d=len(rows)
+    partial=[{} for _ in range(b)]
+    for seq,offset in [(rows,0),(cols,m-d)]:
+        for k,(label,residue) in enumerate(seq):
+            alpha,physical_residue=divmod(offset+k,b)
+            assert residue==physical_residue
+            assert alpha not in partial[residue] or partial[residue][alpha]==label
+            assert label not in partial[residue].values() or partial[residue].get(alpha)==label
+            partial[residue][alpha]=label
+    result=[]
+    for prescribed in partial:
+        remaining=iter(sorted(set(range(h))-set(prescribed.values())))
+        p=[prescribed[i] if i in prescribed else next(remaining) for i in range(h)]
+        assert sorted(p)==list(range(h))
+        result.append(p)
+    for seq,offset in [(rows,0),(cols,m-d)]:
+        for k,(label,residue) in enumerate(seq):
+            assert result[residue][(offset+k)//b]==label
+    assert [r for r,beta in rows[:h]]==list(range(h))
+    assert [c for c,gamma in cols[-h:]]==list(range(h))
+    assert [beta for r,beta in rows[:b]]==list(range(b))
+    assert [gamma for c,gamma in cols[-b:]]==list(range(b))
+    return result
+
+def witness(h,rows,cols,degree):
+    # This witness deliberately differs from the author's square/cubic weights.
+    wa=[(i+1)**degree+1 for i in range(h)]
+    wb=[(2*i+1)**degree+2 for i in range(h+2)]
+    sa,sb=sum(wa),sum(wb);den=sa*sb;d=len(rows)
+    A=[[int(beta==gamma)*wa[c]*sb+int(r==c)*wb[gamma]*sa-wa[c]*wb[gamma] for c,gamma in cols] for r,beta in rows]
+    available=set(range(d));previous=1;pivs=[];vals=[];prefix=[]
+    for i in range(d):
+        col=max(j for j in available if A[i][j]);piv=A[i][col]
+        pivs.append(col);vals.append(str(Fraction(piv,previous*den)));prefix.append(str(Fraction(piv,den**(i+1))))
+        available.remove(col)
+        for k in range(i+1,d):
+            for j in available:
+                value=piv*A[k][j]-A[k][col]*A[i][j]
+                assert value%previous==0
+                A[k][j]=value//previous
+            A[k][col]=0
+        previous=piv
+    assert pivs==expected_pivots(h),(h,degree,pivs)
+    widths=[]
+    for i,j in enumerate(pivs):
+        if i and j==pivs[i-1]+1:widths[-1]+=1
+        else:widths.append(1)
+    assert sorted(t for t in widths if t>1)==[h-6,h-2]
+    assert widths.count(1)==9
+    return {'degree':degree,'left_weights':wa,'right_weights':wb,'pivots':pivs,'pivot_values':vals,'ordered_prefix_determinants':prefix,'widths':widths}
+
+def check_cuts(h,rows,cols,pivots):
+    b=h+2;n=h+b+1;d=len(rows)
+    F=[[int(k==r or k==h+beta or k==n-1) for k in range(n)] for r,beta in rows]
+    G=[[int(k==c or k==h+gamma or k==n-1) for k in range(n)] for c,gamma in cols]
+    records=[]
+    for i,q in enumerate(pivots):
+        preceding=sum(p>q for p in pivots[:i])
+        if 1<=i<=h-2:
+            selected=set(range(i+1,h))|set(range(h+i+1,h+b))|{n-1}
+        elif h+5<=i<=2*h-2:
+            t=i-h-5
+            selected=set(range(3+t,h))|set(range(h+5+t,h+b))
+        else:
+            assert d-q-1==preceding
+            records.append({'row':i,'pivot':q,'trivial_suffix_bound':preceding})
+            continue
+        left=exact_rank([[r[k] for k in range(n) if k in selected] for r in F[:i+1]])
+        right=exact_rank([[r[k] for k in range(n) if k not in selected] for r in G[q+1:]])
+        assert left+right<=preceding,(h,i,q,left,right,preceding)
+        records.append({'row':i,'pivot':q,'selected':sorted(selected),'left_rank':left,'right_rank':right,'preceding_right_pivots':preceding})
+    return records
+
+def main():
+    start=time.monotonic();out={'author_code_executed':False,'cases':[]}
+    for h in [45,53]:
+        rows,cols=corner_labels(h)
+        result={'h':h,'dimensions':[h,h+2],'rows':rows,'columns':cols,'permutation_completions':completion(h,rows,cols),'cuts':check_cuts(h,rows,cols,expected_pivots(h)),'witnesses':[witness(h,rows,cols,degree) for degree in [1,2]]}
+        result['rank_mass']=9+(h-2)+(h-6)+(h*h-2*h-2)
+        assert result['rank_mass']==h*(h+2)-(2*h+1)
+        out['cases'].append(result)
+    out['elapsed_seconds']=time.monotonic()-start
+    (HERE/'INDEPENDENT_CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n')
+    print(json.dumps({'dimensions':[case['dimensions'] for case in out['cases']],'all_pass':True,'elapsed_seconds':out['elapsed_seconds']}))
+if __name__=='__main__':main()
diff --git a/research/skip-strips/scalar-23.json b/research/skip-strips/scalar-23.json
new file mode 100644
index 0000000000..18bea13775
--- /dev/null
+++ b/research/skip-strips/scalar-23.json
@@ -0,0 +1,12 @@
+{
+  "additions": 40329,
+  "all_additions_disjoint": true,
+  "all_partial_outputs_exact": true,
+  "circuit_sha256": "0e309c2023b2832f82badf010649417acfa0cf423b3137c25babf0d7746eb32c",
+  "every_node_has_common_point": true,
+  "h": 23,
+  "inputs": 1771,
+  "merged_additions": 5379,
+  "partial_outputs": 5336,
+  "roles": 45665
+}
diff --git a/research/skip-strips/scalar-25.json b/research/skip-strips/scalar-25.json
new file mode 100644
index 0000000000..900ae4fdcc
--- /dev/null
+++ b/research/skip-strips/scalar-25.json
@@ -0,0 +1,12 @@
+{
+  "additions": 53475,
+  "all_additions_disjoint": true,
+  "all_partial_outputs_exact": true,
+  "circuit_sha256": "52eca99391ab5b7cf85ee80e20846570cfc7b8812e2949990f4f081d6c67bfd4",
+  "every_node_has_common_point": true,
+  "h": 25,
+  "inputs": 2300,
+  "merged_additions": 7008,
+  "partial_outputs": 6925,
+  "roles": 60400
+}
diff --git a/research/skip-strips/skip_graph.py b/research/skip-strips/skip_graph.py
new file mode 100644
index 0000000000..e3ff56e33d
--- /dev/null
+++ b/research/skip-strips/skip_graph.py
@@ -0,0 +1,190 @@
+"""Skip-prefix strip producer, stacked on the PR #48 pipeline.
+
+Only the scalar addition DAG changes. Everything else is inherited from
+Chafik Boukhalfa's PR #48 research/copied-fixed pipeline, which composes
+Rohan Arun's PR #44 weighted carrier matching, exact data-corner recovery
+(PR #46), RaD / hipotures PR #41 alternating pair order and PR #43's
+original-envelope / fixed I+J framework. Those build in turn on icekylinx
+PR #18/#32/#36 and earlier credited work.
+
+New here: every leave-one-out "strip" vector (a vertex's star sums over the
+other groups) uses a skip-prefix layout
+
+    s_j = A_{j-2} + (v_{j-1} + B_{j+1}),
+
+where A_i and B_i are ordinary prefix and suffix sums. In the usual layout
+s_j = A_{j-1} + B_{j+1}, the two consumers of each prefix A_i are A_{i+1} and
+s_{i+1}. Their point covers are incomparable, so the extra use of A_i always
+needs a fresh role. With the skip layout the consumers of A_i are A_{i+1} and
+s_{i+2}, and cover(A_{i+1}) is contained in cover(s_{i+2}). Likewise
+cover(v_{i-1} + B_{i+1}) is contained in cover(B_{i-1}). These nested covers
+are exactly what the inherited carrier matching may join.
