Prove P112 mixed-rank shortening through common support 38

research/p112-mixed-support38/CHECKS.json

@@ -0,0 +1,36 @@
+{
+  "status": "local registered replay, committed export and designated peer check passed",
+  "utc": "2026-10-17T20:01:52.057442+00:00",
+  "baseline": "463113d675af7410ac91d53a4fed51ec6494eb9a",
+  "environment": "CPython3.12.15 macOS arm64; standard library",
+  "registered": "PASS all three commands with -W error; controls under -O",
+  "primary": "3780orthogonalplanes,896live-axis fibers times10795sixspaces; A2 and chamber receipts match RESULT.json",
+  "independent": "788035ambient sixspaces,16support20sets,120fourframes,1920intersections and3780orthogonal-pair orbit",
+  "controls": "7groups passed; includes malformed inventory, corrupted source, symlink, oversized file and absent-puncture control",
+  "diagnostics": "Ruff check and format check passed",
+  "limits": [
+    "Global g(5)=12/g(6)=20 historical censuses inherited, not rerun",
+    "Weyl-chamber theorem is a written dependency",
+    "Independent checker shares authenticated geometry; no independent A2/global-capacity proof",
+    "No Lean, Docker, full selected cold replay, supplier or kappa claim"
+  ],
+  "peer_review": {
+    "reviewer_worker": 1,
+    "reviewed_commit": "fb75f16629b3d664488e9f9a186c9933b0901235",
+    "result": "No blocking issue at stated characteristic-zero commoncut38 ranks5,5,6 scope after singleton-domain clarification.",
+    "resolved": "Singleton sites/J_i explicitly restricted to U outsideS in PROOF.md; proportional recheck accepted SHA25674b3e610a69a2f5e372595efbc17f3215c3f1af53a0f536dbbc12e5abb821327.",
+    "receipt": "REVIEW.json"
+  },
+  "committed_export": {
+    "commit": "fb75f16629b3d664488e9f9a186c9933b0901235",
+    "file_count": 18,
+    "sha256": "1b6780a64e025756c347930849aa2eb25bd5f1ca59cc30e520d1099df2629f30",
+    "result": "All3directcommands passed optimized Python3.13.16 in exact committed read-only Git-free export. Final accepted proof-only correction and review-metadata update do not change executables/sources/RESULT."
+  },
+  "submission_refresh": {
+    "canonical_main": "463113d675af7410ac91d53a4fed51ec6494eb9a",
+    "priority_revision": 6,
+    "cutoff": "2026-10-17T20:03:02Z",
+    "coverage": "Complete311openPRs/2101discussionrecords,no warnings. Worker2 near-synonym duplicate released. No novel-coverage inference from no-match."
+  }
+}

research/p112-mixed-support38/ENGINEERING.md

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+# Verification design
+
+The executable claim is finite geometry supporting the written mixed-rank
+support 38 theorem. No supplier implementation is changed.
+
+Separate Before Serializing Shared State shaped the workspace boundary. This
+package, branch, status file and session cache have one writer. Shared Git
+metadata operations use the batch's OS-released lock. Research arithmetic runs
+without that lock. Peer checks read frozen source and write only reviewer-owned
+receipts.
+
+Prove It Works shaped the publication gate. Run the actual registered commands,
+then replay exact committed export bytes from a read-only Git-free tree under
+a second Python version before freezing the peer-review tree.
+
+## Power of Ten
+
+1. The three scripts use structured loops and explicit ValueError exits. There
+   is no recursion. File contexts and temporary-directory contexts own cleanup.
+2. Root inventories have 120 full/112 live entries. Binary inputs have at most 512
+   labels of at most 18 bits. D6 has 3780 planes,448 minimum planes and 896 live-axis
+   fibers. A fiber has 256 functionals and 10795 dual two-spaces. The alternate
+   ambient census has 788035 three-dimensional annihilators. A2 has 1120 planes
+   and its orbit queue can contain at most 1120 items. The alternate D6 orbit
+   queue has at most 3780 items. Source inventories have 5 rows and 4 retained rows.
+   Registered command timeouts are 120 seconds, sequentially.
+3. CPython requires dynamic allocation. This managed-runtime deviation covers
+   only the bounded inventories above. Source files are at most 1 MB each; the
+   largest retained primary census serialization has 896 rows with bounded
+   histograms. Bit masks have at most 33 bits for D6 fibers. The alternate full
+   inventory retains only 16 maximal supports. No asynchronous work or subprocess
+   remains after a command. Direct allocation instrumentation is unrun.
+4. The largest new function is a 2_receipt at 60 physical lines after Ruff format.
+   Other functions are below 60 lines. It owns one plane classification and
+   orbit/fiber receipt; splitting it further would add a single-use forwarding
+   boundary without a separate ownership requirement.
+5. Runtime require checks validate source pins, exact cardinalities, label
+   ranges, census totals, orbit coverage and arithmetic implications. They
+   remain active with Python optimization. Controls run with -O.
+6. Module globals identify immutable paths and finite expected inventories.
+   Geometry, bit masks, queues and receipts stay local. No peer artifact is
+   writable through the verifier.
+7. Read and parse failures propagate. Output uses exclusive creation. Source
+   inventory lengths and exact allowed paths are checked before reading or
+   importing sources. Hash and Git-blob checks bind retained bytes without Git.
+8. The fixed geometry module is loaded after authentication. No input supplies
+   code, module names or call targets. No generated code or eval is used.
+9. The import loader is the sole dynamic call boundary and loads one fixed
+   authenticated local module. All mathematical helpers otherwise have visible
+   direct call targets. This Python import boundary is a bounded runtime
+   deviation from a ban on indirect calls.
+10. Registered replays enable -W error. Ruff check and formatting checks cover
+    every new Python file. There is no compiled component or compiler warning
+    configuration. The standard-library runtime and inherited geometry internals
+    retain their existing contracts.
+
+The managed-runtime allocation and fixed import-loader deviations stay confined
+to these offline checks. Revisit them if inputs become user-sized, the module
+path becomes configurable, or the verifier becomes a service. Keep the bounds
+and authentication tests when changing the implementation.

