[unpaidmoon]

Change and mathematical scope

Scoped layout lemma for one caller of Cinna Davis's #4100. No new κ. #4100 is not accepted, and the selected frontier stays 0.001672660197076918.

The boundary-arithmetic sentences say that one exposed polynomial row field is padded to the next power of two at a factor below two, moved at O(log p) times the boundary volume, and removed before the next axis, so the pad does not accumulate as 2^g. This package proves those layout claims for every positive integer size in the hypotheses:

• M(a) = 2^{ceil(log2 a)} satisfies a ≤ M(a) < 2a. The factor 2 is not attained.
• When the right-hand-side count R is at least the batch width W, those right-hand sides partition into batches in [W, 2W] with no added cells. If 0 < R < W, the one padded batch adds W − R < W cells. That is not an O(1) record-move bound. Under the caller's superpolynomial R and polynomial W, R ≥ W and the added volume is 0.
• The lockstep transpose of the padded field uses ceil(log2 a) stages and two passes per stage. Forward movement is exactly 2 · ceil(log2 a) · M(a) · N, strictly less than 4 · ceil(log2 a) times the real boundary volume when a ≥ 2. The stage count depends only on a, not on the spectator length.
• Deleting the pad before the next axis keeps every live multiplier below two and returns the final multiplier to 1. Retaining a length-3 pad on each axis multiplies the volume by (4/3)^g, which exceeds 2 at g = 3 and is unbounded in g.
• Factor-page scans equal the batch count. They are not the right-hand-side count R, except when every batch has width 1, and they are not multiplied by the number of axes.

The Strassen count, the carving that produces O(V d W^{τ−1}/B), conditioning, the record header width, and the #4100 ledger remain inherited. #4100 names both τ = 45/16 and τ_eff = 2 + 10^{-64}; this lemma does not choose between them. It is not a replay of #4327's multitape head simulator.

Task and overlap

• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244. Priority revision 7. Prepared head e178c7bdb0aec5930e82a58fa237f21b6d47cf55.
• Card: uniform-transfer-audit, scope refined. The card asks for one quantified implication of Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100. CRT repair and the free-algebra lift already have open audits. This takes the still-open precision/conditioning layout obligation named by the Map independent audit coverage of #4100's obligations (navigation; no new κ) #4311 coverage map as unaudited item 3: GAUSSIAN.md's boundary-arithmetic row-field movement, at the PHASE-LAYOUT caller those sentences cite.
• Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 retained head 337d04939571a637aeab54a0f58f54fde811663a.
• Concrete object: the exposed row field of length a ≥ 1, orthogonal length N ≥ 1, and a batch partition of R right-hand sides at width W ≥ 1.
• Nearest work: Audit literal head travel of #4100's sparse-union schedule: section 2.3 holds with constant 17 (no new κ) #4327 audits UNION-SCHEDULE §2.3 head travel, constant 17, and leaves this batching row open, including the record header width. Map independent audit coverage of #4100's obligations (navigation; no new κ) #4311 is the navigation map and does not prove the factor. Merged pr1731-union-rect-review says partial-batch padding is negligible and checks the operator identity; it does not prove this factor, level count, or removal alternative.
• Package: research/pr4100-row-field-movement.

Validation

CPython 3.12.3, standard library only. The all-size statements are the integer identities in PROOF.md. verify.py recomputes them on an explicit range and executes the schedule:

• padding a ≤ M(a) < 2a for a = 1..65536;
• batch partitions for W = 1..64 and R = 0..512;
• lockstep forward and undo on twelve shapes, with worst forward ratio 2048/144 (length 9, spectator 16);
• a three-axis tensor of length 3, which peaks at 36 records and returns to 27;
• source-sentence binding for PHASE-LAYOUT, GAUSSIAN.md, and UNION-SCHEDULE step 5;
• rejection controls: a pad of length 3 is not 8, and six retained length-3 pads are not a factor below two.

python3 -B scripts/research_package.py --package research/pr4100-row-field-movement --check-only
python3.12 -B scripts/research_package.py --package research/pr4100-row-field-movement
python3 -B scripts/check_research_commit.py --package research/pr4100-row-field-movement --check-source-history

Local results: PASS_PACKAGE_METADATA_AND_PINS, PASS_REGISTERED_RESEARCH_PACKAGE outcome completed, PASS_RESEARCH_COMMIT_BYTES at e178c7bdb0aec5930e82a58fa237f21b6d47cf55, and

PASS_ROW_FIELD_MOVEMENT padding_strictly_below_two=true zero_added_cells_when_R_ge_W=true small_batch_added_cells_lt_W=true forward_passes_per_stage=2 max_forward_moves_over_real_volume=2048/144 stage_count_depends_only_on_row_length=true removal_peak_below_two=true retained_three_length_g6=(4/3)^6>2 page_scans_equal_batch_count=true no_new_kappa=true

Not run: the phase-cell carving, #4327's multitape simulator, the record header width, a proof kernel, the Strassen exponent, and the #4100 ledger. Finite execution corroborates the identities; it does not discharge #4100.

Credit

The layout sentences are from the capped phase-cell audit (Vlado Vojdanovski, PR #1673, and the contributors named there) and from Cinna Davis's #4100 Gaussian and union-schedule text. PR #4327 (Ryanson Jonathan) audited the union head travel and left this batching row separate. PR #1731's rectangular review is a different caller. Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human review and no formal-kernel acceptance are claimed.

Open next questions

1. Carry the move count into the ledger term. The layout bound is O(log a) times the boundary volume. It becomes Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100's O(V d W^{τ−1}/B) only after the exposed length is identified with the carving parameters W and B. First pin |E_i| and μ at the Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 balance and substitute Lemma 3. Blocked here because the carving program is not replayed and τ is not chosen.
2. Fix the boundary exponent. GAUSSIAN.md uses τ = 45/16 (7^16 < 2^45). UNION-SCHEDULE section 4 uses τ_eff = 2 + 10^{-64}. Compare those two sentences with the generated ledger at head 337d04939571a637aeab54a0f58f54fde811663a. Blocked here because that ledger is a different program, and PR Conditional κ = 0.001678123830633547: rectangular (k=5) boundary batching and radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1731's rectangular exponent is a different caller.

The record header width left open by #4327 is a separate tape-accounting obligation and was not taken.

Workflow feedback (optional)

The #4311 coverage map's shortest unaudited list made this obligation easy to separate from #4327's head-travel audit.