PUH Theorem 379 — The Spinning Core: The Lattice Outside Stays Still, Its Field Is T378's Geodesic Shaped by the Flattened Shell, T342 Holds for Every Shape, and Gravity's Twist Needs the Coset

Photonic Universe Hypothesis (PUH) — Theorem paper. THE QUESTION. Cores spin (T136; T332: the spherical reduction "IS NOT AN IDEALISATION"), yet every lattice result from T326 to T378 — T327's first integral, T342's exclusion of the multiplier lambda as gravity, T377's shell, T378's rest — was derived for a core that does not spin. T300 to T304 treated rotation through general relativity's equations (T302: "A ROTATING SUBSTRATE CARRIES TWO FIELDS AND NOT ONE"; T303 located the hyperbolic plane they need, "an identification and not a derivation"), and T329 recorded that the lattice's own rotating case, a two-dimensional calculation, had not been attempted. Every number in T379 is computed on the E8(−24) build of T373, with T377's sign, T238's couplings, T376's metric K_theta and T326's static equations. RESULT 379.1 — THE SPIN STAYS IN THE CORE. With K_theta the lattice's waves have no gap (c^2 = 60J/rho, T337; no potential, T331). A field carries momentum only if it moves, and steady motion with no gap either radiates or fills all space (Rellich's lemma). So a core that spins steadily keeps its spin in itself, and the lattice outside its Shell is static; a core lumpy about its axis radiates lattice waves. Argued, not computed. RESULT 379.2 — THE SAME GEODESIC, RESHAPED. Around a Shell carrying one state, the static field is exactly T378's geodesic of the wall laid out by the Shell's capacitary potential, phi = gamma(h), for any shape; T327's constant becomes C = (L x capacitance)^2. Solved in two dimensions (Newton's method) for the three flattenings T359 found for tension-held spinning cores (f = 0.00375, 0.0338, 0.1734 at spins 0.10, 0.30, 0.68): independent of latitude to 3.3 × 10^-15, within 1.0 × 10^-6 of gamma(h) and converging as the square of the grid step; capacitance 0.9987, 0.9887, 0.9415; far-field C = 0.6384, 0.6256, 0.5673 against 0.64 for a sphere. RESULT 379.3 — CALMER OVER THE POLES. On a flattened Shell the field gradient is lower at the poles than at the equator by (1 − f), and T342's multiplier and the lattice's gradient energy by (1 − f)^2 = 0.9925, 0.9335, 0.6833 — 0.83 and 0.68 at T359's fastest spin. No link to T368's creation of space at the poles is claimed. RESULT 379.4 — T342 HOLDS FOR EVERY SHAPE. Far from any finite-energy static core the field is the vacuum state plus a/r plus 1/r^2 terms, so lambda falls as 1/r^4 in every direction, whatever the Shell's shape and pattern of states. Computed for a Shell whose state varies from pole to equator: near the Shell T327's law fails (r^4|grad phi|^2 from 0.80 to 1.33 across latitude), far away it holds (0.63893 in every direction at r = 48), and r^4 lambda goes from 0.0169–0.0229 near the Shell to 0.00728 in every direction at r = 48, approaching 0.00722. T342's conclusion covers spinning cores; T340's angular escape is closed in general; T378's results hold for any shape. RESULT 379.5 — GRAVITY'S TWIST NEEDS THE COSET. Gravity's sl(2,R) of T303–T304 closes as [Z, X1] = 2X2, [Z, X2] = −2X1, [X1, X2] = −2Z; the massive state x0 = 1.4062 Z lies on the wall, and its moving versions form a surface O (T374's clocks of one rate) that is totally geodesic in the wall. With the Killing form O is a hyperbolic plane (curvature −0.004215 everywhere); with T376's K_theta it is a bowl (+0.004215 at rest, falling toward zero as the state moves); in T376's coset alternative the same plane has curvature −1/30 everywhere. A spinning body's gravity (T302) needs the hyperbolic plane, and the wall's symmetries with K_theta are compact, so with K_theta the lattice cannot carry the twist as T303 proposed; the coset can. T376's OPEN (1) — K_theta or the coset — becomes the deciding question for rotation. Where PUH's frame dragging (measured around the Earth by Gravity Probe B) comes from is open. WHAT CHANGES: T329 CARRIED OUT; T327 EXTENDED; T342 CONCLUSION COVERS EVERY STATIC CORE OF ANY SHAPE; T340 ANGULAR ESCAPE CLOSED IN GENERAL; T332 ANSWERED FOR THE LATTICE OUTSIDE; T359 EXTENDED; T378 EXTENDED (kill-condition iii answered); T303 NARROWED (coset only); T374 OBSERVATION NARROWED; T376 OPEN (1) SHARPENED; T302, T304 USED; T333 PASS STANDS. KILL-CONDITIONS: if the substrate's field moves by T376's cell precession rather than T175's field equations (Result 379.1 must be redone); if T303's identification is dropped (379.5 then narrows T303 and leaves K_theta standing); if a core is lumpy about its axis (it radiates first); T359's flattenings are Newtonian examples; in the coset reading Results 379.2–379.4 must be recomputed. NOT CLAIMED: the radiation rate of a lumpy core; what the Shell's interior holds; any link to T368's pole creation; that the coset reading has been built or reproduces the field outside a spinning body; that frame dragging is explained; that lambda is gravity in any reading. OPEN: K_theta or the coset; the source of frame dragging; the state pattern of real spinning Shells and the spin-down rate of lumpy ones; whether T377's shell and T378's rest structure survive in the coset reading.