PUH Theorem 405 — Which of the Lattice's Vibrations Belong to T175's Field: About a Massive Uniform Vacuum, Only Gapless Waves of Helicity Zero — Not T235's Optical Modes or Light's Polarisations
Year: 2026
Type: preprint
Abstract: Photonic Universe Hypothesis (PUH) — Theorem 405. T404 found that internal modes keeping time with a cell of fixed spacetime volume would have exactly −ρc² and drop out of T403's unimodular equation, but that T366's bounds forbid this to modes of T175's field; it left open whether T235's modes are T175's (T403's OPEN (2), second half) and whether any internal mode lies outside that field. This paper takes every vibration the archive names and asks of each whether it is a wave of T175's field, whether it has a gap, and what helicity it has. Result 405.1: about the archive's vacuum, taken as one massive state at rest everywhere on T175's wall (T377, T378, T381; whether it is massive or a mixture is T378's OPEN (1)), the wall's gradient never vanishes — by Euler's identity its pairing with the state is a sum of positive terms, 57.7 to 97.9 Planck units for T238's couplings — so a uniform state has a multiplier of zero and no direction acquires a mass. With T320's bonds and T376's positive metric in both terms, all 247 directions along the wall (240 root, 7 Cartan) share one dispersion, ω² = (4J/ρ)Σ sin²(k·α/2): gapless, at light's speed for long waves, up to T386's band top of 2.9212 c per lattice unit. The multiplier is quadratic in the waves (T326) and zero for a travelling one, so the wall's normal carries no wave. T175's field has 247 oscillators per wavevector and no internal sector (as T404 found), and the ambient value of its multiplier is zero, so T315's nonzero ambient λ₀ — on which T402's Shell radius and T315's clock rest through X = λ*/λ₀ — is not that value unless the vacuum is not uniform or not static; if T299's tension field is that multiplier, λ₀ = 0 and the Shell radius 2M X/(X − 1) falls to 2M. Result 405.2: each of these waves has its polarisation in the algebra, so it comes back unchanged when turned about its direction of travel — helicity 0, as T383 found for every field in the archive under T317's reading. Light (±1), T385's shear (±1) and T193's two patterns (±2) are therefore not waves of T175's field about the archive's vacuum, in either of T380's readings of the photon (whose parts along the wall are such waves and whose parts along the normal are slaved scalars); T167's phase fluctuation A_μ = ∂_μΛ has a field strength that vanishes identically; and the W and Z, measured at 80.3692 and 91.1880 GeV, are not among them either, since none of those waves has a gap — T189's mass gap for torsion modes does not hold for them. A tie between directions of the algebra and directions in space would change this; T175's Lagrangian contains none, the archive's vacuum supplies none, and the one tie the archive has tried (T382's boost) makes moving matter 1.88 times too heavy. Result 405.3: T235's 3 + 237 per cell is not T175's spectrum. A flat band at ω_P is impossible on T320's lattice; T235's acoustic modes are displacements of the cells, the lattice's own geometry (T383's 'one structure that is not a scalar'), not components of φ; and its 240 is T175's count of root directions without the 7 Cartan directions along the wall, and twice the 120 planes in which T172's leaf turns. T174's oscillators add no internal sector of their own. Result 405.4: of the three dynamics the archive has written for the cells' states (T175's field, T381's rotors, T376's precession on the lattice), only the precession has gapped modes: 56 planes per cell at 2|α·x₀|, of positive energy, with a gapped zero-point energy of 51.0 ± 5.1 per cell in the build's units (Planck energies only if the flow's unit of time is t_P), beside a bond part of 13115 J per cell. They lie outside T175's field, and T381 excludes the precession as the lattice's motion. Granted anyway, their gaps contain no length, so on T210's fixed clock, with the coefficient of T376's energy held, a cell that stretches keeps them (T404's R1, w = 0), and scaling the state to make them grow takes it off the wall; R4 for them needs what it needs for any mode — the cell's own tick as their clock, or an energy coefficient growing with the cell's volume — and nothing in the archive sets either. Result 405.5: if the zero-point energy gravitates, then about a massive uniform vacuum T175's 247 gapless branches, which no clock that keeps light's speed can remove (T404) and which T366's bounds keep at w ≥ −1/3, hold 1.564 Planck energies per Planck cell cut at the Planck rate and 361.0 cut at the cells' density, at w = +1/3, and weigh 1.84 × 10¹²³ to 4.25 × 10¹²⁵ times the measured dark-energy density. T404's least weight, 2.2 × 10¹²¹, counted T235's 3 gapless branches, so it rises 247/3 = 82.3-fold, whatever clock any internal mode keeps. T403's kill-condition (ii) is met for the lattice the archive uses, and the escape T404's kill-condition (v) guarded lapses. As radiation, the 245 gapless directions besides light's would add 6.56 to N_eff had they ever shared the Standard Model plasma's temperature, against the 0.284 Planck allows, unless some 1,018 further degrees of freedom heated the plasma afterwards. What it means: in the lattice the archive uses, about a massive uniform vacuum, T175's field has no internal modes, so T404's escape for the internal zero-point energy has nothing to act on, and if the zero-point energy gravitates all of it weighs; the lattice's waves have no helicity, so what carries light's two polarisations, like gravity's (T383), is not yet found; and the archive's ambient tension λ₀ is not the ambient value of T175's multiplier. Not claimed: that the vacuum is massive rather than a mixture; that the flow's unit of time is fixed, beyond T210's t_P; that the lattice has no field of helicity ±1, only that T175's field about the archive's vacuum is not one; that the cells' displacements exist as a separate field; that the precession's planes are modes of the lattice; their zero-point energy in physical units; that the W and Z, the photon or the gluons are not in E8; that the zero-point energy gravitates; any value for Λ. OPEN: what carries light's polarisations and gravity's; an equation for the cells' displacements; a vacuum or coupling, not yet found, that ties directions of the algebra to directions in space; what gives the W and Z their masses in the lattice; what the other gapless directions are, why none is seen and what kept them from the plasma's temperature; what λ₀ is; whether the zero-point energy gravitates; a mechanism, not yet found, that keeps it out of the expansion; how the E8 lattice's eight dimensions relate to the three of space. Kill-conditions are stated. Scripts and outputs are supplied.
Author: Martell, Brian
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225774337