PUH Theorem 287 (Derivation) — The Core Mass Spectrum from Amalgamation-Only Growth: Horizon-Like Cores Force a Coagulation Kernel of Degree 2, Hence Gelation (the Amalgamated Planck Star, Derived Not Assumed) and a Pre-Gel Spectrum with Exponent 5/2; T285's Kill-Condition Is Satisfied

DERIVATION (T287). Derives the PUH core mass spectrum — the quantity the archive repeatedly listed among the things it explicitly does NOT claim ("a derivation of the rebound mass spectrum"), and which gates the Skeleton Galaxy (T285), the core-to-host-mass relation, and the failed-supernova critical mass. Premise (already filed): T156 proves the Planck Shell is a PERFECT REFLECTOR, so nothing crosses it, so nothing is accreted — AMALGAMATION IS THE ONLY GROWTH CHANNEL T156 LEAVES STANDING. Consequence (new): the core population is a COAGULATION system; for HORIZON-LIKE cores the merger kernel's homogeneity degree is FORCED to 2 by dimensional analysis; hence GELATION (= the Amalgamated Planck Star, derived not assumed) and a gel-point spectrum with exponent tau = (3+lambda)/2 = 5/2 = 2.5. T285's pre-declared kill-condition required alpha > 2. It is 2.5. SATISFIED. Tags: DERIVED / VERIFIED-AGAINST-PRIMARY-LITERATURE / ASSUMPTIONS-STATED / OPEN. === DESCRIPTION (abstract) === Photonic Universe Hypothesis (PUH) — Derivation. ATTRIBUTION NOTE. Arthur Eddington did NOT hold that black holes grow by consuming matter; he did not believe black holes existed. At the Royal Astronomical Society in 1935 he ridiculed Chandrasekhar's collapse result, declaring "I think there should be a law of nature to prevent the star from behaving in this absurd way." What Eddington DID contribute is the EDDINGTON LIMIT on accretion — which, as Section 8 shows, is precisely the constraint PUH escapes. THE PREMISE IS ALREADY FILED. T156 proves the Planck Shell is a perfect reflector: it sits on the C30 = c* Casimir level set and no local process carries anything across it. If nothing crosses, nothing is absorbed, and A CORE CANNOT GROW BY ACCRETION. Amalgamation — core merging with core — is not one option among several; it is the only one T156 leaves standing. This is not a new postulate. It is a consequence of a theorem filed in April 2026 and carried unremarked ever since. THE KERNEL IS FORCED, NOT CHOSEN. A population growing solely by pairwise merging is a COAGULATION system (Smoluchowski, 1916), and its behaviour is fixed by one number: the homogeneity degree lambda of the merger kernel, K(a*m1, a*m2) = a^lambda K(m1,m2). For a horizon-like core (r = 2Gm/c^2) there is exactly ONE length scale, GM/c^2, so ANY merger cross-section — geometric contact, gravitational focusing, or gravitational-wave capture — must be proportional to (GM/c^2)^2 times a dimensionless function of v/c. Hence sigma ~ m^2 and lambda = 2. There is nothing else it could be. (The direct calculation confirms it, with an elegance en route: substituting r = 2Gm/c^2 gives v_esc^2 = c^2 EXACTLY — the escape velocity at a Schwarzschild horizon IS c — so gravitational focusing is always maximal, and K = 4 pi G^2 (m1+m2)^2 / (c^2 v_rel).) FIRST RESULT — GELATION, AND IT IS THE AMALGAMATED PLANCK STAR. Coagulation with lambda > 1 undergoes GELATION: rigorously established (Jeon; Escobedo-Mischler-Perthame; see Fournier, arXiv:2501.03867). Gelation means that in FINITE time a runaway cascade removes a finite fraction of the total mass from the population and deposits it in a single object of divergent mass. The coagulation literature states the astrophysical reading in exactly these terms: gelation "provides a natural description of runaway growth, in which a small number of clusters rapidly decouple from the background population and accumulate a dominant share of the available mass." THEOREM 287.1: horizon-like cores growing only by amalgamation have lambda = 2 > 1 and therefore MUST gel. THAT OBJECT IS THE AMALGAMATED PLANCK STAR. PUH's pre-Rebound cosmology is not an assumption of the framework; it is what a degree-2 coagulation kernel is COMPELLED to do. SECOND RESULT — THE SPECTRUM. At the gel point the distribution is a pure power law with exponent fixed by lambda