[Matarener]

Requested by Mark

Change and mathematical scope

Scoped positive result; proof plus executable certificate; no new κ, no paid-profile change. Target 8 of #5428 asks for compact all-width native identity certificates. This package states the lifting lemma and binds it to the complete selected caller of #5413.

Before: the all-width transfer of the selected caller is argued in prose (WHOLECALLER-PLAN-v1 phase proof) on top of the local-provider results of #5369/#5376, and the port binding to the controller symbols is checked at width one in NORMALIZERS.json.

After: research/allwidth-caller-certificate/verify.py certifies that the complete caller identity is a composition of identities that lift to every positive column multiplicity, and PROOF.md gives the lemma that justifies the census.

Lemma. For a finite group G of exact matrices and fixed coefficients, sum_g c_g g^{(x) f} = 0 for every f >= 1 iff sum_g c_g g = 0 in C[G] (tensor-power homomorphism; Burnside for the converse). Group identities and free-word identities lift; linear relations between distinct elements do not. The census replaces the finite width cutoffs of #5369/#5376 whenever the elements can be canonicalized, which clifford_exact.py does (Pauli tableau plus the image of |0>, fixing the absolute phase).

What is certified.

• Ports: 896 exact group identities with absolute phase, both orientations. On every port the body's physical paths are x->X = D, y->Y = D, y->X = D C_0, and the query operator Q_p = A_y A_x^-1 = C_0. Backward uses the entrywise-conjugate semantics of Verify a complete Clifford provider and exclude 30,328 paid-pair fusions #5376 PROVIDER.md; the mixed reading is a different element on all 112 ports.
• Body: free-word endpoint of NEUTRAL-BACKWARD-v16c with PROVIDER_5402 expanded. All 72 native warp symbols cancel; main rows carry only the port wrappers; the 400 autonomous helpers return under signed alias monomials in their recorded exit frames; all 17 owned fields are zero. The eight mode outputs are C8 z: each cache SWAP moves its transformed row to the owner, and with every symbol bound to its exact group element (backward conjugate charts, output_N, positive C8) the 448 cached provider y terms cancel the 448 y terms through C8 in C[G]. (The first version dropped the cache row and certified C8 (h - Z y); zxv's review found it.)
• Controller: the Laurent replay (semantics of check_wholecaller_v1.py) reproduces the diagonal; the plan's 18 private planes are pairwise disjoint and disjoint from the central axis, so the realization Q = C_0, D_s = fourier(stage axes) is abelian; C_0^2 = X_0, C_0^-1 X_0 = C_0, [C_0, D] = 0 exactly.

Remaining obligations: the physical half of native_all_columns (tape realization of g^{(x) f} with spectators and whole-job adapters at the retained tariff) is not addressed; no independent human or kernel review.

Task and overlap

Base 32890bc5de034ab8fc93ac88ead7279a6a416d04; inputs pinned by blob and SHA-256 in SOURCES.json (#5413 caller/body/plan, #5402 provider and NORMALIZERS.json, the eight-relation word, the vendored exact Clifford compiler). Task card formal-interface-lemma (scope changed; see TASK.json). Concrete object: the selected #5413 caller. Nearest overlap: #5369/#5376 (aliromman) prove width cutoffs and pass the #4610 local provider; this package adds the lemma in census form and the caller level (port binding with phase, body with the #5402 provider, controller realization), which they do not cover. #5438 (ours) shares the replay machinery for a different question (exact grid).

Validation

python3.12 -B scripts/research_package.py --package research/allwidth-caller-certificate

prints PASS_REGISTERED_RESEARCH_PACKAGE research/allwidth-caller-certificate outcome=completed (Windows, CPython 3.12, standard library; verify.py under two minutes). Finite checks: 896 element equalities, 649-row free replay, 32-row Laurent replay, plan geometry. Proof argument: PROOF.md section 2. Lemma controls on dense Q(i) matrices: the Reynolds-type relation and the scalar trap i = iI vanish at width 1 and fail at width 2 while the census counts distinct elements; exponent two lifts; reversed factors differ. test_verify.py rejects ten mutations: input drift, a wrapper phase mutation, the mixed orientation convention, a leftover warp at the endpoint, a cache SWAP that drops the cache row, a sign flip in one mode output word, an owned field not cleared, overlapping private planes, a missing Q_POWER, a tampered result. Not run: the forward body (review only; it does not run in the selected controller), Lean, hosted CI.

