research: classify the fixed L/2 word budget over all linear functionals (no kappa)

[petalas]

Summary

• No new kappa, no f(7) bound, no selected-frontier change. Scoped result: a finite certificate on a stated small domain plus one machine-checked lemma.
• Correct word-census parity budgets; 22 <= f(7) <= 24 in the PR943 model #2973 reports that the retained PR2542/PR2634/PR2691 word engines group prefix states by coordinate bit 0 while applying a collision budget derived for Hamming-weight parity. That report singles out bit 0.
• This PR tests the changed hypothesis: the substitution is not a bit-0 artifact but a property of the L/2 substitution for any functional other than the parity one.
• At (4,10) and (5,12) the substitution falsely prunes a universal word for every nonzero functional of Hamming weight <= n-2, and for none of weight >= n-1. At (4,8) the same holds except one weight-2 functional (1010). Bit 0 is the first member of that class.
• The package also proves and machine-checks the alternation lemma l(q_{i+1}) XOR l(q_i) = v_{w_i}, which identifies the all-ones functional as the only functional whose class value alternates with time, and therefore the only one for which the L/2 substitution has any justification.

Independent recomputation of the retained fixture

Word 01021323, n=4, L=8, recomputed from scratch:

q_0..q_8 = 0, 1, 3, 2, 6, 4, 12, 8, 0

quantity	value
distinct nonclosing states	8
restored (each label twice)	yes
S + S == F2^4	yes
weight split / collision excess	(4,4) / 0
bit-0 split / collision excess	(6,2) / 8
substituted budget 2C(4,2)-7	5
correct bit-0 budget C(6,2)+C(2,2)-7	9

8 > 5 rejects a universal word; 8 <= 9 does not. These match #2973 MATH.md section 4.

Exhaustive functional classification

Every restricted-growth word whose q_0..q_{L-1} are distinct and whose last step returns to q_0, classified against all 2^n - 1 nonzero linear functionals:

n	L	universal words	substituted budget	functionals that falsely prune	firing weights	false prunes
3	6	4	3	none	–	0
3	8	2	9	none	–	0
4	8	22	5	9 of 15	1, 2	44
4	10	110	13	10 of 15	1, 2	190
5	10	0	5	none	–	0
5	12	828	15	24 of 31	1, 2, 3	2856
6	12	0	-1	none	–	0
6	14	0	11	none	–	0
7	14	0	-21	none	–	0

At (4,8) the verifier finds 22 universal simple words and 6 bit-0 false prunes. Both figures agree with the independent #3096 audit, which I did not consult while writing the enumerator.

Scope and what is deliberately not claimed

• The seven-label decision regime is not reached. PR2542's decisive call is cyclic_census(7, 18, budget=9, simple=True); PR2634/PR2691 extend to L = 20. The only seven-label domain here is (7,14), which has zero universal simple words and is vacuous.
• The measured weight >= n-1 safety is not proved. It is an empirical regularity of these two domains.
• Rows (5,10), (6,12), (6,14), (7,14) contain no universal word, so "no false prune" there is a statement about an empty set.
• No claim that any old exclusion is right or wrong; no revival or refutation of the PR2542 counting bound, pair identity, restoration parity, or the later set-level work (KB3c: 17-point cover audit - PR 2542's 'no 17-point set covers F2^7' against its own lemma table (no new kappa) #2773, 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, KB3e: the 18-point cover question for F2^7 narrowed to an exact finite remainder (outcome=partial, no new kappa) #2911, KB3f: the vanishing-character restriction on the 18-point cover question for F2^7 (outcome=partial, no new kappa) #2933, KB3g: the tight classes of the 18-point cover question for F2^7 - (7,1)/(8,2)/(6,0) PROVEN impossible, (11,7)/(9,9) gap pressed (outcome=partial, no new kappa) #2961) on its own hypotheses.
• substituted budget = -1 at (6,12) and -21 at (7,14) is recorded as an observation about where the old substitution is meaningless, not offered as a correction.

Test plan

• python3 -I -O -Werror -B research/f7-bit0-functional-classification/verify.py → PASS_F7_BIT0_FUNCTIONAL_CLASSIFICATION
• python3 -I -O -Werror -B research/f7-bit0-functional-classification/test_verify.py → 18 tests, OK
• python3 -B scripts/research_package.py --package research/f7-bit0-functional-classification --check-only → PASS_PACKAGE_METADATA_AND_PINS
• Maintainer replay under Python 3.12 (python3.12 is not installed on the contributor host; both commands were run under CPython 3.15.0, standard library only, and both pass)

verify.py contains no dependency on repository files and imports no predecessor code. A separately written unpruned enumerator (product(range(n), repeat=L), filtered afterwards) is asserted to agree with the primary pruned enumerator on simple_closed_words and universal_words at (3,6), (3,8), (4,8) on every run. That cross-check is what caught an off-by-one in the walk's starting step during development.

Open next questions

1. The (7,18) / (7,20) functional census — the only domain that changes a maintained decision; needs its own resource budget.
2. Why weight >= n-1 never fires here; and what the two exceptions (mask 1010 at (4,8), mask 11010 at (5,12)) mean.
3. Whether the bit-0 grouping can be repaired by substituting the actual class sizes instead of rewriting the traversal to weight parity — a source-change question for maintainers.

Credit

Nicholas Stone wrote the word-census correction (#2973); James Reed wrote the bit-0 small-domain audit (#3096); Ross Richardson supplied the PR943 singleton prefix-walk model; Douglas Colkitt supplied selected source bindings and review. This package independently re-derives the retained numbers and adds the functional classification and the alternation lemma. It replaces none of them.

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CI replay result (post-push update). After the PR was opened, the repository replay job for this package ran and passed on the canonical host under Python 3.12: check replay (research/f7-bit0-functional-classification) = success, plan = success, mergeable state clean. That closes the interpreter-version caveat on the contributor host; the local run was CPython 3.15.0 and the registered replay is the authoritative 3.12 run.