alone. Verbatim from the literature (arXiv:2606.14768, eq. 3.3): "At the gel point the distribution itself becomes scale-free, decaying as a pure power law, n_k(t_c) ~ k^(-tau), tau = (3+lambda)/2" — and lambda = 2 "reproduces the mean-field exponent tau = 5/2 familiar from Flory-Stockmayer polymerisation theory" (Stockmayer 1943; Flory 1953). THEOREM 287.2: dN/dm ~ m^(-tau) with tau = (3+2)/2 = 5/2 = 2.50. T285'S KILL-CONDITION IS SATISFIED. T285 declared, in advance of any derivation: the light end dominates the mass budget — and un-accreted cores can be the dark matter — if and only if alpha > 2; "a derivation of the rebound mass spectrum yielding alpha < 2 falsifies this hypothesis." alpha = tau = 2.5 > 2. The Skeleton Galaxy survives the test it set for itself, by derivation rather than by fit. A FURTHER CONSEQUENCE — THE REBOUND SPECTRUM MAY BE UNNECESSARY. Smoluchowski scaling solutions are ATTRACTORS: the distribution approaches its self-similar form irrespective of its initial condition. If cores have amalgamated for cosmological times, their present spectrum is set by the coagulation attractor, not by whatever the Rebound produced — the dynamics has forgotten it. The gate that has blocked the Skeleton Galaxy, the core:host relation, and SP-1 is not opened here; it is argued to be BYPASSABLE. (Tagged as a consequence: it requires that enough coagulation time has elapsed, which is not quantified here.) THE SHARP STRUCTURAL TEST — HORIZON-LIKE OR NOTHING. HORIZON-LIKE cores (r ~ m, PUH's own ontology): lambda = 2, tau = 2.50 — above 2, THE SKELETON GALAXY LIVES. CONSTANT-DENSITY cores (r ~ m^(1/3)): lambda = 2/3, tau = 1.83 — below 2, THE SKELETON GALAXY DIES; and lambda < 1 means NO GELATION, hence no Amalgamated Planck Star and no Rebound — the entire PUH cosmology would fail with it. EQUIPARTITIONED rather than virialised velocities: lambda = 5/2, tau = 2.75 — still above 2, the result is ROBUST. PUH's commitment to horizon-like cores is therefore load-bearing and now falsifiable through this route. THE EDDINGTON ESCAPE. The Eddington limit caps growth BY ACCRETION, and it is currently a source of acute difficulty: JWST finds billion-solar-mass black holes a few hundred million years after the Big Bang, which Eddington-limited accretion from stellar seeds cannot reach in time — the literature states the tension directly and the remedies proliferate (direct-collapse seeds, super-Eddington flows, modified cross-sections). MERGERS HAVE NO EDDINGTON LIMIT. If cores grow only by amalgamation — as T156 requires — the constraint strangling accretion models does not apply. This is not a remedy invented to fit the observation; it is a structural consequence of a theorem filed months earlier. Whether PUH's amalgamation RATE suffices is a quantitative question NOT answered here; the claim is narrower: the obstacle is absent by construction. ASSUMED AND STATED PLAINLY: that cores merge on contact as compact objects do; that the population is virialised (the equipartition variant is checked and does not change the verdict); that mean-field Smoluchowski dynamics applies (spatial clustering does not qualitatively alter the kernel); and that cosmological expansion does not prevent the attractor being reached. NONE OF THESE IS DERIVED HERE. OPEN: the effect of cosmological expansion on the gelation time; the coagulation time needed to reach the attractor (on which the bypass argument depends); whether perfectly-reflecting shells merge as assumed; the quantitative early-supermassive-object calculation. FLAGGED, USED NOWHERE: tau = 2.5 lies within 6% of the Salpeter stellar IMF slope, 2.35. IF stellar masses track their seed cores, the IMF would be a direct measurement of this spectrum's heavy tail. That relation is exactly what the archive lists as unclosed. Recorded; leaned upon nowhere. NOT CLAIMED: that the rebound (fragmentation) spectrum is derived — it is argued IRRELEVANT, a weaker and different statement; that dark matter is explained (the Skeleton Galaxy remains a hypothesis, now with its kill-condition survived rather than merely stated); or that any observational test has been performed.