Derive the complete length-22 parity-split domain at seven labels (no census, no f(7))

[petalas]

Summary

Arithmetic reduction only. No seven-label word is enumerated, no length-22 split is excluded, and f(7) is untouched.

This is the first half of the open question recorded in #2973: "derive and audit its complete split domain before a bounded feasibility probe." It derives from scratch, with a separately written standard-library implementation, the complete set of Hamming-weight parity splits a restored universal length-22 word over seven labels could possibly have.

• Within PR Research: sharp four/five-source dirty-connector boundary (no new kappa) #943's prefix-walk model the derived necessary conditions leave 11 split representatives at length 22.
• Importing the standing 17-point set-level exclusions of KB3c: 17-point cover audit - PR 2542's 'no 17-point set covers F2^7' against its own lemma table (no new kappa) #2773 and KB3d: (11,6)/(6,11) remainder rows (5,1),(6,0) of the 17-point cover question impossible by an exact Fourier/Parseval identity (no new kappa) #2870 removes exactly two and leaves 9.
• Nothing further is excluded. A length-22 exclusion remains open, but the census domain is now exact rather than unknown.

The result

Split	|P|	Revisits forced	B0	need	Odd budget
(11,11)	22	0	47	29	121
(11,10)	21	1	37	23	110
(11,9)	20	2	28	18	99
(10,10)	20	2	27	18	100
(11,8)	19	3	20	12	88
(10,9)	19	3	18	13	90
(11,7)	18	4	13	7	77
(10,8)	18	4	10	8	80
(9,9)	18	4	9	9	81
(11,6)	17	5	7	1	66
(10,7)	17	5	3	3	70

Exclusions, with the responsible condition named. The time bound removes (13,5) and (12,6) — counting-feasible at |P| = 18 but impossible in a 22-step walk since s <= 11. The pair identity removes (9,8), whose B0 = 1 cannot reach need = 4. The same conditions independently reduce |P| = 17 to exactly (10,7) and (11,6), which are precisely the two splits the imported set-level result excludes, leaving 9.

Two further consequences: the covering-cardinality floor at n = 7 is 17 points, and self-avoidance is valid at length 22 only at (11,11) — the single split where |P| = L. At the other ten splits 1 to 5 revisits are forced, so a census assuming a simple cycle would silently miss most of the domain.

Verification

python3 -I -B research/f7-l22-split-domain/verify.py
# PASS_F7_L22_SPLIT_DOMAIN length22_self_contained=11 after_17pt_import=9 f7_untouched

About 28 s, standard library only, on macOS arm64 / CPython 3.15.0. Also passes under -O -Werror.

• The 01021323 fixture is reconstructed independently and agrees element for element with check_math.py.
• All restored words at (3,6), (3,8), (4,8) are enumerated; counts and split histograms match the independently published values (for (4,8): 48 (3,4), 48 (4,3), 528 (4,4)). Five further domains are enumerated without a published comparison.
• The published length-18 domain, the five length-20 representatives, and their budgets 27,18,10,9,3 / 36,26,16,17,6 are all recomputed from the definitions.

Control strength, including what did not discriminate

Controls were developed against a 37-mutant battery. Three real gaps it exposed and this PR fixes:

1. The necessity audit inside the enumerator cannot fire on correct code, so deleting it changed nothing. Replaced with a converse control: the four conditions must reject non-covers (all 168 non-universal words at (3,8)).
2. The coverage oracle had no non-vacuity control — an always-true covers initially passed. Now all 256 subsets of the fixture set are cross-checked against an independently written per-mask reference. Note the fixture is not minimal, so the obvious single-point-removal control would have been wrong.
3. The ceiling in need = ceil((s*t - Q)/2) is untested by any length-22 verdict, so a floor-for-ceiling mutant survived. Now checked directly, including odd remainders.

Two mutants survive and are equivalent, not defects: >= universe vs == universe inside covers (the pair-sum set is always a subset), and appending (j,j,j,j) in the padding check (that is exactly two pad steps).

One honesty note: the verifier reports that the even budget is redundant given the identity, since need >= 0 means B0 <= 0 already fails. It is labelled redundant rather than presented as a fourth filter.

Scope

No supplier, tariff, kappa, certificate, frozen input or selected-frontier change. scripts/research_baseline.py still reports PASS_RESEARCH_BASELINE_BINDINGS_ONLY with the same values. Necessary conditions are necessary only: this is an upper bound on what a census must search, not a candidate list of existing words.

Overlap. #3699 independently recomputes the same 01021323 fixture and classifies which linear functionals make the old substitution falsely prune. Different deliverable: it answers which groupings are affected at small n; this answers what the split domain is at n = 7, L = 22. The shared fixture arithmetic agrees exactly.

Not discharged. The 17-point exclusions are imported, not re-proved, so the 9-split figure is conditional on #2773 and #2870. The registered launcher gates on Python 3.12, which this host lacks, so only --check-only ran locally; CI remains the first 3.12 run.

Open next questions

1. Does a restored universal length-22 word exist? The domain is now 9 splits, so this is a finite census rather than an open one. first_step: run the corrected engine at (11,11) alone, where self-avoidance is valid and no revisit handling is needed. blocked_by: no node-count estimate or resource budget; Correct word-census parity budgets; 22 <= f(7) <= 24 in the PR943 model #2973's stop rule was explicitly no L22 census.
2. Which zero-XOR subword lengths can realize the forced revisit counts? Pure combinatorics, no search. blocked_by: interaction with the collision budgets untested.
3. Do KB3e: the 18-point cover question for F2^7 narrowed to an exact finite remainder (outcome=partial, no new kappa) #2911's undecided 18-point splits intersect the reachable ones? (12,6) is outside the length-22 domain entirely, so only three of its four splits are reachable at L = 22.

Credit Ross Richardson (#943 model); Aaron Wang (#2542 word model, #2773/#2870 17-point exclusions); Nicholas Stone (#2691 and the corrected census in research/f7-weight-parity-census-correction); Douglas Colkitt (source bindings and review). Automated research donation session; separate internal self-review is not independent human or proof-assistant review. Apache-2.0.