Research: audit dyadic admission of QQ16 unread minors

research/qq16-dyadic-unread-minors/README.md

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+# Dyadic admission of QQ16 unread minors
+
+This is a scoped scalar-admission result submitted for review, not a supplier
+or a new kappa.
+
+The unread basis used by [PR2443](https://github.com/CrocSwap/integer-mult-bounds/pull/2443),
+`H0=[1,2,3,4,5,6,8,12]`, has determinant `-32` and a dyadic inverse. It passes the
+check below. However, replacing just unread port 2 by port 25 gives a rationally
+invertible minor with determinant `-80` and inverse entry `1/10`. That read set
+cannot support a finite dyadic-linear decoder for all original inputs under the
+same two-cut observations, even when access and retention are free.
+
+This does not contradict PR2443. Its specific basis is dyadic, its theorem is
+rational, and it does not claim native execution. The new contribution is the
+explicit alternative-set obstruction and a complete local census, useful before
+optimizing which ports to leave unread.
+
+## Domain and result
+
+Use the actual integer `8 x 112` query matrix R and ordinary set U96 from the
+retained [QQ16 packet](../joint-query/README.md). For an unread set H of size eight,
+consider exact inversion of
+
+```text
+w = R_H D_(2,H) x1_H,
+x1_H = D_(2,H)^(-1) R_H^(-1) w.
+```
+
+The order is the one in PR2443. Each port's cover map must be undone after scalar
+inversion. No commutation of the different port maps is assumed.
+
+Let `D=Z[1/2]`. An integer square matrix is invertible over D if and only if its
+determinant is a unit of D, namely a signed power of two. Sufficiency follows
+from the adjugate formula. Necessity follows by taking determinants of a left
+inverse. Thus nonsingularity alone is insufficient for dyadic scalar admission.
+The same odd-factor obstruction holds with Gaussian-dyadic coefficients.
+
+The exact census includes every unread set at exchange distance at most two
+from H0. No random sampling is used.
+
+| Distance | Sets | Singular | Dyadic-unit determinant | Nonzero odd-factor determinant |
+| --- | ---: | ---: | ---: | ---: |
+| 0 | 1 | 0 | 1 | 0 |
+| 1 | 704 | 185 | 485 | 34 |
+| 2 | 107184 | 46180 | 58537 | 2467 |
+| Total | 107889 | 46365 | 59023 | 2501 |
+
+The nonzero absolute determinants are `16,32,48,64,80,96`. The odd-factor values
+are `48,80,96`. [RESULT.json](RESULT.json) contains every histogram, both complete
+inverses and the counterexample. This is not a census of all eight-subsets of U96.
+A unit determinant admits the scalar inverse, not a native read or a favorable bill.
+
+## A counterexample for the complete two-cut decoder
+
+The observation contract is exactly PR2443's. Retain `x1_A/T`, `y1_A/T` after
+complete F0, read `x1_(U\H)` there, and retain `y2_A` and `C=x2_U` immediately
+after B1. There is no intervening modification. No `y2_U` or additional original
+input is observed. Here A and T are the retained selected sets.
+
+Use this order for the alternative unread ports and integer vector:
+
+```text
+H = [1,25,3,4,5,6,8,12],
+z = [1,2,1,1,1,1,3,-2].
+```
+
+Exact multiplication gives `R_H z=10 e_1`, where centre coordinates are zero-indexed.
+Set all owner and payload values constant, so every cover map acts as the identity.
+Writing `P_A=R_A^T/4` and `q=10 e_1`, initialize
+
+```text
+x_U = iota_H z,  x_A = -P_A q,  x_T = 0,
+y_U = 0,        y_A =  P_A q,  y_T = 0.
+```
+
+These are valid dyadic inputs, including the selected dirt. Since `R_A P_A=I`,
+the first forward query is zero. The conserved F0 equations give
+
+```text
+x1_U = y1_U = iota_H z,
+x1_A = x1_T = y1_A = y1_T = 0.
+```
+
+At B1, its future logger gives `y2_A=P_A q` and its ordinary target subtraction
+gives `C=0`. All 88 read coordinates vanish. Hence all allowed observations are
+zero except `y2_A=(5/2) R_A^T e_1`. Every observed coordinate belongs to the ideal
+`5D`, but the required original `x_1` equals 1.
+
+A finite D-linear circuit preserves `5D`. Therefore no such circuit can decode
+all original inputs from these observations. The argument also covers finite
+Gaussian-dyadic linear operations and owner permutations. The ideal `5D[i]` is
+proper, as reduction modulo five with `i -> 2` maps 1 to 1 and five to zero. Extra dirty workspace or affine constants do not help a linear
+implementation: compare the displayed input with the zero input using identical
+workspace and subtract the two executions. The observation difference stays in
+the ideal while the required output difference is 1.
