Independent check of the f7 functional classification, plus the (5,14) domain (no new kappa)

[petalas]

Change and mathematical scope

Independent verification (no κ) of a finite certificate, plus new coverage. No f(7) bound, no supplier, no selected-frontier change.

research/f7-functional-independent-check/ re-derives, with a separately written enumerator and classifier, the nine-domain functional classification published in #3699 (research/f7-bit0-functional-classification, exact head 13db47d8d31603c1391572009e1eb7a22675a219): the retained L/2 budget substitution falsely prunes universal words of the PR943 singleton prefix walk, and the question is whether coordinate bit 0 is special.

Measured, reproduced exactly on the first run:

• simple closed word counts 4, 2, 29, 110, 358, 3695, 6525, 123704, 161936;
• universal word counts 4, 2, 22, 110, 0, 828, 0, 0, 0;
• substituted budgets 3, 9, 5, 13, 5, 15, -1, 11, -21;
• false-prune totals 0, 0, 44, 190, 0, 2856, 0, 0, 0;
• all 43 per-functional false-prune counts of the three firing domains;
• the published bit-0 column 6, 20, 248;
• the 01021323 fixture (nine prefix states, both splits, both budgets).

New coverage — three domains the published table does not contain:

n	L	simple closed	universal	budget	false prunes	firing functionals
3	10	0	0	17	0	0 of 7
4	12	314	314	23	150	13 of 15
5	14	33922	24787	27	56686	25 of 31

(5, 14) is the domain #3699 itself names as its open next question 2. There the substitution falsely prunes for exactly the 25 functionals of Hamming weight 1, 2 and 3 and for none of weight 4 or 5, with no exception of the (5, 12) kind. At (4, 12) the unsound class is strictly larger than at (4, 10): three of the four weight-3 functionals fire, leaving only 0111 and 1111 sound — so the unsound set grows with L on the measured range.

Status: finite certificate over ten bounded (n, L) domains. Not a weight-threshold theorem, not the seven-label regime, not a pruning repair. Remaining obligations: the necessity of the exact budget base + d^2 is inherited from #3699 and #2973's MATH.md and is not re-derived here.

Task and overlap

• Base commit 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6.
• Task card census-pruning-audit (adversarial-audit track), scope refined: the card's own first experiment is already delivered by Correct word-census parity budgets; 22 <= f(7) <= 24 in the PR943 model #2973, Record the bit-0 census false prune on 01021323 #3096 and research: classify the fixed L/2 word budget over all linear functionals (no kappa) #3699, whose table this package reproduces; the smallest remaining gap that moves a stated claim is the (5, 14) domain research: classify the fixed L/2 word budget over all linear functionals (no kappa) #3699 names.
• Nearest overlapping work: research: classify the fixed L/2 word budget over all linear functionals (no kappa) #3699 (the checked claim), Record the bit-0 census false prune on 01021323 #3096 / research/f7-bit0-false-prune (the earlier bit-0 audit), Correct word-census parity budgets; 22 <= f(7) <= 24 in the PR943 model #2973 / research/f7-weight-parity-census-correction (the model and MATH.md), Derive the complete length-22 parity-split domain at seven labels (no census, no f(7)) #3708 (a sibling L22 arithmetisation, different object). Added value: an independent route to the published numbers and the first classification at (5, 14).

Validation

Interpreter: ~/git/integer-mult-bounds/.venv/bin/python3.12 (CPython 3.12). Environment: macOS arm64, standard library only, no network, no third-party packages.

python3.12 -B scripts/research_package.py --package research/f7-functional-independent-check
python3.12 -B scripts/check_research_commit.py --package research/f7-functional-independent-check --check-source-history
python3.12 -B research/f7-functional-independent-check/verify.py
python3.12 -B research/f7-functional-independent-check/test_verify.py

Local results: PASS_REGISTERED_RESEARCH_PACKAGE research/f7-functional-independent-check outcome=completed (which itself replayed both registered commands), PASS_RESEARCH_COMMIT_BYTES {..., "source_history_commits": ["20ecf2891d80b5acf18d980b2164fb44ea7faa90"]}, PASS_F7_FUNCTIONAL_INDEPENDENT_CHECK and 26/26 behavioural tests OK. Runtime ≈ 30 s for verify.py, ≈ 16 s for the tests.

Controls run on every invocation: the fixture and its C(s,2)+C(t,2)-(2^(n-1)-1) = base + d^2 identity; two independent collision-excess routes that must agree for every universal word and every functional of every domain; a budget-shift sensitivity control (-2, -1, +1, +2) that must leave every non-firing domain at zero and move the count on each firing domain; a second enumeration route (unconstrained walk, canonicalised afterwards) that must return the same word sets on six domains; and the predicates asserted on every enumerated word.

CI certification. The repository's registered replay job ran on this exact head and passed (replay (research/f7-functional-independent-check), 3m13s, run 39157448322, job 117763977402); its plan job passed in 2m2s. The same two commands pass locally under the 3.12 interpreter at ~/git/integer-mult-bounds/.venv/bin/python3.12.

Not run here: the (7, 18) / (7, 20) domains, any census in the seven-label regime, and any Lean or Docker job. This is finite arithmetic, not a formal proof; no receipt is claimed beyond RESULT.json, which verify.py regenerates and byte-compares.

Credit

The classification question, the model, the nine published domains and the nine pinned numbers are Nick Petalas's, in #3699, including the (5, 14) question this PR answers for universal words. Nicholas Stone wrote the word-census correction (#2973) and its MATH.md; James Reed wrote the bit-0 small-domain audit (#3096); Ross Richardson supplied the PR943 singleton prefix-walk model; Douglas Colkitt supplied the selected source bindings and review. The re-implementation, the three added domains and the controls are by Nick Petalas, produced with AI assistance. Apache-2.0. Independent human and formal proof-kernel review are unclaimed.

Open next questions

In the package README under Open next questions: monotonicity of the firing set in L (first step: classify (4, 14) and (5, 16); blocker: runtime beyond this package's registered 600 s replay budget), a proof of high-weight soundness rather than an observation (blocker: no bound on the collision excess of a universal word), and whether (6, 16) contains a universal word at all (blocker: (6, 14) already has 123704 simple closed words).

Workflow feedback (optional)

Not included; nothing blocked the run.