PUH Theorem 398 — What Form a Hold on a Core's Spin Axis Must Take: In C₃₀'s Frame It Starts at Degree Six, as Klein's f, Lowest on Five-Fold or Three-Fold Axes, Never on Two-Fold

Photonic Universe Hypothesis (PUH) — Theorem paper. THE QUESTION. T396 found that, if spacetime is a half of the frame of the Shell's invariant C₃₀ and time lies along a root's image, a spinning core meets the whole E8 lattice only about the icosahedron's 6, 10 and 15 axes (the cube's 3, 4 and 6 if spacetime is instead a quartet of E8(−24)'s split Cartan), that a tilt of 1° leaves 13 to 15 per cent of a window (ε = 0.01), and left open 'what holds a spin axis on a symmetry axis to a fraction of a degree — a torque not yet found'. The strength of such a torque waits on a coupling of rotation in space to the E8 state that the archive has not written (T352, T381). As in a magnet, whose anisotropy energy is fixed in form by the crystal's symmetry before any mechanism is known, this paper finds what form any such torque must take, where it can hold an axis, how a spinning core moves under it, what its next term changes, and what can unseat an axis once held. RESULT 398.1 — WHAT THE LATTICE'S SYMMETRY ALLOWS. The lattice's turnings act on a spin axis as the icosahedron's 60 rotations in C₃₀'s frame and the cube's 24 in E8(−24)'s, and reversing time, which keeps the lattice in both (computed), reverses the axis; by Neumann's principle an energy the lattice gives a spin axis must be unchanged by all of them. Counted exactly, such energies exist in C₃₀'s frame at degrees 0, 6, 10, 12, 15, 16, 18 and on — none of degree 1 to 5 — and, with time reversal, only at even degrees; in E8(−24)'s frame from degree 4. T397's series for Klein's forms, read at twice the degree, gives the same counts. So in C₃₀'s frame every tensor of rank five or less that the lattice gives space is the same in every direction — as measured for an icosahedral quasicrystal, elastically isotropic within 0.07 per cent where a related cubic phase is not — and if a core's energy depends smoothly on its spin, the spin's direction enters first at the sixth power in C₃₀'s frame and at the fourth in E8(−24)'s. For a spheroid with T359's flattenings, the content of degree six is 0.0019, 0.017 and 0.096 of the content of degree four at spins of 0.10, 0.30 and 0.68 of maximal; for T359's own drop held by surface tension, a quarter to a fifth of that. RESULT 398.2 — THE AIM IS EXACT. Every such energy is stationary, exactly, on every end of every gate axis — 62 in C₃₀'s frame, 26 in the cube's — and on the axes of order three or more its curvature is the same in every direction, so each is a minimum or a maximum, never a saddle, unless the curvature vanishes there. The point about which a lattice torque holds an axis — the energy's minimum or maximum — lies exactly on a gate axis: T396's fraction of a degree is a limit on what disturbs an axis, not on the aim. RESULT 398.3 — THE LOWEST TERM IS KLEIN'S f. The lowest such energy in C₃₀'s frame is Klein's f read on the sphere of directions: F₆ = (φ²x² − y²)(φ²y² − z²)(φ²z² − x²), with three two-fold axes along x, y and z, equals −φ³/21 plus the one invariant harmonic of degree 6, whose Maxwell poles are the six five-fold axes; written on the null cone in T397's spinor variables it is an exact multiple of Klein's f, and the products over the three-fold and two-fold axes are multiples of Klein's H and T. It is −φ³/5, φ³/27 and 0 on the five-fold, three-fold and two-fold ends and has exactly 62 critical points: minima on the 12 five-fold ends, maxima on the 20 three-fold ends, saddles on the 30 two-fold ends. A lattice energy K F₆ is therefore lowest on a five-fold axis if K > 0 (beat 5) and on a three-fold axis if K < 0 (beat 3) — never on a two-fold axis, never between; the five-fold wells are 5.4 times deeper. In E8(−24)'s frame the lowest energy is the cube's quartic, x²y² + y²z² + z²x², lowest on a four-fold axis (beat 4) or a three-fold (beat 3), never a two-fold; iron's easy axis is its cube's four-fold [100] and nickel's the three-fold [111], measured. RESULT 398.4 — HOLDING WITHOUT LOSS, AND WITH IT. A fast-spinning core under such an energy circles the gate axis it starts near, at its starting tilt and at a rate set by the curvature — 13.555 K/L about a five-fold axis and 7.531 K/L about a three-fold one, L the spin's angular momentum — while near a two-fold axis its tilt grows as exp(8.472 Kt/L) and it does not stay. Without loss, five-fold and three-fold axes both hold a spinning axis — it stays near them — whatever the sign of K; with a loss of the Landau–Lifshitz kind only the minima do. RESULT 398.5 — THE NEXT TERM. Klein's H read the same way changes the choice only when it is comparable with the first: in units of equal root-mean-square size the five-fold choice survives a degree-10 term from −0.549 to 4.12 times the degree-6 term, the three-fold choice up to 0.139 times it on the side that lowers the two-fold axes. Beyond, a two-fold axis can be lowest (beat 2), and over 42.3° of the circle of mixtures the lowest point leaves every gate axis, sliding along a mirror arc from a two-fold to a five-fold end: there the gate is shut. The cube's two terms never leave its axes, except at K₁ = 0 with K₂ > 0. RESULT 398.6 — WHAT CAN UNSEAT A HELD AXIS. The wave bath T396 named cannot: by T387's own ceiling, taken per e-fold of frequency at one cycle a year, the cosmic background turns Sgr A*'s axis by at most 1.7 × 10⁻¹⁷ radian in 13.8 Gyr at χ = 0.1, and 4.4 times that over NANOGrav's band. Mergers, tides and arriving matter may; they are not computed. Without a hold, an axis lies within 0.3° of a five-fold axis by chance in one core in 12,159, of any gate axis in one in 2,353. NOT CLAIMED: that the coupling exists or what its size and sign are; that any loss acts; which frame holds spacetime; anything about tides, arriving matter or mergers beyond their naming; terms of degree 12 and above. KILL-CONDITIONS: a spin-axis energy in C₃₀'s frame shown to depend on the axis's direction at degree 2 or 4 (the axis's surroundings then lack C₃₀'s frame's symmetry); the turnings that keep the lattice shown to be other than the 60 (or the cube's 24); a lattice hold shown to keep a slowly spinning axis off every gate axis; should a held core's beat ever be measured, five beats a turn rule out every rational frame and four rule out C₃₀'s, while two, in a slowly spinning core held by a smooth lattice energy, would contradict the lowest term of both. OPEN: the coupling that sets the hold's size and sign, a mechanism not yet found; the loss that brings a tilt down; whether registration itself catches (T244); tides, arriving matter and mergers; terms of degree 12 and above; the rational frames that beat six times a turn; a measurement that reaches a core's beat.