[unpaidmoon]

Change and mathematical scope

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

research/pr4100-row-field-movement proves that exposing one field of length a moves strictly fewer than 8 max(1, ceil(log2 a)) times its real volume. #4100 already writes the Strassen boundary cost

O(V d W^(tau-1)/B polylog(p)),    tau = 45/16,

and the union movement O(V d W/B polylog(p)). This package substitutes the layout factor into that dictionary.

In FINE-PRIME.md §6–§7, a term O(V (d/B) W^alpha polylog p) has saving 1 - epsilon + xi - alpha/2. On the written boundary volume the layout movement is the alpha = 1 shape: ceil(log2 a) = O(log p) both for the polynomial-width row (W = O(sqrt(p) polylog p)) and for the carved line |E_i| = O(L W/B polylog p + W polylog p) with L = 2^{ceil(400 log2 p)}. Its saving exceeds the Strassen saving by

1/2 - (3 - tau)/2 = 13/32

at tau = 45/16. The difference does not depend on epsilon or xi. A polylogarithmic factor in W adds only o(1), which does not close 13/32 for large p. For every integer W ≥ 1, W^{29} ≥ W^{16}, so the movement monomial is absorbed by the boundary big-O. The exposed length is not set equal to W or to B.

The same gap is 5 · 10^{-65} at tau_eff = 2 + 10^{-64}. Both printed exponents absorb the movement. Charging the logarithm on the full tensor volume, dropping the written density O(W/B polylog p), would leave saving 1 - epsilon, which is strictly below the headline epsilon q at the §7 parameters.

Not proved: the Strassen count, conditioning, the row gap, the choice between the two exponents, or an explicit density constant. The carving program is not replayed.

Task and overlap

• Task: uniform-transfer-audit, scope refined to the ledger substitution left open by the layout lemma.
• Base: canonical main fbc9ea0249746acca4f504d4fda2b84e0e36cf0d, priority revision 7. Selected decimal 0.00302015.
• Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 head cited by the family record: 337d04939571a637aeab54a0f58f54fde811663a.
• Ordered eligible list for this session, after dropping obligations this account already delivered: the first still-open successor of each row in the Map independent audit coverage of #4100's obligations (navigation; no new κ) #4311 shortest unaudited list. Item 3 is this substitution. The other four successors are the PR2809 continuation-stack space bound, the row-gap after alias removal, the single-arena CRT evaluator replay, and the free-word shadow of the PR4088 global program.
• PR Prove the boundary-arithmetic row-field pad is removed before the nex… #4376 proves the move count and leaves this substitution as open question 1. This package does not reprove that count.
• PR Audit literal head travel of #4100's sparse-union schedule: section 2.3 holds with constant 17 (no new κ) #4327 audits union head travel and is not this boundary-layout charge.
• PR Audit #4100 BIT.md's local-ring algebra, row stock, depth, recurrence and wrapper: all hold (no new κ) #4432 and Audit #4100 COMPLEX.md's common cover and native warps: |O(73,2)|, X_delta correction, stock arithmetic and Z4-monomial warps hold (no new κ) #4424 close other shortest-list rows and are not used as inputs.

Validation

Python 3.12, standard library only, from the repository root:

python3.12 -B scripts/research_package.py --package research/pr4100-row-field-ledger --check-only
python3.12 -B scripts/research_package.py --package research/pr4100-row-field-ledger

Both passed locally. verify.py checks the rational identities, W^{tau-1} ≥ W for W = 1..256 and two larger widths, the named core-length logarithm, source-sentence binding, and rejecting controls (tau = 2 is not strict; tau = 7/4 makes movement the heavier row; the full-volume charge undercuts the headline).

Not run: the #4100 ledger program, the carving replay, Lean, and Docker.

Credit

The boundary and union sentences are Cinna Davis's #4100. The layout lemma is research/pr4100-row-field-movement. The exponent table is FINE-PRIME.md §6–§7. The carved-line bound is CAPPED-SIGNOFF.md. The coverage map is PR #4311 (Ryanson Jonathan).

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. Which printed boundary exponent, tau = 45/16 or tau_eff = 2 + 10^{-64}, does the ledger use when the layout lemma is copied in? Both absorb this movement. First compare the two sentences with the generated ledger at head 337d04939571a637aeab54a0f58f54fde811663a. Blocked here because that ledger is a different program.
2. What explicit C and k satisfy mu ≤ C (W/B) (log p)^k? That would turn the 13/32 gap into a finite-p inequality. First pin them from the carving argument.

Workflow feedback (optional)

The #4311 coverage map is the ordered list for this card, and it is stale against audits merged later the same day. A one-line successor pointer on each closed row would keep the next question attached to the row.