[petalas]

Summary

Return lengths 4 and 6 coexist in a restored length-22 singleton prefix walk with exactly two revisits. Two new five-label words attain both parity splits, (11,9) and (10,10), with full return spectrum {4,6,18,22}.

22 steps -> 20 distinct prefixes -> 2 revisits
20 distinct states > 2^4         -> at least 5 labels
two explicit 5-label words      -> both bounds attained

Claim type: finite attainers with a scoped counting argument. The least-revisit conclusion combines the new attainers with unpaidmoon's cited one-revisit exclusion in #4120. That exclusion is not replayed. The five-label minimum at two revisits follows directly from the new witnesses and state capacity. This answers the first bounded case in #4120's Open next questions.

Proof, literal words and limitations · generated receipt · advance claim

Evidence and reproduction

Before this contribution, #4120 supplied the one-revisit obstruction and explicitly left the pair 4 and 6 at two revisits open. The new certificate supplies both parity variants and the sharp alphabet size.

From the repository root, using CPython 3.12 and only its standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/revisit-two-return-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/revisit-two-return-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/revisit-two-return-threshold
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/revisit-two-return-threshold --check-source-history --check-modes

A local CPython 3.12 path may replace the displayed interpreter.

All four commands pass locally. The direct marker is PASS_JOINT_FOUR_SIX_RETURN_THRESHOLD. Prefix XOR and a separate coordinate/interval-parity engine agree on the complete observations. Both modes check the two words, 306 symmetry cases and seven semantic rejection controls. Each registered command took about 0.043 seconds locally, below its 60-second limit. These are timeout checks, not comparative performance claims.

Committed bytes, modes and declared source history pass at f69317ece54482d1b01c30b1c4934dc0ebdf0e81. The official export passes integrity/input checks; its normal and optimized replays pass with read-only files, an external working directory and Git absent from PATH. All eight exported files remain unchanged. Three private-copy corruptions, covering a source edit, a boolean receipt substitution and an invalid witness label, reject.

Limitations

The words embed in seven coordinates with two unused labels. They establish feasibility of the declared return constraints; they do not constitute universal seven-label words, a word census, a complete pair spectrum or a value of f(7).

No physical caller, paid-profile improvement, new kappa, supplier admission or selected-frontier change is supplied. The five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. Lean, Docker, predecessor scientific replays and the selected full replay are unrun and outside this claim.

Credit unpaidmoon / #4120, James Reed / #3730 and #3726, Ross Richardson / #943, Nicholas Stone / #2973, and the retained split/revisit lineage. The new attainers, alphabet-size application and checker are produced by Nick Petalas.

Open next questions

• Determine which other even-length pairs in 4 through 18 coexist at exactly two revisits. Start with repeated-position patterns, then construct actual cube embeddings; the position pattern alone is insufficient.
• State a stronger, source-bound joint-return prune that includes actual pairwise-XOR coverage, and test it against these attainers before budgeting a universal-word census.

Related open work

• #3899 audits union-support pruning on the (n,L)=(4,8) domain. This contribution constructs length-22 joint-return witnesses; it neither replays that audit nor computes its pruning counts.

The completion refresh read the eight other open account PRs created since the advance CLAIM: #6796, #6797, #6798, #6799, #6800, #6801, #6802 and #6804. Their dirty-word, cubic-pair, relation-lattice, temporal-cut, query-capacity, tape-arithmetic, NTT and centre-route objects differ from this singleton-prefix return threshold. None computes a competing revisit minimum or alphabet size for this pair. No additional predecessor or peer computation was run.

CI notes

Both hosted checks passed at exact head f69317ece54482d1b01c30b1c4934dc0ebdf0e81: plan and replay (research/revisit-two-return-threshold), in run 42229517835. The completed checks were read once. Hosted replay executes both registered normal and optimized finite checks. Local registered/direct checks, committed bytes/modes/source history, and read-only Git-free export replays also passed. No failed or pending check remains. CI corroborates the registered finite replay and source bindings; independent human and formal proof-kernel review remain unclaimed.

Merge danger

Reversible, isolated research package. Shared code, frozen predecessor bytes and the selected frontier are unchanged.