Width-four exact rank closure: 509 futures and 15-mode scalar (do not merge)

[LightChainr]

Draft. Do not merge. Completes the state-sufficiency request in the latest #636 comment. Does not merge #706 or #707. Additive on main@eb89e942 (merged #705). No runtime dependency on unmerged #707.

What

• Rank-future sufficiency of pinned-boundary connectivity + path gains + global cycle span (interior vertices may be forgotten; completed cycles may not).
• Common integer shear of current-to-pinned gains preserves rank via (h_x,h_y)↦(h_x+c h_y,h_y), not directional/primitive labels.
• Exhaustive width-four closure: 1448 representatives, 23168 transitions, all independently re-derived by graph traversal.
• Continuation refinement 3 → 164 → 509 → 509. All 509 classes reachable at physical m≥2 and pairwise distinguished by suffixes of length ≤2. This is a deterministic rank-language minimum, not a field count.
• M_{4,m}(p)=b(p)^T K(p)^{m-1} c for all m≥2. Not automatically a matrix trace or a pTL sector.
• At p=1/2 only: s_m=16^m M_{4,m}(1/2) has minimal scalar recurrence order 15, square-free over Q, six trace factors. P_{15}(A)c has 82 nonzero entries (not a full-space annihilator); 509 scalar moments vanish; 15×15 Hankel from m=2 is nonzero.

New witness (ordinary summaries still collide after pinning the first row empty):

[0,7,5,7]  → rank 0
[0,13,5,7] → rank 1

Independent Grok check (not just their tests)

Spanning-tree winding rank, rectangular periods:

rows	r
[13,5,13,0]	0
[7,5,13,0]	1
[0,7,5,7]	0
[0,13,5,7]	1
[1,5,4,5]	0
[11,14]	2
[0,15,0]	1

Independent 4×m census:

• 4×2: s_2=-46
• 4×3: s_3=-1464
• 4×4: n0/n1/n2 = 36559 / 19932 / 9045, s_4=-27514, Bernstein P2-P0 matches the known square rung.

These equal first_31_s_m[1], [2], [3]. The claimed P_{15} is exactly (x-2)(x+1)f_2 f_{3a} f_{3b} f_5. Recurrence holds on the published integer sequence. Independent Bareiss determinant of the 15×15 Hankel starting at m=2 matches the JSON and is nonzero. gcd(P,P')=1.

Seven new tests passed after git apply on current main (0.29s). Full-repo CI was not run for this patch. #707 CI 34689086221 is a different check.

Out of this PR

• Do not re-dispatch the same [P1 reserve] Source-visible topology and all-width closure beyond the completed width-four laboratory #636 sufficiency task.
• Next bounded [P1 reserve] Source-visible topology and all-width closure beyond the completed width-four laboratory #636 question (already written in notes/issue-636-width4-completion-and-next-20260912.md): visible spectrum of this operator over Q(p) for P_0, P_2, and P_2-P_0. Not another engine, width scan, GPU job, or histogram.
• [P1 theory-blocked flagship] Original U: two same-source forward maps before further acquisition #275 original-U remains separate.

No docs/STATUS.md edit.

[LightChainr]

Do not merge. Independent Grok check is in the PR body: spanning-tree ranks of the old and new witnesses; 4×2/3/4 census (s_2=-46, s_3=-1464, s_4=-27514, 4×4 ranks 36559/19932/9045); P_{15} factorisation; Bareiss Hankel 15×15 from m=2 matches the JSON and is nonzero; gcd(P,P')=1. Seven tests passed after apply on main@eb89e942. Full-repo CI not run for this patch.

[LightChainr]

Follow-on stacked draft #710 uses this PR’s certificate unchanged (blob match). It lifts the p=1/2 order-15 scalar to all-p identities. Do not merge either PR.