[unpaidmoon]

Change and mathematical scope

Exhaustive finite census, proved by a complete case split. No new κ, and the selected conditional value stays 781451/250000000.

On the emitted local word of PR #5708, move three, four, or five corrections of every A2 conjugation with exactly five contacts onto other frames of the published one-edge route table. The corrections that are not chosen stay on their contact frames. The new frames may be equal. There are 133 intervals and 13,840,035,444,873,750 assignments. Every assignment misses a published edge. None emits, so none lowers the paid-query moment on (0, 1).

The emitter stops at the first missing route. Every assignment has a unique pair of earliest moved corrections and agrees with the two-move word on that pair through the rejecting record. The length-5 two-move census of #6963 is recomputed: 536,289,250 assignments, 4,331 tails, no emission. Each of those 4,331 tails misses an edge before the next correction, at a distance of 2 to 329 records, so every later frame choice is unread. The early extensions are 13,838,927,940,076,445 assignments and the tail extensions are 1,107,504,797,305. Those classes partition the mask count 133 × (10 × 635³ + 5 × 635⁴ + 635⁵).

The least interval in scalar-position order is included. Its outer gates are indexes 1,119 and 8,918, both ADD(572, 22). None of its two-move pairs reaches a tail, so all 104,060,416,878,750 of its three-or-more assignments are early rejections. The first recorded tail is interval 5,673..7,030, corrections 1 and 2 moved to frames 594 and 353. It misses ADD(173, 312) at rewritten index 6,306; the next correction begins at index 6,547.

This is the length-exactly-5 three-or-more census on one word and one route table. It leaves length at least 6, the three-or-more question at 3 or 4 contacts, and any added route untouched. A missing edge is a rejection on this table. It is not a proof of unrestricted nonexistence, and it does not change the selected frontier.

Task and overlap

Base 17469fff78c22941280f04d39dc5a1faee09c840, priority revision 8. Card bit-helper-contract, scope refined. The card says to review the #5679/#5708 chain and stop when a proposed cut does not improve the paid moment. This package answers open question 2 of #6963 on every length-5 interval, which is the question posed in discussion comment #1166 (reply in thread).

The two-move length-5 census is #6963 and is recomputed here as the reduction lemma, not offered as a new theorem. The one-move length-at-least-5 census is #6923. The two-move census at 2, 3 and 4 contacts is #6664. Three or more moves at 3 or 4 contacts remain the open question in research/g20-two-moved-corrections. Two moves at length at least 6 remain question 1 of #6963. No open PR enumerates this three-or-more length-5 domain. Local overlap search covered 10,180 notes at this base and found those predecessors only.

Validation

Python 3.12, standard library, from the repository root. Both registered commands passed locally in about 30 seconds each:

python3 -I -B research/g20-length5-three-move/verify.py
python3 -I -O -B research/g20-length5-three-move/verify.py
python3 -B scripts/research_package.py --package research/g20-length5-three-move

scripts/research_package.py printed PASS_REGISTERED_RESEARCH_PACKAGE research/g20-length5-three-move outcome=negative-result. The direct verifier prints

PASS_G20_LENGTH5_THREE_MOVE intervals=133 assignments=13840035444873750 early_extensions=13838927940076445 tail_extensions=1107504797305 tails=4331 tail_j1=4305 tail_j2=7 tail_j3=19 min_gap=2 max_gap=329 emitted=0 savings=0 first_interval=1119-8918/104060416878750/0 first_tail=5673-7030/1/2/594/353/6306/6547 cold_early=100 cold_tails=4331 cold_maximal=3 one_move_control=3H2-H4 length6_open=86 no_new_kappa=true

Checks raise exceptions, so -O keeps them. Cold controls rebuild every tail from scratch, move one further correction on each of the 4,331 tails, move every later correction on one tail of each later-slot count, and extend 25 early two-move samples from each rejection class. One published length-5 one-move emission from #6923 is replayed as a positive control outside this domain; its histogram difference is 3 H2 − H4, a strict loss. scripts/check_research_commit.py passed for commit b97d14029225dbac298f96ad24e19eceb6d60258.

Not run: the #5708 integration replay, the 18-call caller, adjoint columns, moduli 101 and 10201, ordinary admission, Lean, Docker, and make frontier6030-verify.

Credit

The emitted word, route table, A2 relation, and local-only conjugation count are Ali Romman's research/g20-conjugation-admission, following Dugongue, ai-nanachi, and the retained G20 lineage named there. The length-4-and-above contact-frame obstruction is Nick Petalas's #6305. The two-move classification at lengths 2..4, and the prefix machine reimplemented here, are Nick Petalas's #6664. The one-move obstructions are unpaidmoon's #6337, #6633 and #6923. The length-5 two-move census and the 4,331 tail count are unpaidmoon's #6963.

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

Open next questions

1. Can three or more moved corrections in one interval of length at least 6 emit and lower the paid moment on this word and route table? Length 6 has 86 intervals. The first step is one length-6 interval with three corrections moved on this route table. No length-5 three-or-more assignment emits, so none supplies a saving seed, and the length-5 case split does not cover a longer interval. Two moves at length at least 6 remain question 1 of Record that two moved corrections do not emit on length-5 G20 intervals. #6963.

#6923 question 2, one added route at the first missing edge of a dirty one-move failure, stays on that package. The published route table is held fixed.

Workflow feedback (optional)

The #6963 package layout and the minimal package generator made the reduction checkable. No process change requested. #1200