PUH Theorem 346 (The Accretion Map Is a Distribution, Not a Function) — The Terrestrial Anchor Verified from Geometry Gives core:body = 8.37×10⁻⁴, So a 100 M☉ Star Carries 0.084 M☉ of Core and Would Need 3,750 M☉ to Reach the Failed-Supernova Threshold

Abstract

Photonic Universe Hypothesis (PUH) — Quantification and Structural Result. AN ANCHOR VERIFIED FROM GEOMETRY, A CONSTANT-FRACTION PICTURE EXCLUDED QUANTITATIVELY, AND TWO FILED PAPERS SHOWN TO REST ON AN OBJECT THAT MAY NOT BE A FUNCTION AT ALL. THE GAP. T341 places a threshold at 3.14 M☉ where the Shell reaches the collapse bounce radius; T345 places another at 8.01 M☉ where it overtakes the core's own mass radius. Both state plainly that their boundaries are sharp in CORE mass and unlocated in PROGENITOR mass, because the framework cannot say how much core a star of given mass carries. T291, a corrigendum filed in May, approached the same gap from the other side — correcting an earlier seed-mass estimate 1,250-FOLD and recording that the accretion map from core mass to stellar mass is monotonic but "UNQUANTIFIED". THE SAME MISSING OBJECT, DESCRIBED FROM OPPOSITE ENDS. THEOREM 346.1 (the anchor, checked rather than assumed). The archive's one measured core candidate is terrestrial: ~5×10²¹ kg, stated to derive from "the standard 5 percent inner-core density deficit reinterpreted as Planck core mass fraction". THAT DERIVATION WAS CHECKED RATHER THAN ACCEPTED, because the paper carrying it forward is itself a corrigendum that corrected a figure by three orders of magnitude — a number that has already moved once should not be trusted on citation. COMPUTED FROM GEOMETRY: an inner core of radius 1,221 km at ~13,000 kg m⁻³ gives 9.91×10²² kg, and five percent of that is 4.96×10²¹ kg — REPRODUCING THE ARCHIVE'S FIGURE TO ONE PERCENT. Hence core:body = 5×10²¹/5.972×10²⁴ = 8.37×10⁻⁴. ∎ A CORRECTION RECORDED: an objection raised during this check — that five percent of the inner core is a thousand times smaller than the candidate, so the derivation was impossible — rested on a recalled inner-core mass of 10²⁰ kg, WRONG BY A FACTOR OF 991, and is withdrawn. THEOREM 346.2 (the constant fraction is excluded). On a constant ratio, a 1 M☉ star carries 0.00084 M☉ of core, a 25 M☉ star 0.0209, and a 100 M☉ star 0.0837 — 37× short of the lower threshold and 96× short of the upper. The progenitor masses required are 3.14/8.37×10⁻⁴ = 3,750 M☉ and 8.01/8.37×10⁻⁴ = 9,570 M☉. NO SUCH STARS EXIST; the most massive known are a few hundred solar masses. So on a constant fraction the thresholds are never reached, failed supernovae never occur — AND THEY ARE OBSERVED. ∎ The exclusion is FIRMER than it would have been before Theorem 346.1, precisely because the anchor survived checking: a refutation resting on a doubtful number is weak; one resting on a number reproduced from geometry is not. THEOREM 346.3 (the map is a distribution, not a function). T172 establishes that cores merge and cannot split — splitting would require one set of conserved Casimir values to become two — so a body's core is THE ACCUMULATED SUM OF EVERY PRIMORDIAL SEED IT HAS SWALLOWED. That is a HISTORY rather than a property of present mass, and two stars of equal mass with different histories carry different cores. THE ACCRETION MAP IS THEREFORE NOT A FUNCTION FROM STELLAR MASS TO CORE MASS BUT A DISTRIBUTION OVER CORE MASSES AT FIXED STELLAR MASS. That is why the missing quantity has resisted being written down: IT DOES NOT EXIST IN THE FORM SOUGHT. T291 called the map "unquantified"; on this reading it is unquantifiable AS A FUNCTION, though its distribution may be characterisable. RESULT 346.4 (a prediction that was a necessity). T341 predicted that failed supernovae should track merger history rather than progenitor mass, presenting it as an inference from amalgamation. ON THE PRESENT READING IT IS A NECESSITY: if core mass is a distribution at fixed stellar mass, nothing about the stellar mass alone can determine which side of a threshold a given star falls on. The merger-history prediction is not one option among several; it is the only kind of prediction the structure permits. WHAT THE FRAMEWORK CAN CLAIM: that two classes of collapse outcome exist, separated by core-mass thresholds; that failed supernovae should correlate with binary-interaction signatures AT FIXED PROGENITOR MASS; that the correlation with mass alone should be weaker than the standard picture predicts. WHAT IT CANNOT: which stars fail; a critical progenitor mass; that any particular object exceeded a threshold. THE FRAMEWORK PREDICTS A POPULATION, NOT AN INDIVIDUAL. KILL-CONDITIONS: (i) if the Earth's inner-core density deficit has an ordinary explanation — lighter alloying elements in the iron, which is the standard reading — the anchor is not a core measurement at all and every number here loses its basis; THE DEFICIT IS REAL AND ITS ATTRIBUTION IS THE FRAMEWORK'S CLAIM, NOT A MEASUREMENT; (ii) if amalgamation concentrates seeds by three to four orders of magnitude in some systems, the thresholds become reachable and the distribution must be that broad — a strong requirement the archive supplies no mechanism for; (iii) if cores can split after all, core mass ceases to be a history and the map may be a function; (iv) if failed supernovae are found to track progenitor mass tightly at fixed binary history, Result 346.4 fails and with it T341's prediction. NOT CLAIMED: that the distribution has been characterised, only that a function was the wrong object to seek; that the thresholds are wrong — T341's and T345's derivations are untouched, and only their applicability to named stars is at issue; that amalgamation can supply the required concentration; that the terrestrial anchor is correct, only that its stated derivation reproduces its stated value; or that any observational test has been performed.