+
+The top recursion level applies the layout to the reversed value list. Deeper
+levels apply it to the value list sorted by (support size, support), the
+PR #48 sorting idea: ascending at h=23 and descending support at h=25. Totals
+fold in reverse input order at h=23 (PR #48's rule) and in descending
+(support size, support) order at h=25 (PR #43/#48's rule).
+
+Apache-2.0, like the inherited files. Prepared with assistance from Claude
+(Anthropic).
+"""
+from itertools import combinations
+import sys
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[2]
+sys.path.insert(0, str(ROOT / 'scripts'))
+from partial_swap.paired import PairedExclusionCircuit
+from partial_swap.shared import SharedPointCircuit
+
+
+def alternating_points(h, common):
+    """RaD PR #41 alternating global pair order (as specialized in PR #43/#48)."""
+    pairs = [(a, a+1) for a in range(0, h-1, 2) if common not in (a, a+1)]
+    if common % 2:
+        pairs.reverse()
+    head = [x for pair in pairs for x in pair]
+    return head + [x for x in range(h) if x != common and x not in head]
+
+
+def prefix_suffix(c, vals):
+    """Ordinary leave-one-out: total and s_j = A_{j-1} + B_{j+1}."""
+    n = len(vals)
+    prefix = [0]
+    for x in vals:
+        prefix.append(c.add(prefix[-1], x))
+    suffix = [0]*(n+1)
+    for i in range(n-1, -1, -1):
+        suffix[i] = c.add(vals[i], suffix[i+1])
+    return prefix[-1], [c.add(prefix[i], suffix[i+1]) for i in range(n)]
+
+
+def skip_prefix(c, vals):
+    """Leave-one-out with s_j = A_{j-2} + (v_{j-1} + B_{j+1}) (1-indexed)."""
+    k = len(vals)
+    if k <= 2:
+        return prefix_suffix(c, vals)
+    v = [0] + list(vals)
+    A = [0]*(k+1)
+    for i in range(1, k):
+        A[i] = c.add(A[i-1], v[i])
+    B = [0]*(k+2)
+    for i in range(k, 1, -1):
+        B[i] = c.add(v[i], B[i+1])
+    out = [0]*(k+1)
+    out[1] = B[2]
+    for j in range(2, k+1):
+        out[j] = c.add(A[j-2], c.add(v[j-1], B[j+1]))
+    return c.add(A[k-1], v[k]), out[1:]
+
+
+def skip_prefix_reversed(c, vals):
+    total, out = skip_prefix(c, list(vals)[::-1])
+    return total, out[::-1]
+
+
+def sorted_layout(layout, sign):
+    """Apply a layout to values sorted by (support size, sign*support); restore order."""
+    def apply(c, vals):
+        order = sorted(range(len(vals)), key=lambda i: (c.support[vals[i]].bit_count(), sign*c.support[vals[i]]))
+        total, out = layout(c, [vals[i] for i in order])
+        restored = [0]*len(vals)
+        for position, original in enumerate(order):
+            restored[original] = out[position]
+        return total, restored
+    return apply
+
+
+def leave_one_out(c, values, layout):
+    """Zero entries are omitted; their leave-one-out value is the total."""
+    values = list(values)
+    nonzero = [i for i, x in enumerate(values) if x]
+    if not nonzero:
+        return 0, [0]*len(values)
+    if len(nonzero) == 1:
+        x = values[nonzero[0]]
+        return x, [0 if i in nonzero else x for i in range(len(values))]
+    total, out = layout(c, [values[i] for i in nonzero])
+    result = [total]*len(values)
+    for i, x in zip(nonzero, out):
+        result[i] = x
+    return total, result
+
+
+def circuit_class(h):
+    deep_sign = 1 if h == 23 else -1
+
+    class SkipStripCircuit(PairedExclusionCircuit):
+        base_threshold = 2
+
+        def total(self, values):
+            values = [x for x in values if x]
+            if h == 23:
+                values.reverse()
+            else:
+                values.sort(key=lambda node: (self.support[node].bit_count(), self.support[node]), reverse=True)
+            result = 0
+            for node in values:
+                result = self.add(result, node)
+            return result
+
+        def block(self, points, edges, weights):
+            level = getattr(self, '_level', -1) + 1
+            self._level = level
+            try:
+                return self._block(points, edges, weights, level)
+            finally:
+                self._level = level - 1
+
+        def _block(self, points, edges, weights, level):
+            if len(points) <= self.base_threshold:
+                return PairedExclusionCircuit.block(self, points, edges, weights)
+            layout = skip_prefix_reversed if level == 0 else sorted_layout(skip_prefix, deep_sign)
+            groups = self.grouping(points)
+            ng = len(groups)
+            e = lambda a, b: edges[tuple(sorted((a, b)))]
+            coarse = {(i, j): self.total([e(a, b) for a in groups[i] for b in groups[j]])
+                      for i, j in combinations(range(ng), 2)}
+            wt = {i: self.total([weights[a] for a in g] + [e(a, b) for a, b in combinations(g, 2)])
+                  for i, g in enumerate(groups)}
+            total, outside, far = self.block(list(range(ng)), coarse, wt)
+            strips, sums = {}, {}
+            for i, g in enumerate(groups):
+                other = [j for j in range(ng) if j != i]
+                for a in g:
+                    carry = self.total([weights[u] for u in g if u != a])
+                    vals = [self.total([e(u, v) for u in g if u != a for v in groups[j]]) for j in other]
+                    st, one = leave_one_out(self, [carry] + vals, layout)
+                    strips[a] = {j: z for j, z in zip(other, one[1:])}
+                    sums[a] = st
+            out = {}
+            single = {a: self.add(outside[i], sums[a]) for i, g in enumerate(groups) for a in g}
+            for i, g in enumerate(groups):
+                for a, b in combinations(g, 2):
+                    out[a, b] = outside[i]
+            for i, j in combinations(range(ng), 2):
+                for a in groups[i]:
+                    left = self.add(far[i, j], strips[a][j])
+                    for b in groups[j]:
+                        cross = self.total([e(u, v) for u in groups[i] if u != a for v in groups[j] if v != b])
+                        out[a, b] = self.add(left, self.add(strips[b][i], cross))
+            return total, single, out
+    return SkipStripCircuit
+
+
+def graph(h):
+    assert h in (23, 25)
+    local = circuit_class(h)(h-1)
+    total = local.pair(list(range(h-1)))[0]
+    local.outputs[()] = total
+    stack = [total]
+    while stack:
+        node = stack.pop()
+        if not node or node in local.active:
+            continue
+        local.active.add(node)
+        if local.args[node]:
+            stack.extend(local.args[node])
+    local.additions = sum(local.args[node] is not None for node in local.active)
+    return SharedPointCircuit(h, local, point_order=alternating_points)
diff --git a/research/skip-strips/timeline-23.json b/research/skip-strips/timeline-23.json
new file mode 100644
index 0000000000..75690f9d9a
--- /dev/null
+++ b/research/skip-strips/timeline-23.json
@@ -0,0 +1,64 @@
+{
