[Chrisbryan17]

Summary

This contributes an isolated, exact-rational conditional arithmetic tightening to ? = 0.0030201659769, beyond the currently open #4568 ? = 0.00302016597.

It keeps the selected PR #4470 native supplier and the conditional #4532 critical-root / #4535 all-size transfer interfaces unchanged. It does not claim a new native word, unconditional integer-multiplication theorem, or formally accepted record.

Exact verification

• Reconstructs 10,928 productive fields and 64,620 paid jobs using independent Fraction moment intervals, bracketing the raw ? crossing within (0.0030201659770, 0.0030201659771).
• Shows a positive paid-vs-productive moment at the proposed uniform saving ? + 6?10??? = 0.003020165976906.
• Verifies all 54 strict outer inequalities, four functional guards, and nine cost margins as exact rational inequalities.
• Pays the critical-root logarithmic absorption using fixed rank N = 2^(10^15), with a conservative positive surplus > 68/3.
• Provides a self-contained Python 3.11/3.13 CI workflow, with no additional Python dependencies.

Reproduce

python -I -B research/p16-tight-transfer-20261022/frontier_rational_gate.py
python -I -B research/p16-tight-transfer-20261022/critical_root_tight_transfer.py

Scope and credit

All inherited physical-native, independent all-column, owner/host/precision, tape, analytic and arbitrary-height compiler proof obligations remain inherited and are not independently discharged by this submission. The result is a conditional parameter certificate pending review.

Based directly on current upstream main with a four-file isolated diff. Credit #4470, #4532, #4535, #4564, #4568, Cinna Davis, Dillon, zxv, Jacob Alan Sussman, Ryanson Jonathan, hexapode and earlier contributors. New research prepared for @Chrisbryan17 with GPT-6 assistance. Apache-2.0.

Review welcome, especially the critical-root lemma applicability and outer-ledger transcription.