Prove a sharp 112-call bound for active coded-mode repair

research/coded-active-query-bound/ATTRIBUTION.md

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+# Attribution and inherited scope
+
+Prepared for Dillon with substantial OpenAI Codex assistance. The source-specific coset theorem, its application to chronological input/output cuts, sharp one-sided and rectangular mode bounds, finite-H extension and direct-sum corollary are the additions from this session. General causal/image/row-space rank-cut methods are standard and retain the earlier repository method credit to eumemic's PR288 and Ali Romman's PR468, summarized in `fixed-support-unary-query-bounds/README.md`. That related fixed-carrier theorem is not a dependency or a proof of this physical-code result. Distinct agents derived the proof, audited its mathematical scope, checked an independent binary formula and prepared the portable package. Their agreement is agent corroboration, not independent human or proof-kernel acceptance.
+
+- Dugongue's PR4929 supplies the native center-free code A, its all-input producer contract and 408 independent helper endpoints.
+- PR4933, prepared for Dillon, supplies the actual real signed orthogonal normalizers K_i, physical code C, exact phase conventions and fixed owner recoding.
+- Chris R/Clear1nsight's PR4932 `coded-port-adapter-ceiling` supplies the finite-phase group-algebra formulation, tensor faithfulness theorem, physical adapter interface and the distinct one-parallel-bank grade-two result. The present theorem changes chronology and method; it does not relabel that predecessor result.
+- The retained `passive-saved-query-ceiling` package supplies the explicit uniform finite-alphabet clarification and its separate passive-controller scope. Its owned-reset clarification is respected: no inverse-uncompute or reset-cost lower bound is used here.
+- Cinna Davis/1337isnot1337's PR4694/4698, Cao Yifan/intqwq's PR4610 and round-three native source, zxv's exact Clifford work, and the retained helper, compiler and outer contributors remain credited through the cited packages and source notices.
+
+No predecessor implementation is vendored. The main verifier executes authenticated existing `coded-native-caller/clifford_exact.py` bytes and reads the retained code/normalizer receipts. Its byte origin and Apache-2.0 obligations remain in `coded-native-caller/NOTICE` and `ORIGINS.json`, pinned as provenance in SOURCES.json. The independent binary checker does not import that implementation. All new files are under the repository's Apache-2.0 license.
+
+The native all-input theorem and its physical recoding are inherited premises. This package does not rerun the full source producer, provide a numerical compiler/outer admission, or promote either historical or current source claims to formal theorem acceptance.

research/coded-active-query-bound/EXTENSION-REVIEW.md

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+# Independent mathematical review of the active local-query obstruction
+
+Review by a separate OpenAI Codex agent, 2026-10-23. This is AI-assisted mathematical review and executable corroboration, not independent human acceptance or proof-kernel verification. Repository and GitHub state were not changed by this reviewer.
+
+## Result
+
+I found no remaining mathematical blocker in the fresh-input active-query argument, including the finite-monomial-alphabet strengthening, homogeneous direct sums, canonical rectangular mode maps, and free retraction basis changes described below. The restrictions in this report are essential. In particular this is not a lower bound on the complete p2/p16 caller or an arbitrary multiplication implementation.
+
+The first draft was read with its finite Clifford ambient algebra. I independently proposed the explicit small-width caveat, reviewed the author's stronger finite-H formulation, and checked each extension algebraically. The draft snapshot initially replayed had PROOF.md SHA256 `bdc4e345c2a52bdc1dc4a6caff61ecb5eb84e9909cf8d16515b93ac322f33889` and verifier SHA256 `d21ffd6282cd226080db0fabde57a5e2b96623633d503769a49c6b3f0b85efde`. The extensions below were reviewed from the stated formulas and messages; their final integration may change those hashes. This report does not pretend the final package bytes were already replayed at that point.
+
+## 1. Algebra and tensor faithfulness
+
+Only the free monomial group H needs to be fixed and finite up to its stated finite scalar phases. It may be the full group of one-column coordinate permutations and phases in a fixed root-of-unity group; it need not consist only of Clifford monomials. The group generated by H and Q need not be finite.
+
+Take k=C, quotient the finite H group algebra by each scalar relation [z]=zeta^f, and call the result B_f. It has finite dimension |H/Z|. Every free coefficient belongs to this algebra, and products of free coefficients stay in it. For the endpoint proof, only a finite union of projective cosets is needed: H together with the finitely many g_ij H, or H together with the relevant H g_ij. It is unnecessary to close this support under arbitrary paid products.