+  "additions": 40329,
+  "center_rank": 22,
+  "center_terminal_count": 23,
+  "complete_dirty_basis": {
+    "all_dirty_restore": true,
+    "auxiliary_basis_vectors": 32946,
+    "elementary_word_length": 310810,
+    "inputs": 1771,
+    "orientations": [
+      "forward",
+      "reverse-complement"
+    ],
+    "targets": 1771,
+    "total_basis_vectors": 36488
+  },
+  "containment_cache": {
+    "currsize": 98299,
+    "hits": 170608,
+    "maxsize": 300000,
+    "misses": 98299
+  },
+  "copied_histogram": [
+    10810,
+    41870,
+    36573,
+    6049,
+    16095,
+    7215,
+    4218,
+    1886,
+    3482,
+    833,
+    2865,
+    273,
+    2456,
+    289,
+    3527,
+    829,
+    2257,
+    994,
+    8640,
+    5873,
+    0,
+    165,
+    23,
+    0
+  ],
+  "copied_rank_sum": 758264,
+  "dag_sha256": "5f3b5d5487019f4e3a357a105265a80ddcf5729bafd452fdfa25bbd870a510a1",
+  "designated_outputs": 5336,
+  "full_output_coefficients": 3136441,
+  "h": 23,
+  "native_zero_rank_bookkeeping": 5474,
+  "physical_frame_transitions": 85994,
+  "rational_frame_classes": 36626,
+  "retained_links": 12719,
+  "reverse_complement_frame_incidences": 113604,
+  "roles": 32946,
+  "selected_links_sha256": "77f8a34a65513f89fcf68b91f6cdebf79bbc39ff4e325c0f28f2fc6e4bd2eb55",
+  "status": "independent scalar, rational frames, physical compiler and rank timeline PASS",
+  "v": 1771,
+  "zero_rank_scope": "Explicit output incidences include no-operation frame checks omitted from native histogram bookkeeping; only positive ranks enter the recurrence"
+}
diff --git a/research/skip-strips/timeline-25.json b/research/skip-strips/timeline-25.json
new file mode 100644
index 0000000000..c169d8ac62
--- /dev/null
+++ b/research/skip-strips/timeline-25.json
@@ -0,0 +1,66 @@
+{
+  "additions": 53475,
+  "center_rank": 24,
+  "center_terminal_count": 25,
+  "complete_dirty_basis": {
+    "all_dirty_restore": true,
+    "auxiliary_basis_vectors": 43409,
+    "elementary_word_length": 410536,
+    "inputs": 2300,
+    "orientations": [
+      "forward",
+      "reverse-complement"
+    ],
+    "targets": 2300,
+    "total_basis_vectors": 48009
+  },
+  "containment_cache": {
+    "currsize": 130908,
+    "hits": 225383,
+    "maxsize": 300000,
+    "misses": 130908
+  },
+  "copied_histogram": [
+    14025,
+    54451,
+    47721,
+    7718,
+    21976,
+    9097,
+    4072,
+    3033,
+    5370,
+    893,
+    2724,
+    468,
+    3707,
+    468,
+    2751,
+    168,
+    4431,
+    1536,
+    2945,
+    918,
+    11304,
+    7596,
+    0,
+    221,
+    25,
+    0
+  ],
+  "copied_rank_sum": 1085825,
+  "dag_sha256": "f424f598729643ed7095e8a9c54e747005717070cf9a8aebd78d625092037c90",
+  "designated_outputs": 6925,
+  "full_output_coefficients": 5290000,
+  "h": 25,
+  "native_zero_rank_bookkeeping": 7100,
+  "physical_frame_transitions": 113875,
+  "rational_frame_classes": 48675,
+  "retained_links": 16991,
+  "reverse_complement_frame_incidences": 150359,
+  "roles": 43409,
+  "selected_links_sha256": "01ca09b1a7683d7114d7a8f699e3a0859e57ccb10779c4113e5692377e3993e3",
+  "status": "independent scalar, rational frames, physical compiler and rank timeline PASS",
+  "v": 2300,
+  "zero_rank_scope": "Explicit output incidences include no-operation frame checks omitted from native histogram bookkeeping; only positive ranks enter the recurrence"
+}
diff --git a/research/skip-strips/verify.py b/research/skip-strips/verify.py
new file mode 100644
index 0000000000..642efac75e
--- /dev/null
+++ b/research/skip-strips/verify.py
@@ -0,0 +1,343 @@
+#!/usr/bin/env python3
+"""Skip-prefix strip producer stacked on PR #48's research/copied-fixed pipeline.
+
+Only the scalar producer DAG (skip_graph.py) and its pinned weighted matching
+differ; this is PR #48's verify.py with the new records, constants and an added
+PR #48 control. The adaptation was prepared with assistance from Claude
+(Anthropic). Original docstring follows.
+
+Exact copied-center / fixed-I+J composition, conditional on retained proofs.
+
+Credits: icekylinx PR32/36 (projectors, copied centers), Dominik Scholz PR35
+(two fixed factors and CRT bounds), Zhihao Chen PR29/23, Paureel/Aurel Prosz,
+Swapnil Jain, Rohan Arun PR25/31, RaD/hipotures and all inherited notices.
+Prepared locally with OpenAI Codex assistance; no formal theorem claim.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from functools import lru_cache
+from hashlib import sha256
+from math import comb, factorial, isqrt, prod
+from pathlib import Path
+import argparse
+import importlib.util
+from itertools import combinations
+import json
+import sys
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(ROOT / 'scripts'))
+import copied_centers_network as baseline
+from structured_bulk_assembly import js
+sys.path.insert(0,str(HERE))
+from balanced_assembly import assembly, cutoffs
+
+AB = Q(4498347, 10**11)
+AC = Q(717, 10000000)
+# Strictly below the recomputed balanced assembly margin.
+KAPPA = Q(4498144, 10**11)
+PR48_KAPPA = Q(411862541, 10**13)
+PR47_KAPPA = Q(4105106623,10**14)
+PR46_KAPPA = Q(410062945,10**13)
+PR44_KAPPA = Q(2050314627,5*10**13)
+PR37_KAPPA = Q(3850771033, 10**14)
+PR38_KAPPA = Q(242889, 6250000000)
+PR43_KAPPA = Q(409953,10**10)
+PR42_KAPPA = Q(4099494519,10**14)
+PR41_KAPPA = Q(1008542031,25000000000000)
+PR40_KAPPA = Q(1959367447,50000000000000)
+PR39_KAPPA = Q(971668963, 25000000000000)
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def read(path):
+    def reject(x):
+        raise ValueError('Floating point input forbidden: ' + x)
+    return json.loads(path.read_text(), parse_float=reject, parse_constant=reject)
+
+
+def check_sources():
+    sources = read(HERE / 'SOURCE.json')
+    for name, digest in sources['files'].items():
+        require(sha256((ROOT / name).read_bytes()).hexdigest() == digest,
+                'Pinned source changed: ' + name)
+    return sources
+
+
+def mersenne(exponent):
+    require(exponent >= 3 and all(exponent % d for d in range(2, isqrt(exponent)+1)),
+            'Composite Lucas-Lehmer exponent')
+    p = 2**exponent - 1
+    s = 4
+    for _ in range(exponent - 2):
+        s = (s*s-2) % p
+    require(s == 0, 'Lucas-Lehmer failed')
+    return p
+
+
+def exactness(h):
+    require(h in (23, 25), 'Uncertified dimension')
+    primes = [mersenne(e) for e in (61, 31, 19)]
+    Z = 3*h-7
+    B0 = (4*h-2)*Z + 27*(h+1)
+    D = 3*(h+1)*(h-1)**2
+    B = 2*(h-1)*B0
+    bounds = {r: sum(comb(h,j)*factorial(j)*B**j*D**(r-j)
+                     for j in range(r+1)) for r in (2,3,4)}
+    require(bounds[2] < primes[0], 'Rank-two minor modulus insufficient')
+    require(max(bounds[3],bounds[4]) < prod(primes), 'CRT minor modulus insufficient')
+    require(3*(h+1)*(h-1) < min(primes), 'Denominator may vanish')
+    return dict(h=h, primes=primes, prime_product=prod(primes),
