PUH Theorem 403 — Fixed Cells and the Cosmological Constant: Cells of Fixed Spacetime Volume Allow the Unimodular Equation, in Which the Vacuum Energy Drops Out Only If Its Pressure Is Exactly −ρc²

Photonic Universe Hypothesis (PUH) — Theorem 403. T401 found two field equations that meet the archive's requirements, Einstein's and its unimodular form, and left open which the lattice gives; it noted that a lattice whose cells keep a fixed volume may favour the unimodular form, in which the cosmological constant is a constant of integration rather than an energy of the vacuum. This paper asks what a fixed cell must be for that to follow, what then leaves gravity and on what condition, and what the archive's accounts of Λ become. Fixed cells are a premise here: T210 gives the cell a size l_P³ and a clock t_P, but no archive paper derives a fixed cell volume. Result 403.1: a relabelling changes each cell's spacetime volume by its divergence. If the tension's equation is Lorentz-symmetric, as T401 assumes, a lattice that keeps only its cells' spatial volume, with time labels free, allows Einstein's form alone, while one that keeps each cell's spacetime volume allows a family holding both forms and a mass term for the trace — and the unimodular form alone if, further, the tension's one-number part is that fixed volume. T401's open question is reduced to what the lattice fixes. Result 403.2: in the unimodular form a component enters the expansion only through ρ + p/c², so a vacuum energy of any size drops out if its pressure is exactly −ρc². For T235's zero-point energy, 118.5 Planck energies per cell and 1.05 × 10¹²⁵ times the measured density, a pressure off by one part in 10¹²⁷ lowers today's H² by 9.9 per cent if matched at z = 99 (15.0 per cent from z = 1100), and Planck's measured w₀ allows a departure towards less negative pressure of about 2.9 × 10⁻¹²⁷ at most, at two standard deviations. The archive derives no pressure for that energy, and T366's ρ + 3p ≥ 0 for T175's field points the other way: the fine-tuning moves from the vacuum energy's size to its pressure. While matter is conserved Λ is a constant of integration. Cells of fixed spacetime volume need no new cells; cells that keep l_P³ and t_P separately need new cells at 3H, against T178's conservation of nodes. Result 403.3: on Sorkin's causal-set heuristic, Λ ~ 1/√N, spatial cells miss by about 10³⁰ and the past light cone's cell-events give the right order, but so does any count of Planck-scale cell-events over a Hubble-sized four-volume, and for the Hubble four-volume itself the estimate is exactly the combination T273 quarantined, so the size is no test. The heuristic needs a random count; as a causal diamond's speed through the lattice is scanned from 0 to 0.9c, the largest and smallest variances of a crystal's count in it differ by a factor of 18 ± 2 to 159 ± 16 at the sizes computed (Z⁴), so it picks out the lattice's frame, which a random sprinkling cannot (Bombelli, Henson and Sorkin 2009). Result 403.4: with fixed cells whose volume is the tension's one-number part, the vacuum's pressure exactly −ρc², matter conserved and no evolving lattice energy, dark energy is exactly Λ, as in ΛCDM; DESI's preference for evolving dark energy tests whether Λ is PUH's only dark energy, not fixed cells. In Ahmed, Dodelson, Greene and Sorkin's model of random cell-events, with Das, Nasiri and Yazdi's normalisation, whose one amplitude α is the rms of Λ√V, one element per Planck cell-event is α = 1 and Planck-size elements in Das et al.'s convention α = 4π; at α = 1, and already at α = 0.2, not one of 4,000 histories with fluctuations from the earliest epochs reaches today, and at α = 4π only universes dominated by Λ from their first instants do, and at smaller α none ends with 0.6 < Ω_Λ < 0.8 while meeting nucleosynthesis' bound. With matter not conserved, the cores' whole mass is 4.98 × 10⁻⁶ of ρ_Λ, an upper limit on what they could move (nothing, if all that passes through them leaves as light, as T387 has it), and T369's change of sign, as a change of the constant, would take 8.59 per cent of matter's mass-energy at z = 2.7. What it means: IF PUH's cells keep their spacetime volume, the tension's one-number part is that volume, and the vacuum's pressure is exactly −ρc², the vacuum energy's weight, the archive's 10¹²⁵ problem, no longer arises; while matter is conserved Λ is a constant set once, whose value PUH must still supply, and random Planck-cell events give no universe like ours in this model. Not claimed: that the lattice keeps its cells' spacetime volume; that T235's zero-point energy has p = −ρc²; that the unimodular form solves the cosmological constant problem; any value for Λ; a crystal estimate of Λ for the physical past light cone. OPEN: the lattice mechanism; the vacuum's pressure; what sets the constant; whether any cell-events are random; how new cells arise; matter's conservation at the cores. Kill-conditions are stated. Scripts and outputs are supplied.