PUH Theorem 390 — Spin Opens a Planck Core's Poles but Does Not Speed Its Drain: the Core Never Fills, T244's Gate Averages to Its Geometry, and the Open Polar Caps Grow With Spin

Photonic Universe Hypothesis (PUH) — Theorem paper. THE QUESTION. In the author's picture (T388) a Planck core's Shell unfolds the matter waiting at it into high-energy photons; only they enter the core; and as the Shell rotates, its magnetic field sets up the conditions for vortices to open at the poles, through which some of the light escapes. T389 took the unfolding rate to be one rate per square metre of Shell, the same for every core, with nothing to do with spin or mass. The author asked whether spin could matter after all: a fast-spinning core has a stronger field, so its polar vortices are bigger and let out more — could that permit more unfolding, making the rate contingent on spin? T244 asked the same of its rotational gate and left it open. The author asked that the mathematics decide. RESULT 390.1 — THE CORE NEVER FILLS, SO THE POLES CANNOT SPEED THE DRAIN. Letting more out at the poles could raise the unfolding only if the drain were backed up. T386's count, one quantum per Planck cell (7.4880e78 J per square metre of Shell), says how much a core can hold. In T357's light bulb a core's own mass is light on its Shell: 7.5270e-32 of the count for Earth's core, 1.8771e-40 for a solar-mass core, 4.5341e-47 for Sgr A*, 2.8879e-50 for M87*; the share goes as 1/M, to 1.7154e-2 for a core of one Planck mass. At the top of T389's window 13.8 billion years of unfolding add 7.8590e-52 of the count, the same for every core; filling a Shell would take 1.7559e61 years. The release at the poles and the unfolding at the Shell are two valves, and opening the first does not open the second. RESULT 390.2 — T244'S GATE AVERAGES TO ITS GEOMETRY. Spin could reach the unfolding only through T244's gate. The gate T244 states is a vent: with n registration windows of half-width delta in each turn, it stands open a share n*delta/pi of the time at every spin, since faster spin brings more windows, each shorter by the same factor. An illustrative vent (eight windows of +-0.05 rad, share 0.12732), sampled at the same ten million random times at 1e-3, 1, 1e3 and 1e6 rad/s, gives 0.12747, 0.12727, 0.12720 and 0.12727, within the 1.1e-4 sampling error. If a fold-through takes a time tau, spin can only lower the rate; for a Planck-time event it does so in no core as massive as Earth's unless the windows are narrower than 1.019e-30 rad (the bound grows as 1/M, to 0.2321 rad at one Planck mass). T244's other kind of gate, its threads and gears, passing a set amount at each registration, would rise with the turn rate, which at a given spin falls as 1/M: at spin 0.7 Sgr A* would unfold 2.4155e-7 and M87* 1.5385e-10 of what a solar-mass core does per square metre, against T243 — unless every core turned at one rate, T244's Resolution A, which T387 found at odds with measured spins. Which kind E8 gives is T244's deferred analysis. RESULT 390.3 — THE ARCHIVE'S SPIN-DOWN RELEASE IS NOT THE JET. T370's spin-down power, L = B^2 R^6 Omega^4/(6c^3) with the London field, is the same for every core at the same spin and grows as the sixth power of the spin: 7.7943e7 W at spin 0.7 with electron carriers; 6.6250e8 W (electron) and 2.2336e15 W (proton) at spin 1; at most 1.9577e16 W at the fastest spin (2.9365e16 W for a plasma-filled aligned field, Spitkovsky 2006) — 1.9577e-19 of M87*'s jet at the EHT's conservative 1e42 erg/s. If that jet is light its poles release, it is stored light let out, not spin-down power. RESULT 390.4 — SPIN SETS THE SIZE OF THE POLAR OPENINGS. If the Shell's condensate is charged, as the Shell-Dynamo Theorem postulates, rotation is answered by the London field (measured by Hildebrandt 1964 and Tate et al. 1989) and needs no vortices. If the field closes outside the Shell as a dipole turning with it, the lines that reach the light cylinder stay open (Goldreich and Julian 1969) and leave from two polar caps covering, in flat space, 1 - sqrt(1 - 0.69643 chi) of the Shell — 3.5 per cent at spin 0.1, 19 at 0.5, 41 at 0.94 — whatever the core's size; the clock at the Shell (T299) makes these uncertain by up to about five times either way, though not their growth with spin. A core turning with its planet keeps them all but shut: 9.6486e-19 of Earth's core's Shell. Faster spin, larger openings: the author's chain holds for the release, with the openings set by the spin's light cylinder rather than by the field's strength. RESULT 390.5 — WHAT IT PREDICTS, AND THE CONSTRAINTS. A core's release follows its spin and its unfolding does not. If each square metre of open cap lets out the same light at every core, T389's comparison of the giants' polar shares puts Sgr A*'s spin below about 0.13 to 0.28 for M87* at 0.5 to 0.94, against Sgr A*'s quiet output, if seen (0.0014 to 0.0029 with a steady 1e44 erg/s jet). M87*'s jet requires spin in the EHT's models, which reject all non-spinning ones, and stellar orbits put Sgr A* below about 0.1 (Fragione and Loeb 2020): both fit. The EHT's high-spin models of Sgr A* (above 0.5; Fragione and Loeb 2022) fit with their own luminosity, and not with the quiet output: OPEN, a mechanism not yet found. NOT CLAIMED: that E8 has registration windows, of which kind, or their width; how long a fold-through takes; that the Shell's condensate is charged, beyond the Shell-Dynamo Theorem's postulate; that the field outside the Shell is a dipole turning with it; the caps' size beyond flat-space estimates; how light crosses the Shell at the open caps, or how much leaves per square metre; that the jets' asymmetry comes from the field's direction at the two caps; the spins of Sgr A* and M87*. KILL-CONDITIONS: a core shown to come near T386's count before the rebound; a core's unfolding shown to change with its spin at the same Shell area through a vent; E8's gate shown to pass a set amount at each registration (T243 fails instead); M87*'s jet shown to be light its poles release yet powered by its spin's energy; the field outside a Shell shown not to open at the poles, or its open area not to grow with spin; with each square metre of open cap letting out the same light at every core, Sgr A*'s spin shown above the bound while its quiet output is seen. OPEN: whether E8's rotational symmetry gives T244's gate, and of which kind (T244, T262); the field outside the Shell, what carries it to the light cylinder, and where the light cylinder falls on the Shell's clock (T299); how light leaves through the open caps past T308's non-crossing and T355's missing momentum, and how much per square metre; the jets' asymmetry — an aligned dipole is its own mirror image, so a difference needs a process that tells mirror images apart, or a field or waiting matter that differs between the poles (T388, T242); Sgr A*'s spin; the carrier, electron or proton (T387); the mapping of spin to turn rate at the Shell in curved space.