PUH Theorem 377 — Where Planck Cores Live: The Invariants Must Be Positive on Clocks, Every Core Has a Rest Frame, and at Rest the Massive States Meet the Casimir Wall on a Closed Shell

Photonic Universe Hypothesis (PUH) — Theorem paper. THE QUESTION. T175 built its Casimir constraint, sum of zeta_k I_k = 9E_P, as a wall that stops compression — "the absolute structural limit of the lattice". T342 (Theorem 342.1) and T331 rest on every invariant being non-negative, so that the wall is a closed, bounded surface. In the folding form E8(−24) the invariants change sign from one kind of state to another (T373, OPEN 3), and T376 flagged that the surface is not bounded. Every number in T377 is computed on the E8(−24) build of T373, with T376's positive metric and T238's couplings. RESULT 377.1 — THE SIGN IS FORCED. Each invariant can be written as tr(ad(x)^d) or with the factor (−1)^(d/2). T172's "the Casimir value is the mass squared" (as T314 reads it) and T374's finding that a massive pair is a clock require the sign that makes the invariants positive on clocks: T374's opposite-pair clock then has I'_2 = +480 with all eight invariants positive. This is also exactly T342's formula on massive states (to 3.6 × 10⁻¹⁶). With the other sign no pure clock ever meets the wall. RESULT 377.2 — EVERY ORBIT IS UNBOUNDED. No nonzero state commutes with all of the (56,2), and boosts have real eigenvalues: a wall state boosted three units carries 256 times its energy with every invariant unchanged. T175's surface is never bounded as a set; T342's compactness argument fails as stated. RESULT 377.3 — EVERY CORE HAS A REST FRAME. By the Kempf–Ness theorem (Richardson–Slodowy for real groups), every semisimple state has a state of least energy with its invariants, unique up to E7 × SU(2) rotations, characterised by [x, θx] = 0 — exactly the states that do not precess under T376's dynamics. A seed, being nilpotent, has none. RESULT 377.4 — WHERE CORES LIVE. On the rest-frame slices (m = 0 to 4 descent directions, the real rank being 4): massive states meet the wall exactly once in every direction, on a closed shell of radius 2.002 to 2.288 — T342's ellipsoid, turned by E7 × SU(2) (2.109 to 2.498 with sums over positive roots); pure descent states never meet it; mixtures meet it in 56 to 95 percent of directions and at every size (both signs on spheres out to radius 100). The other sign mirrors all of this. RESULT 377.5 — WHAT SURVIVES. T331's Hessian bound holds on the massive shell (smallest eigenvalue 3.51 above 1.875) and fails at every mixed wall state sampled (239 of 239). T342's ratio |h|/|G|^2, measured with T376's metric in all 248 directions, is at most 0.060 on the massive shell and 0.17 on mixtures at rest, but boosts carry it to 2.7 (15 of 32 random boosts rise above the rest value). Theorem 342.1 holds at rest; for moving states it is unproven. RESULT 377.6 — THE WALL STOPS MASSIVE STATES ONLY. T175's structural limit holds for massive states; along pure descent the constraint never reaches the threshold, and mixtures are walled only at sizes that grow without bound. CORRECTIONS: T376 Section 9 printed the invariants with the other sign — the roles of massive and descent states are reversed, and the unboundedness is wider than stated. T373's OPEN (3) is settled: elliptic (massive) states meet the wall, every direction once. WHAT CHANGES: T373 OPEN (3) SETTLED; T376 Section 9 CORRECTED AND WIDENED; T374 sign CONFIRMED; T175 wall NARROWED to massive states; T342 Theorem 342.1 NARROWED (holds at rest; its exclusion of λ as gravity stands at rest); T331 NARROWED; T332 UNTOUCHED; T238, T339, T314 USED. KILL-CONDITIONS: if PUH's invariants are meant with the other sign; if T238's couplings are replaced (radii and shares change; the closed massive shell and unbounded mixtures need only positive couplings); if the substrate's metric is not T376's; if core states are not semisimple. NOT CLAIMED: that static cores are at rest; that T342's ratio is unbounded on moving states; that a rank drop occurs on mixtures; that the descent is physically unstopped. OPEN: whether a static core's field stays at rest; whether T342's ratio is bounded over moving states; what ends a descent; whether mixtures can host a rank drop.