PUH Theorem 287 (Derivation) — The Core Mass Spectrum from Dual-Channel Growth: Amalgamation and Photon Absorption Independently Force Finite-Time Runaway to the Amalgamated Planck Star; the Gel-Point Exponent 5/2 Survives Because Growth Scales as Mass Squared; T285's Kill-Condition Is Satisfied

DERIVATION. Supersedes an earlier draft withdrawn before circulation, whose premise ("amalgamation is the ONLY growth channel") was too strong. The Planck Shell perfectly reflects GRAVITATIONAL WAVES (bulk substrate ripples, which need internal degrees of freedom to propagate and find exactly zero in the core — T175). PHOTONS ARE NOT BULK RIPPLES: they are E8 SURFACE modes, and they are ABSORBED. This is PUH's position from inception and is how the final Planck Star draws spacetime back into itself. The correction ADDS a growth channel and OVERDETERMINES the conclusion. Tags: DERIVED / SUPERSEDES-WITHDRAWN-DRAFT / VERIFIED-AGAINST-PRIMARY-LITERATURE / ASSUMPTIONS-STATED / OPEN. === DESCRIPTION (abstract) === Photonic Universe Hypothesis (PUH) — Derivation. THE CORRECTED ONTOLOGY. The Planck core is three things at once, and they are distinct. (1) A PERFECT MIRROR to gravitational waves: a bulk substrate ripple propagates only through a medium with internal degrees of freedom to carry it, and by T175 the fully-formed core has EXACTLY ZERO — infinite impedance mismatch, lossless reflection. This is the sea-wall. (2) A PERFECT ABSORBER of photons: photons are E8 SURFACE modes, not bulk ripples; the impedance argument does not transfer, and they cross and are held. (3) NEVER AN EMITTER: by T206 the Phase I to Phase III transition is irreversible. Bulk reflected, surface absorbed, nothing emitted. (Noted honestly: T156's ALGEBRAIC Casimir-level-set formulation, which forbids ANY Hamiltonian trajectory from crossing, is as literally stated too strong — it would forbid photon absorption too. The IMPEDANCE formulation is the correct one and is bulk-specific by construction. A refinement of T156's statement is OWED and is flagged, not performed.) CHANNEL A — AMALGAMATION. Cores merging pairwise are a COAGULATION system (Smoluchowski 1916), whose behaviour is fixed by the homogeneity degree lambda of the merger kernel. For a horizon-like core (r = 2Gm/c^2) there is exactly ONE length scale, GM/c^2, so ANY merger cross-section — contact, gravitational focusing, GW capture — MUST scale as m^2. Hence lambda = 2. (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 focusing is always maximal, and K = 4 pi G^2 (m1+m2)^2 / (c^2 v_rel).) THEOREM 287.1: lambda = 2 > 1 forces GELATION, rigorously established (Jeon; Escobedo-Mischler-Perthame; Fournier arXiv:2501.03867) — in finite time one object decouples and accumulates a dominant share of all the mass. CHANNEL B — PHOTON ABSORPTION. T213 (Equal-Rate Theorem) gives throughput = R_sub x A_shell. With photons absorbed rather than reflected, this IS the core's growth law. For a horizon-like core A = 16 pi G^2 m^2 / c^4, hence THEOREM 287.2: dm/dt = k m^2, which integrates to m(t) = m0/(1 - k m0 t) and BLOWS UP AT THE FINITE TIME t* = 1/(k m0). The blow-up time is INVERSELY proportional to the initial mass: the heaviest core runs away first, and fastest. A runaway cascade to a single dominant object — obtained from T213 alone, with no coagulation theory involved. THE AMALGAMATED PLANCK STAR IS OVERDETERMINED. Two channels, two independent filed theorems, one conclusion — both compelled by the same structural fact: cores are horizon-like, so AREA SCALES AS MASS SQUARED. The merger cross-section is an area; the folding throughput is an area; both carry m^2; both produce runaway. PUH's pre-Rebound cosmology is derived TWICE, and it would take the failure of BOTH channels to remove it. THE SPECTRUM SURVIVES, AND THE REASON IS EXACT. A growth term ordinarily disturbs a coagulation spectrum, so survival must be shown. Solving growth by characteristics: m0 = m/(1+kmt), dm0/dm = (1+kmt)^(-2), and number is conserved along characteristics, so for n0(m) = A m^(-tau): n(m,t) = A m^(-tau) (1 + k m t)^(tau-2). For LIGHT cores kmt << 1 and the correction tends to 1 — THE SLOPE IS UNTOUCHED. A light core, by the m^2 law, is almost not growing at all. HEAVY cores run away