PUH Theorem 358 (Λ Is the Unsettled Fraction) — The Two-Boundary Structure Caps the Standing Entropy at Half a Nat per Cell, Which T165's ln(240) Exceeds by 2ln240; the Bound Is Satisfied Because Λ/ρ_P = 1.3×10⁻¹²³ Measures How Little of the Lattice

Photonic Universe Hypothesis (PUH) — Constraint and Reconciliation. THE OCCASION. Bekenstein's argument requires that an object absorbing entropy must carry entropy, or the second law fails. T272 answered this for the Planck Shell by calling it "a mirror, not a memory" — it reflects everything, so it stores nothing. THAT ANSWER IS NO LONGER AVAILABLE: T307 established that the Shell ABSORBS, and T308 showed the Casimir argument proves NON-CROSSING rather than reflection. Absorbed entropy has to be somewhere. RESULT 358.1 (the sandwich doubles the capacity and locks it). T240 describes this universe as a lattice region between an inner boundary — the aggregate of all Planck Shells — and an outer conical edge. A REGION WITH TWO BOUNDARY COMPONENTS IS CONSTRAINED BY BOTH: S ≤ (A_inner + A_outer)/(4l_P²) = 4.657×10¹²³ k_B, so the allowed standing entropy is HALF A NAT PER PLANCK CELL, not a quarter. And T179 forces the two areas to be EQUAL, so the capacities are locked: neither can grow without the other, and as the universe expands both rise together — which is why the second law has room to operate. RESULT 358.2 (T165's ln(240) exceeds the bound exactly). T165 gives each cell settling into optimal packing ΔS = k_B ln(240), the 240 being the E8 KISSING NUMBER. Applied to all 9.314×10¹²³ boundary cells that is 5.105×10¹²⁴ against a capacity of 4.657×10¹²³ — AN OVERSHOOT OF EXACTLY 2ln(240) = 10.961, and 4ln(240) = 21.92 against a single boundary. THE VIOLATION IS EXACT, NOT APPROXIMATE: it is the kissing number and the holographic quarter, nothing else. So ln(240) CANNOT be a standing entropy per cell — no adjustment of scale removes a factor fixed by two constants. RESULT 358.3 (two numbers that follow, neither fitted). If the release really is ln(240), the bound holds only if not every cell is unsettled at once: f·N·ln(240) ≤ N/2 gives f ≤ 1/(2 ln 240) = 9.12 PER CENT. And in steady state, with f = t_P/τ, a cell must take τ = 2ln(240) = 10.96 PLANCK TIMES to settle. BOTH FOLLOW FROM THE HOLOGRAPHIC BOUND AND THE E8 KISSING NUMBER WITH NOTHING ADJUSTABLE. Eleven Planck ticks for a cell to find its place among 240 neighbours is physically reasonable, and it is forced rather than chosen. The steady-state relation is assumed, not derived. RESULT 358.4 (Λ measures the unsettled fraction). T165's OWN FOURTH RESULT supplies the reconciliation and had not been connected to the entropy question: "the cosmological constant Λ is the residual entropic force from INCOMPLETE E8 packing in the expanding cone." SO THE UNSETTLED FRACTION IS NOT FREE — IT IS MEASURED: f = ρ_Λ/ρ_P = 6.0×10⁻¹⁰/4.634×10¹¹³ = 1.295×10⁻¹²³. That is 1.42×10⁻¹²² of what the bound permits, and THE STANDING ENTROPY FALLS FROM 5.1×10¹²⁴ TO ABOUT SIXTY-SIX k_B. The apparent violation was an artefact of assuming every boundary cell is unsettled; ALMOST NONE ARE. WHICH REFRAMES THE SMALLNESS OF Λ: it is not a puzzle requiring explanation but A STATEMENT ABOUT COMPLETION — THE LATTICE IS 1 − 10⁻¹²³ CRYSTALLISED. The same number measures the vacuum energy and carries the standing entropy: ONE QUANTITY, TWO ROLES. RESULT 358.5 (black holes occupy 1.33×10⁻¹⁹ of the inner capacity). A CORRECTION TO AN EARLIER DRAFT, WHICH ALSO COMPARED THIS WITH THE UNIVERSE'S OCCUPANCY OF THE TOTAL CAPACITY AND CALLED THE AGREEMENT A CONSISTENCY. IT IS NOT — the two differ by exactly two because the total is twice the inner, so the comparison is trivial AND THE CLAIM IS WITHDRAWN. The observed entropy of the universe, 3.1×10¹⁰⁴ k_B, is dominated by supermassive black holes — WHICH IN THIS FRAMEWORK ARE PLANCK CORES WITH SHELLS, so that entropy already sits on the inner boundary. It occupies 1.33×10⁻¹⁹ of the inner capacity, while the universe overall occupies 6.66×10⁻²⁰ of the total. THE TWO BOUNDARIES ARE EQUALLY UNSATURATED TO WITHIN A FACTOR OF TWO, WHICH IS EXACTLY WHAT T179's AREA BALANCE REQUIRES, and neither was arranged to match the other. The universe runs at one part in 10¹⁹ of what its boundaries could hold. RESULT 358.6 (the historical ledger is a FLOW, not a total). An earlier draft recorded it as an unresolved failure: summed over cosmic history, full ln(240) release gives 4.63×10¹⁸⁵ k_B against an observed 3.1×10¹⁰⁴, an overshoot of 10⁸¹. THAT FRAMING WAS WRONG, AND T178 SAYS WHY. T178 LOCKS CREATION TO DRAINAGE: lattice appears at the outer boundary and is CONSUMED at the inner one as cores pull spacetime back, unfold it, and take in the released photons, with r₁(dr₁/dt) = r₂(dr₂/dt) and both fractional rates equal to H₀. SO ENTROPY DOES NOT ACCUMULATE IN THE LATTICE — IT PASSES THROUGH. Production at the outer boundary, scaled by f, is 1.23×10⁴⁵ k_B/s; absorption capacity grows at 2H₀ × capacity = 1.02×10¹⁰⁶ k_B/s. THE CORES ABSORB 8×10⁶⁰ TIMES FASTER THAN THE LATTICE PRODUCES. Nothing accumulates and no bound is threatened. AND THE ACCUMULATED ENTROPY IS NOT THE INTEGRAL OF PRODUCTION — IT IS WHAT THE CORES HAVE TAKEN IN AND STILL HOLD, WHICH IS THE BLACK-HOLE ENTROPY WE MEASURE. That also explains why the observed entropy is BOUNDARY entropy rather than LATTICE entropy: the lattice entropy does not stay in the lattice, because the cores take it. THE DRAIN MAKES IT A CONSEQUENCE RATHER THAN A COINCIDENCE. AND T179's BALANCE IS FORCED, NOT ASSUMED. Substituting dr/dt = H₀r into the continuity equation gives r₁²H₀ = r₂²H₀, so r₁ = r₂: THE INNER AND OUTER EFFECTIVE RADII ARE EQUAL BY THE CONTINUITY EQUATION, and the area balance is a structural requirement of the drain rather than an independent assumption. Which changes how an apparent shortfall should be read: measured against our horizon, cumulative black-hole area falls 1.33×10⁻¹⁹ short of the outer area, but ON THIS READING THAT IS NOT A FAILURE OF THE BALANCE — IT IS EVIDENCE THAT OUR HORIZON IS NOT THE OUTER BOUNDARY. The size that would close it is 7.5×10¹⁸ horizon radii, about e⁴³, and standard inflation already requires at least e⁶⁰, so the figure sits INSIDE what mainstream cosmology assumes. THIS IS OFFERED AS A READING, NOT AS SUPPORT: the universe's size is inferred FROM the balance rather than measured independently, which is close to circular and is flagged as such. KILL-CONDITIONS: (i) T165's SIGN IS AMBIGUOUS AND EVERYTHING ABOVE DEPENDS ON IT. It writes ΔS = +k_B ln(240) for a settling cell, defending the positive sign with a dice analogy about the packed ENSEMBLE; but settling is ORDERING, and an individual cell finding one optimum has FEWER states, not more. If the per-cell change is negative, the lattice is crystallising and EXPELLING entropy rather than acquiring it, and Sections 3–4 must be re-read as bounds on expelled rather than standing entropy. THIS IS A ONE-LINE CLARIFICATION IN A FILED PAPER AND SHOULD BE MADE; (ii) if the holographic bound does not apply to a region with an inner boundary in the form assumed, Result 358.1 fails and the capacity halves; (iii) if T178's continuity equation does not hold, T179's balance is NOT forced and the Section 8 reading collapses — the shortfall would then be a genuine failure of the balance rather than evidence about the horizon; (iv) if Λ has any contribution other than incomplete packing, Result 358.4's identification is partial and f is an upper bound rather than a measurement. NOT CLAIMED: that the entropy budget is solved — Section 7 states plainly it is not; that the settling time is derived, since f = t_P/τ is assumed; that Λ has been explained, only that its value is read as a completion fraction; that the sign question is resolved; or that any of this derives the second law — IT CONSTRAINS WHERE ENTROPY CAN SIT, NOT WHY IT INCREASES.