PUH Theorem 347 (The Concentration Mechanism Was Already Filed) — T287's dm/dt = km² Blows Up at t* = 1/(km₀), So Growth Is Superlinear and the Core Fraction Cannot Stay Fixed; T346's Kill-Condition Is Closed and Its Distribution Acquires the Exponen

Photonic Universe Hypothesis (PUH) — Gap Closed. THE GAP AS FILED. T346, filed yesterday , verified the terrestrial core anchor from geometry, confirmed core:body = 8.37×10⁻⁴, and showed that on a constant fraction a 25 M☉ star carries 0.0209 M☉ (150× short of the lower threshold) and a 100 M☉ star 0.0837 M☉ (37× short of 3.14 M☉ and 96× short of 8.01 M☉). It concluded the constant-fraction picture is excluded and the accretion map must be a distribution rather than a function — and filed a kill-condition stating that reaching the thresholds requires seed concentration by three to four orders of magnitude, "a strong requirement the archive supplies no mechanism for". THAT LAST CLAUSE WAS FALSE. THE MECHANISM, DERIVED IN JULY. T287, filed 13 July, derives DUAL-CHANNEL GROWTH. CHANNEL A (amalgamation): cores merging pairwise form a coagulation system, and because a horizon-like core has r = 2Gm/c² as its only length scale, the capture kernel scales as m². CHANNEL B (absorption of incident radiation): an earlier draft of T287 was withdrawn for asserting the Shell reflects everything — a premise "too strong". A CORRECTION IS APPLIED HERE. T287 (14 July) states the Shell perfectly reflects gravitational waves; T307 (27 July) SUPERSEDES THAT, resolving a two-month disagreement between two filed papers — an impedance argument making the Shell a perfect reflector, a blackbody argument making it a perfect absorber — which traced to reading the same Lagrangian in opposite senses: frozen with zero residual degrees of freedom, against the full mode spectrum ACTIVATED at the same threshold. THE NULL GRAVITATIONAL-WAVE ECHO SEARCH DECIDED IT: barrier transmissivity 0.468 at the ringdown frequency means a perfectly reflecting Shell predicts a first echo at roughly half the ringdown amplitude, signal-to-noise 5–14 for the loudest catalogued events, and NO SUCH ECHOES WERE FOUND. THE SHELL ABSORBS AND THE CORE REFLECTS; the echo prediction is withdrawn entirely. And the correction IMPROVES this note's argument: T287's surface/bulk assignment inverts for gravitational waves, which in T307's words "makes it more uniform rather than less" — both photons and gravitational waves are absorbed, so Channel B is BROADER than T287 claimed, and the quadratic law is unaffected since it follows from the AREA scaling rather than from what is absorbed. And that channel obeys the same law, dm/dt = R_sub·A = km² with k = 16πG²R_sub/c⁴, obtained "from T213 alone, with no coagulation theory involved whatever". TWO INDEPENDENT CHANNELS, THE SAME QUADRATIC LAW. THEOREM 347.1 (quadratic growth breaks the proportionality). T346 argued that if cores grew in proportion to the matter they gather, the fraction stays fixed and no threshold is reached — SO SOMETHING MUST BREAK THE PROPORTIONALITY. T287's law breaks it: dm/dt = km² integrates exactly to m(t) = m₀/(1 − km₀t), DIVERGING AT THE FINITE TIME t* = 1/(km₀). Linear growth is exponential and preserves the fraction at 8.37×10⁻⁴; quadratic growth is a runaway and destroys it. The blow-up time is INVERSELY proportional to the initial mass, so in T287's words "the heaviest core runs away first, and fastest. A population evolving under this law does not spread — it produces a single object which decouples and diverges." T346 ASKED FOR THREE TO FOUR ORDERS OF CONCENTRATION; SUPERLINEAR GROWTH CONCENTRATES WITHOUT BOUND IN FINITE TIME. ∎ THEOREM 347.2 (the distribution acquires its exponent). T346's Theorem 346.3 held that the accretion map is a distribution over core masses rather than a function of stellar mass, and that it "does not exist in the form sought". THAT IS CONFIRMED — AND T287 SUPPLIES ITS SLOPE. The surviving spectrum has gel-point exponent τ = 5/2, so abundance above mass m falls as m^(1−τ) = m^(−3/2): a factor of ten in mass costs 3.16×10⁻² in abundance, a hundred costs 1.00×10⁻³, a thousand costs 3.16×10⁻⁵, ten thousand costs 1.00×10⁻⁶. CORES ABOVE A THRESHOLD ARE RARE BUT NOT ABSENT. And the exponent is DOUBLY DETERMINED: coagulation sets 5/2 at the light end and photon absorption PRESERVES it, because both channels scale as m². RESULT 347.3 (three papers strengthened, none corrected). T346's structural claim is confirmed with a derived slope of −3/2 and its kill-condition closed. T341 predicted that failed supernovae should be RARE and should track MERGER HISTORY rather than progenitor mass — the spectrum now justifies the rarity and the runaway makes history decisive. T345's two regimes become populations with computable relative abundance once the spectrum is normalised. NO FILED RESULT REQUIRES CORRECTION. THE LESSON. This is the FOURTH occasion this month on which a gap declared open was already closed elsewhere in the archive: the mass-location question was settled by a June note; the coupling-formula conflict dissolved on reading both papers; the winding-number premise rested on a single correspondence paper; and now a kill-condition asserts no mechanism exists when one was derived five weeks earlier. A KILL-CONDITION OF THE FORM "THE ARCHIVE SUPPLIES NO MECHANISM FOR THIS" IS A CLAIM ABOUT THE WHOLE CORPUS, NOT ABOUT THE PAPER MAKING IT, and should be searched before it is filed. The archive is now large enough that its author cannot hold it in mind, which is precisely why the search discipline exists. KILL-CONDITIONS: (i) if the quadratic law does not hold at the relevant masses — if the capture kernel departs from m² once cores are large, or if the substrate rate falls with mass — the runaway is slower and the concentration may be bounded; (ii) if the gel-point exponent is not 5/2 for this system, the abundance slope changes and the rarity prediction with it; (iii) IF THE RUNAWAY IS TOO EFFICIENT, THE FRAMEWORK HAS THE OPPOSITE PROBLEM — a law that concentrates without bound in finite time must be prevented from consuming everything, and what stops it is not addressed here; (iv) if incident radiation does not cross the Shell after all — if T307's resolution is itself superseded — Channel B fails, though Channel A alone gives the same law. NOT CLAIMED: that the spectrum has been normalised, so no absolute abundance follows; that any core has been shown to reach a threshold, only that the mechanism permits it; that T287's derivation has been re-verified here, since it is read and applied rather than rechecked; that the runaway timescale is short enough to matter astrophysically, which needs the substrate rate at stellar conditions and remains underived; or that T346 requires correction.