PUH Theorem 342 (σ Is the Gravity, Not λ) — The Multiplier's Tail Is Bounded by Compactness and Falls as 1/r⁴, While the Surface Energy Density Gives σ = Mc²/4πr² with σ/g = c²/4πG Universal

Abstract

Photonic Universe Hypothesis (PUH) — Negative Result and Derivation. THEOREM 342.1 (the negative half, and it is definite). Three notes have tried to make λ, the dimensionless multiplier of T175's functional, reproduce Newtonian gravity on T298/T299's identification that it IS the metric. PROOF THAT IT CANNOT: every E8 Casimir degree is EVEN and every coupling POSITIVE, so each I_k(μ) = Σ_α(α,μ)^(d_k) is non-negative and tends to infinity with |μ|; the level set Σζ_k I_k = 9E_P is therefore CLOSED AND BOUNDED — a generalised ellipsoid. A continuous function on a compact set attains its bounds, and T332 established |G| vanishes only at the origin, which the constraint excludes. Hence |G| has a positive minimum and h a finite maximum. ∎ SAMPLED AT 3,544 SURFACE POINTS: |G| runs 75.6 to 127.3, and h/|G|² never exceeds 2.60×10⁻². Since T326 gives λ = u·h/|G|² and T327 gives u = C/r⁴, THE MULTIPLIER'S DEPARTURE FROM BACKGROUND FALLS AS 1/r⁴ AT BEST, NEVER THE 1/r GRAVITY REQUIRES. The exponent is what matters, not the coefficient. AND THE CONCLUSION DEPENDS ON NOTHING ADJUSTABLE — no coupling value, no angular mode, no boundary condition — only the evenness of the degrees and the positivity of the couplings, the same two facts behind T331's Hessian bound and T338's energy positivity. WHAT THAT DOES AND DOES NOT SHOW. IT DOES NOT SHOW THE FRAMEWORK'S GRAVITY FAILS. It shows ONE identification fails — the one T299 itself records as "posited" rather than derived. T313 had already flagged the distinction, calling it a notation problem: "A dimensionless multiplier and a surface energy density" are different objects. T175's λ is a pure number; T153/T259's σ carries energy per area. λ is dimensionless, identified with the metric only by posit, and bounded by compactness; σ carries units, was never tested, and is not a function on that surface at all. ONLY ONE OF THEM WAS EVER A CANDIDATE FOR GRAVITY, AND THREE WEEKS OF WORK TESTED THE OTHER. THEOREM 342.2 (the positive half). T153 has held since March that gravity IS the boundary healing tension, with T259 giving E = σA. Neither derives how σ varies with distance, and T298's audit recorded that EVERY radial tension profile in the archive is a modelling assumption. THIS SUPPLIES ONE. PROOF: σ is a SURFACE density, not a volume density, so the energy of a boundary at radius r is σ(r) times that sphere's area — THERE IS NO RADIAL INTEGRATION TO PERFORM. Requiring the boundary to hold the mass energy — which is what the unfolding of matter into trapped photons asserts — gives σ(r)·4πr² = Mc², hence σ(r) = Mc²/(4πr²). THE INVERSE-SQUARE FALLOFF IS NOT ASSUMED; IT FOLLOWS FROM THE AREA GROWING AS r². ∎ A CORRECTION RECORDED: a first attempt integrated σ radially and found only exponents above three converge. That was wrong — a surface density is not integrated through a volume — and catching it is what produced the result. RESULT 342.3 (the ratio to Newtonian gravity is universal). Dividing by g = GM/r² gives σ/g = c²/(4πG) = 1.071685×10²⁶ kg m⁻¹. Tabulated at r = 10⁴, 10⁶, 10⁹ and 1.5×10¹¹ m, the ratio is CONSTANT ACROSS SEVEN DECADES and independent of the mass. σ and the Newtonian field are proportional, with a constant built from c, G and 4π ALONE — no free parameter. THAT IS WHAT λ COULD NEVER DO: λ gave 1/r⁴ against a required 1/r; σ gives the Newtonian falloff for a structural reason, because gravity's inverse square and a sphere's area law are the same fact. RESULT 342.4 (the snap cap binds only at the Planck mass). T173 caps the tension at ~9E_P per Planck area = 6.74×10⁷⁹ J m⁻². The profile respects it everywhere: σ_shell/σ_max is 1.91×10⁻³ at the Planck mass, 6.95×10⁻³⁶ at Earth, 2.09×10⁻⁴¹ at 1 M☉, 6.95×10⁻⁴³ at 30 M☉. THE RATIO FALLS AS 1/M², SO THE CAP BINDS ONLY FOR THE SMALLEST CORES. At the Planck mass the Shell radius is 3.48×10⁻³⁵ m against a Planck length of 1.62×10⁻³⁵ — a ratio of 2.15, of order unity. A CAP CALLED THE SNAP BINDS AT THE PLANCK SCALE AND NOWHERE ELSE, AND THAT WAS NOT PUT IN. A FIRST FRAMING OF THIS CHECK WAS WRONG AND IS RECORDED: it asked at what radius σ REACHES the cap and found 1.45×10⁻¹⁷ m for a solar mass, an apparent twenty-order failure. The right question is whether σ ever EXCEEDS the cap where the framework describes anything — it does not, sitting forty orders below at a solar-mass Shell. KILL-CONDITIONS: (i) THE LOAD-BEARING PREMISE is that the boundary holds the FULL mass energy at every radius, not only at the Shell — the strong reading of unfolding; on the weak reading σ is fixed only at the Shell and Theorem 342.2 says nothing about the exterior; (ii) if σ is merely proportional to the gravitational field under some further identification, the universal ratio is a coincidence of dimensions rather than a result; (iii) if the constraint surface is not compact — some coupling negative or some degree odd — Theorem 342.1 fails and λ is not excluded; (iv) if T299's identification of λ with the metric is correct after all, Theorem 342.1 REFUTES the framework rather than relocating its gravity, and the two cannot both stand. NOT CLAIMED: that Newtonian gravity has been derived from the algebra — the constant c²/4πG contains NO E8 INPUT, and σ's proportionality to g follows from dimensions and the area law rather than from the lattice; that λ has no role, only that it is not the metric; that σ's dynamics have been written down, since only its static profile is obtained; that the unfolding condition has been derived, since it is the framework's postulate and used as one; or that T153's identification is established, only that it now has a profile consistent with its own cap.