Credit

#5413 (Clear1nsight), #5402 (dillona), #5383/#5398/#5369/#5376 (aliromman; width-cutoff theorems and the orientation semantics), the Clifford compiler lineage (Cao Yifan and predecessors), #5428 (zxv) for the target. Apache-2.0. Prepared by Mark (Matarener) with substantial Claude Code assistance.

Open next questions

Listed in README.md: reduce the distinct exact group elements used by the caller (wrappers, D, C_0, X_0, the plan's J routes) to a finite list of fixed-tape primitives with their cost at width f (the physical half; target 10 of #5428), and the same census for the forward body.

Workflow feedback (optional)

The orientation convention (entrywise conjugate for backward) is documented only in #5376 PROVIDER.md; a note next to NORMALIZERS.json would save the next reader a detour.

🤖 Generated with Claude Code

[zxv]

Reviewed 19e8f0452c09cbfe99511ea697fa239c1f457314. The registered package replay and all eight controls pass on CPython 3.12.8 / Darwin arm64. The group-algebra lifting argument and the 896 port identities are useful, but the complete-body certificate has a blocking execution error.

[P1] Preserve the nonzero provider caches and certify the selected caller's actual mode endpoint. At verify.py:273–277, self.rows[cs], self.rows[o] = self.rows[o], {} deletes the transformed cache instead of swapping it into o. The preceding assertion guarantees that the old owner is zero, so both resulting rows are zero. This is the same transfer error reported in the review of #5438.

Replacing only that assignment with the literal swap

self.rows[cs], self.rows[o] = self.rows[o], self.rows[cs]

leaves 113 free-word terms in each of the eight mode outputs, including 56 terms without the supplementary G symbol, and body_endpoint() fails with FAIL: mode output: own input plus 56 y inputs.

Those terms are essential: the selected source proof, lines 26–30 requires S(-DZy + D(Sz+Zy)) = C8 z. The complete caller's mode outputs must be autonomous C8 z, rather than the coupled C8(h-Zy) currently asserted in this package. The present 57-term acceptance check therefore certifies an altered computation and a different endpoint.

Minimal reproduction from this PR checkout, without editing submitted files:

from pathlib import Path
p = Path('research/allwidth-caller-certificate/verify.py').resolve()
s = p.read_text().replace(
    'self.rows[cs], self.rows[o] = self.rows[o], {}',
    'self.rows[cs], self.rows[o] = self.rows[o], self.rows[cs]')
ns = {'__file__': str(p), '__name__': 'review_probe'}
exec(compile(s, str(p), 'exec'), ns)
ns['build']()  # FAIL: mode output: own input plus 56 y inputs

Please retain the cached terms, bind their physical operators to the supplementary D/C8 routes as exact phase-bearing group elements, prove the cancellation of all y contributions, and check the autonomous positive C8 outputs. Then update PROOF.md, RESULT.json, and the claim that the complete caller's algebraic obligation has been discharged. The eight supplied controls do not detect this error; a cache-transfer/mode-output regression would.

This finding concerns the new certificate. It does not refute the selected caller's previously reviewed neutral completion or the lifting lemma. The physical fixed-tape implementation remains outside this review's scope.

Review performed with OpenAI Codex assistance.

[Matarener]

Thanks, confirmed and fixed in 4d5abe0. The SWAP now moves the cache row, and each mode owner has the 113 free-word terms you observed. mode_outputs binds every symbol on them to an exact element of G:

• monomials to their literal words;
• G to the positive C8 on axes 0..7;
• Nin_f to the input wrapper followed by the inverse canonical chart of the entrance frame;
• N_o to the chart of the recorded mode frame followed by the literal output_N word.

Charts use the backward conjugate convention, as for the ports. Summed in C[G], all 448 y pairs cancel and each owner is exactly C8 z, with coefficient +1 and absolute phase. That is the source's S(-DZy + D(Sz+Zy)) = C8 z.

There are three checks against false positives. The unconjugated reading leaves all 896 y terms. A sign flip in one output_N word is rejected. The drop-the-row mutation is rejected. PROOF.md §3B, README, TASK, VERIFY and RESULT are updated.