+                single_numerator_bound=B0, common_denominator_bound=D,
+                correction_numerator_bound=B, minor_bounds=bounds,
+                modulus_gaps={2: primes[0]-bounds[2],
+                              3: prod(primes)-bounds[3],4: prod(primes)-bounds[4]})
+
+
+def tree(edges, n):
+    require(len(edges) == n-1 and len(set(edges)) == n-1, 'Wrong tree edge count')
+    seen = {0}
+    while True:
+        new = seen | {v for u,v in edges if u in seen} | {u for u,v in edges if v in seen}
+        if new == seen:
+            break
+        seen = new
+    require(len(seen) == n, 'Disconnected restriction graph')
+
+
+def data_corners():
+    c = read(HERE/'data-corners.json')
+    prime = c['prime']
+    require(prime == 1000003 and all(prime % j for j in range(2,isqrt(prime)+1)),
+            'Data witness prime not proved')
+    require(c['dimensions'] == [23,25], 'Data dimensions')
+    require([tuple(r[:3]) for r in c['records']] == list(combinations(range(23),3)),
+            'Incomplete first triple coverage')
+    require(all(0 <= r[3] <= 2300 and 0 < r[4] < prime for r in c['records']),
+            'Invalid modular product record')
+    failures = c['fallback_pairs']
+    require(len({tuple(r[:6]) for r in failures}) == len(failures), 'Duplicate fallback')
+    counts = Counter()
+    for aa,bb,cc,u,v,w,row in failures:
+        require(0<=aa<bb<cc<23 and 0<=u<v<w<25 and 0<=row<47,'Invalid fallback pair')
+        counts[aa,bb,cc] += 1
+    require(all(r[3]+counts[tuple(r[:3])]==2300 for r in c['records']), 'Pair coverage gap')
+    require(c['good'] == sum(r[3] for r in c['records']), 'Wrong good count')
+    require(c['fallback'] == len(failures) and c['pairs'] == c['good']+c['fallback']==4073300,
+            'Wrong exhaustive pair count')
+    require(c['nonzero_pivots'] == 47*c['good']+sum(r[6] for r in failures),
+            'Wrong complete nonvanishing coverage')
+    from data_recovery import recover
+    recovery=recover(tuple(tuple(pair) for pair in failures))
+    require(len(recovery)==len(failures)==10,'Incomplete exact recovery')
+    require(all(len(row['pivots'])==47 and all(Q(x) for x in row['pivots'])
+                for row in recovery),'Incomplete rational pivot proof')
+    return dict(c, primary_good=c['good'],primary_fallback=c['fallback'],
+                good=c['pairs'],fallback=0,exact_recovery=recovery)
+
+
+def geometry():
+    spec=importlib.util.spec_from_file_location('fixed_both_reversed_geometry',HERE/'reversed/geometry.py')
+    inherited=importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(inherited)
+    g=inherited.run()  # Exact all-weight cuts and actual common-basis contractions.
+    c=data_corners()
+    require(g['dimensions']==[23,25] and g['profile']==[1]*9+[21,17,481], 'Reversed family')
+    require(g['rows']==[(r,i%25) for i,r in enumerate(list(range(23))+[22]+list(range(23)))],
+            'Rational recovery row geometry mismatch')
+    require(g['columns']==[(r,(528+i)%25) for i,r in enumerate(list(range(23))+[0]+list(range(23)))],
+            'Rational recovery column geometry mismatch')
+    coordinates={}
+    for h in (23,25):
+        primal=[Q(3),Q(4)]
+        dual=[-Q(5,3*(h+1)),Q(1,2)-Q(5,3*(h+1))]
+        require(all(primal) and all(dual), 'Zero fixed line coordinate')
+        coordinates[h]=dict(primal=primal,dual=dual)
+    return dict(dimensions=[23,25],inherited=g,modular=c,line_coordinates=coordinates,
+                row_tree=[(r,23+be) for r,be in g['rows']],
+                column_tree=[(r,23+be) for r,be in g['columns']],
+                accepted_profile=dict(singletons=9,blocks=[21,17,481]),
+                fallback_profile=dict(singletons=47,blocks=[481]),
+                scope='Nonzero modular pivots certify rational nonvanishing at the actual two fixed bases; '
+                      'zeros follow separate exact rank cuts. Ten primary-prime failures are replayed '
+                      'with exact rational elimination, yielding the same profile for every pair.')
+
+
+@lru_cache(None)
+def logs(x):
+    x = Q(x)
+    require(x >= 1, 'Log domain')
+    k = 0
+    while x > 2:
+        x /= 2
+        k += 1
+    def small(y):
+        z = (y-1)/(y+1)
+        lo = 2*sum((z**(2*j+1)/Q(2*j+1) for j in range(32)), Q())
+        return lo, lo+2*z**65/(65*(1-z*z))
+    lo,hi = small(x)
+    lo2,hi2 = small(Q(2))
+    lo,hi = lo+k*lo2,hi+k*hi2
+    scale = 10**30
+    return Q((lo*scale).numerator//(lo*scale).denominator,scale), \
+        Q(-(-(hi*scale).numerator//(hi*scale).denominator),scale)
+
+
+def moment(m,W,rows,saving):
+    require(type(saving) is Q and 0 < saving < 1, 'Exact positive saving required')
+    lower,upper = Q(),Q()
+    enclosure = {}
+    for t,n in sorted(rows.items()):
+        require(type(t) is int and type(n) is int and 0<t<m and n>0, 'Invalid child')
+        lo,hi = logs(Q(m,t))
+        u,v = saving*lo,saving*hi
+        require(0 <= u <= v < 1, 'Exponential enclosure domain')
+        weight = Q(t*n,m*W)
+        lower += weight*(1+u+u*u/2+u*u*u/6)
+        upper += weight*(1+v+v*v/(2*(1-v/3)))
+        enclosure[t] = dict(lower=lo,upper=hi)
+    return dict(saving=saving,exponent=1-saving,lower=lower,upper=upper,
+                strict_gap=1-upper,logarithms=enclosure)
+
+
+def profile():
+    a,b = 23,25
+    m = a*b
+    rows = [read(HERE / f'original-{h}.json') for h in (a,b)]
+    N = comb(a,3)*comb(b,3)
+    W = 2*N+sum(N//r['v']*r['R'] for r in rows)
+    L = sum(N//r['v']*r['loss'] for r in rows)
+    data = data_corners()
+    good,bad = data['good'],data['fallback']
+    parts = {'data':Counter({1:2*(9*good+47*bad),21:2*good,17:2*good,481:2*N}),
+             'paid_endpoint_copy':Counter({1:N})}
+    axes = []
+    for row in rows:
+        h = row['h']
+        require(row['v'] == comb(h,3), 'Triple count')
+        require(row['R'] == row['c']+row['q']-row['matched'], 'Role allocation')
+        require(row['loss'] == h*(h-1), 'Retained center loss')
+        require(sum(r*n for r,n in enumerate(row['histogram'])) == h*row['R']+2*row['loss'],
+                'Original frame rank mass')
+        f = read(HERE / f'profiles-{h}.json')
+        for key in ('h','v','R','loss','rank_sum'):
+            require(f[key] == row[key], 'Fixed/original mismatch: '+key)
+        require(f['crt_disagreements'] == 0 and f['field_prime'] == 2**61-1,
+                'Invalid CRT profile')
+        blocks = f['blocks'][:]
+        require(len(blocks) == h+1 and all(type(n) is int and n>=0 for n in blocks),
+                'Invalid fixed blocks')
+        require(sum(t*n for t,n in enumerate(blocks)) == f['rank_sum'], 'Fixed profile mass')
+        require(blocks[h] == h, 'Exactly h full center cleanup calls expected')
+        # Preserve every transformed copy. Replace only its original's full cleanup.
