[petalas]

Summary

Prove a scoped fractional-moment improvement for the unchanged selected #4470 mathematical nested compiler:

#5067: p=1        -> K/N < 3.070
this:  p=511/512  -> K/N < 3.0694

The uniform source-logarithm envelope decreases from 197115103572543 to 197100824008381. The new proof bounds both productive and paid fractional moments, including all completion and split terms, by 2^18291. It uses JoyboyBrian's #5067 root/nonroot source census as a cited premise and answers its Open next question 1. It does not audit that predecessor.

Claim type: scoped mathematical proof with exact finite corroboration. No new kappa or physical admission. The first proved improvement is shipped without an exponent search.

Evidence

• Before: Sharpen selected nested-width coefficient to K/N < 3.070 via truncated sources #5067's first-moment envelope gives K/N < 3.070.
• After: exact integer/rational power enclosures and a fully charged split bound prove K/N < 3.0694; 458 new tiny source-moment fixtures, 32 split-moment fixtures and three discriminating examples pass. No product histogram or integer of size N is materialized.
• Warning-strict normal and optimized direct commands pass, timed at 0.03 seconds each locally on macOS with CPython 3.12.15. Both registered commands pass. Generated receipts bind all five package files.
• Committed-byte/source-history preflight passes at b609781074e982ab89fb54c0f897b17ec2789bb6. Official export integrity and read-only Git-free normal/optimized replay pass; all eight exported files remain unchanged. Five additional invoice/domain/source mutations reject.

From the repository root:

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

Any CPython 3.12 installation can replace the interpreter path. Expected marker: PASS_FRACTIONAL_SOURCE_WIDTH K/N<3.0694 no_new_kappa.

Limitations

The #5067 census and retained compiler semantics are cited premises. Physical stock, routing, native/all-column realization, owned storage, whole-job adapters, precision/tape and analytic/recovery obligations remain inherited. No Lean, Docker, selected operator replay or complete multiplication theorem is claimed. All registered checks ran locally. Independent human and formal proof-kernel review are unclaimed.

Related open work

The refreshed open-PR and claim search found no other open contribution targeting this fractional source bound or the selected complex nested-width coefficient. Closest predecessors #5067 and #5026 are merged. Other stock and compiler proposals have different physical or batching contracts; this result preserves the selected nested recurrence and its invoice.

Open next questions

• Can a boundary-sensitive moment or another fixed exponent tighten the mathematical bound while retaining every completion and split term?
• Can the actual stock/ownership/routing caller realize the permitted width at its complete tariff?

CI notes

CI pending at b609781074e982ab89fb54c0f897b17ec2789bb6. Hosted certification is not yet claimed. No CI failure has been inspected or attributed.

Merge Danger

Door: two-way. Blast radius: research. Only the new research/fractional-source-width package is added; no selected-frontier or predecessor files change.

Produced by Nick Petalas. Credit JoyboyBrian #5067, Sagar Thakrar #5026, James Reed/jwr404, Mark/Matarener, zxv #4436, Cinna Davis/zxv #4440/#4470, Jacob Alan Sussman, Ryanson Jonathan, mihir-s-05 and the retained compiler lineage. Advance claim.