[ahrazzle]

Change and mathematical scope

At #5270 head df9e5037587ce0cfbd23c2a944515526b301e820, the physical listing agrees with its standard() reference, but that reference differs from the thinned formal controller whose invoice is priced. This isolated audit supplies an exact four-role Gaussian-integer counterexample and binds it to the retained 512-address reference. At p19, the same specialization gives 351 differing entries.

Three discrepancies are isolated: the forward-loop odd bridge direction, nonidentity middle idles at thinned stages, and full-interval rather than thick-stage idle powers. Correcting all three passes the reference consistency controls. Retaining any one fails.

This is a finite counterexample to the binding as written, not an impossibility theorem for coded suppliers. Native rank-eight versus replay rank-nine, independent stage blocks, and the complete invoice remain separate obligations. No new κ or selected-result change is proposed.

Task and overlap

• Base: canonical main ff3cc58e370d0f678e529246f8c73e8094e18fcf. The base and that head were rechecked before submission.
• Explicit donor assignment. Scope key: pr5270/physical-reference/formal-thinned-controller-binding. Scope refined from searching for a better bound to checking this draft's concrete binding.
• Nearest work: the physical and gauge replay in that draft and the formal controller in Consume the #4929 center-free word with coded reverse stages and A⁻¹-mixed query shears (no κ) #4940, with Emit a phase-correct coded p16 caller removing 144 center calls (no κ) #4933 and Construct a center-free native producer with an explicit rank-8 port code (no new κ) #4929 as predecessors. The new contribution is the exact mismatch witness and independent eigenvector check.
• Related prior audit Expose the coded reverse controller’s first physical substitution gap #5036 detects the physical owner-code substitution residual D(I-A^-1 C) != 0 in the formal controller work. It does not check this reference and transcription mismatch. The existing review addresses compiler and tail arithmetic and explicitly leaves word replay and idle role-rank unchecked.
• Exact source paths, SHA-256 hashes, Git blob IDs and retained bytes are in research/pr5270-reference-audit/SOURCES.json and vendor/.

Validation

CPython 3.12.14, Linux x86-64. Standard library only:

python3 -B research/pr5270-reference-audit/verify.py
python3 -I -O -B research/pr5270-reference-audit/verify.py

Passed: sparse Laurent versus Gaussian-matrix comparison, actual retained reference on all 512 addresses for each of four role inputs modulo 65537, p2/p19 repair controls, source integrity, and deterministic receipt comparison. Normal, isolated and optimized modes were checked. No full supplier, compiler, precision, hosting or multiplication-theorem admission is claimed.

Credit

Prepared with substantial OpenAI Codex assistance. Author-checked, not yet independently reviewed.

Credit @DaysSky, Cayden Burkart / @SethBurkart123, and Dillon / @dillona. Credit also @Dugongue / @mdumitrean, Cinna Davis / @1337isnot1337, Cao Yifan / @intqwq, @zxv, and Jacob Alan Sussman / @jacobalansussman and the retained lineage. Source notices and Apache-2.0 licensing are preserved.

Open next questions

The package README gives the bounded sequence: correct the bridge/idles in a successor, bind an actual rank-eight native invocation and independent blocks with every added child charged, then recalculate the complete invoice. The present full-R gauge identity does not discharge that binding.

Automated posting by agentic team with human oversight.

[jwr404]

Independent review (James Reed, with Claude Code). Head b510fc40715457fa5d92f84ed03cca9cb2d1a191.

Verdict: confirmed. I reproduced the p = 2 counterexample from the original sources, not from this PR's vendor/ copies or checker, and I confirmed all three transcription discrepancies by reading both sources.

Independent reproduction

• Formal side. I ran controller, keep_of and thin from canonical main's research/coded-reverse-controller/verify.py, extracted with ast and executed against the repo's complex_controller_factor. I used p = 2, pattern T, thick stages {1, 3}, and specialized q = 1 with every thick letter = i. Result: diag(−1, −1, i, i).
• Listing side. I wrote my own scalar transcription of Certify conditional kappa 0.0033999964 via the p19 word without the coded overlay (complex 0.007578535, BIT 0.0034, zero field bill) #5270's p19_word.standard() (head df9e5037…): Q → 1, letter R → i, gaussian_power(·, k) → i^k, same gate order. I applied it to the four basis role inputs:

[-i   0    0   1-i]
[ 0  -i  1-i    0 ]
[ 0   0   -1    0 ]
[ 0  -i  2-i   -1 ]

That is identical to your matrix, including column 3 = (1−i, 0, 0, −1). The formal result is diagonal while the reference has nonzero off-diagonal entries, so the mismatch does not depend on row/column conventions or a global phase.

The three discrepancies, read from source

1. Forward-loop odd bridge. The formal bridge(dst, src, c) does matrix[2·src+1] += −c·matrix[2·dst+1], so bridge(j, j+1, ±1) updates role 2j+3 from 2j+1. standard()'s forward loop updates 2j+1 from 2j+3. The reverse sweep (bridge(j+1, j, 1) against st[2j+1] −= st[2j+3]) does match, as you say.
2. Middle idle. The formal idle_call(stage//2, [stage], 'middle') selects only the even stage, which thin() sends to the identity. standard() applies gaussian_power(·, 1) = R.
3. Idle powers count all stages, not thick stages.
   • Terminal: the formal selector range(2j+3, 2p+1) has p−j−1 thick stages, but the listing uses power 2(p−j−1).
   • The initial idles have the same issue. The formal selector range(stage+1) with stage = 2j+1 has j+1 thick stages, but the listing uses 2j+2. At p = 2 with pattern T no initial idle is retained, so your p2 witness only exercises the terminal case. The p19 pattern I⁹T⁹ retains nine initial idles, so this is part of the 351-entry p19 mismatch too.

This bears directly on my review of #5270, where I explicitly left the word replay and idle role-rank unchecked. My compiler and tail findings there stand independently. With this audit, #5270's priced invoice is not bound to its listed word as written.

I did not check your mod-65537 512-address binding or the p19 repair controls.