q=5 cubic leader closure, exact decoder radius, and root-Cayley spectral geometry.
For a cut-minimal 18-chamber leader, the sharp adjacent-pair cap is 27. Pairwise Delsarte alone only forces apartment weight 320. The all-q triple law supplies the missing term. If N1 is the number of adjacent selected chamber pairs, selected three-edge Levi paths satisfy P3 ≥ max(0,4(N1−18)). Each such path has gallery signature (1,1,2) and contributes 25 common apartments at q=5. Combining the parity minorant odd(r) ≥ r−2C(r,2)+(6/17)C(r,3) with exact Delsarte maxima for every N1≤27 gives the worst shell at N1=27: 17 wt ≥ 10840, hence wt ≥ 638 > 625. Therefore every sub-625 q=5 word now requires chamber leader at least 19.
For three distinct chamber stars, the common-apartment count is determined entirely by the sorted gallery-distance triangle:
(1,1,2) → q², (1,2,3) → q, (1,3,4) → 1, (2,2,4) → 1, (2,3,3) → 1; every other signature gives zero. Full rooted censuses verify the law at q=2,3,4,5, including GF(4). At q=5 the fixed-base histogram is 25^75, 5^750, 1^7500, 0^428320.
Local K6 cut weights 5,8,9 carry selected theta-triangle counts 10,8,6. Hence T_type=2500−8h8−12h9. The smallest exotic coarea profile (h8,h9)=(2,1) has A_type=248 and T_type=2472, a defect of 28 from the chamber-star value.
Pass5131 proves all errors of weight at most five are corrected by the equivariant max-vote rule. Pass5136 supplies a weight-six counterexample: errors {1,2,3,6,27,54} make false apartment 0 tie at maximum vote 3, so the first correction introduces false bit 0. Thus this decoder's guaranteed radius is exactly five.
The q²-dimensional linear-character sector has eigenvalues 4(q−1), 3q−4, 2q−4 with multiplicities 1, 2(q−1), (q−1)². Full exact spectra are certified for q=2,3,4,5 by integer annihilator polynomials and exact trace moments; all four anchors have spectral gap q. The state/program polynomial compiler is a filtered group-algebra isomorphism preserving the augmentation ideal and every Jennings layer.
The chamber-dependency code is [(q+1)²(q²+1), 2(q+1)(q²+1)−1, q+1]₂ for every finite q, and every minimum word is a point- or line-panel star. The root-direction Cayley metric has diameter at most four; exact odd-prime anchors q=5,7,11,13 obey shell polynomials 1, 4(q−1), 8(q−1)², (q−1)²(10q−21), (q−1)²(q−4)². That polynomial shell family remains conjectural beyond the anchors.
Boundary. The q=5/all-q apartment-code minimum distance is still open for leaders at least 19. Leader 18 is closed by the cubic minorant; higher shells may require fourth and higher intersections. The decoder statement concerns this hard-decision rule only. The full all-q nonlinear theta spectrum and uniform gap-q proof remain open. No controller/compiler statement is a hardware-performance claim.