PUH Theorem 363 (Elimination) — Nine Routes to the Flat Rotation Curve Closed, Modified Inertia Identified as the Survivor, and Its Exponent Solved by a Force-Per-Cell Threshold

Photonic Universe Hypothesis (PUH) — Elimination Record. THIS PAPER SOLVES NOTHING. It records nine mechanisms tested against the flat rotation curve and closed, the single constraint that closed most of them, one methodological trap that invalidated several apparent successes, and the one direction that survives. IT IS FILED SO THAT THE GROUND IS NOT RE-TRODDEN. THE NINE, WITH MARGINS: primordial cores as missing mass fail by 10⁴ AND are mislocated (the curve is flat where no stars are); cores in the halo are capped at 10–30% by EROS-2 and OGLE microlensing; a steep core mass spectrum (τ = 5/2) concentrates the population at exactly the mass where those limits are tightest; clustering trades microlensing limits for wide-binary and dwarf-galaxy dynamical limits with no net gain; spacetime flow into cores is short by 10³⁹; aligned polar emission by 10³⁰ (right geometry, hopeless magnitude); gravitational time dilation by 10⁶ and it cancels from the measurement besides; polar-lobe mass, magnetism and frame-dragging by 10⁵ or more, and the tracers are electrically neutral; and inherited rotation from an earlier epoch fails structurally — the disc has completed 18 orbits and remains bound, which requires a force acting NOW rather than a spin imparted long ago. RESULT 363.1 (the constraint is acceleration, not geometry). Globular clusters sit 13× above a₀ and show NO missing mass; ellipticals 4.6× above, none; the Milky Way at 20 kpc 0.17×, yes; DWARF SPHEROIDALS 0.13×, THE MOST OF ANY CLASS; LSB outskirts 0.01×, yes. EVERY SYSTEM ABOVE a₀ SHOWS NONE; EVERY SYSTEM BELOW IT DOES — ROTATING OR NOT. Dwarf spheroidals are pressure-supported, have no axis and no disc, and need the effect most. SO ANY MECHANISM REQUIRING AN AXIS, DISC, RIM, SPIN OR BOUNDARY IS EXCLUDED BEFORE ANY ARITHMETIC — covering the twist, cylindrical tension, line source, polar jets, rim boundary and vortex, all tested and all geometric. RESULT 363.2 (a circularity that invalidated several apparent successes) — THE MOST TRANSFERABLE FINDING HERE. Several attempts produced magnitudes agreeing with observation to a factor of a few, each treated as encouraging. THEY WERE NOT INDEPENDENT: ρ_Λ ≈ c²H₀²/G to within order unity, and a₀ ≈ cH₀/5.7. SO ANY ARGUMENT TAKING THE VACUUM ENERGY AS INPUT AND RETURNING THE CHARACTERISTIC ACCELERATION HAS PERFORMED ARITHMETIC, NOT PHYSICS. A genuine derivation must not use ρ_Λ as an input. One non-circular attempt was made and failed cleanly: setting the Unruh temperature of a local accelerating screen equal to the de Sitter temperature of the horizon gives a = cH₀ exactly, with ln(240) AND 2π appearing on both sides and CANCELLING. Any universal lattice constant does the same, so THE RESIDUAL FACTOR OF 5.7 CANNOT BE ALGEBRAIC. RESULT 363.3 (the measurements are clean). Rotation curves are read from the DIFFERENCE between approaching and receding sides at the same radius, so gravitational redshift cancels EXACTLY — and is 65 m/s against a 106,000 m/s discrepancy in any case. Distances are geometric (parallax, angular size) or affected at 2×10⁻⁵ per cent (Cepheids), seven orders below the measurement error. NO OBSERVATIONAL ARTEFACT IS HIDING. And the deeper version is not a loophole: A CLOCK RATE DIFFERING FROM RELATIVITY'S IS EXTRA GRAVITY BY DEFINITION. RESULT 363.4 (the survivor). Three weeks were spent on the force side of F = ma; the other side was never examined. With m·(a/a₀)·a = GMm/r²: a = √(GMa₀)/r and v² = √(GMa₀) = constant. It depends on ACCELERATION ALONE (works for round spheroidals), and v⁴ = GMa₀ — the baryonic Tully-Fisher relation — IS THE SAME EQUATION rather than an added assumption. AND THE FRAMEWORK ALREADY IMPLIES IT: T316 holds that a particle IS a fold in the lattice with no sphere underneath, so inertial mass is a property of the LATTICE'S STATE, not of the particle. A lattice phase change is therefore an inertia change. AND THE ETHER OBJECTION IS ALREADY ANSWERED — d'Alembert's paradox gives steady motion zero cost, and the modification multiplies ACCELERATION not velocity, so Lorentz invariance is untouched. Most modified-inertia proposals struggle precisely there. RESULT 363.5 (the exponent, solved by discreteness). An earlier draft recorded the exponent as the blocking item: a yield-stress medium engages R ~ √(ma/S), giving volume as R³ and added mass as a^1.5 where a^1.0 is required. THE ERROR WAS IN THE CRITERION, NOT THE MAGNITUDE. A stress criterion asks whether force per unit AREA exceeds a threshold; A LATTICE CAN ASK WHETHER THE FORCE PER CELL DOES: force per cell = ma·l_P³/R³ > f_th gives R³ < ma·l_P³/f_th, so VOLUME ∝ a EXACTLY. THE EXPONENT COMES OUT RIGHT WITH NO FITTING AND NO ASSUMED DIMENSIONALITY. And the criterion is specifically a LATTICE criterion — a continuum has no cells to divide the force among and CANNOT MAKE THIS DISTINCTION AT ALL. So the modification is a consequence of DISCRETENESS rather than of elasticity or density, and discreteness is this framework's central claim: the mechanism is inherited, not imported. RESULT 363.6 (the threshold cannot be a lattice constant). Fixing the threshold from the required scaling gives f_th = ρ·l_P³·a₀, which feeds in the vacuum energy and is CIRCULAR by Result 363.2's own standard. It is not presented as a derivation. BUT THE ATTEMPT PRODUCED A STRUCTURAL RESULT: the threshold is proportional to a₀, and a₀ tracks H₀, WHICH CHANGES WITH COSMIC TIME — at z = 2 it was 2.97× today's value, at z = 10 twenty times. A QUANTITY THAT DIFFERED AT REDSHIFT TEN CANNOT BE BUILT FROM THE KISSING NUMBER, THE COXETER NUMBER, OR ANY FIXED PROPERTY OF E₈, since those are the same now as they were then. THE SEARCH FOR AN ALGEBRAIC DERIVATION WAS NEVER GOING TO SUCCEED, AND THAT IS A RESULT RATHER THAN A FAILURE. It follows that the threshold must track the lattice's EVOLVING state, and the archive has such a quantity in the unsettled fraction. ONE ATTEMPT IN THAT DIRECTION IS RECORDED AND REJECTED: a settling law falling as t⁻² reproduces the observed fraction to 8 per cent with the exponent emerging at 2.0006 — striking, but it RESTATES the coincidence problem rather than solving it, and it implies a DECAYING cosmological constant that observation excludes at w = −1. KILL-CONDITIONS: (i) if the force-per-cell criterion cannot be justified from the lattice's own dynamics, Result 363.5 is a coincidence of exponents and the modification is unmotivated; (ii) if a nonlocal formulation is required and cannot be made consistent — the difficulty Milgrom's own attempts met — the direction closes; (iii) if the Bullet Cluster cannot be addressed, the approach fails where all baryon-tracking theories fail; (iv) if any eliminated route is shown to have been miscomputed, this paper is wrong about that route. NOT CLAIMED: that the flat rotation curve is explained — IT IS NOT, AND THIS PAPER EXISTS TO SAY SO PRECISELY; that modified inertia works, only that it survives where nine others did not; THAT THE THRESHOLD VALUE IS DERIVED — IT IS NOT, AND RESULT 363.6 EXPLAINS WHY NO FIXED LATTICE CONSTANT COULD SUPPLY IT; or THAT ANY OF THE NINE ELIMINATIONS IS NOVEL — microlensing limits, the Bullet Cluster and the acceleration discriminant are standard, and the contribution is applying them systematically to one framework's own mechanisms.