Research: exclude single-child absorption with fixed stream encoders

research/idle-single-child-absorption/PROOF.md

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+# A single existing phase child cannot absorb this idle-axis defect
+
+This is a source-bound obstruction to a precise proposed cost change. It is
+not a barrier to changed words or arbitrary encodings, and gives no new kappa.
+
+## 1. The replacement interface
+
+Use the retained PR3148 T740 allocation and the literal source recipe at
+owner replica zero, port zero. The independent stream inputs are arbitrary
+complex arrays on `F2^73`, including arbitrary entering dirt. Transfer axis
+`v=16` from residual child 10 to residual child 12. Their ranks change from
+`48,48` to `47,49`. All hosted calls, other residuals, interactions, translations
+and the exact target operator stay fixed.
+
+Permit replacing exactly one of the twenty residual phase-child positions.
+At that position the new completed child is Fourier diagonal and acts on
+one scalar stream. Its eigenvalue `z(x)` at character `x` can be *any* nonzero
+complex number. This grants more freedom than a legitimate quarter-turn
+phase child, regardless of its new address axes or rank.
+
+Before and after that child, grant arbitrary invertible constant scalar
+stream matrices, at zero cost. Their action is explicitly
+`L_stream tensor I_address` and `R_stream tensor I_address`: they apply the
+same stream map at every address and do not mix addresses with stream labels.
+They need not be reciprocal before normalization. Arbitrary constant global
+phases and arbitrary dirty spectator streams restored exactly are allowed.
+The one-child operator, rather than its private recursive implementation,
+is the interface being replaced.
+
+This assumption is essential. A character-dependent controlled permutation,
+an address-changing chart, two child uses or two changed sites belongs to a
+different interface. Scalar coefficients alone do not imply stream-only action.
+No such broader encoding is excluded or assigned a zero implementation cost.
+
+## 2. A common projector is necessary
+
+Let `E=e_s e_s^T` select the child's stream. With a choice of adapter order,
+the local replacement at character `x` is
+
+```
+R(x) = L [I+(z(x)-1)E] K.
+```
+
+At character zero the required local operator is `R(0)=I`. Since `z(0)!=0`,
+the bracket is invertible, so
+
+```
+K = [I+(z(0)-1)E]^-1 L^-1,
+R(x) = I + (z(x)/z(0)-1) P,
+P = L E L^-1,   P^2=P,   rank(P)=1.
+```
+
+The normalization also removes a constant global phase. In particular:
+
+1. `rank(R(x)-I)<=1` at every character;
+2. all nonzero flattened matrices `R(x)-I` span the same one-dimensional
+   space, because the same `P` is used at every character;
+3. a nonzero rank-one defect has nonzero trace, since its trace is the scalar
+   multiplying `P`. A nonzero rank-one nilpotent is insufficient.
+
+Conversely, on any supplied finite character set containing zero, rank at
+most one, common one-dimensional span, nonzero trace for nonzero defects,
+and `1+trace(R(x)-I)!=0` characterize this favorable interface on that set.
+For a nonzero rank-one matrix `D`, `D^2=trace(D)D`, so
+`P=D/trace(D)` is the required idempotent. Every rank-one idempotent is similar
+to `E`. If all defects vanish, choose any `P` and all eigenvalues one.
+The converse is a useful control; no finite passing set proves a full-domain
+replacement or a legal physical implementation.
+
+Adding arbitrary dirty restored scalar spectator streams cannot avoid the
+necessary condition. On the extended space the same argument gives a rank-one
+projector. The full required endpoint is `R(x) direct-sum I_spectator`; any
+nonzero defect therefore forces that projector to vanish on, and have no
+cross terms with, all spectator rows and columns. Restriction to the eight
+visible streams gives the same condition. A clean ancillary subspace or
+unrestored garbage is a different promise.
+
+## 3. Bind the condition to literal cuts
+
+All matrices below are in the Fourier-character representation; global phase
+is retained. At a candidate site, write the altered chronological word as
+`B(x) D_site(x) A(x)`, with earlier factors on the right. Omit only that site's
+residual child phase. Retain its separate address translation, the other
+child at the same event, every hosted phase and every other literal event.
+The required replacement is unique:
+
+```
+R_site(x) = B(x)^-1 T(x) A(x)^-1,
+T(x) = i^popcount(x) I_8.
