All-p width-four spectrum and diamond 3L−1 onset (stacked on #708; do not merge)

[LightChainr]

Draft. Do not merge. Stacked on #708 (f782061). Needs that certificate; does not duplicate it. No dependence on #706/#707/#709. Geometry onset is logically independent of #708 but lives in this same patch.

Width four, all p

(1+t)^{4m} P0 = tr(B5^m) - tr(B16^m) + 2 t^{2m}
(1+t)^{4m} P2 = tr(B15^m) - tr(B16^m) + 2 t^{2m}
M_{4,m}(p) = (tr(B15^m) - tr(B5^m)) / (1+t)^{4m}

t=p/(1-p), m≥2. Common strong merge 3→35→94→94 (not a positive-realization minimum). 126 exact polynomial identities + Cayley–Hamilton, not a parameter scan.

channel	generic min order	p=1/2	p=2/3
P0	17
P2	23
M	16	15 (#708)	14

Shared B16 relative decay 0.7739 cancels in M; visible ρ0.25175. Using a probability channel's slowest mode to bound matching-function error is the wrong mode.

q4=0.59141717085313848… = Jacobsen 2015 Table 2 n=4. Not a new pc. All finite p_{4,m}<q4. Fixed-width only; no L^{-4} claim.

Thermal ∂p M(q4) contains m λ_*^m. Semisimple crossing, not a Jordan block. diag(λ+ε,λ-ε) is the toy.

Diamond / axis onsets (#673)

Two-cycle core: wedge or theta, dumbbell excluded. Axis 2L-1 with L^2 crosses. Diamond 3L-1 with 4L^2 minimizers (full 2L-line plus straight L-1 plug). Diamond L=5 is k=14, count 100 by theorem, not 2^50.

Replaces the invalid #692 “shared vertex ⇒ |union|≥2L-1” direction.

Independent Grok check

• Jacobsen CSV n=4 equals q4_diagnostic (abs diff 0 at printed digits).
• At t=1 (p=1/2): tr(B15^m)-tr(B5^m) equals independent census s_m for m=2,3,4 (-46,-1464,-27514).
• p=1/2 Hankel 15×15 determinant equals the Width-four exact rank closure: 509 futures and 15-mode scalar (do not merge) #708 independent Bareiss value.
• Axis L=2,3: 0 below onset, L^2 minimizers. Diamond L=2,3: 0 below onset, 4L^2 minimizers.
• d/dε[(λ+ε)^m-(λ-ε)^m]|_{ε=0}=2mλ^{m-1} (semisimple).
• 10 tests passed after apply on Width-four exact rank closure: 509 futures and 15-mode scalar (do not merge) #708.

Full-repo CI not run. No GPU, no width scan, no new MC.

Do not re-dispatch the same #636 visible-spectrum task or the same #673 diamond census. Next genuinely larger question is all-width observable weights, not more roots. #275 still needs candidate forward maps; this only tightens why mλ^m is not a Jordan test.

[LightChainr]

Do not merge. Stacked on #708. Independent checks in the PR body: q4 = Jacobsen Table 2 n=4; tr(B15^m)-tr(B5^m) at t=1 equals census s_m for m=2,3,4; p=1/2 Hankel matches the #708 Bareiss value; axis/diamond L=2,3 exhaustive counts; semisimple mλ^{m-1} toy. 10 tests passed on the #708 branch. Full-repo CI not run.

[LightChainr]

Follow-on stacked draft #716 uses this PR’s parametric definition unchanged (SHA-1 match). Do not merge either.