research/p112-mixed-support38/PROOF.md

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+# Mixed-rank shortening through common support 38
+
+Let K be any characteristic-zero field. On the fixed 112 nonframe E8 root
+lines, let C be the root-evaluation code, using doubled roots r_t:
+
+```
+C = {(r_t dot h)_t : h in K^8}.
+```
+
+Binary support rank is the rank over F2 of the pinned nine-bit labels in a
+vector's support. Let pi_S delete ONE common coordinate set S.
+
+**Theorem.** If |S| <= 38, and three residual pieces q_1,q_2,q_3 have binary
+support ranks at most 5,5,6, then
+
+```
+dim_K(pi_S(C) intersect span_K(q_1,q_2,q_3)) <= 1.
+```
+
+The pieces can be dependent and have arbitrary coefficients. Rank constraints
+apply only outside S. The theorem is sufficient; sharpness is not claimed.
+PR3260 proves support 37 and leaves the 30-root D6 classification branch
+open. This proof closes that branch at 38. It supplies no new historical
+profile exclusion, full C5, decoder-floor theorem, supplier or kappa.
+
+## Dependencies and exact finite coverage
+
+The source-pinned root-simplex proof and receipt supply global P112 capacities
+12 and 20 for binary ranks five and six. Their complete historical flat
+censuses are inherited, not rerun here. The standard real Weyl-chamber theorem
+and the rational-nullspace transfer below are written dependencies.
+
+The new replay checks the 28 six-node parabolics. Only D6 and E6 have at
+least 30 root lines, with counts 30 and 36. All 56 five-node parabolics have
+at most 20 root lines. The normal planes of the two qualifying six-node
+parabolics contain respectively two orthogonal root lines and an A2 triple.
+Thus their Weyl images are covered by orthogonal root pairs and A2 planes.
+The source module also checks all 960 simple reflections.
+
+The replay checks all 1,120 A2 planes, their live union spectrum 78/80,
+and transitivity under eight simple reflections. Each reflection induces
+an invertible linear map of the nine-bit labels. The three representative
+27-line projective fibers each span a binary seven-space. Their 2,667
+five-spaces have fiber capacity ten. Their 127 hyperplanes have the following
+complete full-E8 support / fiber-support counts:
+
+```
+(0,0):1, (12,11):27, (12,12):36, (16,15):36, (16,16):27.
+```
+
+These joint counts follow PR3262's support-plus-singleton argument; they are
+rechecked here. The independent D6 check does not independently re-prove
+these A2 premises or the global 12/20 capacities.
+
+## Classify a two-dimensional intersection
+
+Suppose the displayed intersection has dimension at least two. Choose two
+independent restricted codewords and lifts c_h,c_k. Their full support union U
+lies in S together with the three piece supports. Extend each residual piece
+by zero on S. The global capacities give
+
+```
+|U| <= |S| + 12 + 12 + 20 <= 82.
+```
+
+The lifts are independent, since their restrictions are independent. Therefore
+the rational span Z of all full E8 roots perpendicular to h and k has rank
+at most six. At least 112-82=30 live roots belong to Z.
+
+Every root in the rational span of Z is also perpendicular to h,k. A generic
+real vector perpendicular to Z therefore has exactly those full E8 roots as
+its perpendicular root set. Indeed, finitely many additional hyperplanes can
+be avoided. Move this real vector into a closed Weyl chamber. Its perpendicular
+root system is a parabolic subsystem of rank at most six. The checked counts
+force rank six and type D6 or E6. Its two-dimensional normal plane equals the
+K-span of h,k, since rational matrix rank is unchanged by characteristic-zero
+extension. In particular it has a rational basis.
+
+This argument uses the full perpendicular root system, even though the count
+30 comes from live roots. It does not assume any individual piece is a
+restricted codeword or any quotient-basis normalization.
+
+## Exclude E6 using joint incidence capacity
+
+In this branch the normal plane is an A2 root plane. Replace h,k by a rational
+basis of that plane. Their live support union has at least 78 coordinates.
+On the complement of S there is a fixed two-by-three matrix A with
+
+```
+(c_h(t), c_k(t))^T = A (q_1(t), q_2(t), q_3(t))^T.
+```
+
+At a coordinate in U outside S where exactly one piece q_i is nonzero,
