PUH Theorem 366 (Corrigendum to T365) — The Substrate Tensor, Written Explicitly, Satisfies ρ + 3p ≥ 0; Λ Is Not in T175, and T178 Says It Need Not Be

Photonic Universe Hypothesis (PUH) — Corrigendum. WHAT IS WITHDRAWN. T365 Result 365.1 — "the cosmological constant is the gradient energy of an incompletely relaxed lattice" — IS WITHDRAWN. T365 Result 365.4 — the prediction of w slightly greater than −1 — IS WITHDRAWN WITH IT. The error was not in the reading of T175 but in FAILING TO COMPUTE THE EQUATION OF STATE. A claim about Λ must be checked against w before anything else, and it was not. RESULT 366.1 (the tensor, written). T364 named the substrate stress-energy tensor as the outstanding work. IT REQUIRES NOTHING THAT WAS MISSING. Two facts collapse it: differentiating the constraint along any field variation gives n·∂_μφ = 0, so EVERY FIELD DERIVATIVE IS TANGENT to the surface and the induced metric G_AB = δ_AB − n_A n_B acts as the identity on exactly the objects the tensor contains; and the constraint VANISHES ON SHELL, so the multiplier term drops. What remains is T^μν = ∂^μφ·∂^νφ − g^μν(½ ∂φ·∂φ). IT DOES NOT REQUIRE THE EIGHT COUPLING CONSTANTS ζ_k, which T175 leaves as "coupling constants with appropriate mass dimensions" — THEY DECIDE WHICH CONFIGURATIONS SOLVE THE EQUATIONS, NOT WHAT THE TENSOR IS. RESULT 366.2 (and it refutes the claim it was meant to support). ρ = T⁰⁰ = ½|φ̇|² + ½|∇φ|²; p = T^ii/3 = ½|φ̇|² − ⅙|∇φ|²; therefore ρ + 3p = 2|φ̇|², WHICH IS NEVER NEGATIVE. The field satisfies the STRONG ENERGY CONDITION in every configuration, and a source satisfying it DECELERATES expansion. Static isotropic gradients give w = −1/3; motion raises w towards +1; THE OBSERVED Λ AT w = −1.00 ± 0.03 IS UNREACHABLE. Twenty thousand random tangent configurations in 248 dimensions were tested numerically; the minimum of (ρ+3p)/ρ found was 1.15×10⁻⁹ — positive at the level of floating-point zero. The numerical check confirms the algebra rather than establishing it. RESULT 366.3 (Λ is not in this Lagrangian by any route). NOT AS A POTENTIAL — V is gravitational and vanishes where M = 0. NOT AS GRADIENT ENERGY — ρ + 3p ≥ 0. NOT AS KINETIC ENERGY — w = +1. AND THE

