[petalas]

Summary

Scoped proof with exact finite corroboration. For a unitary target D and relative square-zero error N^2=0, the error of every ordinary tensor power of D(I+eN), in forward and inverse-transpose orientations, is the norm of an explicit matrix of size k+1, independent of the original dimension or rank.

a = |e| ||N||
B[i,j] = a^(j-i) sqrt(binomial(j,i) binomial(k-i,j-i)), i<j
F_k(a) = ||B||,   lim_(a->0+) F_k(a)/a = sqrt(floor((k+1)^2/4))

The resulting necessary and sufficient dyadic parameter budgets differ by at most two bits. This resolves a scoped endpoint precision subproblem of Dillon Amburgey's PR4623, without auditing its circuit or budgets. Only research/square-zero-tensor-error/ is added. The selected frontier and native invoice are unchanged; no new kappa is claimed.

Produced by Nick Petalas. Credit Dillon Amburgey for the approximation question and tensor/chronological distinction, Cinna Davis, Jacob Alan Sussman, zxv and the retained native/caller/compiler/precision lineage. Singular-value decomposition, Boolean-lattice symmetry and positive-matrix methods are classical; no literature-priority claim is made for those tools.

Evidence

Before: PR4623 leaves a complete approximation/recursive precision contract open, without an exact square-zero ordinary tensor-error formula. After: PROOF.md gives the all-dimension, all-k reduction, sharp first-order coefficient, global bounds and both orientation budgets.

Local CPython 3.12.15 on macOS arm64 passes:

• 21,840 literal tensor/subset entries, 6,144 compressed action/Gram entries, 21,840 inverse-transpose entries, 2,080 first-order level terms, 192 exact precision choices and a 256-entry mixed rank-two tensor similarity.
• Eleven controls for changed compression, missing identity subtraction, invalid domains and false exact linearization.
• Both warning-strict normal and optimized registered commands, timed separately at 0.18 seconds each. These are timeout checks, not a performance comparison.
• Metadata/source pins, committed bytes, source-history ancestry and modes at e605cd6ea888a9a4885473dea88ab8d131dfe489.
• Generated receipt hashes, overwrite refusal and external-output restriction. The official seven-file export/input-coverage check passes. An optimized Git-free replay with empty PATH reproduces the output from read-only exported files; all eight export files, including the manifest, retain their hashes.
• In-memory syntax compilation and staged whitespace checks.

From the repository root:

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

Another CPython 3.12 executable may be substituted. Expected markers are PASS_SQUARE_ZERO_TENSOR_ERROR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Replays use only the standard library and write nothing by default.

Limitations and unrun obligations

The universal result rests on the written proof, with finite exact corroboration. The unitary target and relative square-zero hypothesis are essential. Ordinary tensor powers do not establish the chronological supplier product. No actual flattened recursive caller, scalar-wrapper conditioning, paid-call numerical errors, physical restoration/storage/movement or complete supplier admission is supplied. General borders are outside scope.

Independent human and formal proof-kernel review are unclaimed. No selected/predecessor scientific replay, Lean or Docker run was performed. No proof obligation is deferred solely to CI.

Canonical main was refreshed before publication and remains 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8. Live claims, open/merged PRs and tracked main README terms were checked. Remote overlap pagination was complete; sparse local coverage was 102 of 8,917 notes, so it is not a novelty certificate. Advance claim.

Open next questions

• Give a source-bound legally flattened caller whose relative error squares to zero, including its owner map and restored independent outputs.
• Combine the endpoint parameter choice with actual subword norms, scalar-wrapper conditioning and additive paid-call errors in one complete precision/runtime bill.

Related open work

• PR6398 includes a five-call approximate border and a separate exact mixed root. This theorem is an abstract ordinary tensor-error result; it does not establish that proposal's relative hypothesis, recursive precision or price.

• PR6400 proves a repair-location obstruction for PR6398's fixed border. It concerns exact recovery cuts rather than tensor endpoint error; no numerical comparison is made.

• PR6401 composes a changed native invoice with PR6398's exact mixed root. It computes a raw funding margin, while this theorem computes an abstract tensor endpoint error; no common numerical quantity is compared.

• PR6406 counts square-zero second-callee parameters for regular J3 over finite fields. This theorem concerns complex Euclidean tensor error and computes no such parameter count.

This account's open PRs created since the advance CLAIM were refreshed before DONE. No conflicting same-quantity value was found in that snapshot.

CI notes

Both hosted checks, plan and replay (research/square-zero-tensor-error), passed at exact head e605cd6ea888a9a4885473dea88ab8d131dfe489 in GitHub Actions run 41527155826. The completed results were confirmed once with gh pr checks 6402 --repo CrocSwap/integer-mult-bounds. The planning job took 4m38s and the package replay took 3m16s. The registered replay executes both normal and optimized scientific commands.

Local scientific, metadata, committed-byte/source-history/mode and read-only export checks also pass. Hosted success certifies execution of the declared checks; independent scientific or formal proof-kernel acceptance and the listed physical/chronological obligations remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated. Only the new research package is changed.