[petalas]

Summary

Scoped proof with finite exact corroboration: the minimum sufficient integer rank exponent in the published two-inequality compiler model can be found by testing two integer candidates around one explicit stationary point. Each result includes a rational separator witness. This answers the compiler-size question in ai-nanachi’s #4814 while retaining Cinna Davis’s compiler induction and native/tail premises.

merged BIT parameters:
  midpoint H1   25986965159937
  optimum H1    25986964913670     reduction 246267

PR4814 parameters:
  midpoint H1   262208150916063859466241
  optimum H1    262208150916063859220003     reduction 246238

Only research/integer-compiler-separator/ is added. Produced by Nick Petalas. Credit Cinna Davis, Jacob Alan Sussman, zxv and ai-nanachi for the cited compiler, strictification, native and backend lineage; README and PROOF give the exact scope and predecessor links.

Evidence

The proof eliminates the separator exactly, proves convexity of the reduced rational objective, and retains the strict final inequality when rounding. The separate interval oracle enumerates 36 small parameter cases and 18,942 possible intermediate integers. Six invalid-input controls and three scientific controls reject. Both large witnesses satisfy the original inequalities exactly.

From the repository root, using CPython 3.12.15 on macOS 27.0.1 arm64:

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

All passed locally. Each registered command took 0.12 seconds; the complete registered runner took 0.51 seconds. Normal and optimized scientific output and package hashes match. Committed-byte/source-history preflight passes for seven files and the pinned canonical base. The direct verifier emits a receipt bound to the six package-file hashes and writes no files.

Limitations

Optimality is for the stated sufficient inequalities. It is not an optimality theorem for other compilers, a practical speedup or a smaller eventual multiplication-input threshold. No native invoice or tail is audited or recomputed. Complete compiler installation, native/owned-erasure/whole-job implementation, stock, precision, tape, analytic and recovery premises remain inherited. No formal-kernel or independent human review is claimed. No selected-frontier or kappa change is proposed. The pending PR4814 code is not a replay dependency; only its explicitly cited parameter values are used.

Open next questions

• Characterize the integer H0 interval tying the minimum H1 after rounding. Start with the exact quadratic inequality F(H0)<H1_best.
• Bind one rational separator to a complete concrete compiler interface and identify any additional separator restrictions before installation.

Related open work

• Ordinary BIT saving 0.003475458 via three finite leaf backends #4814 supplies the second parameter fixture and its compiler-size question. Its midpoint H1 agrees with the direct evaluation here; this theorem lowers that fixed sufficient bound by 246238 without changing its ordinary saving or replaying its native/tail certificate.
• Ordinary BIT saving 0.0034597 (+0.67939%) via compact partial helpers #4760 supplies Ordinary BIT saving 0.003475458 via three finite leaf backends #4814’s productive BIT/native predecessor. This package changes no word, helper invoice or ordinary exponent from that construction.
• Prove the sharp integer job-splitting charge (no new κ) #4832 sharpens a splitting coefficient in the retained higher-moment estimate. This package keeps every such coefficient fixed and optimizes a different auxiliary variable in the compiler inequalities; no combined improvement is claimed.
• Extract optional ordinary B2 bootstrap with canonical-source replay #4338 studies completed ordinary backend composition and stock in a different B2 construction. It does not provide the separator optimization proved here.

Completion refresh reviewed all eight other open PRs this account created after the CLAIM: #4874, #4875, #4876, #4878, #4879, #4880, #4881 and #4882. None studies these separator inequalities or computes the same H0/H1 quantity; no additional related entry or numerical disagreement was found.

CI notes

Both hosted checks passed for head f6e6173e41b579332f4961e8101fb1830b339a20: planning and registered package replay. The checks were read once at completion. Local normal/optimized checks and committed-byte/source-history preflight also passed. Hosted replay certifies execution of the registered checks, not independent mathematical acceptance. Formal-kernel and full physical/compiler installation remain outside the claim.

Merge danger

Door: two-way. Blast radius: isolated. This adds a self-contained research package and opt-in replay, with no shared code or selected record changes.