[unpaidmoon]

Change and mathematical scope

Scoped sentence proof. No new κ. #4100 is not accepted, and the selected conditional result stays 0.00302015.

On the one numeric balance printed for the #4100 union schedule, the boundary row is tau_eff = 2 + 10^{-64}. It is not tau = 45/16.

UNION-SCHEDULE §4 gives S_b - S_u = c - 1/2 at a common (eps, xi). GAUGEFREE-INTERIORS §6 prints

eps = 0.99999437577296, xi = 0.50167412128176, beta = 0.99664613320944

and UNION-SCHEDULE §5b prints that point rounded half-up to seven decimals, at tau_eff. The union saving on those 14 digits is exactly 1049840943/625000000000 = 0.0016797455088, matching the displayed kappa prefix through 0.00167974550880.

At tau_eff the boundary and union rows differ by 5 · 10^{-65}, inside the stated joint margin of about 10^{-64}. At tau = 45/16 they differ by 13/32, and the boundary saving is -0.4045702544912. The joint-binding sentence is false at 45/16 for this printed point.

tau = 45/16 remains the dense Strassen exponent in GAUSSIAN.md and FINE-PRIME §6, which needs no structured inverse. The short-field layout sentence sits on the later union-cost paragraph. UNION-SCHEDULE §6 assumes that movement and does not change the boundary matrix exponent, so copying the layout lemma in keeps tau_eff.

Task and overlap

• Base: ff3cc58e370d0f678e529246f8c73e8094e18fcf (canonical CrocSwap/integer-mult-bounds main at the 2026-10-23 refresh). Priority revision 7. Selected κ 0.00302015.
• Card: uniform-transfer-audit, scope refined. This answers open question 1 of research/pr4100-row-field-ledger.
• Item taken: item 3 of the ordered list in the package README.
• Nearest work: Show the short-field movement does not tighten the Strassen boundary … #4645 proves the movement does not tighten the Strassen row and stops before choosing an exponent. Prove the boundary-arithmetic row-field pad is removed before the nex… #4376 proves the layout count. Neither names the ledger row.
• The fork default main is behind canonical main. The scientific change is research/pr4100-ledger-boundary-exponent/.

Validation

python3 -B scripts/research_package.py --package research/pr4100-ledger-boundary-exponent --check-only
python3 -B scripts/research_package.py --package research/pr4100-ledger-boundary-exponent
python3 -B scripts/check_research_commit.py --package research/pr4100-ledger-boundary-exponent --check-source-history

Observed: PASS_PACKAGE_METADATA_AND_PINS, PASS_LEDGER_BOUNDARY_EXPONENT, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES.

Not run: outer/check_ledger.py (absent from the #4100 tree), the carving replay, Lean, Docker, and the selected cold replay.

Credit

The Strassen sentence and the union schedule are Cinna Davis's #4100. The gauge-free parameter point is the vendored GAUGEFREE-INTERIORS note. The layout count and the substitution that left this choice open are unpaidmoon's research/pr4100-row-field-movement and research/pr4100-row-field-ledger. The coverage map is PR #4311 (Ryanson Jonathan).

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human or formal-kernel review is claimed.

Open next questions

1. Does a byte-identical outer/check_ledger.py regenerate this tau_eff point? Section 5b cites that script, and the Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 tree does not contain it.
2. What explicit C and k satisfy mu ≤ C (W/B) (log p)^k? Naming tau_eff does not supply the constant.