PUH Theorem 318 (The Convergence Window Has No Mechanism) — A Planck Core's Area Goes as m², Not m^(2/3), So Surface-to-Mass RISES with Mass and T198's Theorem 4 Inverts; No Candidate Mechanism Supplies the Required Scaling

Abstract

Photonic Universe Hypothesis (PUH) — Defect Located. THE CLAIM AT ISSUE. T198 derives a planetary-formation timeline from rebound momentum sorting, whose central relation v = p₀/m gives lighter cores higher velocity and sorts primordial cores into concentric mass shells. That relation is NOT in question. What is in question is how the shells are brought back together. T198 states — in its text and again as THEOREM 4 — that lighter cores decelerate faster than heavier ones BECAUSE THEY PRESENT A LARGER SURFACE-TO-VOLUME RATIO to the E8 condensate. That difference produces its convergence window at roughly 1.5–3 Gyr, during which stellar-mass cores overtake the decelerating planetary-seed shells and capture them. The window is how that paper forms solar systems; the mechanism is load-bearing. THEOREM 318.1 (the surface-to-mass ratio inverts). For an ordinary body of fixed density, m = (4/3)πρr³ gives r ∝ m^(1/3), A ∝ m^(2/3), and A/m ∝ m^(−1/3) — area per unit mass FALLS with mass, which is the reasoning Theorem 4 invokes and which is correct for such bodies. But a Planck core has the Schwarzschild radius r_s = 2Gm/c², so r ∝ m EXACTLY, hence A ∝ m² and A/m ∝ m^(+1) — area per unit mass RISES in direct proportion to mass. The scalings differ by m^(4/3). COMPUTED: as Planck cores, A/m runs 1.39×10⁻³¹ (Earth-seed, 5×10²¹ kg), 5.54×10⁻²⁶ (Jupiter-seed), 5.51×10⁻²³ (solar-mass), 3.31×10⁻²¹ (60 M☉) — RISING by a factor 2.4×10¹⁰. The same masses at rock density give 1.36×10⁻⁹, 1.85×10⁻¹¹, 1.85×10⁻¹², 4.72×10⁻¹³ — FALLING by a factor ≈2,900. Same objects, opposite trends. ON THEOREM 4'S OWN REASONING THE ORDERING IS REVERSED: heavier cores present more surface per unit mass and would decelerate faster. THEOREM 318.2 (no candidate supplies the scaling). What matters is the DECELERATION a = F/m, not the force: a force proportional to mass gives identical deceleration for every body. Tabulating: surface drag on an ordinary body gives F ∝ m^(2/3), a ∝ m^(−1/3) — the only case delivering lighter-faster, and precisely the one that does not apply. Surface drag on a horizon-scale core gives F ∝ m², a ∝ m^(+1) — reversed. Gravitational dynamical friction gives F ∝ m², a ∝ m^(+1) — reversed. Cosmological decay (v ∝ 1/a) gives F ∝ m, a ∝ m⁰ — mass-independent. T198's own exponential form gives F ∝ m, a ∝ m⁰ — mass-independent. EVERY MECHANISM THAT APPLIES YIELDS EITHER THE REVERSE ORDERING OR NO MASS DEPENDENCE. Without a mass dependence in the required direction the shells never converge: identical deceleration preserves the velocity ordering for all time, and reversed deceleration increases the separation. In neither case do stellar cores overtake planetary-seed shells, and the capture event does not occur. A SEPARATE DIFFICULTY WITH THE COEFFICIENT. T198 writes F = −η·m·v/t_P with η dimensionless, giving decay timescale t_P/η. The coefficient is not derived and the required value is extreme: η ≈ 1.7×10⁻⁶⁰ for a gigayear decay, 1.2×10⁻⁶¹ for a Hubble time, while η of order unity would halt all motion within a Planck time. By contrast the cosmological decay carries NO free coefficient: the geodesic equation in a Friedmann background contains a term Hv, so peculiar velocity falls as the inverse scale factor — matter's momentum redshifts exactly as a photon's does, at a rate fitted to nothing. It supplies deceleration for free; what it does not supply is mass dependence. WHAT SURVIVES, AND IT IS MOST OF THE PAPER. The momentum-sorting relation v = p₀/m is untouched, so lighter cores travel further and the mass-radius inversion holds. The planet-free early universe follows from that inversion ALONE, without any appeal to deceleration, and stands independently of Theorem 4. The observational anchor is unaffected, testing the timeline's prediction rather than its mechanism. WHAT FAILS is Theorem 4 specifically — the claim that lighter cores decelerate faster, its surface-to-volume justification, and the convergence window following from it. The capture mechanism for solar-system formation is therefore without a driver. THIS NOTE DOES NOT SUPPLY A REPLACEMENT: it shows the stated mechanism inverts and the obvious alternatives do not help. Whether some other process — one not considered here, or one specific to the lattice rather than borrowed from fluid or gravitational dynamics — can produce the required scaling is left open. KILL-CONDITIONS: (i) if primordial cores are not horizon-scale, Theorem 318.1's scaling does not apply and Theorem 4's reasoning may be restored; (ii) if a mechanism is exhibited whose deceleration falls with mass and applies to horizon-scale objects, Theorem 318.2 fails and the window reopens; (iii) if convergence arises from something other than differential deceleration — capture cross-sections, wake structure, or the sorting itself — this note addresses a mechanism the paper does not require; (iv) if η is derived and found mass-dependent in the required direction, both Section 4's difficulty and Theorem 318.2 are answered at once. NOT CLAIMED: that the planetary-formation timeline is wrong (only its convergence mechanism); that a replacement was sought exhaustively beyond the four tabulated cases; that η cannot be derived, only that it has not been; that cosmological decay is inadequate for deceleration (it is adequate and free — inadequate only for mass DEPENDENCE); or that momentum sorting, the mass-radius inversion, or the planet-free early universe require revision.