[unpaidmoon]

Change and mathematical scope

Exhaustive finite census. No new κ. The selected conditional exponent stays 781451/250000000 (0.003125804).

On the emitted local word of #5708, every length-at-least-5 one-move assignment that #6923 classifies as a dirty failure was replayed with exactly one route added at its first missing edge. The added route has an integer rank in 1..7. Of the 686,779 assignments, 686,386 still miss a different edge. The other 393 emit. Each emission pays the new edge one, two, or three times, and each is a strict loss on (0, 1) at every rank in 1..7. None is a strict saving.

The 686,779 failures use 22,709 missing frame pairs. The multiplicity table has SHA-256 8f38eb2f6e526ede87a62c1fc88c35fd5d3ba87a16f755df690a76cea5a68494. The maximum multiplicity is 1,841, attained by 303 edges into frame 1 (frames 326..635 except 507 and 519–524), accounting for 557,823 failures. Every first missing hop pays.

The lex-first emission is scalar positions 5,760–6,924, length 13, outer gate ADD(89, 19), correction 6 moved from frame 575 to frame 429. Its missing edge is (429, 575). At rank 1 the paid difference is 3 H2 − H4.

Task and overlap

Base 17469fff78c22941280f04d39dc5a1faee09c840. Card bit-helper-contract, scope refined. This is open question 2 of #6923. #6963 settled two moved corrections on length-5 intervals and left this added-route question on the #6923 package.

#6305, #6337, #6633, and #6664 screen other conjugation clauses. This census does not rerun them, does not move a second correction, and does not add a second route. It does not change the selected frontier.

Validation

Python 3.12, standard library, from the repository root. Both registered commands passed, and scripts/research_package.py printed PASS_REGISTERED_RESEARCH_PACKAGE with outcome=negative-result.

python3 -I -B research/g20-dirty-one-move-added-route/verify.py
python3 -I -O -B research/g20-dirty-one-move-added-route/verify.py
python3 -B scripts/research_package.py --package research/g20-dirty-one-move-added-route

PASS_G20_DIRTY_ONE_MOVE_ADDED_ROUTE dirty=686779 pairs=22709 later_missing=686386 emitted=393 savings=0 mismatches=0 first_hop_pays=686779 max_multiplicity=1841 pair_sha256=8f38eb2f6e526ede87a62c1fc88c35fd5d3ba87a16f755df690a76cea5a68494 split=382 elementary=11 witness=5760-6924/6/429/429-575/uses1 no_new_kappa=true

Checks raise exceptions, so -O does not disable them. The census walk is the #6923 route rule resumed at the contact, with the first missing edge inserted. Every emission was replayed independently by paid_histogram at ranks 1..7. One still-rejected word in each of the 28 lengths was replayed the same way and still misses an edge. The classifier and the walk agree on all 686,779 first missing pairs.

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 the 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 is Nick Petalas's #6664. The length-1..3 one-move obstruction is unpaidmoon's #6337. The length-4 one-move obstruction is unpaidmoon's #6633. The length-at-least-5 one-move obstruction, including the dirty-failure count and the classifier replayed here, is unpaidmoon's #6923. The length-5 two-move census that left this question open is unpaidmoon's #6963.

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

Open next questions

1. Does adding the second missing edge as well turn one of the 686,386 still-rejected words into a strict saving? The first step is the lexicographically first still-rejected word: record the edge that rejects it after the first route is added, and replay that word with both edges present at rank 1. This package inserts only the first missing edge.

Workflow feedback (optional)

The #6923 classifier already isolates the dirty class, so the added-route walk only has to resume at the contact. Reclassifying the whole one-move domain was the slow step because the missing frame pair is not stored. Recording that pair in the one-move census output would make this follow-up a direct replay. Feedback thread: #1200