PUH Theorem 404 — The Pressure of the Lattice's Zero-Point Energy: Internal Modes on the Clock of a Cell of Fixed Spacetime Volume Have Exactly −ρc²; Gapless Modes Redshift, With a Positive Pressure

Photonic Universe Hypothesis (PUH) — Theorem 404. T403 found that in the unimodular form of the field equation a vacuum energy drops out of gravity only if its pressure is exactly −ρc², and that for T235's 118.5 Planck energies of zero-point energy per cell a departure of a few parts in 10¹²⁷ already shows; it left the pressure itself open (its OPEN (2), first half). This paper computes that pressure, mode by mode, from the archive's own spectra, under both of T403's premises for fixed cells: cells that keep their spacetime volume and stretch, and cells kept at l_P³ and t_P with new cells made as space grows (T364). Result 404.1: for a fixed number of cells that stretch, w = p/ρc² = −d ln E_cell/d ln v exactly; for oscillators in their ground state, which a slow change preserves (checked through the Ermakov–Pinney equation), w is the zero-point-weighted mean of the modes' Grüneisen parameters, so w = −1 means each cell's energy grows in exact proportion to its volume. T366's bounds, −1/3 ≤ w ≤ +1, hold in every state of T175's field, and its tensor in the free ground state gives w = 1/3 exactly. Result 404.2: the modes are of two kinds. Gapless modes keep light's constant speed (T229 for light, T317 for the acoustic branches), so their frequencies fall as the lattice stretches, and their zero-point half-quanta carry a momentum flux that makes w + 1 the ratio of the zone-boundary mean frequency to the zone mean: 1/3 for T235's three acoustic branches and for T175's field; for T320's exact dispersion on E8, 0.0178 with new cells and 1/8 for a lattice of its form that stretches; 93.5 per cent of E8's modes lie within 5 per cent of 4.00 c/b. Internal modes — T235's 237 flat optical modes — carry no momentum flux, and their pressure is set by the clock their frequency keeps: w = 0 if the medium's constants hold it, 1/3 if it follows the light-crossing time, and exactly −1 if it keeps time with a cell of fixed spacetime volume (T403's own premise), at the cost of T210's fixed minimum process time; that −1 lies outside T366's bounds, so it is open only to internal modes that are not modes of T175's field. Result 404.3: if the zero-point energy gravitates, T235's linear acoustic branches, with w = 1/3, weigh (4/3)ρ in the unimodular form. For Planck cells they hold 0.0190 Planck energies per cell cut at the Planck rate, or 4.385 cut at the cells' density — not an amount of order their lowest frequency — and weigh 2.2 × 10¹²¹ to 5.2 × 10¹²³ times the measured dark-energy density, 7.4 × 10¹²² to 1.7 × 10¹²⁵ times what Planck's measured w₀ allows. Cells that stretch cannot dilute this enough: they would have to be at least 8.4 × 10⁻⁵ m wide today, a lattice that could carry no visible light, far outside T372's limit on the lattice unit. With new cells the weight stands unless the making of each cell exerts a pressure of −(4/3)ρ_g c², as −ρ_int c² would remove the internal modes' weight; the archive does not say how a cell is made. T320's 240 branches are all gapless. Result 404.4: a lattice at rest in its own equilibrium weighs its whole energy, static and zero-point; a constraint holding the cells' spacetime volume reacts with a pure trace, which drops out; a static tension could cancel the gapless weight only by matching it to 1.3 × 10⁻¹²³, and a tension-stiffened bond's modes average −1 only within 2 × 10⁻¹²⁸ of one stretch. Result 404.5: T403's kill-condition (ii) is met for the gapless sector, for the internal modes under three of the four readings, and for every mode of T175's field; under the fourth, with the cells' spacetime volume held by a constraint, internal modes outside T175's field drop out, which adds to T403's OPEN (1) the requirement that the internal frequencies keep the cell's tick and, with cells that stretch, gives T174's C174.3 its non-diluting hum, which then drops out as well; with new cells the hum holds whatever the clock. What it means: IF the zero-point energy gravitates and the cells stretch with their spacetime volume held, internal modes on the cell's clock leave gravity — if they are not modes of T175's field, whose every state T366's bounds keep at w ≥ −1/3 — but the gapless sector cannot by any clock that keeps light's speed: the archive's 10¹²⁵ problem shrinks at best to 10¹²¹ for Planck cells, and not at all if T235's modes are T175's, the route by which T350's unsubtracted source makes them gravitate. What remains needs the gapless zero-point energy not to gravitate (the suppression T235's Refinement hands to T170, whose rate T273 found computed nowhere), or a stress beside its modes, such as a static tension of the lattice tuned to 1.3 × 10⁻¹²³ of it or, with new cells, a stress of their making. Not claimed: that the internal modes keep the cell's tick, or that any internal mode lies outside T175's field, as that needs; that the photon rides a root pair (T292, used conditionally); that the cosmological constant problem cannot be solved in PUH; that T170's account is wrong; that T320's eight-dimensional modes are T235's; any value for Λ. OPEN: which clock the internal modes keep, and whether any internal mode lies outside T175's field, by a mechanism not yet found; how a new cell is made, the stress of its making and who pays; the static energy and tension of the vacuum lattice; T170's Γ_rebound; whether the zero-point energy gravitates; how T320's modes relate to T235's and which carry light; whether T235's modes are T175's; a mechanism, not yet found, that keeps the gapless zero-point energy out of the expansion. Kill-conditions are stated. Scripts and outputs are supplied.