PUH Theorem 320 (The Missing Coupling Term) — Symmetry Forces the Nearest-Neighbour Energy; the E8 Lattice Is Isotropic to SIXTH Order Because It Has No Degree-4 or Degree-6 Casimir; and Directional Lorentz Violation Falls to (a/λ)⁶

Abstract

Photonic Universe Hypothesis (PUH) — Postulate and Consequences. THE GAP IS SINGULAR, NOT PLURAL. T298's audit recorded that T175's Lagrangian contains NO SPATIAL DERIVATIVES — no gradient term, no dependence on x — and is constrained optimization at a point in the 248-dimensional algebra. That was filed as a gap in one paper. It is better understood as THE gap: a lattice theory that has never written down what it costs for two neighbouring cells to differ has no local dynamics, and SIX separate open problems trace to that one absence. (i) The wave equation is assumed, not derived — T317's chain begins at the wave equation rather than the lattice. (ii) The elastic constants do not exist. (iii) T317's Section 6 dispersion is a placeholder borrowed from solid-state physics and labelled as such. (iv) The Shell's reflectivity cannot be computed, and since T307 and T311 governs two observational channels. (v) The block scale fixing the void size has no derivation. (vi) The planetary convergence mechanism requires lattice-specific dynamics that T318 and T319 showed no borrowed mechanism supplies. ONE ABSENCE, SIX SYMPTOMS. THE TERM IS VERY NEARLY FORCED. Three requirements pin it down: it must be invariant under the algebra (so the Killing form is the only available inner product); it must vanish when neighbours agree (so it depends on their difference); it must be a scalar (hence quadratic at leading order). With φ(x) the algebra-valued field and the 240 neighbours at the root vectors α: E_bond = (J/2)·Σ_x Σ_α ⟨φ(x) − φ(x+α), φ(x) − φ(x+α)⟩. ONE FREE PARAMETER, J. The archive already supplies the geometry — 240 nearest neighbours, the kissing number, each pair sharing a seven-dimensional Voronoi facet. THIS IS A POSTULATE AND IS STATED AS ONE: symmetry forces the form; writing it is a physical decision. THEOREM 320.1 (isotropy at leading order, proved in full). Expanding φ(x) − φ(x+α) ≈ −α·∇φ gives Σ_α|φ(x)−φ(x+α)|² ≈ ∇φ·(Σ_α α⊗α)·∇φ, so everything turns on M_ij = Σ_α α_i α_j. If M were not proportional to the identity the medium would have preferred directions and no relativistic limit. Evaluated in an orthonormal frame from the Cholesky factorisation of the Cartan matrix: M = 60·I₈, exactly, to 10⁻¹⁰. TWO INDEPENDENT CONFIRMATIONS: trace M = 240 roots × |α|² = 480, and 480/8 = 60; and 60 = 2×30 is twice the dual Coxeter number, the SAME 60 appearing in K(h_i,h_j) = 60·A_ij, because both are the same root-system sum. The continuum energy density is (J/2)·60·|∇φ|² — an ordinary isotropic gradient term with the geometry contributing exactly one number. THEOREM 320.2 (isotropy survives to SIXTH order). Moments of order 2n compared against the isotropic form, whose ratios are fixed combinatorially: ORDER 2 — Σα₁² = 60, Σα₁α₂ = 0 ✓. ORDER 4 — Σα₁⁴ = 36, Σα₁²α₂² = 12 (isotropic 12 ✓), Σα₁α₂α₃α₄ = 0 ✓. ORDER 6 — Σα₁⁶ = 30, Σα₁⁴α₂² = 6 (isotropic 6 ✓), Σα₁²α₂²α₃² = 2 (isotropic 2 ✓). ORDER 8 — Σα₁⁸ = 39, Σα₁⁴α₂⁴ = 3.000 against isotropic 3.343 ✗, Σα₁⁶α₂² = 3.000 against 5.571 ✗. EXACTLY ISOTROPIC AT 2, 4 AND 6; ANISOTROPIC AT 8. Residuals against the fully isotropic tensor below 10⁻¹². AND THE REASON IS ALGEBRAIC: a rotationally invariant polynomial of degree 2n is built from the basic