+```
+
+Both checkers verify `B D_site A` equals the altered word and
+`B R_site A=T` at the tested cuts. At character zero all twenty required
+replacements equal identity. The original unaltered source word equals `T`
+at each character used for a witness. Thus each finite witness is an exact
+counterexample to a necessary full-domain identity, not an extrapolation from
+sampling.
+
+For nineteen sites, `x=16` alone gives a nonzero two-by-two minor of
+`R_site(16)-I`. Their row/column indices and exact determinants are in
+`RESULT.json`; every determinant is `2` or `-2`. This excludes all one-stream
+phase replacements at those sites, even with free oblique scalar encoding.
+
+Site 12 exposes why checking each character separately is too weak. With
+zero-based matrix indices, the exact required defects are
+
+```
+R_12(16)-I       = (i-1)(E_(6,4)+E_(6,6)),
+R_12(33554432)-I = (i-1)E_(0,0).
+```
+
+Each parenthesized matrix is a rank-one idempotent. They are different,
+and even commute (both products are zero). Flattening positions `(6,4)` and
+`(0,0)` across those two character defects gives determinant
+`(i-1)^2=-2i`, which is nonzero. They cannot share one fixed projector.
+This excludes the remaining site. Commutativity alone would not have sufficed.
+
+The finite computation covers exactly these twenty sites for this one
+transfer, owner and port. The algebraic exclusion quantifies over every
+nonzero eigenvalue function, every invertible *stream-only* scalar adapter
+and every proposed child rank, because none can pass the necessary witnesses.
+It does not quantify over other transfers or changed intervening words.
+
+## 4. Why this tests recursive cost
+
+Let `p=1-a`, `0<a<1`. Holding every other invoice term, width and stock
+normalization fixed, strict concavity gives
+
+```
+(47/73)^p+(49/73)^p < 2(48/73)^p.
+```
+
+Thus the proposed transfer has a favorable *recursive child* profile at
+unchanged call count and rank mass. PR3220 ruled out a separately added
+whole repair child under its bounded fixed-mass tariff. This proposal instead
+tries to replace an existing call and avoid that added tariff; PR3229's
+abstract direct-sum shared-repair premise does not settle it.
+
+The optimistic test here grants the replacement arbitrary eigenvalues and
+all adapter, movement, source-access, helper, restoration, precision and stock
+costs free within the stated stream-only interface; no additional address
+operator is granted. It already fails exact operator equality at every site. Hence no
+implementation within this one-site stream-only contract can realize the
+claimed profile saving. This is an algebraic rejection of the stated proposal,
+not a runtime lower bound, a universal cost obstruction or a stronger kappa.
+A passing broader construction would need its own complete invoice and
+both supplier/outer interfaces; unknown costs are not zero.
+
+## 5. Evidence, controls and credit
+
+`verify.py` authenticates the retained input bytes and executes the exact
+hashed interpreter source via `compile(..., optimize=0)`, bypassing bytecode
+caches. It reuses that source's data-event interpreter, with separate
+Gaussian-rational prefix/suffix inversion. It authenticates all 2,717,280 T740
+fill rows, reconstructs the actual owner-zero/port-zero residuals, and finds
+all witnesses after 27 deterministic characters.
+
+`independent.py` is a separate elementary-matrix reconstruction and reverse
+factor inverse implementation. It reuses only authenticated source/data
+parsing, not the retained `run_data_event` or `replay` functions, and checks a
+larger explicitly declared finite character domain. Its receipt is separate.
+These are independently tasked AI-assisted implementations, not independent
+human review. `VERIFY.json` declares both complete local commands.
+
+Controls include an accepted non-coordinate projector, an unchanged literal
+child, a pair of individually valid but inconsistent projectors, a nonzero
+rank-one nilpotent and a rank-two defect. The actual receiver witness is also
+a source-bound counterexample to accepting only per-character rank. Full
+arbitrary stream input columns and exact phase are retained.
