[pinterbanget]

Summary

This adds research/f7-firing-n7-census/. It answers #4322's open question 2: which functionals of the fixed L/2 substitution fire at (7, 18) and (7, 20)? #1166 comment 19021170 names this census as the one that can change a maintained exclusion.

Claim type: an exhaustive finite census. No new κ.

Result

n	L	closed canonical words	universal	firing masks
7	16	4,578,096	0	0
7	18	133,421,988	0	0
7	20	3,857,028,444	0	0
6	16	2,262,380	6,744	54 of 63
6	18	39,195,780	1,854,738	59 of 63

No functional fires at (7, 18) or (7, 20). A mask can fire only on a universal word, and neither domain has one, so the substitution cannot have falsely pruned an exclusion at those lengths.

Controls.

• Positive control: seeded random closed walks do find universal words at (7, 26), so the test can fire at n = 7.
• Corroboration: random walks find none at (7, 20).

The universal domain at n = 7 starts between 22 and 26.

Evidence

• Enumerator. census.c extends my Answer #4322's open question 1: at n = 5 the first all-sound length is 26; the firing set drops from 30 to 28 masks at L = 24 (no new κ) #4348 enumerator to 128-bit masks.
• Agreement. It reproduces every overlapping published row exactly: research: classify the fixed L/2 word budget over all linear functionals (no kappa) #3699's (6,12), (6,14) and (7,14); Independent check of the f7 functional classification, plus the (5,14) domain (no new kappa) #3733's (5,14); Show the fixed L/2 firing set is not monotone in length. #4322's (6,16) closed count.
• Registered check. verify.py, standard library, about 9 s, reruns (6,12), (6,14) and (7,14) in Python against research: classify the fixed L/2 word budget over all linear functionals (no kappa) #3699 on main. It checks that the recorded runs were made with this census.c by SHA256, and runs the random controls.
• Local evidence. The large runs are not repeated by CI. (7, 20) took 36 minutes.

python3.12 -B scripts/research_package.py --package research/f7-firing-n7-census --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/f7-firing-n7-census
  PASS_REGISTERED_RESEARCH_PACKAGE research/f7-firing-n7-census outcome=completed

Head: eaa39d7439821a408f4c4ef03a71fd4c0399028e.

Not covered

• (7, 22) and (7, 24). No universal word appeared in random samples there, but that is not a proof.
• (6, 20) and longer.

Task and overlap

• Card: census-pruning-audit, refined, priority revision 7.
• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244.
• Claimed in Handoffs and requests for independent checks #1166 before starting (comment 19103485). It builds on my Answer #4322's open question 1: at n = 5 the first all-sound length is 26; the firing set drops from 30 to 28 masks at L = 24 (no new κ) #4348 and Prove the fixed L/2 firing set's first all-sound length for every n >= 5: L*(n) = 2^n - 2 floor(sqrt(2^(n-1)-1)), with explicit firing words one step below (no new κ) #4357.

Credit

The model and question are unpaidmoon's (#4322), building on Nick Petalas's #3699 and #3733. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance. Apache-2.0.

🤖 Generated with Claude Code