[unpaidmoon]

Change and mathematical scope

Negative result on payload transport, with an exact in-record schedule. No new κ. The selected conditional exponent stays 605457647/200000000000 (0.003027288235).

research/crt-reservoir-interchange-bill shows that six ordinary width-H calls meet O(V H^{1-a2}+V), and that the literal call multiplies the volume by (q^E/2^H)^2, between 2^{33} and 2^{34} at the minimal layout. It leaves open whether a direct exchange of H single-bit swaps meets the same bill on fixed tapes without that padding.

Two readings are not the same operation.

• In-record rewrite. One fixed pass, with a two-cell buffer, swaps stored bits i < j in exactly 2j+16 actions and restores both heads. An H-bit field exchange is H such passes, total H(2β+H+15) per record. Six exchanges cost at most 18 · 2^{1/20} V H^{1-a2} when V is a full scan of the address bits, 1 ≤ H < 2^{25} and L ≥ 15. The volume factor is 1. Rewriting two-bit labels [0,1,2,3] to [0,2,1,3] leaves the payloads in their slots.
• Payload transport. BIT.md moves the record at address x to the address with the two bits swapped. On a linear tape, with one payload crossing one record boundary per action, the Audit CRT reservoir placement and split-axis coordinate descriptors CrocSwap/integer-mult-bounds#4174 front field [0,H) forces more than 2^{L-4} scans. At the minimal layout (L = 1310720, H = 10752) that is more than 2^{1310716} scans, against an in-record allowance of 193536. At production (L = 1540096, H = 15360) it is more than 2^{1540092} scans, against 276480. No layout-uniform constant, in the sense BIT.md requires of its O, absorbs this. The transport lower bound is also larger than the 2^{33} padding, so this schedule does not remove that factor.

This is not BIT.md's theorem, not a complete multiplication theorem, and it does not change the selected frontier.

Task and overlap

Base commit 32890bc5de034ab8fc93ac88ead7279a6a416d04 (upstream CrocSwap/integer-mult-bounds main). Dependency pins are in research/crt-binary-field-exchange/SOURCES.json.

Task card uniform-transfer-audit, priority revision 7. Scope status is refined. The card asks whether one quantified step of the proposed CrocSwap#4100 construction holds at its actual caller. This package answers open question 1 of research/crt-reservoir-interchange-bill and stops. BIT.md is not reproved.

Session r43, item 11 of the ordered eligible list recorded in the package README. Open handoffs from the 19 October index come first, then priority-table rows whose next decision is still open, then still-open uniform-transfer-audit successors. Dropped before the cut: work this account already delivered, results on current main, questions covered by an open pull request, and topics already on a fork cursor/* branch. The fresh T385 wrapper, the retiming handoff, the four-provider rebind, the matched park/restore bill (open CrocSwap#4167 and CrocSwap#4172), the CrocSwap#4091 ceiling transfer, and the blocked Beneš-evaluator tape row were excluded before this item. Items 1–10 on that list stay open for the sibling sessions.

python3 scripts/find_overlap.py on 2026-10-25, terms direct binary exchange, single-bit swap, q^E/2^H. Coverage: local notes complete at 32890bc5 (7404 records, 7402 at head), 398 open PRs fetched at 2026-10-25T07:21:39Z, 4819 discussion records fetched at 2026-10-25T07:22:12Z, both paginations complete. The six hits are the predecessor README and TASK, this package, and discussion comments 19137011 and 19137065, which announce the six-call bill and its padding. No open PR prices the direct single-bit schedule. Fork cursor/* branches and draft PRs through this session's start do not use this topic.

Validation

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

python3 -B scripts/research_package.py --package research/crt-binary-field-exchange --check-only
python3 -B research/crt-binary-field-exchange/verify.py
python3 -O -B research/crt-binary-field-exchange/verify.py
python3 -B scripts/research_package.py --package research/crt-binary-field-exchange

The verifier imports the pinned CrocSwap#4174 allocator and executes the first plan of each layout. The big-integer scan bounds are comparisons of exponents and one subtraction of powers of two. Observed:

PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
PASS_CRT_BINARY_FIELD_EXCHANGE swap=2j+16 in_record_C=18*2^(1/20) minimal_L=1310720 minimal_H=10752 production_L=1540096 production_H=15360 transport=GT no_padding=1 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/crt-binary-field-exchange outcome=negative-result

The same verifier line is printed under python3 -O. Not run: Lean, Docker, the selected cold replay, and BIT.md's elementary q-digit algorithm.

Credit

Cinna Davis's CrocSwap#4100 supplies BIT.md and CRT.md. Hiroki KUMAZAKI's CrocSwap#4174 allocator supplies the field geometry. JoyboyBrian's research/crt-reservoir-interchange-bill states the padding and leaves the direct schedule open. James Reed's interchange-threshold values are used there and are not recomputed here. The executed key tapes in research/crt-transported-key-tape and research/crt-paid-key-service are a different geometry and are not this bill.

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. Can one adjacent address-bit transposition be charged as a block move, with a buffer of 2^k payloads counted as one action, and still stay inside O(V H^{1-a2})? The one-payload cut bound does not apply to that model. CRT.md charges one-cell moves, so this is a different tape.
2. What is the stage-0 constant of BIT.md's elementary q-digit swap? It is still the whole callee constant on these layouts, and BIT.md does not state that algorithm. This package does not reconstruct it.

Workflow feedback (optional)

The useful cut was to price the in-record rewrite and the BIT.md payload move as different operations. Treating any finite cost at one fixed H as O(V H^{1-a2}) would hide the layout-uniform constant BIT.md actually requires.