[petalas]

Summary

Scoped algebraic proof with exact finite corroboration: dyadic helper capacity is governed by the full prime-rank profiles under tensor composition.

c(tensor_i K_i) = min_{odd primes p} product_i rank_Fp(K_i mod p)
product_i c_i <= c(tensor_i K_i) <= min_i c_i product_{j != i} r_j

c(diag(1,3)) = c(diag(1,5)) = 1
c(diag(1,3) tensor diag(1,5)) = 2

Both bounds are attained for every prescribed positive capacity/rational-rank tuple. In particular, t rank-n factors of capacity one can have tensor capacity n^(t-1), while repeated equal factors retain the product capacity. The proof also gives the exact common-prime equality criterion and an explicit two-channel unit-block witness.

This builds on Ali Romman's #5615 single-cut theorem. That theorem and classical Smith/rank facts are cited, not audited or replayed. The new step is the sharp composition envelope in the helper interface. No physical tensor cut, extra productive target, new kappa or selected-frontier change is claimed.

Evidence

• Separate capacity summaries give one channel per factor in the displayed example. Literal X K Y = I_2 and the modular upper bound prove the joint value two. Individual capacities therefore cannot be multiplied as an upper bound.
• Exact new-domain verification checks all 6,561 ordered pairs of 2-by-2 matrices over {0,1,3} and {0,1,5}; 784 have a strict composition gain. A determinant-ideal oracle independently computes every tensor capacity, while field elimination supplies the prime profiles.
• There are 177 prescribed rank/capacity tuples and 354 attaining unit blocks, with 6,165 checked identity entries; 121 rectangular/oblique dyadic cases and 16 boundary, rejection and mutation controls also pass.
• Warning-strict normal and optimized checks pass locally on CPython 3.12.15, macOS arm64, taking 2.76 and 2.84 seconds respectively. Both registered commands pass. Metadata/pins, exact committed-byte/source-history preflight and the official six-file package export pass at 9b51d4bc57d3ecbc19994cc19644b2fde894b664.
• A read-only Git-free optimized export replay with empty PATH passes in 2.85 seconds. All seven exported files, including the export manifest, retain their bytes and modes. The verifier emits a generated source-hash receipt and writes no files.

Exact reproduction from the repository root:

PY=/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12
"$PY" -W error -B research/tensor-helper-capacity/verify.py
"$PY" -W error -B -O research/tensor-helper-capacity/verify.py
"$PY" -B scripts/research_package.py --package research/tensor-helper-capacity
"$PY" -B scripts/check_research_commit.py --package research/tensor-helper-capacity --check-source-history

Any CPython 3.12 interpreter can replace that local path. Expected markers are PASS_TENSOR_HELPER_CAPACITY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The scientific replay uses only the standard library and imports no predecessor executable. The package pins the exact predecessor proof and README as provenance in the reachable base history.

Limitations

The all-size result is a written proof conditional on the cited single-cut criterion; finite checks corroborate only their enumerated domains. The minor oracle is bounded to 4-by-4 matrices; executed diagonal extremizers have at most 12 entries and three factors. No physical prefix/suffix, saturated tensor eligible module, shared chart, carrier/timing contract or complete paid/target invoice is supplied. Basis work, coefficient growth, native execution, dirty restoration, storage and compiler/outer admission remain unpriced. No selected cold replay, Lean, complete rounded multiplier, independent human review or formal proof-kernel review was run.

Open next questions

• Exhibit one named physical pair of cuts with complementary prime-rank losses, complete saturated bases and legal tensor carriers/timing. Then account for delivery and restoration before crediting targets.
• Compile and price the displayed two-channel X,Y in an admissible caller, including inverses, coefficient growth and every restoration operation.

Related open work

• Add a mathematical atlas of shared queries, helper hubs and compiler routes #5803 studies algebraic query sharing and dyadic response factorization; this instead counts normalized dyadic unit channels under tensor composition. It does not alter that atlas's query-count claims or claim a physical query saving.

• Execute Gaussian Smith circuits and prove sharp precision requirements #5821 executes Gaussian Smith response circuits at finite precision; this concerns helper pairing matrices over Z[1/2], with a different coefficient ring and unit-channel quantity.

• Prove sharp clean storage for reversible Gaussian overwrite (no κ) #5880 proves clean-bit capacity for destructive reversible Gaussian overwrite; its information-retention quantity differs from this dyadic helper unit-block capacity. No common numerical value is compared.

• Construct a 22-channel monotone delivery counterexample (no new κ) #5881 supplies a finite immutable-channel delivery certificate and a monotone tariff comparison; it does not compute dyadic tensor unit-block capacity.

• Prove odd-axis Clifford tensor Gram lower bound above one half #5883 bounds a Clifford tensor Gram spectrum; its spectral constant is a different quantity from the dyadic helper-capacity formula here.

The completion refresh inspected this account's nine other open PRs created since the CLAIM. No additional submission computes a common dyadic tensor-capacity value, so no numerical agreement/disagreement claim or extra comparison computation is made.

CI notes

At head 9b51d4bc57d3ecbc19994cc19644b2fde894b664, run 40491513039 finished with plan failed; replay skipped. The only failing step is Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22). The two errors are missing follow_up_steps and catalog_first_step; the eight failures concern frozen-card/current-command resolution, missing historical references and sparse omission handling. This is the reported shared onboarding failure on recent PRs based on main de5ec2e7a; maintainers will handle that repair. No shared tests or files were changed in response.

The replay job was skipped. Hosted replay has not certified this package; only its local scientific checks have run. The exact local checks and limitations remain recorded above. No CI rerun, shared-code repair or new scientific claim is made.

Merge Danger

Door: two-way. Blast radius: package. Only research/tensor-helper-capacity/ is added; shared implementations, predecessor artifacts and the selected record are unchanged.

This contribution was produced by Nick Petalas. Credit Ali Romman #5615 for the single-cut theorem, #5602/#5609/#5611 for causal pairing/completion, and #5599 with the retained helper/controller/native lineage. The predecessor also credits Nick Petalas #5115 for a different Euclidean row preparation. Smith normal form, Bezout identities and rank multiplicativity are classical. Apache-2.0.

Advance claim.