[petalas]

Summary

Scoped proof with exact finite corroboration. For fixed nonnegative K,C, the largest integer g satisfying the inherited heavy tensor-error envelope obeys

M = g p^(10g) (1+p^-4) exp(Kp-p^2) < 2^(-Cp)
g_max(p) ln(p) / p^2 -> 1/10.

The proof gives the exact monotone-root characterization, explicit root brackets, and an integer-only sufficient gate 10gL+2L+1+(2K+C)p <= p^2, with L=ceil(log2 p). At p=64, K=C=0, the exact maximum is 98 and 99 fails. The integer gate certifies 68; James Reed's #4695 stated gate certifies through 10 there. This comparison substitutes into the predecessor's hypothesis and does not replay its computation.

Credit James Reed #4695/#4693/#4526/#4524 for the heavy-slice envelope, unequal-tail factor and sufficient gates, and Jose Manuel #4100 for the Gaussian/HALO setting. This contribution is produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed.

Evidence

Before: the retained sufficient gate is g ceil(log2 p) <= p, with additional parameter conditions. After: a quadratic-scale sufficient gate and a sharp leading threshold for this scalar envelope are proved in research/heavy-tensor-dimension/README.md.

CPython 3.12.15, standard library only. Both warning-strict commands passed locally in 1.65 and 1.67 seconds; the registered pair passed in 3.61 seconds. Each reports

PASS_HEAVY_TENSOR_DIMENSION boundaries=207 gates=175 power_comparisons=11 controls=17

The 207 cases certify neighboring strict signs using rational logarithm intervals; eleven signs also use a separate rational e-power comparison. The source-bound receipt, metadata/pins, committed-byte/source-history preflight, official eight-file export integrity, and optimized Git-free replay all pass. Local corruption controls reject altered proof bytes, a Boolean receipt mutation and altered predecessor input. An existing receipt cannot be overwritten.

From the repository root, reproduce with

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

Another CPython 3.12 installation may replace the absolute interpreter path. No predecessor scientific verifier is imported or run.

Limitations

The tail formula and operator norm ceilings remain hypotheses. A failed upper envelope does not imply that an actual operator fails. Header, storage, geometry, movement, restoration and physical-cost restrictions may still require g log p=o(p). No caller at the larger dimension, supplier admission, new kappa or complete multiplication theorem is claimed. No selected cold replay, Lean, Docker, independent human acceptance or proof-kernel verification was run. The selected frontier is unchanged.

Open next questions

• Identify the first actual caller restriction blocking quadratic-scale dimensions, starting from its explicit header-size and storage inequalities.
• Determine whether a concrete ordered list of unequal per-axis ceilings improves on the worst-slice envelope. Actual caller data and a complete physical contract are absent here.

Related open work

• Audit source-driven Gaussian halo copy enumeration (no new kappa) #4870 studies source-driven Gaussian halo copy enumeration in the same wider caller. This PR studies the scalar tensor-error dimension restriction; it supplies no copy enumerator or physical schedule. No shared computed quantity is compared.
• Prove sharp conditioning of the complete complex port (no new κ) #4828 studies conditioning of the complete complex port. This PR assumes norm ceilings and studies their scalar error envelope; it does not estimate that port's conditioning or extend its open questions.

The completion refresh inspected all 79 open PRs on this account, including eight created since the CLAIM and this PR among them. The seven other new contributions study different objects or quantities; none reports a competing heavy-envelope dimension threshold. Related work remains as listed above. No additional computations were run to audit those submissions. Searches omit arbitrary diffs and unpublished work.

CI notes

Both hosted checks passed for exact head ea68638b4d0d86f1cbf0b0be846b050ca6230359 in run 39873361876: planning in 3m05s and the registered package replay in 1m19s. Run metadata confirms the matching head and successful conclusion. The registered replay includes warning-strict ordinary and optimized commands; both also passed locally. Hosted replay certifies execution of the bounded verifier, with independent human and formal proof-kernel review unclaimed. No CI failure was observed.

Merge danger

Reversible addition of one isolated research package. Scope is the package and its explicit replay registration; shared code, frozen evidence, predecessor credit and the selected record are unchanged.