+the nonzero evaluation pair is proportional to column i of A. Such singleton
+sites for q_i lie in ONE projective fiber. A zero column has no such sites. Besides three large
+27-line fibers, the A2 plane has three one-line fibers.
+
+Let M_i be total residual support of piece i and let J_i count its singleton
+sites in U outside S.
+For i=1,2, the global capacity 12 and fiber capacity ten give M_i+J_i<=22.
+For the rank-six piece, M_3+J_3<=32. To see the latter, a one-line fiber gives
+the stronger bound 21. For a large fiber in a seven-space W, let V be the
+actual binary support span of q_3. If dim(V intersect W)<=5, then J_3<=10
+and M_3<=20, giving at most 30. Otherwise V is a six-space contained in W.
+The complete hyperplane inventory above gives M_3+J_3<=32. Supports can be
+proper subsets of their binary closures; this only lowers these quantities.
+
+Let N=|U minus S|. Each union coordinate has at least one piece incidence;
+a nonsingleton has at least two. Thus
+
+```
+2N <= sum_i(M_i+J_i) <= 22+22+32 = 76.
+2N >= 2*(78-38) = 80.
+```
+
+This contradiction permits cancellations, arbitrary columns of A, and piece
+support outside U. Extra support only increases M_i. No common shift is
+assumed modal.
+
+## Exclude the D6 equality case with the actual puncture
+
+Every D6 normal plane is spanned by two orthogonal E8 roots a,b. The full E8
+nonzero evaluation union has 90 lines. Its four projective fibers have sizes
+33,33,12,12. The two large fibers are the two coordinate axes. The exact live
+union histogram over all 3,780 orthogonal root pairs is
+
+```
+82:448, 84:2688, 85:448, 86:168, 88:28.
+```
+
+Since |U|<=82, only the 448 minimum planes can occur. Then |U|=82 and |S|=38
+are forced. Their residual union has at least 44 coordinates, while the
+three pieces have at most 12+12+20=44 incidences in total. Equality follows
+throughout. The pieces have disjoint supports of sizes 12,12,20, every support
+coordinate lies in U outside S, and every coordinate is a singleton.
+
+Consequently the rank-six piece's entire support of size 20 lies in ONE
+projective fiber. A small fiber has only twelve full E8 coordinates. Each
+large fiber in a minimum plane has 29 live labels of binary rank eight.
+For each of the 448 planes and both large fibers, the primary replay checks
+all 10,795 six-spaces in that eight-space via unique two-dimensional dual
+spaces. Their maximum live intersection is always 18. A binary space of rank
+at most six meets the fiber in a subset of one such six-space. This contradicts
+support size 20 and proves the theorem.
+
+The puncture is essential to this finite premise. A full 33-line axis has
+six-space capacity 20, with sixteen maximizing supports. A separate checker
+enumerates all 788,035 ambient six-spaces, rather than using the primary
+fiber coordinates, and reconstructs those sixteen sets. It checks all 120
+orthogonal four-root subsets inside the axis. Each support 20 meets each such
+four-frame in two or four roots, with multiplicities 1,440 and 480. All 896
+actual minimum-plane axis punctures are such four-frames. A separate orbit
+check covers all 3,780 orthogonal pairs under the simple reflections.
+
+This alternate check excludes a surviving support 20. It does not by itself
+establish the stronger punctured capacity 18 for subsets of smaller full
+intersections; the primary complete census supplies that stronger value.
+
+## Credit and limits
+
+Nick Petalas's PR3260 supplies the mixed 5,5,6 question and support 37 proof.
+PR3262 supplies the joint A2 support/singleton argument. The source-pinned
+root-simplex work, PR3249 and PR3255 supply the capacity and geometry lineage.
+DreamingOfClouds/PR3154 supplies the TWOSIDE motivation; the retained E8/P112
+lineage includes joaqcoo. Worker1's simultaneous uniform 5,5,5 support 46 audit
+identified the same D6 prerequisite and has a distinct equality obligation.
+The present theorem uses no unpublished sibling result.
+
+The finite certificate and written characteristic-zero proof do not constitute
+proof-kernel or independent human verification. Support39, global sharpness,
+a fresh caller, the original C5 reductions, physical/recursive realization,
+restoration, the complete consumer bill and multiplication transfer remain
+separate obligations. No selected result is changed.

