PUH Theorem 319 (Freeze-Out Makes the Mass-Radius Law Permanent) — Peculiar Displacement Converges in Matter Domination, So m·r_final = 3p₀t₀ Is a Fixed Configuration Rather Than an Epoch-Dependent Coincidence; T318's Gap Remains

Photonic Universe Hypothesis (PUH) — Repair and Negative Result. THE DIFFICULTY. T198 derives its mass-radius law from momentum sorting: equal rebound impulse gives v = p₀/m, distance r = (p₀/m)·t, hence m·r = p₀·t. T198 STATES THE QUALIFICATION ITSELF — the product is constant "for fixed cosmic time t." That is honest, and it is also the difficulty: a law of this form describes a configuration that never stops growing. The solar-mass shell would sit at 8 kpc at one moment and elsewhere at every other, and the galactic mass gradient the paper explains — heavy stars toward the centre, light stars in the outer disk and halo — would be a coincidence of the present epoch rather than a structure. Since that gradient is the paper's principal observational support, the epoch dependence is not a technicality. A fossil structure must be fossilised. THEOREM 319.1 (proved in full). In a matter-dominated expansion the total comoving displacement is finite and equals 3v₀t₀/a₀. PROOF: the linear law is the non-expanding form. In an expanding universe the geodesic equation gives dv/dt + Hv = 0, so v_pec = v₀(a₀/a) — standard general relativity, no free parameter. With proper distance r = a·x, the comoving displacement accumulates as dx/dt = v_pec/a = v₀a₀/a². With a = a₀(t/t₀)^(2/3), ∫dt/a² = (t₀^(4/3)/a₀²)∫t^(−4/3)dt = (t₀^(4/3)/a₀²)·[−3t^(−1/3)] = 3t₀/a₀², so Δx = v₀a₀·(3t₀/a₀²) = 3v₀t₀/a₀. ∎ The core coasts, decelerates, and STOPS at a definite comoving radius. Verified symbolically. THE CONVERGENCE IS NOT UNIVERSAL: tabulating the three eras, radiation domination (a ∝ t^(1/2)) gives a LOGARITHMIC, formally divergent integral; matter domination gives 3t₀/a₀²; exponential expansion gives 1/(2Ha₀²). The radiation-era contribution is bounded in practice by the logarithm of that era's duration — about 28 from one second to matter-radiation equality, about 102 from the earliest plausible start — so the total remains finite. The conclusion is unaffected; the accounting is stated correctly rather than glossed. RESULT 319.2. Substituting v₀ = p₀/m gives r_final = 3p₀t₀/m, hence m·r_final = 3p₀t₀ = CONSTANT. T198's equation is recovered and PROMOTED: what was a relation holding at one moment becomes a permanent feature of the configuration — the same form, no longer carrying the epoch. The mass gradient becomes a fixed structure rather than a passing coincidence, and the paper's central observational argument is repaired. AND THE ANCHOR FIXES THE LADDER: with the solar-mass shell at ~8 kpc, 0.1 M☉ lands at 80 kpc (outer halo), 0.5 M☉ at 16 kpc (outer disk), 3 M☉ at 2.7 kpc (inner disk), 10 M☉ at 0.80 kpc (bulge), 60 M☉ at 0.13 kpc (galactic centre). The observed IMF is top-heavy toward the centre and bottom-heavy in the outer disk and halo — the ordering this ladder produces. The agreement is DIRECTIONAL ONLY; no fit is performed and none is claimed. THE FREEZE-OUT IS EARLY. A fossil gradient requires the motion to stop before the structures carrying it have formed. The fraction of total displacement accumulated by time t is 1 − (t₀/t)^(1/3): 0.206 at t/t₀ = 2, 0.415 at 5, 0.500 at 8, 0.536 at 10, 0.785 at 100, 0.900 at 1000. Half the displacement accumulates within a factor of eight in time and ninety percent within a thousand. The sorting completes early and the configuration locks — which the linear law, under which motion continues undiminished forever, cannot provide. WHAT THIS DOES NOT DO, AND IT IS WHY THE WORK WAS UNDERTAKEN. Freeze-out was examined as a candidate replacement for the convergence mechanism whose absence T318 established. IT IS NOT ONE. The deceleration is MASS-INDEPENDENT in exactly the way T318 found the cosmological decay to be: every core halts at a radius inversely proportional to its mass, preserving the velocity ordering through the entire process. Nothing overtakes anything, no shell crosses another, the stellar cores never pass through the planetary-seed shells, and the capture event remains without a driver. T318's conclusion stands unaltered — of the mechanisms available, only surface drag on a body of fixed density gives lighter-decelerates-faster, and that is precisely the case a horizon-scale core does not realise. FREEZE-OUT ADDS A FOURTH MASS-INDEPENDENT MECHANISM TO THAT LIST RATHER THAN REMOVING THE DIFFICULTY. KILL-CONDITIONS: (i) if primordial cores are not free-streaming — if bound into structures early enough that peculiar-velocity decay does not apply — Theorem 319.1 governs only the pre-binding phase and the frozen radius is set by when binding occurred; (ii) if the expansion history departs from matter domination over the relevant interval, the integral must be redone, and radiation domination does not converge; (iii) if the mass-radius ladder is contradicted by the observed mass distribution at any radius, the law is falsified independently of freeze-out; (iv) if the galactic mass gradient arises from stellar dynamics rather than primordial sorting, the observational support is removed even though the mathematics stands. NOT CLAIMED: that the convergence mechanism has been supplied (Section 5 states the contrary — this is the paper's principal negative result); that the ladder has been fitted to data (no fit; agreement directional only); that subsequent dynamical evolution — bar migration, scattering, mergers — preserves the frozen radii, a separate question not addressed; that p₀ has been determined; or that T198's other results require revision.