[unpaidmoon]

Change and mathematical scope

A restored length-22 seven-label word of split (11,10) has exactly two extra prefix visits. None of the 231 closed position patterns contains both return distances 4 and 6. Explicit words attain {2,20,22} and {4,18,22}.

This does not decide f(7), census universal words, or change κ. The selected frontier stays 836330098538459/500000000000000000.

Task and overlap

• Base: bb4bccbf42844d2801487450727c2f63e10d1d0f, priority revision 6.
• Task: census-pruning-audit, scope refined. The even-length existence cover is open PR Record that every even length from 4 through 18 meets the positive length-22 splits #3730; this package only records the one-word gap at revisit 1.
• Not redone: the length-22 domain (Derive the complete length-22 parity-split domain at seven labels (no census, no f(7)) #3708), the length-2 witnesses (Record a length-2 return witness for every positive length-22 split #3726), the even-length existence cover (Record that every even length from 4 through 18 meets the positive length-22 splits #3730), the parity note (Record that a prefix-walk return has even length #3719), and the universal census probes.

Validation

python3.12 -B research/revisit-one-return-gap/verify.py
python3.12 -B scripts/research_package.py --package research/revisit-one-return-gap

Both passed on CPython 3.12.3. Expected marker:

PASS_REVISIT_ONE_RETURN_GAP split=11,10 incompatible=4+6 patterns=231 no_new_kappa

No Lean, Docker, or selected cold replay was run.

Credit

The revisit count and the positive splits come from the length-22 split-domain note. Lengths 2, 20 and 22, and the question for lengths 4 through 18, come from the later witness notes. The existence cover is open PR #3730. The one-bit parity fact is the preceding return note. The prefix-walk model is Ross Richardson's. The corrected parity census is Nicholas Stone's.

Prepared by unpaidmoon with substantial Cursor agent assistance. No independent human review is claimed.

Open next questions

Which pairs of even distances from 4 through 18 can share a restored word at revisit 2? The first step is to enumerate the three-extra-visit patterns and test the pair 4 and 6.