research/p112-mixed-support38/README.md

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+# P112 mixed-rank shortening at common support 38
+
+Three residual P112 pieces of binary support ranks at most 5,5,6, after ONE
+common deletion set of size at most 38, meet the restricted root-evaluation
+code in dimension at most one over every characteristic-zero field. This
+improves PR3260's sufficient threshold 37. It does not prove sharpness or a
+new historical profile exclusion. [PROOF.md](PROOF.md) gives the complete
+argument and inherited dependencies.
+
+The new step closes the D6 boundary. A putative equality requires a rank-six
+piece supported on 20 singleton coordinates in one projective fiber. All 896
+relevant live axis fibers have rank-six capacity 18. An independent ambient
+annihilator/puncture check also excludes support 20. The E6 branch uses
+PR3262's rechecked joint incidence argument, with allowance76 versus80.
+Exact coverage and full-census digest are in [RESULT.json](RESULT.json).
+
+## Reproduce
+
+Python 3.12 or later, standard library only, from the repository root:
+
+```sh
+python3.12 -W error -B scripts/research_package.py \
+  --package research/p112-mixed-support38
+```
+
+The three registered commands run sequentially with 120-second limits. Direct
+Git-free export replay uses these commands, from the exported root:
+
+```sh
+python3 -W error -B research/p112-mixed-support38/verify.py
+python3 -W error -B research/p112-mixed-support38/independent_check.py
+python3 -W error -O -B research/p112-mixed-support38/test_verify.py
+```
+
+The export needs the package and its three declared baseline inputs at their
+original relative paths. Pending PR proof/receipt bytes are retained locally
+with exact commit, Git blob and SHA256 pins in [SOURCES.json](SOURCES.json).
+No pending Git objects or Git subprocesses are needed for direct replay.
+Authentication bounds inventory length, paths and source sizes before import.
+`verify.py --output NEW_EXTERNAL_PATH` creates an external receipt and refuses
+to overwrite an existing file. Set TMPDIR to an external session directory to
+keep control-fixture files outside the checkout.
+
+## Open next questions
+
+Does support 39 admit a counterexample or a larger sufficient theorem? The D6
+incidence equality used here no longer follows at 39. First bind the actual
+puncture and mixed support-overlap patterns before another census. This is
+unrun and not evidence that 38 is sharp. A new TWOSIDE application also needs
+a named caller, current public-exclusion reconciliation and its exact retained
+code dimension. [TASK.json](TASK.json) records these questions and blockers.
+
+## Sources and verification limits
+
+Credit Nick Petalas/PR3260/3262, DreamingOfClouds/PR3154 and the retained
+root-simplex/E8/P112 lineage including joaqcoo. Worker1's concurrent uniform
+rank-five support 46 theorem shares the D6 prerequisite but proves a different
+predicate. This package uses no unpublished sibling theorem. All copied
+predecessor bytes stay unchanged. New work and retained sources are Apache-2.0.
+
+The global five-/six-flat capacities 12/20 and standard Weyl-chamber theorem
+are inherited; their historical global censuses are not rerun. The independent
+checker shares authenticated root/label reconstruction and does not independently
+establish A2 capacities or characteristic-zero classification. Internal review
+is not independent human or proof-kernel review. [CHECKS.json](CHECKS.json)
+records executed commands, export evidence and review limits.
+
+No full C5, decoder floor, native or physical supplier, restoration, complete
+consumer, all-size multiplication theorem, new kappa or selected-frontier
+update is claimed. Lean, Docker and the selected full cold replay are unrun.