PUH Theorem 345 (Three Regimes, Two Failure Modes) — The Shell Crosses the Bounce Radius at 3.14 M☉ and the Core's Own Mass Radius at 8.01 M☉, Bracketing a Regime in Which the Shock Cannot Form but Mass Remains Outside; and T267's 0.66 Is the Density Law's Area Exponent

Abstract

Photonic Universe Hypothesis (PUH) — Reconciliation and New Structure. AN APPARENT THIRD RADIUS LAW RECONCILED, A GENUINE SECOND CROSSOVER FOUND, AND A REGIME BRACKETED THAT HAD NOT BEEN NAMED. THE PROBLEM. The PUH archive carries three exponents for what looks like the same object's size: R_mass ~ M^(1/3) from the density law (T259/T260), r_shell ~ M¹ from the Shell placement (T298/T300), and an empirical heat-flux fit at M^0.66 (T267). Across a factor of thirty in mass these spread by a factor of ten — 3.11, 9.44 and 30.00 — and they cannot all describe the same radius. T267's own text says its composite exponent "matches neither" the volumetric 0.33 nor the surface-concentrated 0.50. THEOREM 345.1. The empirical exponent is an AREA, not a radius. T267 itself identifies 2/3 as "its clean structural value", and 2×(1/3) = 0.6667 agrees with the fitted 0.66 to one percent. A HEAT FLUX IS ENERGY PER UNIT AREA PER UNIT TIME, so fitting it against mass returns the AREA scaling: for the density law R ~ M^(1/3), area ~ R² ~ M^(2/3). T260 already writes exactly this — "the density radius-law R ~ M^(1/3), area ~ M^(2/3)" — and T267 explicitly declines to read its exponent as a statement about where shell mass sits. ∎ THE MEASUREMENT AGREES WITH THE DENSITY LAW ONCE ONE NOTICES WHAT IS BEING MEASURED, AND THERE ARE TWO RADIUS LAWS, NOT THREE. THEOREM 345.2. The two surviving laws have different mass dependence, so their ratio grows as M^(2/3) and the Shell must eventually overtake the mass radius. At nuclear density ρ ≈ 2.3×10¹⁷ kg m⁻³: at 1 M☉ the Shell is 3.18 km against a mass radius of 12.73 km; at 3.14 M☉, 9.99 against 18.65; at 5 M☉, 15.91 against 21.77; at 8.01 M☉ both are 25.48 km; at 15 M☉, 47.72 against 31.40. THE SHELL OVERTAKES AT 8.01 M☉ — below it the mass distribution extends beyond the Shell, above it the Shell encloses the mass entirely. ∎ RESULT 345.3. T341 found a first crossing at 3.14 M☉, where the Shell reaches the 10 km bounce radius. THE TWO CONDITIONS ARE BOTH ABOUT NUCLEAR DENSITY AND DIFFER ONLY IN WHOSE: T341 uses the radius of the INFALLING STELLAR CORE, about 0.5 M☉ compressed, fixed at ~10 km and independent of the Planck core's mass; this note uses the radius of the PLANCK CORE'S OWN mass, which grows as M^(1/3). They are distinct transitions, not one quantity computed twice — the earlier is a condition against an external fixed scale, this one relates two internal scales. AND THEY DO NOT AGREE: 8.01 against 3.14 is a factor of 2.6. THAT IS THE POINT RATHER THAN A DISCREPANCY — two different conditions crossing at two different masses BRACKET A REGIME BETWEEN THEM. RESULT 345.4 (three regimes, and the middle one is new). REGIME I, M < 3.14 M☉: Shell inside the bounce radius, collapse stiffens, shock forms, SUPERNOVA SUCCEEDS. REGIME II, 3.14–8.01 M☉: Shell outside the bounce radius but inside the mass — THE SHOCK CANNOT FORM, BUT MASS REMAINS OUTSIDE AND RADIATES. REGIME III, M > 8.01 M☉: Shell encloses the mass entirely, DIRECT COLLAPSE WITH NOTHING LEFT OUTSIDE. REGIME II HAS NOT BEEN NAMED IN THE ARCHIVE, which has treated failed supernovae as a single class. It predicts two failure modes rather than one: a failed supernova WITH a residual envelope, and one WITHOUT. RESULT 345.5 (the observational difference). Both regimes give no explosion; they differ in whether mass lies outside the Shell, and therefore in whether a residual infrared source is expected. THE LEADING CANDIDATE FITS REGIME II: N6946-BH1 left a persistent infrared source at roughly a tenth of its progenitor's luminosity — mass still outside the Shell, still radiating. REGIME III WOULD LEAVE NOTHING WHATEVER. Searches for disappearing stars have generally treated a persistent infrared counterpart as supporting the failed-supernova reading; on this account ITS ABSENCE WOULD BE EQUALLY DIAGNOSTIC, of a different and more extreme case. So the framework predicts two classes of disappearing star, separated at about eight solar masses of core. KILL-CONDITIONS: (i) THE CAVEAT T341 CARRIED IS UNCHANGED AND IS NOT SOFTENED — these are CORE masses, not progenitor masses, and the framework still cannot say how much core a given star carries, so the boundaries are sharp in core mass and unlocated in progenitor mass; (ii) if the Planck core's mass does not follow the density law at nuclear density — IF ITS INTERNAL DENSITY IS FAR HIGHER, AS "PLANCK CORE" MIGHT SUGGEST — the mass radius is much smaller and the crossover moves to lower mass or vanishes; (iii) if the bounce radius differs materially from 10 km, T341's boundary moves and the middle regime narrows or widens; (iv) if disappearing stars are found with no residual source but low inferred core mass, the regime ordering is wrong. NOT CLAIMED: that N6946-BH1 is in Regime II, only that its residual source is consistent with it; that any star has been assigned a regime, since core masses are not measurable; that the density law applies inside the Shell, which is an extrapolation; that 8.01 M☉ is precise, since it scales as ρ^(−1/2); or that T267's fit is thereby validated, only that its exponent is reconciled rather than competing.