+        blocks[h] -= h
+        blocks[1] += h
+        require(sum(t*n for t,n in enumerate(blocks)) == h*row['R']+row['loss'],
+                'Copied fixed profile mass')
+        rep = N//row['v']
+        bank = rep*row['R']
+        parts[f'internal_{h}'] = Counter({t:n*rep for t,n in enumerate(blocks) if t and n})
+        parts[f'exterior_{h}'] = Counter({h:bank,m-2*h:bank})
+        parts[f'data_growth_{h}'] = Counter({1:2*N,h-2:2*N})
+        axes.append(dict(original=row,fixed=f,copied_blocks=blocks,exactness=exactness(h)))
+    hist = sum(parts.values(),Counter())
+    s = sum(t*n for t,n in hist.items())
+    require(s == W*m-N+L == 92458003025, 'Complete rank mass')
+    require((m,N,W,L) == (575,4073300,160799739,2226400), 'Physical constants')
+    require(max(hist) == 529 and all(0<t<m and n>0 for t,n in hist.items()), 'Proper children')
+    return dict(m=m,N=N,W=W,L=L,total_rank=s,deficit=W*m-s,maxchild=max(hist),
+                child_multiplicities=dict(sorted(hist.items())),parts=parts,axes=axes)
+
+
+def run():
+    require(not sys.flags.optimize, 'Assertions must remain enabled')
+    sources = check_sources()
+    prior = baseline.certificate()
+    geo,p = geometry(),profile()
+    exact = moment(p['m'],p['W'],p['child_multiplicities'],AB)
+    require(exact['upper'] < 1, 'Bit characteristic failed')
+    old = prior['bit']['counts']
+    old_at_new = moment(old['m'],old['W'],old['child_multiplicities'],AB)
+    require(old_at_new['lower'] > 1, 'Old construction not excluded at new saving')
+    forward=read(HERE/'comparison-pr40.json')
+    forward_rows={int(t):n for t,n in forward['child_multiplicities'].items()}
+    require(sum(t*n for t,n in forward_rows.items())==108202762275, 'PR40 rank mismatch')
+    forward_at_new=moment(forward['m'],forward['W'],forward_rows,AB)
+    require(forward_at_new['lower']>1,'PR40 network not excluded at new saving')
+    rad=read(HERE/'comparison-pr41.json')
+    rad_rows={int(t):n for t,n in rad['child_multiplicities'].items()}
+    require(sum(t*n for t,n in rad_rows.items())==rad['total_rank']==102565738275,'PR41 rank mismatch')
+    rad_at_new=moment(rad['m'],rad['W'],rad_rows,AB)
+    require(rad_at_new['lower']>1,'PR41 alternating network not excluded at new saving')
+    newest=read(HERE/'comparison-pr42.json')
+    newest_rows={int(t):n for t,n in newest['child_multiplicities'].items()}
+    require(sum(t*n for t,n in newest_rows.items())==newest['total_rank']==102565738275,'PR42 rank mismatch')
+    newest_at_new=moment(newest['m'],newest['W'],newest_rows,AB)
+    require(newest_at_new['lower']>1,'PR42 network not excluded at new saving')
+    previous=read(HERE/'comparison-pr43.json')
+    previous_rows={int(t):n for t,n in previous['child_multiplicities'].items()}
+    require(sum(t*n for t,n in previous_rows.items())==previous['total_rank']==102565738275,'PR43 rank mismatch')
+    previous_at_new=moment(previous['m'],previous['W'],previous_rows,AB)
+    require(previous_at_new['lower']>1,'PR43 network not excluded at new saving')
+    frontier=read(HERE/'comparison-pr44.json')
+    frontier_rows={int(t):n for t,n in frontier['child_multiplicities'].items()}
+    require(sum(t*n for t,n in frontier_rows.items())==frontier['total_rank']==102565738275,'PR44 rank mismatch')
+    frontier_at_new=moment(frontier['m'],frontier['W'],frontier_rows,AB)
+    require(frontier_at_new['lower']>1,'PR44 network not excluded at new saving')
+    parent=read(HERE/'comparison-pr46.json')
+    parent_rows={int(t):n for t,n in parent['child_multiplicities'].items()}
+    require(sum(t*n for t,n in parent_rows.items())==parent['total_rank']==102565738275,'PR46 rank mismatch')
+    parent_at_new=moment(parent['m'],parent['W'],parent_rows,AB)
+    require(parent_at_new['lower']>1,'PR46 network not excluded at new saving')
+    latest=read(HERE/'comparison-pr47.json')
+    latest_rows={int(t):n for t,n in latest['child_multiplicities'].items()}
+    require(sum(t*n for t,n in latest_rows.items())==latest['total_rank']==latest['W']*latest['m']-1846900,'PR47 rank mismatch')
+    latest_at_new=moment(latest['m'],latest['W'],latest_rows,AB)
+    require(latest_at_new['lower']>1,'PR47 network not excluded at new saving')
+    pr48=read(HERE/'comparison-pr48.json')
+    pr48_rows={int(t):n for t,n in pr48['child_multiplicities'].items()}
+    require(sum(t*n for t,n in pr48_rows.items())==pr48['total_rank']==102078397025,'PR48 rank mismatch')
+    pr48_at_new=moment(pr48['m'],pr48['W'],pr48_rows,AB)
+    require(pr48_at_new['lower']>1,'PR48 network not excluded at new saving')
+    phase = prior['complex']['counts']
+    phase_row = read(ROOT / 'certificates/copied-centers-complex-input.json')
+    bridge = baseline.finite_bridge(p,phase,[phase_row,phase_row])
+    require(all(js(bridge[k])==js(prior['finite_bridge'][k]) for k in ('complex','semantic','rows')), 'Unexpected phase or stock change')
+    final = assembly(bridge,AB,KAPPA,a_complex=AC)
+    eventual = cutoffs(bridge,final)
+    require(KAPPA > PR48_KAPPA > PR47_KAPPA > PR46_KAPPA > PR44_KAPPA > PR43_KAPPA > PR42_KAPPA > PR41_KAPPA > PR40_KAPPA > PR39_KAPPA > PR38_KAPPA > PR37_KAPPA > prior['kappa'] > Q(1,2**15), 'Comparison failed')
+    require(KAPPA < Q(1,2**14), 'Unexpected power-of-two bracket')
+    own_files = sorted(HERE.glob('*.py'))+sorted(HERE.glob('*.cpp'))+[HERE/'PROOF.md',HERE/'SOURCE.json']
+    own_files += [HERE/f'{kind}-{h}.json' for h in (23,25) for kind in ('profiles','original','scalar')]
+    own_files += [HERE/'data-corners.json',HERE/'comparison-pr40.json',HERE/'comparison-pr41.json',HERE/'comparison-pr42.json',HERE/'comparison-pr43.json',HERE/'comparison-pr44.json',HERE/'comparison-pr46.json',HERE/'comparison-pr47.json',HERE/'comparison-pr48.json',HERE/'links-23.uses',HERE/'links-25.uses']
+    return dict(status='Conditional exact arithmetic witness; general transfer proofs are dependencies',
+                kappa=KAPPA,bit=dict(counts=p,**exact),complex=prior['complex'],
+                geometry=geo,finite_bridge=bridge,assembly=final,eventual_bounds=eventual,sources=sources,
+                comparison=dict(PR48=PR48_KAPPA,ratio_PR48=KAPPA/PR48_KAPPA,PR47=PR47_KAPPA,ratio_PR47=KAPPA/PR47_KAPPA,PR46=PR46_KAPPA,ratio_PR46=KAPPA/PR46_KAPPA,PR44=PR44_KAPPA,ratio_PR44=KAPPA/PR44_KAPPA,PR36=prior['kappa'],PR37=PR37_KAPPA,PR38=PR38_KAPPA,PR39=PR39_KAPPA,PR40=PR40_KAPPA,PR41=PR41_KAPPA,PR42=PR42_KAPPA,PR43=PR43_KAPPA,
+                    ratio_PR36=KAPPA/prior['kappa'],ratio_PR37=KAPPA/PR37_KAPPA,
+                    ratio_PR38=KAPPA/PR38_KAPPA,ratio_PR39=KAPPA/PR39_KAPPA,ratio_PR40=KAPPA/PR40_KAPPA,ratio_PR41=KAPPA/PR41_KAPPA,ratio_PR42=KAPPA/PR42_KAPPA,ratio_PR43=KAPPA/PR43_KAPPA,
+                    dyadic_corollary='2^-15',next_dyadic_not_reached='2^-14'),
+                controls=dict(PR48_bit_at_new_lower=pr48_at_new['lower'],PR47_bit_at_new_lower=latest_at_new['lower'],PR46_bit_at_new_lower=parent_at_new['lower'],PR44_bit_at_new_lower=frontier_at_new['lower'],old_bit_at_new_lower=old_at_new['lower'],PR40_bit_at_new_lower=forward_at_new['lower'],PR41_bit_at_new_lower=rad_at_new['lower'],PR42_bit_at_new_lower=newest_at_new['lower'],PR43_bit_at_new_lower=previous_at_new['lower']),
+                local_sha256={str(p.relative_to(ROOT)):sha256(p.read_bytes()).hexdigest() for p in own_files},
+                scope='Skip-prefix leave-one-out strips (skip_graph.py) with RaD alternating point order, newly pinned PR44-weighted carrier matching and fixed-basis internal blocks at (23,25); '
+                      'all actual data pairs checked for 9 singletons +21+17+481; '
+                      'ten failed modular tests are recovered by exact rational elimination. Exact scalar/label/'
+                      'matching/profile reconstruction is producer.py. OpenAI base theorem, '
+                      'written residual compiler, fixed-tape recursion, analytic/semantic/'
+                      'routing/recovery and eventual-threshold interfaces remain assumed.')
