PUH Theorem 341 (A Structural Criterion for Failed Supernovae) — The Throughput Criterion Fails by Forty Orders, but Requiring the Shell to Reach the Bounce Radius Gives M_crit = 3.14 M☉; and the Framework Predicts Failed Supernovae Track MERGER HIST

Photonic Universe Hypothesis (PUH) — Mechanism Replaced. THE OLD CRITERION IS ABANDONED. The Stellar Planck Core Observational Lead Note interprets vanishing stars as an absorption-dominated regime in which accumulated material exceeds what the Shell can process. T328 showed that criterion INVERTS: a Planck core is horizon-scale, so capacity grows as M², making larger cores MORE able to process, not less. THIS NOTE ESTABLISHES SOMETHING WORSE. Using the surface flux the framework itself supplies (measured at Pluto, adopted in T330), a 1.4 M☉ core processes 6.9×10⁵ W while a core-collapse supernova releases ~10⁴⁶ J in ~10 s, a power of 1.0×10⁴⁵ W. THE RATIO IS 6.9×10⁻⁴⁰ — the Shell handles one part in 10³⁹, NOT FOR LARGE CORES BUT FOR EVERY CORE. A core would need 5×10¹⁹ M☉ before its throughput mattered, and there is no such star. A criterion never satisfied cannot distinguish an explosion from an implosion, and explosions are observed. T328's escape (b) — a larger stellar surface rate — would require a factor of 10⁴⁰, which is not a correction but a different physical quantity. THEOREM 341.1 (the structural criterion). A collapsing stellar core halts when its inner region reaches nuclear density and STIFFENS, at a radius near 10 km enclosing roughly 0.5 M☉; the shock forms there and propagates outward. IF THE PLANCK SHELL ALREADY OCCUPIES THAT RADIUS, THERE IS NOTHING TO BOUNCE OFF — the infalling material meets an absorbing boundary where it needed a stiffening one. Setting r_shell(M) = (2GM/c²)·X/(X−1) = 10 km with X ≈ 14 gives M_crit = 3.14 M☉. Tabulated: 0.10 M☉ → 0.32 km; 0.50 M☉ → 1.59 km; 1.00 M☉ → 3.18 km; 3.14 M☉ → 10.0 km (threshold); 3.40 M☉ → 10.8 km. Below the threshold the bounce proceeds normally; above it, the core has already swallowed the region where the shock must form. ∎ RESULT 341.2 (the number lands where it should, unfitted). M_crit = 3.14 M☉ falls between the heaviest observed neutron star (~2.1 M☉) and the lightest confirmed black holes (~3–5 M☉) — THE MASS GAP, precisely where failed supernovae are expected to leave remnants. THE SHELL PLACEMENT WAS FIXED BY T298 AND T300 IN JULY FOR ENTIRELY DIFFERENT REASONS — the coordinate-invariant tension identification and the accretion floor — AND NEITHER HAD FAILED SUPERNOVAE IN VIEW. Contrast: the throughput criterion predicts 5×10¹⁹ M☉, an error of nineteen orders; the structural criterion lands within a factor of two. The difference is that tension and geometry are quantities this framework derives, whereas the surface flux was measured at one small body and extrapolated forty orders to a regime nothing constrains. THEOREM 341.3 (the core fraction cannot be universal). The framework's one measured core-mass candidate is the Earth's: the inner-core density deficit suggests ~5×10²¹ kg, a fraction of 8.4×10⁻⁴ (0.084 percent); the Moon's is constrained below 1.5 percent. On that fraction the Sun would carry 0.0008 M☉ of core and a 25 M☉ progenitor 0.021 M☉ — 150× SHORT of the threshold. ON A UNIVERSAL FRACTION NO STAR EVER REACHES IT AND FAILED SUPERNOVAE NEVER OCCUR. Since they are observed, the fraction must vary. ∎ AND THE FRAMEWORK SAYS WHAT MAKES IT VARY: AMALGAMATION. T172 establishes that cores merge and cannot split — splitting would require one set of conserved Casimir values to become two — so a star's core is not a fixed fraction of its mass but THE SUM OF EVERY PRIMORDIAL SEED IT HAS SWALLOWED. The question is not what fraction but how many seeds, and that is A HISTORY RATHER THAN A CONSTANT. RESULT 341.4 (a prediction contradicting the standard expectation). Conventional astrophysics holds that failed supernovae track PROGENITOR MASS, the heaviest stars collapsing directly. THIS FRAMEWORK PREDICTS THEY TRACK MERGER HISTORY, with mass mattering only because massive stars are more often in interacting binaries. THESE SEPARATE OBSERVATIONALLY: if failed supernovae occur preferentially in stars carrying binary-interaction signatures AT FIXED PROGENITOR MASS, the amalgamation picture is favoured; if they track mass alone, it is not. A 25 M☉ star with an ordinary history does not reach the threshold on these numbers; one that has swallowed a companion's core might. That is a population-level test, and the samples of disappearing stars are now large enough to attempt it. ONE DETAIL ON THE LEADING CANDIDATE: the 2026 reanalysis of N6946-BH1 had to exclude a stellar-merger interpretation for the REMNANT; whether the PROGENITOR had a merger history is a different question, and the data may already bear on it. KILL-CONDITIONS: (i) if the bounce radius differs materially from 10 km — it depends on the nuclear equation of state and progenitor structure — M_crit scales linearly with it; (ii) if X is much larger than the accretion floor, the Shell approaches the horizon and M_crit rises; (iii) if failed supernovae track progenitor mass at FIXED merger history, Result 341.4 fails; (iv) if the Earth's inner-core density deficit has an ordinary explanation, the framework has no measured core-mass candidate at all and Theorem 341.3's starting point disappears — THE DEFICIT IS A REAL OBSERVATION BUT ITS ATTRIBUTION TO A PLANCK CORE IS THE FRAMEWORK'S CLAIM, NOT A MEASUREMENT. NOT CLAIMED: that N6946-BH1 contains a Planck core, which no observation demonstrates; that its failed-supernova interpretation is established, since the 2026 analysis states a black hole is not proved; that the amalgamation history of any particular star has been computed; that M_crit = 3.14 M☉ is precise, since it scales with an uncertain bounce radius; or that the throughput criterion was wrong in every form — only that it cannot work on the surface flux the framework currently supplies.