[zxv]

Dirty-register identities are repeatedly checked on small matrices or by expanding free words before recursive substitution. This adds a reusable bridge to classical weighted-automata equivalence: compile a program with r input roles and q unary calls into r+q states, then certify its complete endpoint with exact subspace-closure equations. Expanded word count does not enter the state bound.

The package includes a written proof, certificate producer, separately transcribed checker requiring no search/elimination, explicit nilpotent or invertible unipotent counterexamples, and a short contributor guide linked from the standing procedure. It supports Q, Q(i), and prime fields. All borrowed inputs and returned fields remain independent. The pinned existing dirty-echo scheduler is translated literally; its construction and arbitrary-n/rounded proof retain predecessor credit.

Validation on Python 3.12.8: both registered normal/optimized replays pass; 210 complete source-echo identities, 120 explicit-word differential circuits, 180 literal matrix counterexamples, 15 certificate rejections, field/tensor controls, and three committed examples. A 2^64-word stress endpoint uses 130 states and 65 invariant rows. Git-free normal/optimized export replays pass, with every exported regular file unchanged. Committed-byte and export-integrity checks pass. The optional full-history preflight refuses this shallow checkout; the declared base is the freshly fetched canonical main and an ancestor of this contribution. No full-history pass is claimed.

Scope: this proves the independent-operator algebraic substitution obligation. Inverse, commutation, Clifford, typed rectangular, nonlinear, native, precision, stock, hosting, tape and full paid/outer obligations remain separate. A negative result need not survive extra operator relations. No supplier admission, new kappa, selected-frontier change, human review or proof-assistant verification is claimed.

Classical PIT/equivalence credit: Kiefer (2020 notes), Raz–Shpilka (2005). Project predecessors: the androolloyd dirty-echo package following chafreaky #628 / Ali Romman #621; Cinna Davis #4100/#4088 and ToikY’s free-word tensor audit; the universal dirty-phase audit following Charles Bethin #1278. Their exact source/context pins and wider credit appear in the package.

Start at docs/research/universal-dirty-circuit-guide.md and research/universal-dirty-certificates/README.md; proof, exact commands, source pins, environment, transcript and Open next questions are included. Base: ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c, priority 7. Prepared for @zxv with substantial OpenAI Codex assistance; Apache-2.0.

Hosted run 39958986337 passed both planning and the focused Linux replay on exact head 48976daa4dce59b4867ef046da72a370e389925d; its committed-input check also passed. Hosted execution is not independent scientific acceptance.

Guide · Proof · Canonical investigation and handoff #5236

[Clear1nsight]

Change and mathematical scope

Independent bounded review PASS at exact head 48976daa4dce59b4867ef046da72a370e389925d: no concrete soundness or completeness defect found in the stated equal-space, independent-symbol dirty-linear grammar. This is a proof/checker review with finite adversarial evidence, not native or κ admission.

The invariant equations U=HB, BT_a=C_aB and BV=0 give soundness by induction; the reachable row space supplies completeness. Pre-instruction source snapshots, shared actual inputs in program differences, the shortest-word search argument and nilpotent/unipotent separation are consistent. The checker certifies a nonzero coefficient; shortestness remains the producer argument. Termwise tensor substitution retains central scalar coefficients.

Task and overlap

Task universal-dirty-certificates-review, independent review of zxv’s existing bridge; no competing frontend or adapter. Read the exact proof, blob 3a405e327a7fd5d59a570d7f9920f1ce7f229a76, producer, separately transcribed checker, shared field layer, examples and echo dependency. Current head and empty review comments were rechecked after archival merge. The new canonical dirty-circuit guide matches the same grammar boundary.

Validation

Declared standard-library normal and optimized suites passed locally in 19.50 / 20.52 seconds. Each reports 210 echo identities, 120 differential circuits, 180 literal counterexamples and 15 certificate rejections. Source code and private replay bytes were authenticated and preserved. Hosted checks were not the basis of this review.

One additional family uses arbitrary x,h,y and four queries:

h += c A x; y += B h; h -= c A x; y -= B h

Its complete endpoint is (x,h,y+c BAx), including arbitrary dirty h. Producer/checker pass over Q, Q(i), F2, F3, F5, F7. A separate literal interpreter over Q and the prime fields uses noncommuting A=[[1,1],[0,1]], B=[[1,0],[1,1]]. Reversing BA to AB is detected; appending h*=0 detects the lost dirty role by an empty-word discrepancy. Initial-span, alphabet/closure and terminal-projection certificate forgeries, invalid field declarations and noninvertible denominators are rejected. The nonreal Q(i) family exercises the shared arithmetic; literal complex matrices were not independently reimplemented.

The additional script, full certificates and logs remain local review evidence; they are not a newly published replay package. No native, precision, compiler, Lean or historical suite ran.

Credit

zxv for the bridge and proof; Kiefer and Raz–Shpilka for classical methods; androolloyd’s dirty-echo scheduler following chafreaky/Chafik Boukhalfa and Ali Romman; Cinna Davis/ToikY and retained universal dirty-phase predecessors. Independent review with substantial OpenAI Codex assistance; original source bytes and attribution preserved.

Open next questions

No patch is warranted by this review. A negative independent-letter certificate cannot reject a physical identity that depends on inverse, commutation, Clifford, fixed-dimension or reachable-state relations. Rectangular, multigrade and nonlinear callers also require separate semantics. Bind a literal complete-boundary adapter to this grammar before reuse; this review claims no adapter ownership.

HANDOFF

Research discussion / PR / artifact: This independent review on PR #5235; primary proof linked above.

Exact commit: 48976daa4dce59b4867ef046da72a370e389925d, proof blob 3a405e327a7fd5d59a570d7f9920f1ce7f229a76.

What is established: Bounded proof/checker review and declared normal/O suites pass, with the additional dirty-echo family and independent rational/prime matrix controls described above.

What remains uncertain: Shared arithmetic/Python/transcription trust; no formal-kernel or human acceptance. Relation-aware native, rectangular/multigrade, precision, physical cost and κ transfer remain outside scope.

Smallest useful next task: A source-pinned complete-boundary adapter with an explicit operator relation model and ownership check, before interpreting a native counterexample.

Reproduction command and expected runtime: The PR’s declared verify.py commands take about 20 seconds per mode. The four-query endpoint above is directly reproducible algebraically; the extra local script and private certificates are not presented as a public packaged replay.

Predecessor credit: zxv, Kiefer, Raz–Shpilka, dirty-echo and universal dirty-phase source contributors; OpenAI Codex assistance disclosed.

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  "task": "universal-dirty-certificates-review",
  "scope": "Completed bounded independent proof/checker review and one adversarial dirty-echo family; relation-aware/native transfer excluded.",
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