PUH Theorem 376 — The Substrate's Energy Is Positive: The Killing Form Cannot Be Its Metric, the Cascade Descends Only Because the Metric Is Positive, and the Field's Kinetic Symmetry Is E7 × SU(2)

Photonic Universe Hypothesis (PUH) — Theorem paper. THE CONFLICT. T264 makes the Killing form the metric of the substrate's configuration space, and T320 builds the bond energy between neighbouring cells from it ("the only available inner product is the Killing form"). T175's kinetic term is a sum over all 248 components, and T338 and T350 rest on the energy never being negative. In the folding form E8(−24) (T265) the Killing form is positive on 112 directions and negative on 136, so these cannot all hold. T374 (Result 374.7) found the bond between two massless seeds negative and left the question open (OPEN 2). Every number in T376 is computed on the E8(−24) build of T373. RESULT 376.1 — NO POSITIVE FORM HAS ALL OF E8(−24). The only quadratic form invariant under the whole algebra is the Killing form (one solution; singular values 1.2 × 10⁻¹² against 1.2 × 10³), with 112 positive and 136 negative directions. A quadratic energy both E8(−24)-invariant and never negative does not exist; only the compact form allows one. RESULT 376.2 — THE KILLING FORM FAILS ON EVERY READING. With T175's positive kinetic term and T320's Killing bond, 136 directions grow instead of oscillating (a ripple of one metre e-folds in 5.31 × 10⁻¹⁰ s; of ten Planck lengths in 8.58 × 10⁻⁴⁴ s). With the Killing form in both terms, 136 directions carry negative energy, and a texture in which every cell keeps every invariant has a bond energy with no floor (−0.128, −0.543, −2.762, −9.068 per bond at boosts of 0.5, 1, 2, 3 per cell). RESULT 376.3 — DESCENT NEEDS A POSITIVE METRIC. T247 writes the cascade as a gradient flow with mobility M ≥ 0; in many dimensions dE/dt = −∇E·G⁻¹∇E. With the Killing metric the energy climbs whenever the gradient weighs more in the 136 compact directions: 86% of random directions (exactly 86.04%; 86.1% of 200,000 draws). T264's argument for an indefinite metric is inverted. The freeze exponents of T251, T295 and T296 use the invariants and the flat volume measure, which is unchanged, so they stand. RESULT 376.4 — THE POSITIVE TWIN. The substrate's metric is K_θ(x, y) = −K(x, θy), with θ the Cartan involution: positive in all 248 directions (eigenvalues 60 to 120; 60 per unit Cartan length, T320's 60), with the Killing form's volume measure (log |det| = 1181.752775 for both), invariant under exactly E7 × SU(2) (136 generators), and equal to the archive's own kappa_ab = 60 delta_ab of T189 and T192 (60 × identity to 1.4 × 10⁻¹⁴ in the Chevalley-normalised basis, where the Killing form is ±60). It keeps T320's 60, isotropy and dispersion, T337's c² = 60J/ρ, T323–T324 and T372, and makes T338's sum of squares exact. RESULT 376.5 — A CELL NOW MOVES. Under the Killing energy (the quadratic Casimir) a cell's Lie–Poisson flow is zero. Under K_θ the compact part is conserved and the noncompact part precesses about it (closed form matched to 3.7 × 10⁻¹⁵; energy and invariants conserved). A lone massless seed turns into T374's opposite seed at t = π/4 and back at π/2; its generator is exactly T374's opposite-pair total, at rate 4 = √(240/15) — the rate of T374's first clock. RESULT 376.6 — E8(−24) KEPT, FOR BETTER REASONS. The compact form (Killing form negative on all 248 directions) is still excluded — no massless state, no light cone, every adjoint eigenvalue imaginary, so no descent direction, no hyperbolic plane (T302) and no gravitational sl(2,ℝ) (T303–T304) — rather than by any argument about the metric. T295, T296 and T353 cite T264 only for this exclusion. RESULT 376.7 — WHAT POSITIVITY COSTS. The 112 noncompact directions are no longer symmetries of the energy: a seed's energy becomes e^(4s) × 240 under a boost (0.081 to 13,103.556 for s = −2 to +1) while its mass stays zero. T374's Results 374.2–374.6 stand; 374.1 holds for invariant observations; 374.7 holds only for the Killing metric — the positive bonds (+240, +180, +240, +300) change under a common boost that leaves the relational number fixed (240 → 6,554.0 at s = ±1). T320's "very nearly forced" becomes a minimal choice, and T321's uniqueness of the quartic term becomes conditional. THE ALTERNATIVE, named and left open: a field on the coset E8(−24)/(E7 × SU(2)), whose invariant metric is positive and unique up to scale, keeps all of E8(−24) at the price of rebuilding T175's 248-component field. A FLAG. In E8(−24) the invariants change sign on elliptic states (− + + − − + + − for degrees 2 to 30 along a compact Cartan direction), so the Casimir constraint surface is not bounded; T342's compactness holds only for states with real pairings (T373 OPEN 3). WHAT CHANGES: T374 OPEN (2) SETTLED; T264 ARGUMENT REPLACED, CONCLUSION KEPT; T265 RE-GROUNDED; T320 CORRECTED; T175 CONFIRMED; T338/T350 MADE EXACT; T247 CONFIRMED; T251/T295/T296/T353 and T337/T323/T324/T372 UNTOUCHED; T321 CONDITIONAL; T189/T192 IDENTIFIED; T342/T331 FLAGGED; T172 EXTENDED. KILL-CONDITIONS: if PUH intends the Killing form as the energy metric and accepts negative energy (then T338 and T350 fall); if the energy must keep the full E8(−24) symmetry (then the coset is required); if the substrate's field is not valued in the algebra. NOT CLAIMED: that the coset alternative is excluded; that the full lattice equations have been solved; that the precession is observable; any new constant. OPEN: K_θ or the coset; the quartic term under E7 × SU(2); the unbounded constraint surface; seeds and pairs under the full lattice equations.