+
+The tensor-power Gram argument is valid on this finite set of projectively distinct unitary representatives. Distinct projective unitaries have normalized inner product of magnitude strictly below one, so diagonal dominance establishes injectivity at every sufficiently large f. Products of free coefficients and the target by a free kernel vector stay inside that finite support. This is sufficient for the contradiction even if a paid query is an arbitrary one-field linear operator outside the finite support.
+
+The theorem must be phrased as a uniform/all-width obstruction, or pointwise for every sufficiently large width. It is false as a 112-query assertion at every individual width: at f=1 the central Gaussian Q=((1+i)/2)I+((1−i)/2)X already lies in the free monomial algebra. Width-dependent scalar coefficients are permitted because the proof works separately at each sufficiently large width. A width-dependent alphabet or arbitrary noncolumnwise address maps are not permitted by this argument.
+
+## 2. Actual source cosets and the eight-mode issue
+
+For g_ij=Q^−1 K_i^−1 K_j Q, the ratio g_ij^−1 g_ik is Q^−1 K_j^−1 K_k Q. With the signed orthogonal binary normalizers, its Z-to-X block is G e e^T+e e^T G. It has rank zero precisely when G e=e and rank two otherwise. Distinct source central vectors therefore give distinct right monomial cosets, including after enlarging H to all monomials: a genuinely nonmonomial unitary cannot become monomial by changing its allowed phase alphabet.
+
+The source's zero diagonal and nonempty off-diagonal support in every column imply that (J−F)w=0, w in B_f^112, forces w=0 for every free F. The module convention is consistent: matrix coefficients act on the left, and the column domain is a right B_f-module. The kernel argument uses only finite-dimensional k-linear algebra, so zero divisors or noncommutativity in B_f cause no difficulty.
+
+The eight-mode factorization is not a contradiction. It gives low rank over the larger operator algebra containing the conjugated Gaussian maps. It does not supply a nonzero kernel column over the smaller free algebra B_f. The separated-coset proof establishes exactly that distinction.
+
+## 3. Chronological feedback and bypasses
+
+The normal form y_t=T_t(b_t x+sum_(s<t)e_ts y_s) is valid for a fixed finite linear circuit with data-independent free coefficients. Saved banks, COPY, fanout, overwritten registers and clean buffers can be retained symbolically without adding paid calls. Arbitrary independent dirty auxiliary inputs may be set to zero when testing an all-input identity.
+
+If r<112, stacking b_t gives a k-linear map B_f^112→B_f^r with a nonzero kernel. Substituting x=wz makes every paid response zero inductively, irrespective of the ranks and internal linear operators T_t. Faithfulness on the finite endpoint support then turns exact physical equality into the forbidden free-kernel equality. A final free bypass is already included in F and does not evade the proof.
+
+The phrase “adaptive” must mean chronological linear feedback here. Input-dependent coefficient selection, nonlinear decoding/bit packing, and free operations outside B_f are not covered. Independent dirty auxiliaries differ from Gaussian-dependent saved observations of these same x owners: the latter cannot be set to zero independently and do evade the fresh-input normalization. Retaining a preimage of an already Gaussian-transformed bank is also additional correlated information, even if the bank itself ranges over all values.
+
+## 4. One-sided operators and homogeneous direct sums
+
+For PQ, fixed-row entries K_i^−1 K_j Q occupy distinct right monomial cosets, all outside H. The nonzero diagonal P_ii=1/14 ensures support in each input column. Thus the same input cut gives 112 calls.
+
+For QP, fixed-column entries Q K_i^−1 K_j occupy distinct LEFT monomial cosets. If a free row w annihilates QP modulo a free operator, left-coset projection forces w=0. The final free output matrix L has a nonzero left kernel whenever it has fewer than 112 response columns. This gives the dual output cut. Both bounds are attained by 112 rank-one Gaussian calls with the free P program before or after them.
+
+For every fixed finite m, each homogeneous direct-sum family diag(J_s), diag(P_sQ), and diag(QP_s) has the corresponding trivial free kernel on all 112m fresh owners, despite arbitrary free cross-block mixing. Projecting one target block at a time forces each kernel block to vanish; the other blocks contribute only to the free comparison term. The dimension cut is therefore 112m. The one-sided families attain this with 112m H1 calls. This does not assert a bound for an arbitrary mixture of left- and right-sided endpoint types.
+
+All blocks must inhabit one fixed ambient one-column address space, with equal field widths at the counted calls. The large-width threshold may depend on that fixed ambient space and on m. The claim does not establish a bound uniform in a growing m/alphabet.
+
+## 5. Canonical rectangular modes and all free retraction bases
+
+Let U=diag(K_i^−1)M and V=M^T diag(K_i)/56, so P=UV and VU=I8. Every row of the source M has four nonzero entries.