+
+A frame change is covered only if its coefficients are separately shown to lie
+in the proved ring. Normalized Clifford coefficients such as `1/sqrt(2)` are
+outside `D[i]` and are not covered here. No binding of the complete native gate
+alphabet to this ring is supplied, and no claim that all native operations are
+Gaussian-dyadic is made.
+
+This is not an information-loss witness over Q. The rational inverse exists,
+and its row 0, column 1 entry is `1/10`. Nor is it an obstruction to restoring
+only the selected original inputs. In this displayed family those selected
+inputs are already determined dyadically by `y2_A`. The claim concerns complete
+input recovery, as in PR2443, under the fixed observations and coefficient ring.
+Nonlinear exact integer division, odd-denominator gates, new observations or a
+changed first pair are outside the result.
+
+## Reproduce
+
+Python 3.12 or later, standard library only:
+
+```sh
+python3.12 -I -B scripts/research_package.py --package research/qq16-dyadic-unread-minors
+python3.12 -I -O -B research/qq16-dyadic-unread-minors/verify.py
+```
+
+The registered replay checks source pins and runs:
+
+- `verify.py`, using the one-/two-column determinant lemma, exact Fraction
+  elimination, two inverse orders and the constant-owner two-cut witness;
+- `oracle.py`, importing no primary verifier or pending PR code, using direct
+  fraction-free elimination on all 107889 matrices and adjugate cofactors for
+  both inverses;
+- `test_verify.py`, checking optimized replay, a Git-free source export and
+  rejection of source corruption, receipt type corruption, a missing receipt
+  and an altered histogram.
+
+The six mathematical controls include a singular minor, a wrong inverse sign,
+an odd denominator, treating every nonzero determinant as a dyadic unit, a
+changed modular vector and a duplicate unread port. Default replay writes no
+artifacts. `verify.py --emit` explicitly emits a new complete receipt to stdout.
+
+[SOURCES.json](SOURCES.json) pins retained main inputs at
+`0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`. Pending PRs are attribution and
+contract context, not runtime dependencies. Native replay, chronological
+retention, child hosting, precision, tape, all-size admission and the complete
+supplier invoice are not established. The separate retained QQ16 local replay
+was run as a baseline, not as verification of a new native decoder.
+
+## Engineering limits
+
+All loops have fixed finite domains: matrix dimension at most eight, 112 ports,
+and 107889 streamed candidates. Python uses dynamic allocation and arbitrary-size
+integers. This is a managed-runtime deviation from fixed steady-state allocation.
+The exact hash-bound source, fixed matrix sizes and streamed census bound live
+state; no list of all candidates is retained. Parser sizes are checked before
+hashing/decompression. Functions stay below 60 physical lines. Direct and
+optimized replays, warning-enabled tests and a second determinant algorithm are
+the available interpreted-code checks. A future broader domain must add explicit
+input bounds and recheck resource limits rather than reuse this bounded receipt.
+
+## Open next questions
+
+Can a dyadic-unit unread basis reduce a complete native decoder's cost? First
+for a dyadic-linear implementation, filter candidate read sets by the determinant
+gate, then compare actual endpoint frames and the full read, chart-map, retention
+and cleanup invoice. The blocker
+is physical access and sustainable retention, not nonsingularity. The original
+PR2443 basis already passes the dyadic check. No broader search is justified by
+this finite census alone. [TASK.json](TASK.json) records the structured handoff.
+
+## Credit
+
+Gareth Davidson's PR2432 supplies the unread-column criterion and H0, pinned at
+`cbede349fc764aa38379aa533befe3215916cfcd`. starship-s's PR2443 supplies the
+arbitrary-input rational decoder and explicit two-cut contract, pinned at
+`545c05e17243c962b112b91e7f84b4c1a41de7b8`. thakrarsagar's PR2470,
+`1b2eee3bf1ec67e68c0b0fa7f47c7b2a66febe56`, is the separate retention census.
+None is adopted as a supplier or represented as independently reviewed here.
+
+Ali Romman and the retained QQ16 contributors supply the matrix, selected
+sections, conserved maps and cover semantics. Adam Gall, commdata2338,
+kyzer1023 and the coordinate-read predecessors retain their information-bound
+credit. Rohan Arun, Jeremy Kothe, DreamingOfClouds and the earlier supplier and
+compiler lineage retain their attribution through the unchanged
+[QQ16 notice](../joint-query/NOTICE.md). No predecessor executable is imported or
+vendored. New work is Apache-2.0. No independent human or proof-assistant review
+is claimed.