PUH Theorem 331 (The Rank Drop Is Forbidden, and the Escape Is Shut) — All E8 Casimir Degrees Are Even and All Couplings Positive, So the Hessian Is Bounded Below by ζ₁·60·I; and T175's Potential Contributes EXACTLY ZERO Because It Does Not Depend on φ at All

Photonic Universe Hypothesis (PUH) — Obstruction. HOW THIS AROSE. T326 varied the core-formation functional as a field theory, T327 reduced it in spherical symmetry, and T329 integrated it outward. THOSE INTEGRATIONS BEGAN AT THE HIGHEST ROOT θ WITH ARBITRARY TANGENT DIRECTIONS — a regular configuration where the constraint gradient is large and the Hessian well-conditioned. THERE IS NO CORE ANYWHERE IN THAT CALCULATION. Asked whether the profile had been built with the core in mind, the answer was no, and looking for the correct inner boundary condition produced this note. WHAT THE CORE IS SUPPOSED TO BE. T175 is explicit, verbatim: "The Hessian of the Lagrangian at φ* satisfies rank(H(L)(φ*)) = 8 − 8 = 0. Each of the 8 … rank drops from 8 to 0. Zero degrees of freedom. The field is frozen. This is the Planck Core." T325 established that this is the Hessian of the LAGRANGIAN and not the Poisson tensor — the two had been conflated. So the core is, by definition, the configuration on the constraint surface at which the Hessian is maximally degenerate, which makes "where is the core?" a well-posed computational question. THE SEARCH. Minimising the smallest Hessian eigenvalue over the constraint surface from fourteen independent starting configurations bottomed out at 6.95×10⁻² against a largest eigenvalue of 9.99×10², with zero near-null directions and a constraint residual of −2.6×10⁻⁷. A SEARCH PROVES NOTHING ON ITS OWN — the surface is seven-dimensional and the sampling local. What follows is an argument, and it explains the search result rather than resting on it. THEOREM 331.1. On T175's constraint surface, with the root power sums as invariants and T238's coupling pattern, H ≥ ζ₁·60·I as an operator, everywhere; the rank drop 8 → 0 is unattainable. PROOF: for the root power sums, ∂²I_d/∂μ² = Σα>0 d(d−1)(α,μ)^(d−2)·α⊗α. EACH OUTER PRODUCT α⊗α IS POSITIVE SEMI-DEFINITE, so the sign of every term is the sign of (α,μ)^(d−2). EVERY CASIMIR DEGREE OF E8 IS EVEN — 2, 8, 12, 14, 18, 20, 24, 30 — so every exponent (d−2) is even, every power non-negative, EVERY TERM POSITIVE SEMI-DEFINITE. And every coupling is positive: T238 gives ζ_k = S·d_k/I_k^max with S, d_k, I_k^max all positive, computed as running from 2.34×10⁻³ down to 1.97×10⁻⁹, all strictly positive. So H is a positive combination of positive semi-definite operators. AND THE QUADRATIC TERM IS SPECIAL: for d = 2 the exponent is zero, so the sum reduces to the second moment of the root system, which T320 proved equals 60·I EXACTLY — verified here as an operator, eight eigenvalues all exactly 60.000000. That term is a CONSTANT positive multiple of the identity, independent of configuration, and nothing in the remaining sum can cancel it because nothing in the remaining sum is negative. Hence H ≥ ζ₁·60·I everywhere. ∎ The search could never have gone below that floor, and it did not. THEOREM 331.2 (the potential contributes exactly zero — the escape is shut). The obvious escape is that T175's Hessian is of the FULL Lagrangian L = T − V − λ·g, and that V might contribute a negative-definite term cancelling the floor. T175 §3.1 writes V(φ) = −∫d³x GMρ_P/|x − x₀|. LOOK AT WHAT IT DEPENDS ON: position x, the source mass M, and the Planck density ρ_P. THE FIELD φ DOES NOT APPEAR IN IT ANYWHERE. So ∂²V/∂φ² = 0 identically — V is not a potential energy function of the field but an EXTERNALLY SPECIFIED GRAVITATIONAL POTENTIAL, a source term depending on position only. ∎ The same applies to the kinetic term: T(φ) = (1/2)∫d³x|∂_μφ^A|² depends on the field's GRADIENT, not the field, so its Hessian in φ also vanishes. THEREFORE H(L) = −λ·H(g), and Theorem 331.1's bound applies to the full Lagrangian's Hessian directly, with |H(L)| ≥ |λ|·ζ₁·60 in every direction. THE RANK DROP THEREFORE REQUIRES λ = 0 — which is the unconstrained problem T175 itself states has no minimum at finite φ. THE ESCAPE IS SHUT FOR A REASON SIMPLER THAN A CANCELLATION FAILING: THERE WAS NEVER ANYTHING TO CANCEL WITH. A Lagrangian whose only φ-dependence is a gradient term and a constraint cannot have a degenerate Hessian in φ unless the multiplier vanishes. The structure forbids it before any numbers are computed. WHAT THIS DOES AND DOES NOT ESTABLISH. IT DOES NOT SHOW THE FRAMEWORK IS WRONG. It shows that a defining property stated in a foundational paper is inconsistent with the Lagrangian that same paper writes down, and the inconsistency is structural. TWO ESCAPES REMAIN, NAMED RATHER THAN DISMISSED. (a) The invariants may not be the root power sums — that basis is proved valid (Jacobian rank 8 at a generic point) but validity is not uniqueness; a basis with odd-degree elements would break the evenness argument, though E8 has no odd Casimir degrees. (b) The couplings may not all be positive — T238's pattern makes them so, but was derived at saturation under a closure condition; a negative coupling would permit cancellation. WHAT WOULD RESOLVE IT IS A GENUINE SELF-INTERACTION V(φ), WHICH THE ARCHIVE HAS NEVER WRITTEN DOWN. A CORRECTION RECORDED. The first version of this argument computed the degree-2 Hessian in SIMPLE-ROOT COORDINATES and compared its eigenvalues against T320's result, which is proved in an ORTHONORMAL FRAME. The coordinate matrix has eigenvalues spread from 0.66 to 239, none equal, and quoting the smallest as "the floor" was meaningless. Raising the index correctly — forming A⁻¹H rather than H — returns exactly 60·I, as T320 requires. The conclusion is unchanged; the numbers in the first pass were not. A matrix and an operator are different objects, and eigenvalues belong to the second. KILL-CONDITIONS: (i) if T175's Lagrangian is not the one written in its §3.1 — if some other potential, depending genuinely on φ, is intended — Theorem 331.2 does not apply and the escape reopens; the framework would then need a self-interaction it has never written down; (ii) if T175's invariants are not the root power sums, the evenness argument does not apply; (iii) if any coupling is negative, cancellation is possible; (iv) if the saturation reference is not θ, the couplings change and the floor moves, though positivity would survive. NOT CLAIMED: that T175 is wrong — Section 5 names an escape leaving it intact; that the Planck core does not exist, since its existence rests on more than this characterisation; that a self-interaction potential is impossible, only that T175 does not contain one; that the root power sums are the unique valid basis; or that the earlier tension-profile results are affected, since they concern the constraint gradient rather than the Hessian's degeneracy.