PUH Theorem 396 — The Gate's Beat: A Spinning Core Can Meet the Whole E8 Lattice Five Times a Turn in the Frame of the Shell's Invariant C₃₀, and Four Times a Turn in E8(−24)'s Rational Frame

Abstract

Photonic Universe Hypothesis (PUH) — Theorem paper. THE QUESTION. T244 conjectured that the E8 folding cascade through a Planck core's boundary is pulsed, open only when the spinning core's geometry registers with the lattice, and asked first whether E8's rotational symmetry gives 'sharp, well-separated gating windows'; T245 made that its Step 2. A spin happens in three dimensions and E8 in eight, so the question needs a frame: where spacetime's four directions sit among E8's eight. Chloe J. Tully's recent exact-arithmetic check of Conway and Sloane's icosian construction of E8 showed how a spin could be tested exactly. What can we do to help PUH derive this from within? This paper derives a frame from PUH's own C₃₀, the Shell's invariant (T156), finds that PUH's own folding real form E8(−24) (T265) offers a second, and computes the gate's beat in each, exactly. RESULT 396.1 — THE SHELL'S INVARIANT FIXES A FRAME. E8's Coxeter element, the symmetry of order 30 behind C₃₀, has eight complex eigenvectors on which every invariant whose degree is not a multiple of 30 vanishes, so that among E8's eight basic invariants only C₃₀ survives there (Springer). Built from four unit icosians, its characteristic polynomial factors over the golden field — x⁴ + (1 − φ)(x³ + x² + x) + 1 on one half, times its golden conjugate — and splits E8's eight directions into two halves, uniquely up to E8's symmetry, which also exchanges them: C₃₀'s frame. For this Coxeter element one half turns by 12° and 132° and is exactly the icosian 'physical' half; the other turns by 84° and 156°. Neither half holds a lattice vector or lies in a rational subspace short of the whole. RESULT 396.2 — THE BEAT IN C₃₀'S FRAME. If spacetime is a half of C₃₀'s frame and time lies along a root's image — two stated postulates — the rotations of space that carry the whole lattice onto itself are exactly the icosahedron's 60. A core spinning about one of 6 five-fold axes meets the whole lattice every 72°, about one of 10 three-fold axes every 120°, about one of 15 two-fold axes every 180°, and about any other axis never: a gate that opens 5, 3 or 2 times a turn, or stays shut. Between whole windows part of the lattice registers — 19.7 per cent at 36° on a five-fold axis, 27.2 per cent at 44.48° on a three-fold and at 90° on a two-fold, out to norm 8 (26,640 vectors). RESULT 396.3 — THE RATIONAL FRAME PUH ALSO HOLDS. E8(−24) has a maximally split Cartan whose two quartets — its split half, four commuting directions of its (56,2), and the Cartan of the compact so(8) that commutes with it — are rational and exchanged by a symmetry of E8. Seen in the split half the roots form F₄: 24 long ones each the image of 1 root, 24 short ones each the image of 8, while the 24 roots of so(8) vanish (28 + 4 + 24 + 192 = 248). With spacetime in either quartet the beat is the cube's: every 90° about 3 four-fold axes, every 120° about 4 three-fold, every 180° about 6 two-fold. RESULT 396.4 — THE BEAT TELLS THE FRAMES APART. In C₃₀'s frame every rotation that keeps the lattice and fixes a direction turns by 72°, 120°, 144° or 180° (all 7,200 rotations of H₄'s group examined), never 60° or 90°. In any rational frame 2 cos θ must be a whole number — 60°, 90°, 120° or 180°, never 72° or 144° — and in E8(−24)'s frame only 90°, 120° or 180°, while other rational frames, one inside an E6 Cartan, beat six times a turn. A five-fold beat would mark C₃₀'s frame; a four- or six-fold beat a rational one. RESULT 396.5 — HOW SHARP THE WINDOWS ARE. With a positional tolerance ε, a window's half-width in C₃₀'s frame is close to ε over the points' median distance from the axis — 0.17°, 0.57° and 1.68° for ε = 0.003, 0.01 and 0.03, out to norm 8 — so a five-fold gate is open 0.47, 1.58 and 4.67 per cent of a turn; a four-fold gate in E8(−24)'s frame is a little narrower, 0.27, 0.89 and 2.69 per cent. The axis must be held: tilted by 0.3° it keeps 89 per cent registered (ε = 0.01), by 1° only 13 to 15 per cent. T244's first step is answered. RESULT 396.6 — WHAT C₃₀'S FRAME BRINGS, AND WHAT IT COSTS. A golden ladder: multiplying by φ maps the lattice onto itself and stretches spacetime by φ, a step no rational frame allows. The Casimir degrees, T245's candidate source for its ladder, and the gate's frame sit in one object, the Coxeter element — a structural candidate for T245's Conflict 1, not an answer. It costs two placements: the photon's root (T284, T294) has 72.4 or 27.6 per cent of its squared length in spacetime, never wholly internal, and the placement, in the corrigendum to T167, of the 'four directions including gravity' inside E6's Cartan cannot hold, while its count, rank/2 = 4, stands. NOT CLAIMED: that spacetime is a half of C₃₀'s frame or a quartet of E8(−24)'s split Cartan; that time lies along a root's image; that the gate exists (T244 remains a conjecture); any tolerance, depth, throughput or pulse rate; that the Coxeter element acts as a rotation in space; that the golden ladder or the Casimir degrees are T245's ladder of tensions. KILL-CONDITIONS: PUH's spacetime shown to be a rational half (five-fold beats then cannot occur); the Shell's invariant shown to place spacetime in C₃₀'s frame (every rational placement then fails); time shown not to lie along a root's image (the beat is cut to a subgroup); should a core's gating ever be measured, a 72° beat rules out every rational frame and a 90° or 60° beat rules out C₃₀'s. OPEN: which frame — a mechanism not yet found that places spacetime among E8's eight directions; why time would lie along a root's image; the gate's tolerance, the depth of the lattice it sees and its throughput; what holds a spin axis on a symmetry axis to a fraction of a degree; which root carries the photon; how spacetime's four lattice directions connect to a Lorentz structure in the algebra, whose boost reading T382 found fails for ordinary matter.