invariants, and E8's Casimir degrees are 2, 8, 12, 14, 18, 20, 24, 30 — THERE IS NO DEGREE-4 AND NO DEGREE-6 INVARIANT, so at those orders the only invariant is a power of |k|² and anisotropy has nowhere to live. The first degree admitting a second invariant is 8, precisely where the departure appears. THIS WAS PREDICTED FROM THE DEGREE LIST BEFORE THE MOMENTS WERE COMPUTED. For contrast, an ordinary cubic crystal has the degree-4 invariant x⁴+y⁴+z⁴ and is anisotropic at fourth order — where the familiar elastic anisotropy of solids lives. RESULT 320.3 (exact dispersion, direction by direction). ω²(k) = (4J/ρ)·Σ_α sin²(k·α/2), exact. With s(k) = ω/(c|k|), c² = 60J/ρ, evaluated along five inequivalent directions (a root, a sum of two roots, a coordinate axis, the full diagonal, a random direction): at |k| = 0.10 the spread is 3.84×10⁻¹²; at 0.30, 2.79×10⁻⁹ (ratio 728, k⁶ predicts 729); at 0.60, 1.78×10⁻⁷ (ratio 63.7, predicts 64); at 1.00, 3.77×10⁻⁶ (ratio 21.2, predicts 21.4). THE SIXTH-POWER SCALING IS CONFIRMED TO A FRACTION OF A PERCENT, independently of the moment tensors. The two effects separate cleanly: the common departure of s from unity is ISOTROPIC dispersion growing as k², which every lattice has; the disagreement BETWEEN directions is anisotropy growing as k⁶. At |k| = 0.01 the isotropic effect is ~10⁸ times larger. RESULT 320.4 (Lorentz-violation consequence). T317's Section 6 used the standard nearest-neighbour dispersion, stating it was an illustration. Replaced by the framework's own: optical light 1.0×10⁻⁵⁷ → 1.1×10⁻¹⁷¹; gamma ray 2.6×10⁻⁴⁶ → 1.8×10⁻¹³⁷; LHC scale 2.6×10⁻³² → 1.8×10⁻⁹⁵; highest cosmic ray 2.6×10⁻¹⁸ → 1.8×10⁻⁵³. THIRTY-FIVE ORDERS FURTHER SUPPRESSED, and now a prediction rather than a borrowed figure. ONE DISTINCTION MUST BE KEPT: what falls to the sixth power is the DIRECTION-DEPENDENT effect; isotropic dispersion remains at (a/λ)², as in any discrete medium. Since most experimental tests constrain directional dependence, the sixth-power suppression is the relevant one — but the claim should not be overstated into a statement about dispersion generally. AND THE COEFFICIENTS ARE FIXED, NOT FITTED: the axis moments are 60, 36, 30, 39 at orders 2, 4, 6, 8 — pure root-system data. With the single parameter J, every Lorentz-violation coefficient at every order is determined. KILL-CONDITIONS: (i) if directional Lorentz violation is detected exceeding (a/λ)⁶, either the spacing exceeds the Planck length or the coupling is not nearest-neighbour; (ii) if the coupling is anharmonic at leading order, the moment analysis does not apply; (iii) if the field is not algebra-valued, the Killing form is not the correct inner product; (iv) if a degree-4 or degree-6 invariant of E8 is exhibited, Theorem 320.2's explanation fails and the isotropy would be accidental. NOT CLAIMED: that the coupling has been DERIVED (it is a postulate whose form symmetry constrains); that J has been determined or connected to a measured quantity, which needs the field normalisation the archive does not fix; that the field equation follows — varying the full Lagrangian gives a Laplacian equation of the same TYPE as T301's, but deriving that specific form requires eliminating the field in favour of the multiplier, not attempted here and NAMED AS THE SUCCESSOR; that isotropic dispersion is improved, since it is not; or that the six problems of Section 1 are solved, since only the machinery they need has been supplied.