Conclusion

IS ROBUST TO THE ONE STRUCTURAL AMBIGUITY: if the vacuum sits ON the constraint surface the tangency condition applies; if it sits OFF it — the constraint being the "absolute structural limit" rather than a state — the multiplier is zero and the field is free. SAME TENSOR, SAME INEQUALITY, EITHER WAY. RESULT 366.4 (T178 places expansion somewhere else entirely). T178 (April 2026) models the universe as a BOUNDED ANNULAR REGION between two surfaces — an outer rebound-cone edge pushing into the Super Polariton Field at r₁ and an inner cumulative core surface at r₂ — with topological conservation forcing r₁(dr₁/dt) = r₂(dr₂/dt) and H₀ = (dr₂/dt)/r₂. AND IT STATES THE CONSEQUENCE IN ITS OWN ABSTRACT: "Dark energy is not required. Cosmic expansion is not driven by a repulsive vacuum pressure but pulled from the inside by the cumulative growth of Planck cores swallowing Phase I lattice." SO THE LATTICE FIELD IS NOT MEANT TO ACCELERATE ANYTHING, AND ITS SATISFYING THE STRONG ENERGY CONDITION IS WHAT THE FRAMEWORK REQUIRES RATHER THAN A DEFECT. T365's ERROR WAS THEREFORE DEEPER THAN PICKING THE WRONG COMPONENT: it was accepting that Λ needs a component at all — IMPORTING A FRIEDMANN ASSUMPTION THE FRAMEWORK HAD ALREADY REJECTED FIVE MONTHS EARLIER. And T358's unsettled fraction is a reading of the MEASURED NUMBER ρ_Λ, not a claim that a Λ fluid exists; the two are compatible once said plainly. ONE OBLIGATION FOLLOWS AND BELONGS TO T178: if expansion is boundary-driven rather than fluid-driven, that picture must reproduce what the Friedmann equations get right — nucleosynthesis, the microwave background, structure growth. THIS NOTE NAMES THAT BURDEN WITHOUT DISCHARGING IT. A CAVEAT ON THE TENSOR, FROM T177. The Planck Shell is a physical boundary — the degree-30 Casimir level set separating Phase I from Phase II, and a perfect absorber and emitter. A boundary carries surface terms, and T342 supplies one: σ = Mc²/4πr². THE TENSOR IN 366.1 IS THE BULK TENSOR AND OMITS THEM. That does not affect Results 366.2 to 366.4, which concern the vacuum far from any Shell, but the tensor as written is not the whole stress-energy of a region containing a core. THE ONE INTERNAL ESCAPE, AND ITS SIGN PROBLEM. If the multiplier were a FIXED

Background

TENSION rather than a quantity enforcing the constraint, the constraint term would act as a genuine potential V_eff = λ₀[Σζ_k I_k − 9E_P], and a potential CAN give w = −1. BUT T331 establishes that every E₈ Casimir degree is even and the couplings positive, so the minimum sits at φ = 0 where V_eff = −9E_Pλ₀ — for positive tension that is NEGATIVE vacuum energy, THE WRONG SIGN. And T326 establishes the multiplier is ALGEBRAICALLY SLAVED to the field, the opposite of a fixed background. Recorded as the only internal escape, NOT as a candidate. WHAT SURVIVES IN T365: the correction of T364's "V at the minimum" (survives — V is gravitational); V vanishes in the vacuum (survives); 365.2, the sigma model derived via T326 (survives — independent of Λ); 365.3, core gradient ≠ vacuum gradient (survives but moot). FOUR OF SIX STAND; the two that fall are the paper's central claim and its prediction. KILL-CONDITIONS: (i) IF T178's BOUNDARY PICTURE CANNOT REPRODUCE STANDARD COSMOLOGY — nucleosynthesis, the CMB, structure growth — then dark energy IS required after all and Result 366.4 collapses; (ii) if T175's vacuum Lagrangian contains a term not in its Section 3.1 — an intrinsic potential the core-formation problem did not need — Result 366.3 fails; (ii) if the multiplier can act as a background tension without contradicting T326, the escape reopens, though the sign problem remains; (iii) if the strong energy condition can be evaded by a non-canonical kinetic term, the inequality does not hold — BUT NO SUCH TERM IS IN THE ARCHIVE; (iv) if the tangency argument fails because the vacuum is not a solution of the field equation at all, the tensor's form is not what is written. NOT CLAIMED: that the framework has a gap here — AN EARLIER DRAFT OF THIS NOTE SAID SO AND WAS WRONG; T178 supplies the mechanism and the correct statement is that T365 imported an assumption the framework does not make. Not claimed that the framework's account of Λ is wrong — T165 AND T358 ARE UNTOUCHED BY THIS, and what is shown is that their mechanism is not T175's field theory, a statement about CONNECTION rather than correctness. Not claimed that the tensor is complete: its form is written and the couplings remain unspecified. Not claimed that the two sectors cannot be connected — only that they are not. And not claimed that T365 should be withdrawn entirely: FOUR OF ITS SIX

Results

STAND.