Photonic Universe Hypothesis (PUH) — Bounds and Structure. THE KEYSTONE. Four open items converge on the substrate equation of state: Rₛub (T213/T287), the Snap threshold (T288/T175), sigmaₙ (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 Id (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 I₂ ~ 1e3 vs grad I₃0 ~ 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 Id (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, lambdaₖ ∝ s^ (2-dₖ) ; 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 = C₃0 identification (previously an identification without a reason), and the EoS two-regime skeleton: weak field C₂-dominated (elastic/GR, T193), near-saturation C₃0-dominated (Snap). Tagged: ray-restricted homogeneity estimate under quadratic-H, a rationale not a solved constrained problem. RESULT 294. 4 (SIGMAN SUPPRESSION): with H ∝ C₂ weak-field, the 240 tangent modes are iso-energetic (C₂ constant on leaves) ; only the 8 transverse modes carry density. Equipartition: sigmaₙ = sqrt (8/248) ·sigmaₗattice = 0. 1796·sigmaₗattice. T159/T293's sigmaₙ = 0. 05 requires only sigmaₗattice ≈ 0. 28 — moderate disorder. Assumption DOWNGRADED, not removed; any future sigmaₗattice derivation converts directly. RESULT 294. 5 (FIRST BOUND ON RSUB): Channel B dm/dt = k m², k = 16 pi G² Rₛub, eff/c⁴; Earth's core (5e21 kg, terrestrial anchor) persisting 4. 5 Gyr forces k 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) ; sigmaₗattice; numeric lambda*; dispersion; an Rₛub floor.
Brian Martell (Fri,) studied this question.