+
+
+if __name__ == '__main__':
+    parser = argparse.ArgumentParser()
+    parser.add_argument('--output',type=Path)
+    args = parser.parse_args()
+    result = run()
+    if args.output:
+        args.output.write_text(json.dumps(js(result),indent=2,sort_keys=True)+'\n')
+    print('PASS conditional kappa='+str(KAPPA)+'; bit saving='+str(AB))
+    print('47 strict inequalities; seven margins; all-exact enclosures; inherited balanced assembly')
diff --git a/tests/test_skip_clones.py b/tests/test_skip_clones.py
new file mode 100644
index 0000000000..7c76dfbc00
--- /dev/null
+++ b/tests/test_skip_clones.py
@@ -0,0 +1,215 @@
+"""Adversarial controls for pinned whole-chain, original-envelope cloning."""
+from collections import Counter
+from copy import copy, deepcopy
+from pathlib import Path
+from fractions import Fraction as Q
+from unittest.mock import patch
+import importlib.util
+import sys
+import unittest
+
+ROOT = Path(__file__).resolve().parents[1]
+HERE = ROOT/'research/skip-clones'
+sys.path.insert(0, str(HERE))
+spec = importlib.util.spec_from_file_location('skip_clones_tests_module', HERE/'cloned_graph.py')
+clones = importlib.util.module_from_spec(spec)
+spec.loader.exec_module(clones)
+
+
+class WholeChainCloneTests(unittest.TestCase):
+    @classmethod
+    def setUpClass(cls):
+        cls.axes = {}
+        cls.real_supports = staticmethod(clones.scalar_supports)
+        cls.support_cache = {}
+        for h in (23, 25):
+            base = clones.base_graph(h)
+            document = clones.read(HERE/f'clone-jobs-{h}.json')
+            base_digest = clones.digest(base)
+            stages = []
+            current = base
+            # Cache only previously verified immutable graph instances. Mutated
+            # result graphs still receive the real finite scalar check.
+            def cached(circuit, verify_outputs=True):
+                key = id(circuit)
+                if verify_outputs and key in cls.support_cache:
+                    return cls.support_cache[key]
+                result = cls.real_supports(circuit, verify_outputs)
+                if verify_outputs:
+                    cls.support_cache[key] = result
+                return result
+            with patch.object(clones, 'scalar_supports', cached):
+                cached(current)
+                for stage in document['rounds']:
+                    result = clones.replay_round(current, stage)
+                    stages.append((current, stage, result))
+                    current = result
+            cls.axes[h] = (base, document, current, stages, base_digest)
+        cls.base, cls.document, _, _, _ = cls.axes[23]
+        cls.stage = cls.document['rounds'][0]
+
+    def reject(self, document, message):
+        original = type(self).real_supports
+        supports = self.support_cache[id(self.base)]
+        def checked(circuit, verify_outputs=True):
+            if circuit is self.base and verify_outputs:
+                return supports
+            return original(circuit, verify_outputs)
+        with patch.object(clones, 'scalar_supports', checked):
+            with self.assertRaisesRegex(ValueError, message):
+                clones.replay_round(self.base, document)
+
+    def test_invalid_parent_and_provider_are_rejected(self):
+        for field in ('parent', 'provider'):
+            for bad in (True, -1, len(self.base.args), 1):
+                stage = deepcopy(self.stage)
+                if field == 'parent':
+                    stage['jobs'][0]['parent'] = bad
+                else:
+                    stage['jobs'][0]['providers'][0] = bad
+                with self.subTest(field=field, bad=bad):
+                    self.reject(stage, 'Invalid integer|Out-of-range|Inactive or source gate')
+
+    def test_provider_capacity_and_matching_collisions_are_rejected(self):
+        stage = deepcopy(self.stage)
+        stage['jobs'][0]['providers'][1] = stage['jobs'][0]['providers'][0]
+        self.reject(stage, 'Providers must be distinct')
+        stage = deepcopy(self.stage)
+        stage['matching_before'].append(stage['matching_before'][0][:])
+        self.reject(stage, 'Prior matching collision')
+        stage = deepcopy(self.stage)
+        stage['jobs'].append(deepcopy(stage['jobs'][0]))
+        self.reject(stage, 'Conflicting clone parent|Provider capacity already used|Clone chains collide')
+
+    def test_named_provider_operand_must_partition_the_parent_exactly(self):
+        stage = deepcopy(self.stage)
+        job = stage['jobs'][0]
+        provider, old_child = job['providers'][0], job['source_children'][0]
+        # The replacement remains an actual operand of its named provider;
+        # only the required disjoint partition of the parent is broken.
