PUH Theorem 294 (The Keystone by Bounds) — Exact Casimir Gradients at the Snap Ray, grad I_d(θ) = d(28+2^(d-1))θ; θ the Deepest Degeneration (Order 63); the Stiffness Hierarchy; the σ_n Suppression Factor √(8/248); and the First Bound on R_sub

Photonic Universe Hypothesis (PUH) — Bounds and Structure. THE KEYSTONE. Four open items converge on the substrate equation of state: R_sub (T213/T287), the Snap threshold (T288/T175), sigma_n (T159/T293), and the dispersion relations (T292's sequel). The EoS is gated on closed forms for the degree-8..30 Casimir invariants of E8 — unavailable. The attack is oblique: extract what homogeneity, degeneration theory (Jacobian of basic invariants ∝ product of positive roots; Steinberg/Humphreys), and T292's 240/8 mode split give WITHOUT closed forms. Working basis: root power sums I_d(mu) = sum over positive roots of (alpha,mu)^d at d in {2,8,12,14,18,20,24,30} — validity PROVED, not assumed: exact Jacobian rank 8 at a verified-regular integer point. THE NUMERICAL HAZARD, CAUGHT IN-RUN. The first floating-point run produced a SELF-CONTRADICTORY output: all eight gradients verified exactly parallel to theta while the rank routine printed 2, and the basis check printed rank 7 at a generic point. The contradiction was the alarm. Diagnosis: gradient rows span ~forty decades (grad I_2 ~ 1e3 vs grad I_30 ~ 1e43); small rows sit below the float-error floor of large ones. Resolution: exact rational arithmetic (big-integer Gaussian elimination, no tolerances) — generic rank 8 EXACT (basis valid); theta rank 1 EXACT; subregular wall rank 7 EXACT. Every float anomaly an artifact; every theoretical expectation exact. Principle filed: a self-contradictory diagnostic is itself the alarm; where magnitudes span decades, rank is a question for exact arithmetic. THEOREM 294.1 (CLOSED FORMS AT THE SNAP RAY): grad I_d(theta) = d·(28 + 2^(d-1))·theta for ALL EIGHT degrees including 30 — exactly where closed forms were "unavailable," at theta they are trivial. Proof: the 63 charge-0 roots contribute nothing; the 56 charge-1 positive roots sum to 28·theta (W(E7)-invariance; cross-check (2rho,theta) = 58 verified); the theta pair contributes 2^(d-1)·theta. Verified integer-exactly: 60; 1,248; 24,912; 115,080; 2,359,800; 10,486,320; 201,327,264; 16,106,128,200 (d=2 independently reproduces the dual-Coxeter identity, 2h∨ = 60). COROLLARY (TENSION COLLAPSE): at theta the eight-gradient system has EXACT rank 1 — the eight independent Casimir tensions collapse to ONE effective tension along theta; T175's rank-drop in the constraint gradients. HONEST SCOPE, in full: by equivariance the rank-1 collapse occurs at EVERY fundamental ray (each gradient is fixed by the ray's Weyl stabilizer, whose fixed space is the ray; verified at all eight); theta-specific are the closed form (the {0,±1,±2} charge structure exists only relative to theta) and the DEPTH. THEOREM 294.2 (THETA IS THE DEEPEST DEGENERATION; THE THIRD CONVERGENCE): vanishing orders of det(Jacobian) on the eight fundamental rays = {14,16,22,23,28,37,42,63}, each equal to |Delta+| of the corresponding maximal Levi (A2×A1×A4, A4×A3, A1×A6, D5×A2, A7, E6×A1, D7, E7) — all eight verified. MAXIMUM 63 = |Delta+(E7)| on the theta ray, half again deeper than the runner-up (42, D7), with the largest stabilizer |W(E7)×W(A1)| = 5,806,080. Third independent convergence on theta: T290 (bracket connectivity), T292 (coadjoint geometry), now invariant-theoretic degeneration. THE PHOTON DIRECTION IS THE COLLAPSE CHANNEL OF THE CASIMIR GEOMETRY — the group-theoretic ground of T288's memory leg: why photons, specifically, fire the Snap. THEOREM 294.3 (STIFFNESS HIERARCHY; bound-level): along any ray with quadratic H, lambda_k ∝ s^(2-d_k); exponents 0, -6, -10, -12, -16, -18, -22, -28 — the degree-30 tension diverges fastest approaching saturation: THE STIFFEST CONSTRAINT BINDS FIRST. Variational RATIONALE for T156's Shell = C_30 identification (previously an identification without a reason), and the EoS two-regime skeleton: weak field C_2-dominated (elastic/GR, T193), near-saturation C_30-dominated (Snap). Tagged: ray-restricted homogeneity estimate under quadratic-H, a rationale not a solved constrained problem. RESULT 294.4 (SIGMA_N SUPPRESSION): with H ∝ C_2 weak-field, the 240 tangent modes are iso-energetic (C_2 constant on leaves); only the 8 transverse modes carry density. Equipartition: sigma_n = sqrt(8/248)·sigma_lattice = 0.1796·sigma_lattice. T159/T293's sigma_n = 0.05 requires only sigma_lattice ≈ 0.28 — moderate disorder. Assumption DOWNGRADED, not removed; any future sigma_lattice derivation converts directly. RESULT 294.5 (FIRST BOUND ON R_SUB): Channel B dm/dt = k m^2, k = 16 pi G^2 R_sub,eff/c^4; Earth's core (5e21 kg, terrestrial anchor) persisting 4.5 Gyr forces k < 1.41e-39 /kg/s, hence R_sub,eff < 5.1e13 kg m^-2 s^-1 — NINETY-ONE DECADES below the causal Planck flux rho_P c ≈ 1.5e105. The first number ever attached to the substrate throughput; efficiency folded in and stated; upper bound only. Falsifiable context: demonstrated Earth-core growth above the ceiling breaks the Channel-B/terrestrial-core coexistence. LEDGER. Delivered: validated basis; closed forms at the Snap ray with the tension collapse; the deepest-degeneration theorem and third convergence; the stiffness hierarchy and Shell rationale; the sigma_n factor; the R_sub ceiling — five downstream papers (T156, T287, T288, T292, T293) each received something they lacked. Still gated, without euphemism: F(C_2..C_30) itself; sigma_lattice; numeric lambda*; the dispersion relations; any R_sub floor. KILL-CONDITIONS: Shell at any level set other than the fastest-diverging Casimir breaks 294.3's rationale; measured sigma_n requiring sigma_lattice > 1 breaks 294.4; Earth-core growth above the ceiling breaks 294.5; the theta-convergence is retroactively falsifiable through any of its three routes. NOT CLAIMED: the EoS functional form; uniqueness of the rank-1 collapse to theta (stated, not buried); sigma_lattice; numeric lambda*; dispersion; an R_sub floor.