PUH Theorem 315 (λ₀ Is Dynamical) — The Outer Boundary Makes the Domain Finite, So the Threshold Ratio Is a Cosmological Clock: Shells Move Outward as the Universe Ages, and the Accretion, Buchdahl and Imaging Bounds Are Epochs Rather Than Rivals

Abstract

Photonic Universe Hypothesis (PUH) — Reframing. THE SETTING. The Snap threshold is now pinned twice. T238 fixes it structurally: the coupling pattern and the work-function constraint give λ* = 9E_P divided by the sum of the eight E8 Casimir degrees {2,8,12,14,18,20,24,30}, which is 128. T313 fixes the same value dimensionally by an independent route, showing both source Lagrangians force λ dimensionless and the expression was never an energy: λ* = 9/128 = 0.0703, a pure number. What remains undetermined is λ₀, the ambient tension of empty lattice, and with it X = λ*/λ₀ — which appears in the Shell radius (2GM/c²)·X/(X−1), in the corrected echo delay, and in the compactness ratio. EVERY ROUTE HAS FAILED. The couplings determine λ* and the core's location at simultaneous Casimir saturation but say nothing of the ambient value. The Machian inertia relation introduces two further unknowns per particle rather than removing one. The entropic route through the kissing number gives λ₀ = ln(240) = 5.48 in Planck normalisation — SEVENTY-EIGHT TIMES λ* — which would place empty space far beyond its own tearing point; that route is excluded, and its failure shows λ is not a surface energy density in Planck units. WHY THEY FAIL. λ* is a Lagrange multiplier evaluated at a single configuration, and T298's audit established that the Lagrangian supplying it contains NO SPATIAL DERIVATIVES — constrained optimisation at a point. λ₀ is the asymptotic value of a FIELD, introduced when T299 made the tension identification coordinate-invariant. A MULTIPLIER AT A POINT AND THE ASYMPTOTICS OF A FIELD ARE NOT THE SAME KIND OF OBJECT, and no equation in the archive connects them. The question was being asked of the wrong kind of quantity. THEOREM 315.1. The framework's own boundary structure removes the question as posed. The Three Boundaries paper identifies an OUTER boundary — the conical edge of the expanding universe, at which the lattice, stretched by the rebound expansion, will eventually reach maximum tension and unfold; recorded there as a conjecture and explicitly NOT YET QUANTIFIED. That boundary is FINITE. There is therefore no spatial infinity at which to impose a condition, and λ₀ is not a constant awaiting derivation: IT IS A DYNAMICAL VARIABLE, RISING AS THE EDGE IS STRETCHED. This is a change in the problem's type, not a weakening: a quantity that evolves cannot be derived as a number, only given an equation and an initial condition. The failures of Section 1 are explained — they were attempts to compute a constant that does not exist. RESULT 315.2 (the cosmological clock). λ* is fixed; λ₀ rises; therefore X falls monotonically with cosmic time. The Shell radius in Schwarzschild units is X/(X−1), which INCREASES as X falls and DIVERGES as X → 1. EVERY PLANCK SHELL MOVES OUTWARD RELATIVE TO ITS OWN HORIZON AS THE UNIVERSE AGES. The ladder: X = 1000 gives 1.0010 r_s (early); X = 14 gives 1.0769 r_s with λ₀ = 0.0050 (PRESENT, from the accretion floor); X = 9 gives 1.1250 (compactness limit); X = 3 gives 1.5000 (imaging bound); X → 1 gives divergence and NO SHELL ANYWHERE. AND THAT TERMINAL CONDITION IS NOT A SEPARATE EVENT: X → 1 means λ₀ has risen to λ*, the ambient lattice reaching its own tearing threshold — precisely the condition the Three Boundaries paper gives for the outer boundary to unfold. THE INNER AND OUTER BOUNDARIES FAIL TOGETHER, at the same moment, for the same reason. Neither the Shell papers nor the boundary paper anticipated this; it follows from putting them side by side. RESULT 315.3 (the bounds are epochs). Three constraints are on record and conflict if read as constraints on a constant: imaging X ≥ 3 (T298); high-spin accretion X ≥ 14 (T300); compactness X ≤ 9 (T312, Buchdahl). Under 315.2 they are a TEMPORAL SEQUENCE. The framework sits above fourteen today and therefore on the anisotropic side of Buchdahl, which T312 resolves through gravity-as-surface-tension. A universe that ages passes through nine, after which its Shells would satisfy Buchdahl even as isotropic bodies, and later through three. THIS REFRAMES T312 WITHOUT CORRECTING IT: its squeeze is a genuine tension only if X is constant, and its resolution remains necessary and correct — but describes THE PRESENT ERA rather than a permanent condition. SUPPORT AND ITS LIMIT. The dynamical reading assumes interior ambient tension tracks the edge. STATIC: T301 gives ∇²(λ^(−1/2)) = 0, so that function is harmonic and attains extrema on the boundary; in a source-free region uniqueness forces interior = boundary, and with sources it is the edge value plus core contributions falling off with distance, so far from every core the ambient value IS the edge value. ROTATING: T302 gives ∇²α = (h_φφ/2α)|∇ω|², whose right side is NON-NEGATIVE, making the same function SUBHARMONIC — maximum on the boundary, tension bounded BELOW in the interior. ROTATION ADDS INTERIOR TENSION rather than relieving it; the direction is definite, the magnitude negligible at observed rotation rates. THE LIMIT, STATED PLAINLY: T301's equation is derived for the STATIC VACUUM case and says so, while the cosmological domain is neither static nor vacuum. What is shown is that the dynamical reading is CONSISTENT WITH the framework's own field equation IN THE REGIME THAT EQUATION COVERS — not that the equation applies cosmologically. That gap is T302's finding in another guise: a non-static setting requires more than one field. KILL-CONDITIONS: (i) if the outer boundary is shown not to exist, or the domain infinite, Theorem 315.1 fails and λ₀ returns to being a constant awaiting derivation; (ii) if the edge tension does not rise, X is constant, Result 315.3's reframing collapses, and T312's squeeze becomes a permanent tension requiring the anisotropic resolution forever; (iii) if interior ambient tension does NOT track the edge, Section 6's support fails and X's evolution is indeterminate even if λ₀ evolves; (iv) if a compact object is observed whose Shell compactness is inconsistent with the present-epoch X, the ladder is falsified at that epoch. NOT CLAIMED: a value for λ₀ at any epoch including the present (the accretion floor bounds it above; nothing determines it); a RATE for its rise, which requires the outer boundary to be quantified and the Three Boundaries paper states it is not; that the field equation applies cosmologically; that the terminal condition is reached in any particular time; that T312, T300 or T298 require correction, since none do; or that X's evolution has been derived rather than argued from the finiteness of the domain.