+
+Credit remains with Rohan Arun's idle-axis proposal, Gonzalo-Mier's PR3199
+physical interpreter/census, Nick Petalas's PR3187 physical counterexample,
+Gareth Davidson's PR1387 obstruction, Jeremy Diviney's PR3148 allocation,
+and the retained dcharb78, Marius-Paul Dumitrean, Jeremy Kothe, geometry/native
+and selected-supplier lineage. Ross Richardson's PR3220 motivated absorption;
+PR3229's abstract shared-repair follow-up is explicitly distinguished rather
+than repeated. Actual runtime source bytes and base are pinned in `SOURCES.json`.
+This proof, the one-site test and its separate oracle were prepared by Ross
+Richardson with substantial OpenAI Codex assistance. New material Apache-2.0;
+source archives and notices are unchanged. No formal proof kernel, full native
+supplier, complete outer multiplication theorem or new selected result is claimed.

research/idle-single-child-absorption/README.md

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+# One-child absorption fails for the concrete idle-axis transfer
+
+**Scoped exact negative result; no new kappa or supplier admission.**
+
+Moving axis16 from T740 child10 to child12 gives the favorable concave ranks
+`48,48 -> 47,49`, but changes its eight-stream physical operator. Can the
+defect be absorbed into one existing child, avoiding an additional repair call?
+
+For owner replica zero / port zero, the answer is **no at all twenty residual
+sites** under the declared interface: one completed Fourier-diagonal phase on
+one stream, surrounded by arbitrary constant invertible scalar *stream-only*
+encoding and decoding (`L_stream tensor I_address`). The eigenvalue function
+and all added implementation costs are granted free. Arbitrary dirty inputs,
+restored dirty scalar spectators and exact global phase remain required.
+
+Nineteen sites fail a rank-two minor at character16. The remaining receiver
+site needs two different rank-one projectors at characters16 and33554432;
+their flattened minor is `-2i`. Separate per-character rank checks, and even
+commutativity, miss this failure. [PROOF.md](PROOF.md) gives the normalization
+argument, exact witnesses, cost connection and covered domain.
+
+This tests a different mechanism from PR3220's added repair tariff and
+PR3229's abstract independent-group sharing. It does not cover two changed
+sites, a changed interaction word, a character-dependent encoder or an encoder
+mixing addresses with stream labels. It does not validate or refute another
+PR's complete allocation, supplier or kappa claim.
+
+## Reproduction
+
+```
+python3.12 -B scripts/research_package.py --package research/idle-single-child-absorption
+```
+
+Both commands in `VERIFY.json` are standard-library-only. `verify.py` uses
+authenticated retained event semantics plus exact Gaussian-rational cut
+inversion; `independent.py` separately expands elementary matrices and reverses
+their factors. They use all independent stream columns and retain exact phase.
+The full T740 payload authenticates; only one owner/port/transfer is investigated.
+`RESULT.json` and the independent committed receipt preserve exact coverage
+and semantic controls. `--output /new/path.json` writes an exclusive fresh receipt.
+These are finite exact checks supporting a written interface proof, not a
+formal-kernel proof or full selected cold replay. Hosted CI is separate.
+
+Base main: `463113d675af7410ac91d53a4fed51ec6494eb9a`, priority revision6.
+Selected record unchanged. Source/input pins, original card, concrete scope,
+stopping rule and attribution are in `SOURCES.json` and `TASK.json`.
+
+## Open next questions
+
+- A character-dependent stream/address adapter can change the projector.
+  First give one actual encoder/inverse and its arbitrary-dirt endpoint,
+  then price all child, routing/volume, helper, precision and restoration work.
+  This result provides no affordable adapter.
+- A joint rewrite of at least two existing sites is outside this theorem.
+  First specify a concrete factorization and complete recursive invoice.
+  The known additive two-child repair merely restores the original factors;
+  this result does not justify an automatic wider search.
+
+Structured relevance, first steps and blockers are in `TASK.json`.
+
+## Credit
+
+Ross Richardson with substantial OpenAI Codex assistance and two separately
+tasked mathematical specialists. No independent human review is claimed.
+The retained physical interpreter/geometry and predecessor credit remain with
+Gonzalo-Mier/PR3199, Nick Petalas/PR3187, Gareth Davidson/PR1387, Jeremy
+Diviney/PR3148, Rohan Arun, dcharb78, Dumitrean, Jeremy Kothe and the retained
+native/geometry/selected construction lineage. PR3220 and PR3229 retain their
+distinct results and credit. New material Apache-2.0; frozen sources unchanged.