PUH Theorem 329 (The Lattice Does Not Reproduce T301) — Integrated Solutions Give |G|/√h ∝ r^(−0.59) Where the Static Field Equation Requires r^(−2), a 56 Percent Residual Against Invariants Conserved to 10⁻¹⁰; and T327's First Integral Is Spherical-Only

Photonic Universe Hypothesis (PUH) — Negative Result and Correction. THE QUESTION. T326 varied T175's functional as a field theory, obtaining ∇²φ^A = −λG^A with the constraint holding everywhere and λ = Q/|G|² slaved to the field. T327 reduced that system in spherical symmetry, proving |∇φ|² = C/r⁴ exactly and showing T301's field equation holds IF AND ONLY IF |G|/√h = (a + b/r)/r², with h = t^A t^B H_AB on the unit tangent. T327 named the evaluation of |G| and h as its successor. THIS NOTE PERFORMS IT. THE MACHINERY, AND ITS VERIFICATION. One archive paper records that the higher Casimir invariants have NO SIMPLE CLOSED FORM as invariant polynomials and that computing with them is research-level work. But the archive also supplies a working basis whose validity is PROVED rather than assumed: the root power sums I_d(μ) = Σα>0(α,μ)^d at the eight fundamental degrees, with Jacobian rank 8 established at a generic point. Both statements are true and do not conflict — no closed form as invariant polynomials, but directly computable as root sums. VERIFICATION: building these sums and their derivatives reproduces the archive's independently computed gradient coefficients at the highest root INTEGER-EXACTLY FOR ALL EIGHT DEGREES — 60; 1,248; 24,912; 115,080; 2,359,800; 10,486,320; 201,327,264; 16,106,128,200 — across ten orders of magnitude, and the quadratic case independently returns the dual-Coxeter identity I₂(θ) = 60 = 2h∨ with h∨ = 30. THE INTEGRATION, AND TWO CHECKS IT WAS NOT TOLD ABOUT. The system was integrated outward from θ as sixteen coupled first-order equations in (φ, φ′), with couplings in T238's pattern ζ_k = S·d_k/I_k^max and S = 9/128 from the degree sum. Two quantities the integrator had no knowledge of were monitored: the constraint g(φ) = 0, and the product r⁴|φ′|², which T327 proved constant. RESULTS: constraint drift 0 at r = 1, −6.4×10⁻¹¹ at r = 2, −8.8×10⁻¹¹ at r = 5, −1.4×10⁻¹⁰ at r = 12; orthogonality |G·φ′| at 10⁻¹¹ throughout; and r⁴|φ′|² = 1.000000 at every radius. T327'S FIRST INTEGRAL IS THEREBY CONFIRMED NUMERICALLY TO SIX FIGURES ACROSS A TWELVEFOLD RANGE, by a solver that knew only the second-order equation. THEOREM 329.1 (the negative result). Along integrated solutions, |G|/√h does not take the required form. METHOD: (a + b/r)/r² is equivalent to (|G|/√h)·r² being AFFINE in 1/r — a linear fit with no freedom, whose residual measures the failure directly. Three independent random tangent directions at θ: seed 11 gives a = 1.3159×10³, b = −2.5492×10³, residual 5.53×10⁻¹; seed 3 gives 1.4118×10³, −2.7388×10³, 5.59×10⁻¹; seed 7 gives 1.4934×10³, −2.9030×10³, 5.62×10⁻¹. THE RESIDUAL IS 56 PERCENT, reproduced to within half a percent across all three. Against an integration whose own invariants hold to 10⁻¹⁰, THE FAILURE EXCEEDS NUMERICAL ERROR BY TEN ORDERS OF MAGNITUDE. ∎ AND THE SHAPE IS INFORMATIVE: fitting a free power over the outer decade gives |G|/√h ∝ r^(−0.5871) against the required r^(−2) — NOT A DIFFERENT COEFFICIENT BUT A DIFFERENT POWER, and a far shallower one. WHAT THIS MEANS AND WHAT IT DOES NOT. T301 obtained its equation from general relativity — the Schwarzschild slice, on which the relevant Laplacian vanishes identically, combined with T299's coordinate-invariant tension identification — and its own NOT CLAIMED records that nothing in it is derived from the algebra. THIS NOTE DOES NOT CONTRADICT THAT PAPER'S ACCOUNT OF ITSELF; IT ESTABLISHES THAT THE ALGEBRA DISAGREES. T326 named this as the alternative outcome to convergence, and it is arguably more informative: agreement would show two derivations meeting, while disagreement LOCATES WHERE THE FRAMEWORK AND GENERAL RELATIVITY PART COMPANY — specifically, in the radial falloff of the constraint geometry, r^(−0.59) against r^(−2). THREE CAVEATS, AND THEY ARE REAL: the solutions start at θ with random tangent directions, and the physical solution is selected by boundary conditions not imposed here — three seeds agreeing to two decimals weakens that escape without closing it; the reference configuration for the couplings is θ, which is a choice; and T327's second kill-condition applies throughout, since the reduction fails where G vanishes, which is the core. RESULT 329.2 (a correction carried in the same note). T327's kill-condition (i) states that if the core carries the circulation sector — intrinsically axial — the reduction does not apply. BUT ITS ABSTRACT ASSERTS MORE: that the law "holds for every core the framework describes, whatever the algebra turns out to do." THAT PHRASING IS TOO BROAD AND IS NARROWED HERE. In spherical symmetry ∇²φ = φ″ + (2/r)φ′, and contracting with φ′ annihilates the right side by orthogonality, giving u′ = −(4/r)u. UNDER ROTATION the Laplacian acquires ∇²φ = φ_rr + (2/r)φ_r + (1/r²)[φ_θθ + cotθ·φ_θ], and although orthogonality holds in BOTH directions, the terms φ_r·φ_θθ and φ_r·φ_θ are inner products between DIFFERENT derivative directions which the constraint does not annihilate. The derivation gains a source and the inverse-fourth-power law FAILS. The law is a theorem of spherical symmetry, not of the framework. THIS DOES NOT RESCUE T301: that equation is the static one, obtained from a non-rotating slice, so testing it against a spherically symmetric lattice solution is the correct comparison. WHAT ROTATION CHANGES IS RELEVANCE, NOT VERDICT — if real cores rotate, as T136 and T300 hold, both T301 and this test may describe an idealisation. The successor is therefore T302's rotating equation, not a re-test of the static one. KILL-CONDITIONS: (i) if the physical solution is selected by boundary conditions picking out a curve unlike the three tested, the exponent could differ — the principal escape, not closed, though three directions agreeing to two decimals constrains it; (ii) if the reference configuration for the couplings is not θ, the whole evaluation must be redone; (iii) if the root power sums are not a valid Casimir basis at the configurations traversed — the archive proves rank 8 at a generic point, not everywhere — the constraint surface computed here is not the framework's; (iv) if the physical regime is small-radius, T327's second kill-condition says the reduction fails there and this test is silent. NOT CLAIMED: that T301 is wrong — it was derived from general relativity and is correct there; what is shown is that the lattice does not reproduce it; that the framework is refuted, since the disagreement may reflect the idealisation of spherical symmetry; that the rotating case has been tested, which needs a PDE on a 2D grid and the circulation sector parametrised; that the physical solution has been identified; or that −0.59 is exact, being a fit over a finite range.