and LEAVE the population — precisely what gelation already said. Therefore coagulation SETS the light-end exponent and photon absorption PRESERVES it. At the gel point (arXiv:2606.14768 eq. 3.3): tau = (3+lambda)/2, and lambda = 2 reproduces the Flory-Stockmayer mean-field exponent tau = 5/2 (Stockmayer 1943; Flory 1953). THEOREM 287.3: dN/dm ~ m^(-tau), tau = 5/2 = 2.50. T285'S KILL-CONDITION IS SATISFIED. T285 declared in advance: 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. Satisfied — and satisfied at the LIGHT end, exactly the population the Skeleton Galaxy needs and exactly the population the new growth channel leaves alone. WHERE THE PHOTONS GO — the apparent conflict is Schwarzschild geometry. PUH's Earth-core work has the lattice tearing around the core, releasing gamma photons that thermalise in the surrounding iron (photons going OUT, powering the geodynamo). This paper has photons going IN. Both are correct: THE TEAR ZONE LIES OUTSIDE THE SHELL. A photon emitted at radius r escapes only within a cone with sin(theta) = (3sqrt3/2)(r_s/r) sqrt(1 - r_s/r). At the shell (r -> r_s) the escape fraction tends to ZERO — at 1.001 r_s, 99.8% are captured; at 1.1 r_s, 85%. At the photon sphere (1.5 r_s) it is exactly 50/50. At 3 r_s, 85% escape. Photons born DEEP in the tear zone feed the core; photons born in its OUTER reaches escape and heat the host body. A computable geometric factor, not a contradiction. NOTHING IN THE EARTH-CORE WORK REQUIRES RETRACTION, and the m^2 scaling on which Theorem 287.2 depends is unaffected by the efficiency factor. PREDICTION P287.1 — PUH FORBIDS HAWKING EVAPORATION. Evaporation requires a NET OUTWARD energy flow reducing the core's mass. T206 forbids it: the Phase I to Phase III transition is irreversible; cores are topologically permanent; the core absorbs and never emits. Sharpest test: primordial black holes near 1e15 g should, on the standard picture, be reaching the end of their evaporation and exploding NOW with a characteristic gamma-ray signature. None has been observed. PUH PREDICTS NONE EVER WILL BE. The prediction is weak as evidence (the non-detection is equally consistent with no such objects existing) but it is a genuine standing commitment and it is FALSIFIABLE: a confirmed detection of black-hole evaporation of any kind falsifies T206 and with it this paper's second growth channel. It also DISPOSES of the information paradox rather than solving it: the paradox arises because Hawking radiation is thermal AND the hole eventually vanishes. In PUH the hole never vanishes. Information falls in, is held, and is returned at the Rebound — the same statement as exact photon conservation, read from the other end. THE SHARP STRUCTURAL TEST — HORIZON-LIKE OR NOTHING. Horizon-like (r ~ m, PUH's ontology): sigma ~ m^2 AND A ~ m^2; Channel A gives lambda = 2 -> gelation, tau = 2.50; Channel B gives dm/dt ~ m^2 -> blow-up. BOTH FIRE; the Skeleton Galaxy LIVES. Constant-density (r ~ m^(1/3)): sigma ~ m^(2/3), lambda = 2/3 — BELOW 1, so NO GELATION; and dm/dt ~ m^(2/3) — sub-linear, so NO BLOW-UP EITHER; tau would be 1.83, below 2. BOTH CHANNELS FAIL TOGETHER — no Amalgamated Planck Star, no Rebound, no Skeleton Galaxy; the entire PUH cosmology falls. Equipartitioned velocities: lambda = 5/2, tau = 2.75 — still above 2; ROBUST. PUH's horizon-like core ontology is the load-bearing assumption of its own cosmology and is attackable through this route. ASSUMED AND STATED: cores merge on contact as compact objects do; the population is virialised (equipartition variant checked, verdict unchanged); mean-field Smoluchowski applies; cosmological expansion does not prevent the attractor being reached. NONE OF THESE IS DERIVED HERE. OPEN: the numerical value of R_sub (gated on the substrate equation of state; it sets the blow-up TIMESCALE but NOT the m^2 scaling Theorem 287.2 uses); the geometric efficiency factor; expansion's effect on the gelation time; the exact joint coagulation-with-growth problem; and the OWED refinement of T156's algebraic statement. FLAGGED, USED NOWHERE: tau = 2.5 lies within 6% of the Salpeter IMF slope 2.35 — suggestive IF stellar masses track their seed cores, which is exactly the unclosed relation. Leaned upon nowhere.