[petalas]

Summary

Scoped mathematical proof: the fixed logarithmic compiler guard from Richard Hoekstra's PR #5787 has exact minimum positive integer height 12040587444441491.

(H+1)^322 < 2^(H/700000000000)
H < 12040587444441491: fails
H >= 12040587444441491: passes

This removes 112555558509 height units from the cited sufficient grid height 12040700000000000, with the exponent and gap fixed. The separate H>8192 guard holds. The written proof uses strict decrease of ln(1+H)/H and exact opposite signs at adjacent integers.

New theorem and checkers produced by Nick Petalas. Credit Richard Hoekstra for the complete carrier application and explicit unclaimed integer-height boundary, zxv for the logarithmic compiler, Jacob Alan Sussman for mixed flow, and the retained #5413 caller, native, provider and controller lineage. Two interface sources from exact predecessor head 08f0e58 are retained verbatim with Git blobs and SHA256 hashes. Predecessor executables and scientific results are not replayed as the deliverable.

Advance claim. Delivered scope matches the claim.

Evidence

• Before: PR Reduce PR5413 carrier height at fixed κ and certify size-saving tradeoffs #5787 proves a sufficient rational-gap grid and explicitly leaves the minimum integer H unclaimed.
• After: the new proof settles every positive integer H. The primary exact positive-series checker certifies -36/10^15 < D(H-1) < -35/10^15 and 1354/10^15 < D(H) < 1355/10^15, where D(h)=h/700000000000-322 log2(h+1). A separately implemented alternating-series checker proves the same two strict signs.
• Local CPython 3.12.15 warning-strict checks pass 1024 exact small integer-power comparisons, 65 exact power-of-two logarithms and 17 rejection controls. All registered commands pass locally. Their single-run durations are 0.50 seconds and 0.14 seconds, below the 120-second timeouts; these are timeout checks, not speedup measurements.
• Package metadata/pins, committed bytes, executable modes and declared source history pass. Official committed export integrity and normal/optimized read-only Git-free replay with empty executable PATH pass. All eleven exported files retain their bytes and modes. A generated receipt binds all ten package files. AST/compilation diagnostics pass; no external linter was available.

From the repository root:

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

Any CPython 3.12 executable can replace the local interpreter path. Expected markers are PASS_INTEGER_CARRIER_HEIGHT minimum=12040587444441491 reduction=112555558509 small_exact=1024 controls=13 and PASS_INDEPENDENT_INTEGER_HEIGHT minimum=12040587444441491 adjacent_signs=2 exact_powers=65 controls=4.

Limitations and unrun obligations

This is an exact threshold theorem for one fixed logarithmic inequality. The full changed-height finite-template, precision, conjunction, integer-layout and physical application remains open. It does not certify a smaller complete conditional caller, a global compiler optimum, a new kappa or a complete multiplication theorem. No predecessor supplier replay, selected cold replay, Lean/kernel proof or independent human review is claimed. No registered scientific command is left for CI alone. The selected frontier is unchanged.

Open next questions

Can the full PR #5787 application be instantiated at this exact height? First derive its height-dependent counts, rounding, precision and conjunction guards at H=12040587444441491 with every paid tail retained. The complete changed-N application and any necessary layout integrality are the blockers.

Related open work

• #5787 supplies the complete Complete coupled-output caller: conditional κ = 0.003027288235 (+5e-12 over #5400) #5413 carrier application and sufficient grid height 12040700000000000. This package proves the minimum integer height of its fixed logarithmic guard.
• #5899 determines a different PR5756 tail-gate block-length threshold. It does not compute this Complete coupled-output caller: conditional κ = 0.003027288235 (+5e-12 over #5400) #5413 integer-height guard and is not a predecessor of this result.

Merged #5462 and #5468 retain sufficient grids while refining the #5413 saving and gap-parametric precision. This package changes neither result.

Completion refresh inspected this account's nine other open PRs created since the advance CLAIM: #5919 and #5921 through #5928. Their titles and result scopes cover Gaussian-reader coefficients, binary pattern pools, entropy defects, descendant circuits, affine targets, Tensor rounding, Fano cuts, sign-lifetime identity tests and shared-block spectra. None studies this fixed logarithmic integer-height guard. No extra computations were run to validate or compare those results.

CI notes

At head e07ed3d60124234d9805b789dbddd8c1badad3aa, run 40513510103 finished with plan failed; replay skipped. The only failed 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 KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures concern frozen-card/current-command metadata, the missing-current-helper receipt, missing historical-reference rejection and sparse omitted-path expectations. The failure/error headers, both KeyErrors and the final summary exactly match the inspected PR #5919 run, also based on canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a.

Maintainers own this shared failure. This contribution changes no shared tests or helpers; no repair or rerun was attempted. Hosted replay has not certified this package. Only the local scientific and publication checks listed above have run. The CI package replay was skipped.

Merge Danger

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

Blast Radius: package. Shared code, selected records and frozen predecessor sources are unchanged.