+        job['source_children'][0] = next(x for x in self.base.args[provider] if x != old_child)
+        self.reject(stage, 'Clone partition does not equal parent with disjoint sources')
+
+    def test_clone_replaces_the_whole_continuation_chain(self):
+        position = next(i for i, job in enumerate(self.stage['jobs']) if len(job['chain']) > 1)
+        for mutation in ('prefix_only', 'duplicate_use', 'boolean_use'):
+            stage = deepcopy(self.stage)
+            chain = stage['jobs'][position]['chain']
+            if mutation == 'prefix_only':
+                chain.pop()
+            elif mutation == 'duplicate_use':
+                chain.append(chain[-1])
+            else:
+                chain[0] = True
+            with self.subTest(mutation=mutation):
+                self.reject(stage, 'Clone must replace one whole continuation chain')
+
+    def test_wrong_first_chain_use_and_stale_graph_pins_are_rejected(self):
+        stage = deepcopy(self.stage)
+        stage['jobs'][0]['first'] = stage['matching_before'][0][1]
+        self.reject(stage, 'Clone must start an actual parent chain')
+        for key, message in (('source_graph_sha256', 'Clone source graph differs from pin'),
+                             ('output_graph_sha256', 'Clone output graph differs from pin')):
+            stage = deepcopy(self.stage)
+            stage[key] = '0'*64
+            with self.subTest(key=key):
+                self.reject(stage, message)
+
+    def test_declared_addition_and_role_counts_cannot_override_actual_graph(self):
+        for h, (_, _, result, _, _) in self.axes.items():
+            altered = copy(result)
+            altered.additions -= 1
+            with self.subTest(h=h), self.assertRaisesRegex(ValueError, 'Wrong addition count'):
+                type(self).real_supports(altered, verify_outputs=False)
+
+    def test_original_scalar_frames_are_preserved_in_every_round(self):
+        for h, (base, _, _, stages, original_digest) in self.axes.items():
+            self.assertEqual(clones.digest(base), original_digest)
+            for old, stage, result in stages:
+                old_support = self.support_cache[id(old)]
+                new_support = self.support_cache[id(result)]
+                expected = Counter((old_support[n], old.core[n], old.union[n]) for n in old.active)
+                expected.update((old_support[job['parent']], old.core[job['parent']],
+                                 old.union[job['parent']]) for job in stage['jobs'])
+                actual = Counter((new_support[n], result.core[n], result.union[n]) for n in result.active)
+                with self.subTest(h=h, source=stage['source_graph_sha256']):
+                    self.assertEqual(actual, expected)
+                    self.assertEqual(clones.digest(result), stage['output_graph_sha256'])
+
+    def test_rebuilt_finite_fixtures_have_expected_clone_and_role_counts(self):
+        # These are independent expected values, not values read from the
+        # fixture under test. Old pre-clone fixtures must fail this control.
+        for h, expected_clones, expected_roles in ((23, 253, 32693), (25, 353, 43056)):
+            base, document, result, _, _ = self.axes[h]
+            original = clones.read(HERE/f'original-{h}.json')
+            scalar = clones.read(HERE/f'scalar-{h}.json')
+            profile = clones.read(HERE/f'profiles-{h}.json')
+            with self.subTest(h=h):
+                self.assertEqual(sum(len(r['jobs']) for r in document['rounds']), expected_clones)
+                self.assertEqual(result.clone_count, expected_clones)
+                self.assertEqual(result.additions, base.additions+expected_clones)
+                self.assertEqual(scalar['cloned_additions'], expected_clones)
+                self.assertEqual(scalar['circuit_sha256'], clones.digest(result))
+                self.assertEqual(original['c'], result.additions)
+                self.assertEqual(original['q'], len(result.outputs))
+                self.assertEqual(original['R'], original['c']+original['q']-original['matched'])
+                self.assertEqual(original['R'], expected_roles)
+                self.assertEqual(profile['R'], expected_roles)
+                self.assertEqual(profile['rank_sum'], sum(t*n for t, n in enumerate(profile['blocks'])))
+
+
+
+
+class SkipCloneCertificateTests(unittest.TestCase):
+    @classmethod
+    def setUpClass(cls):
+        specification = importlib.util.spec_from_file_location('skip_clone_witness_tests', HERE/'witness.py')
+        cls.witness = importlib.util.module_from_spec(specification)
+        specification.loader.exec_module(cls.witness)
+        cls.result = cls.witness.run()
+
+    def test_exact_certificate_and_strict_pr53_improvement(self):
+        w, result = self.witness, self.result
+        self.assertEqual(w.js(result), w.read(HERE/'certificate.json'))
+        self.assertGreater(result['kappa'], Q(4498144, 10**11))
+        self.assertEqual(len(result['assembly']['constraints']), 47)
+        self.assertEqual(len(result['assembly']['margins']), 7)
+        self.assertTrue(all(value > 0 for value in result['assembly']['constraints'].values()))
+        self.assertTrue(all(value > result['kappa'] for value in result['assembly']['margins'].values()))
+        self.assertLess(result['bit']['upper'], 1)
+
+    def test_profile_mass_and_compensated_negative_blocks_rejected(self):
+        w = self.witness
+        original_read = w.read
+        for mutation in ('negative_blocks', 'synchronized_rank_sum'):
+            def changed_read(path):
+                row = deepcopy(original_read(path))
+                if mutation == 'negative_blocks' and Path(path).name == 'profiles-25.json':
+                    row['blocks'][1] += 2*(row['blocks'][2]+1)
+                    row['blocks'][2] = -1
+                elif mutation == 'synchronized_rank_sum' and Path(path).name in ('profiles-25.json', 'original-25.json'):
+                    row['rank_sum'] += 1
+                return row
+            with self.subTest(mutation=mutation), patch.object(w, 'read', changed_read), self.assertRaises(ValueError):
+                w.profile()
+
+    def test_clone_jobs_and_graph_code_require_current_source_pins(self):
+        import clone_io
+        original_read = clone_io.read
+        for missing in ('cloned_graph.py', 'clone-jobs-23.json'):
+            def missing_pin(path):
+                row = deepcopy(original_read(path))
+                if Path(path).name == 'SOURCE.json':
+                    row['files'].pop('research/skip-clones/'+missing)
+                return row
+            with self.subTest(missing=missing), patch.object(clone_io, 'read', missing_pin), self.assertRaises(ValueError):
+                clone_io.check_sources(clone_io.CERTIFICATE_INPUTS)
+        original_bytes = Path.read_bytes
+        def stale_bytes(path):
+            return original_bytes(path)+(b' changed' if path == HERE/'cloned_graph.py' else b'')
+        with patch.object(Path, 'read_bytes', stale_bytes), self.assertRaises(ValueError):
+            clone_io.check_sources(clone_io.CERTIFICATE_INPUTS)
+
+
+if __name__ == '__main__':
+    unittest.main()
diff --git a/tests/test_skip_strips.py b/tests/test_skip_strips.py
new file mode 100644
index 0000000000..36b8d1f496
--- /dev/null
+++ b/tests/test_skip_strips.py
@@ -0,0 +1,186 @@
+"""Adversarial controls for the skip-prefix strip producer (adapted from test_copied_fixed.py, PR #43/#48)."""
