[petalas]

Summary

Prove a scoped obstruction for the complete P104 Gram pair: one persistent Gaussian-dyadic invertible code cannot transform it to the spectral-projector pair with alpha*beta=169, even when the code mixes both banks and any finite number of identically restored dirty scratch coordinates. No new kappa is claimed.

PR3979: bank-separate free spectral normalization excluded
new:    joint-bank codes + arbitrary restored scratch also excluded
        endpoint correction rank >=832; balanced case >=1664

The new block calculation gives ranks 1664, 128 and zero for F-I and its next two powers modulo 13. The correction bounds are sharp over F13. Exact dyadic attainment is open. Matrix rank is not a native-call tariff or a complete paid invoice.

This contribution was produced by Nick Petalas. Credit Chris R / Clear1nsight's PR3979, at a8a1d744811cd7d1f9c43ec7ccbae85ef9695b61, for the tensor-code model and the cited premises A²=13A, n=104 and rank(A mod13)=8. Its original code, invoice and proofs are not replayed or audited. The P104/native/caller lineage keeps its credit. Jordan classification and rank inequalities are classical.

Evidence

Before, PR3979 explicitly left general bank mixing outside its obstruction. After, the complete-operator invariant excludes every common joint code and finite identity-restored stabilization in the stated ring.

Local CPython 3.12.15 on macOS arm64 passes:

• 20 literal atomic tensor-block cases over five prime fields and 80 small assemblies.
• All 9216 canonical P104 components and all 21632 Jordan coordinates, with an explicit projector-arrow permutation attaining the modular rank-832 correction.
• 75 stable fixtures covering every zero-product coefficient residue pair, plus rational odd-denominator, omitted-stage, unreturned-scratch and invalid-parameter controls.
• Both warning-strict registered commands, timed locally at 0.038 and 0.037 seconds. Their generated hash-bound receipts agree. The complete registered package command passes.
• Exact seven-file publication bytes, modes and declared source ancestry. Official export integrity and input coverage pass. Normal and optimized Git-free replays pass with an empty PATH and eight read-only exported files, including export metadata; bytes and modes remain unchanged.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p104-joint-spectral-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/p104-joint-spectral-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-joint-spectral-obstruction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-joint-spectral-obstruction --check-source-history --check-modes

A different CPython 3.12 executable can replace the local interpreter path. Expected markers are PASS_P104_JOINT_SPECTRAL_OBSTRUCTION modular_repair=832 balanced=1664 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limits and unrun obligations

The quantified claim rests on PROOF.md and the cited Gram premises. Finite canonical-block corroboration is not a reconstruction of the predecessor's original matrix. One common endpoint code, the Gaussian-dyadic ring and identity return of the whole scratch space are essential. Distinct endpoint codes, interleaved new operations and changed targets remain outside the theorem.

No exact dyadic rank-832 correction, native realization, selected caller/compiler/precision/storage/tape/analytic replay, full multiplication theorem, independent human review or formal proof-kernel acceptance is claimed. No registered scientific command is left for CI alone. The selected frontier and frozen packages retain their bytes.

Open next questions

Can an exact Gaussian-dyadic correction attain rank 832? Begin with the n=2, r=1 integral example A=[[13,1],[0,0]] in the proof, one common dyadic-invertible code and a rank-two correction. A modular solution alone does not lift the endpoint. A physical construction must then price native access and complete restoration.

Related open work

No open PR was found on this exact two-stage spectral-normalization question. Related work has distinct contracts or quantities:

• #6166 studies the P104 ternary decoding code; this proof concerns the complete Gram pair modulo 13.

• #6173 studies two-channel support in that code; no support optimum is computed here.

• #6171 reduces tensor-frame rank moments under fixed boundaries; it does not identify this dyadic spectral endpoint.

• #6172 studies variable-target signed tensor moments in an abstract contact graph; no common invoice or numerical optimum is compared.

• #6144 changes P104 owned-word trajectories; this package supplies no word edit or kappa comparison.

• #6167 retains an operand to change another P104 root stop; no trajectory or kappa comparison is computed here.

• #6169 resynthesizes P104 scalar events; this obstruction supplies no event or scalar-unit saving.

• #6170 supplies P104 design tooling; no tool or predecessor checker is changed here.

• #6175 localizes ADD-coefficient faults in P104 words; this package studies a different algebraic endpoint invariant.

• #6178 bounds costs of functional decoder pairings over fields; its pairing alphabet is not the persistent joint-code contract here.

• #6180 certifies rectangular dirty circuits; it does not compute this spectral correction rank.

• #6183 studies real contractive approximation of the P104 Gram channel; this exact Gaussian-dyadic conjugacy and correction-rank obstruction computes a different quantity. No approximation optimum is computed here.

The finish-line refresh checked this account's open PRs created since the CLAIM and added #6183 as new related Gram-channel work. The two results compute different quantities, so no numerical agreement or disagreement is asserted. No extra computations were run to validate these other contributions. Advance claim. Canonical main remained 3f0152a32eeebac0df11946dbd50cabe5a7f7839 at the submission refresh.

CI notes

Both hosted checks passed at 65704df0ff4e7cad336694d39ff3492df48e0c04: plan and replay (research/p104-joint-spectral-obstruction), in run 40881221129. The checks were read once at completion; run metadata confirms this exact head and a completed, successful conclusion. Hosted CI certifies execution of the registered finite checks and metadata gate. Independent scientific review and the exact dyadic/physical obligations above remain unclaimed.

Merge danger

Door: two-way. One isolated research package can be reverted independently.

Blast radius: research. The package adds a scoped theorem and opt-in replay.