[pinterbanget]

Summary

This adds research/f7-firing-threshold-all-n/. For every n ≥ 5, it proves the first even length at which the fixed L/2 substitution of #4322 fires on no universal canonical word:

L*(n) = 2^n - 2 floor(sqrt(2^(n-1) - 1))      (26, 54, 114, 234, 482, 980, ... ; 2^n − L* ≈ 2^((n+1)/2))

From L*(n) up to 2^n, every functional is sound on every L-point subset. At L*(n) − 2, a weight-one mask fires on an explicit word.

This answers #4322's open question 3: the collapse recurs at every n. It also settles #4322's remark that neither of its results shows "the firing set eventually becomes empty for every n". The value at n = 5 matches the census in #4348.

Claim type: a proof for every n ≥ 5, with finite checks and explicit witnesses. No new κ.

Proof outline (README)

1. Min-sumset cap. By Kneser's theorem in F_2^k, |A+A| ≥ 2^⌈log₂|A|⌉. So a k-point side of a level set has excess at most e(k) = C(k,2) − 2^⌈log₂k⌉ + 1.
2. The binding split. It is a full level set against the rest. With s = 2^(n−1) and L = 2(s−j), base − cap = (s−1) − j². For n ≥ 6 both parts lie in (2^(n−2), 2^(n−1)] at every L ≥ L*, and the cap is convex there, so no other split binds.
3. Sharpness.
   • Traverse all of {x_1 = 0} along a reflected Gray cycle.
   • Make a "ladder" detour through t = s − 2j points of {x_1 = 1}.
   • Then e_1 has excess exactly e(s) + e(t), which lies in (base, base + j²], so it fires.

Evidence

Check	Result
Min-sumset lemma and cap	exhaustive in F_2^k, k ≤ 4 (all 65,536 subsets at k = 4); the cap is attained
Full split scan	6, 14, then exactly the closed form for every 5 ≤ n ≤ 22
Explicit words at L*(n) − 2, n = 5..11	simple, closed, universal and firing, also after canonical relabeling
Controls of length L*(n)	nothing fires
#3733's (4,12) on main	still fires, so the n = 4 threshold 14 is attained

Standard library only; runs in about 6 seconds.

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

Head: bc720388557ac1c2f7f8fe89807ccb7769102676.

Not covered

• Which masks fire below L*(n) − 2.
• (7, 18) and (7, 20), which lie far below L*(7) = 114.
• Any retained exclusion.

Task and overlap

• Card: census-pruning-audit, refined, priority revision 7.
• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244.
• This 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. No other PR or discussion treats the all-n threshold; Prove codimension-one L/2 substitution soundness on double transversals. #4148 proves a different double-transversal statement.

Credit

The model and the n = 4 theorem are unpaidmoon's (#4322), building on Nick Petalas's #3699 and #3733. Kneser's theorem is classical. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance. Apache-2.0.

🤖 Generated with Claude Code