+from copy import deepcopy
+from fractions import Fraction as Q
+from pathlib import Path
+import importlib.util
+import sys
+import unittest
+from unittest.mock import patch
+
+ROOT=Path(__file__).resolve().parents[1]
+HERE=ROOT/'research/skip-strips'
+spec=importlib.util.spec_from_file_location('skip_strips_verify',HERE/'verify.py')
+v=importlib.util.module_from_spec(spec)
+spec.loader.exec_module(v)
+
+
+class SkipStripTests(unittest.TestCase):
+    def test_frozen_certificate(self):
+        self.assertEqual(v.js(v.run()),v.read(HERE/'certificate.json'))
+
+    def test_all_strict_constraints_and_dyadic_bracket(self):
+        result=v.run()
+        a=result['assembly']
+        self.assertEqual(len(a['constraints']),47)
+        self.assertEqual(len(a['margins']),7)
+        self.assertTrue(all(x>0 for x in a['constraints'].values()))
+        self.assertGreater(v.KAPPA,Q(1,2**15))
+        self.assertLess(v.KAPPA,Q(1,2**14))
+
+    def test_independent_cost_rows(self):
+        result=v.run();a=result['assembly'];p=a['parameters']
+        e,c,r,delta=p['epsilon'],p['c'],p['alpha_squared_power'],p['delta']
+        costs=[e,p['tau'],1-e*(1-p['lambda_prime']),p['tau'],
+               max(e+delta,1-r+delta),e+delta,1-e]
+        self.assertEqual([1-x for x in costs],list(a['margins'].values()))
+        self.assertTrue(all(1-x>v.KAPPA for x in costs))
+        self.assertEqual(p['q'],v.AB*(1-2*p['h']))
+        self.assertEqual(a['margins']['g1']-a['minimum_margin'],p['h'])
+
+    def test_real_lower_bound_exclusions(self):
+        result=v.run()
+        self.assertGreater(result['controls']['old_bit_at_new_lower'],1)
+        self.assertGreater(result['controls']['PR40_bit_at_new_lower'],1)
+        self.assertGreater(result['controls']['PR41_bit_at_new_lower'],1)
+        self.assertGreater(result['controls']['PR42_bit_at_new_lower'],1)
+        self.assertGreater(result['controls']['PR43_bit_at_new_lower'],1)
+        self.assertGreater(result['controls']['PR44_bit_at_new_lower'],1)
+        self.assertGreater(result['controls']['PR46_bit_at_new_lower'],1)
+        self.assertGreater(result['controls']['PR47_bit_at_new_lower'],1)
+        self.assertGreater(result['controls']['PR48_bit_at_new_lower'],1)
+
+    def test_recovery_and_every_pair_charged(self):
+        c=v.data_corners()
+        self.assertEqual((c['good'],c['fallback'],c['pairs']),(4073300,0,4073300))
+        p=v.profile();data=p['parts']['data']
+        self.assertEqual(data[1],2*(9*c['good']+47*c['fallback']))
+        self.assertEqual(data[21],2*c['good'])
+        self.assertEqual(data[17],2*c['good'])
+        self.assertEqual(sum(t*n for t,n in data.items()),2*p['N']*528)
+
+    def test_skip_strips_reduce_physical_roles(self):
+        p=v.profile()
+        self.assertEqual([axis['original']['R'] for axis in p['axes']],[32946,43409])
+        self.assertEqual([axis['original']['matched'] for axis in p['axes']],[12719,16991])
+        # PR #48 had R=(36432,48329) and W=177530859.
+        self.assertEqual(p['W'],177530859-(36432-32946)*2300-(48329-43409)*1771)
+        self.assertEqual(p['deficit'],1846900)
+
+    def test_skip_layout_is_exact_leave_one_out(self):
+        import skip_graph
+        class Adder:
+            support={}
+            def add(self,a,b):
+                if not a: return b
+                if not b: return a
+                assert not a&b, 'cancellation'
+                return a|b
+        c=Adder()
+        for k in range(1,14):
+            vals=[1<<i for i in range(k)]
+            full=(1<<k)-1
+            for layout in (skip_graph.skip_prefix,skip_graph.skip_prefix_reversed,skip_graph.prefix_suffix):
+                total,out=layout(c,vals)
+                self.assertEqual(total,full)
+                self.assertEqual(out,[full^(1<<j) for j in range(k)])
+
+    def test_skip_layout_nests_prefix_consumers(self):
+        # Covers of the two consumers of every prefix A_i are nested in the
+        # skip layout (A_{i+1} inside s_{i+2}) and incomparable in the
+        # ordinary layout (A_{i+1} versus s_{i+1}).
+        for k in range(4,12):
+            for i in range(1,k-1):
+                prefix_next=set(range(1,i+2))
+                skip_consumer=set(range(1,k+1))-{i+2}
+                ordinary_consumer=set(range(1,k+1))-{i+1}
+                self.assertTrue(prefix_next<=skip_consumer)
+                self.assertFalse(prefix_next<=ordinary_consumer or ordinary_consumer<=prefix_next)
+
+    def test_pinned_links_use_distinct_donors_and_uses(self):
+        import struct
+        raw=(HERE/'links-23.uses').read_bytes()
+        n,count=struct.unpack_from('<2I',raw)
+        pairs=list(struct.iter_unpack('<2I',raw[8:]))
+        self.assertEqual(count,len(pairs))
+        self.assertEqual(len({d for d,_ in pairs}),count)
+        self.assertEqual(len({u for _,u in pairs}),count)
+
+    def test_exact_recovery_explains_primary_modular_failures(self):
+        c=v.data_corners();prime=c['prime']
+        self.assertEqual((c['primary_good'],c['primary_fallback']),(4073290,10))
+        self.assertEqual(len(c['exact_recovery']),10)
+        for row in c['exact_recovery']:
+            self.assertEqual(len(row['pivots']),47)
+            self.assertEqual(row['ordered_zero_checks'],346)
+            for i,value in enumerate(row['pivots']):
+                q=Q(value)
+                self.assertNotEqual(q,0)
+                if i<=row['primary_failed_row']:
+                    self.assertNotEqual(q.denominator%prime,0)
+                    self.assertEqual(q.numerator%prime==0,i==row['primary_failed_row'])
+
+    def test_zero_recovered_pivot_rejected(self):
+        import data_recovery
+        bad=deepcopy(v.data_corners()['exact_recovery']);bad[0]['pivots'][0]='0'
+        with patch.object(data_recovery,'recover',return_value=bad):
+            with self.assertRaisesRegex(ValueError,'Incomplete rational pivot proof'):v.data_corners()
+
+    def test_corrupted_pair_coverage_rejected(self):
+        original_read=v.read
+        def corrupt(path):
+            result=deepcopy(original_read(path))
+            if path.name=='data-corners.json':result['records'][0][3]-=1
+            return result
+        with patch.object(v,'read',side_effect=corrupt):
+            with self.assertRaisesRegex(ValueError,'Pair coverage gap'):v.data_corners()
+
+    def test_mass_corruption_rejected(self):
+        original_read=v.read
+        def corrupt(path):
+            result=deepcopy(original_read(path))
+            if path.name=='profiles-23.json':result['blocks'][1]+=1
+            return result
+        with patch.object(v,'read',side_effect=corrupt):
+            with self.assertRaisesRegex(ValueError,'Fixed profile mass'):v.profile()
+
+    def test_omitted_paid_copy_detected(self):
+        p=v.profile();rows=p['child_multiplicities'].copy()
+        rows[1]-=p['N']
+        self.assertNotEqual(sum(t*n for t,n in rows.items()),p['total_rank'])
+        self.assertEqual(sum(t*n for t,n in p['parts']['paid_endpoint_copy'].items()),4073300)
+
+    def test_mismatched_source_hash_rejected(self):
+        original_read=v.read
+        def corrupt(path):
+            result=deepcopy(original_read(path))
+            if path.name=='SOURCE.json':result['files']['scripts/copied_centers_network.py']='0'*64
+            return result
+        with patch.object(v,'read',side_effect=corrupt):
+            with self.assertRaisesRegex(ValueError,'Pinned source changed'):v.check_sources()
+
+    def test_assembly_boundary_rejected(self):
+        result=v.run()
+        with self.assertRaises(AssertionError):
+            v.assembly(result['finite_bridge'],v.AB,result['assembly']['minimum_margin'],a_complex=v.AC)
+
+    def test_balanced_transfer_negative_controls(self):
+        r=v.run()
+        for kw in ({'original_prefix':True},{'old_guard':True},{'old_exposures':True}):
+            with self.assertRaises(AssertionError):
+                v.assembly(r['finite_bridge'],v.AB,v.KAPPA,a_complex=v.AC,**kw)
+
+    def test_float_saving_rejected(self):
+        with self.assertRaises(ValueError):v.moment(2,2,{1:1},0.01)
+
+    def test_restriction_tree_mutation_rejected(self):
+        g=v.geometry();edges=g['row_tree'][:];edges[-1]=edges[0]
+        with self.assertRaises(ValueError):v.tree(edges,48)
+
+    def test_deterministic_prime_and_minor_gaps(self):
+        for h in (23,25):
+            e=v.exactness(h)
+            self.assertTrue(all(x>0 for x in e['modulus_gaps'].values()))
+            self.assertEqual(e['prime_product'],2596143476298333157846630848266239)
+
+
+if __name__=='__main__':unittest.main()