+
+For QU:8→112, fixed mode-column coefficients have words Q K_i^−1 in distinct LEFT monomial cosets. Every owner row appears in at least one mode column. Consequently QU has no nonzero free left relation modulo any free 112×8 operator, and the output cut requires 112 unary calls even though there are only eight input fields.
+
+For VQ:112→8, fixed mode-row coefficients have words K_j Q in distinct RIGHT cosets. Every owner column participates, so the input cut requires 112 calls even though only eight mode fields are returned. Both upper bounds are the literal 112-H1 implementations. These rectangular statements extend blockwise to fixed direct sums.
+
+The result is independent of the chosen free retraction basis. If P=U'V' and V'U'=I8 with U',V' free, put W=VU' and W^−1=V'U. Then
+
+WW^−1=V P U=I,   W^−1W=V' P U'=I,
+U'=UW,          V'=W^−1V.
+
+Thus QU' differs from QU by an invertible free input basis change, and V'Q differs from VQ by an invertible free output basis change. The same proof applies to a joint 8m-mode free retraction of the direct-sum projector, including free basis changes that mix stages. It does not establish the cost of the compressed sandwich VQU, PQP, or an unrelated changed output contract.
+
+## 6. Isolated shear tariff and exact sign
+
+Since C=I−14P, the physical coded backward block is
+
+B_s=diag(D_s,D_s) U(−Q+14QP).
+
+Therefore the exact correction is on the RIGHT:
+
+B_s U(−14QP)=diag(D_s,D_s) U(−Q).
+
+Rightmost-first composition means the isolated shear is applied before that FULL. The negative sign is necessary. This uses U(A)U(B)=U(A+B), which holds without assuming the blocks commute.
+
+A standalone implementation of U(−14QP) on fresh arbitrary X,Y contains an implementation of QP by restricting X=0 and rescaling its first output. Hence it costs at least 112 paid same-width unary calls. Under the Gaussian moment tariff r^(1−C), with integer ranks r≥1 and 0≤C<1, its price is at least 112, whereas the deleted eight rank-eight centers cost 8·8^(1−C)≤64. The isolated repair therefore loses by at least 48 moment units before additional finite scalar work. This conclusion assumes fresh standalone repair; it is not a lower bound after merging the correction into the native FULL, its internal observations, or an existing IDLE.
+
+The phrase “exactly 112 H1 calls” should mean minimum unary call count 112, attained using 112 H1 jobs; it does not mean every optimal-count implementation must use that exact rank profile.
+
+## 7. Executed checks and remaining limits
+
+On CPython 3.14.7, I executed:
+
+```
+python3 -I -B /tmp/global-query-bound/verify.py
+python3 -I -O -B /tmp/global-query-bound/verify.py
+```
+
+Both exited successfully and produced byte-identical JSON, checked with cmp. Outputs are retained as independent-primary-replay.json and independent-optimized-replay.json beside this report. This replay authenticates the three pinned inputs, reconstructs all 12,544 ordered transport tests by binary formula and the retained exact tableau implementation, obtains 112 grade-zero self pairs and 12,432 grade-two unequal pairs, checks all 5,824 supported pairs, confirms 52 supported entries per code column, and rejects the three supplied source controls.
+
+These were independent executions of the author's checker, not a separately written Clifford engine. The subsequently added rectangular-M Gram/factorization checks and final packaging replay need their own final execution; they were not part of the initial verifier hash recorded above. The review of the extension proofs is separate from those executable source checks.
+
+No full native producer replay, p2/p16 paid-word replay, finite-precision implementation, outer multiplier, Lean/kernel audit, or independent human review was performed. The contribution relies on the retained physical normalizer/source identities and their exact attribution. The entrance-adapter equivalence M_i=U_i K_i Q^−1 must retain its actual physical wrappers and phase convention; replacing U_i by a bare Fourier along q_i would not be justified.
+
+## Addendum: finite projective nonunitary monomial alphabets
+
+The author's further extension from a finite-phase unitary H to any fixed **finite projective group of invertible monomial maps** is valid. Choose one nonzero representative h for each projective class; multiplication has nonzero scalar structure constants, and their f-th powers define the width-f twisted algebra over C. Its dimension remains the finite number of projective classes. For the finite endpoint support, normalize each representative individually by its Hilbert–Schmidt norm before the Gram argument. Cauchy–Schwarz gives magnitude strictly below one for projectively distinct representatives, tensor norms multiply, and independent rescaling of columns does not affect linear independence. Unitarity was therefore sufficient, not necessary. Source nonmonomial transport still prevents projective coset coincidences with such monomial maps. This does not extend to a width-dependent projective universe or to arbitrary invertible monomials whose generated projective group is infinite. The final version's new M-factorization checks were reported passed by their author; they were not executed by this reviewer before this addendum.