PUH Theorem 316 (Inertia Is Added Mass, Not Drag) — T185's Own Equation Multiplies Acceleration, So d'Alembert's Paradox Answers the Ether Objection; the Kinetic Energy Is Lattice Motion Exactly; and Equation (1) Carries a Spurious c²

Photonic Universe Hypothesis (PUH) — Correction and Derivation. THE OBJECTION THE NAME INVITES. T185 resolves Mach's principle by identifying inertia as E8 LATTICE DRAG on a localised vortex pattern, and derives F = ma from it. The result is good and this note defends it. But "drag" invites an objection that would be fatal if it applied: a drag force is VELOCITY-dependent, F = −ζv, so a coasting body would slow without external influence, Newton's first law would fail, and uniform motion relative to the lattice would be detectable — which Michelson-Morley and every subsequent Lorentz-invariance test exclude. Against a genuine drag the objection succeeds outright, and it is the first thing a reader will raise. THEOREM 316.1. T185's force law multiplies its coefficient by ACCELERATION, not velocity. No velocity appears anywhere in it. A force proportional to acceleration is by definition inertial, and its constant of proportionality is a mass. SO THE PHYSICS OF T185 IS CORRECT AND ONLY ITS NAME IS WRONG. The correct term is ADDED MASS (induced mass) — standard fluid mechanics, not a coinage: a body accelerating through a medium must accelerate the medium around it and so behaves as though heavier. RESULT 316.2 (the correct term carries the defence). A body in STEADY motion through an ideal medium experiences NO FORCE WHATEVER. That is d'Alembert's paradox — a theorem, not an approximation or a small-velocity limit. So acceleration costs energy, uniform motion costs nothing, and no experiment on a uniformly moving body can detect the medium. THE ETHER OBJECTION IS ANSWERED BY A CLASSICAL RESULT OF THE EIGHTEENTH CENTURY, not by any assumption introduced for the purpose. The lattice is not an ether in the sense the null experiments exclude, because the quantity those experiments bound — resistance to uniform motion — is exactly what an ideal medium does not produce. Acceleration, by contrast, is absolute and is felt: precisely the asymmetry the framework requires and general relativity also observes. THEOREM 316.3 (where the kinetic energy is). The case is exactly solvable. For a sphere of radius a moving at speed v through an ideal medium of density ρ, the induced flow has velocity potential φ = −(a³/2r²)v·cosθ, and integrating the medium's kinetic energy over all space outside the sphere gives KE = πa³ρv²/3 = ½·m_add·v², with m_add = ½ρV. THE KINETIC ENERGY IS NOT IN THE OBJECT — IT IS IN THE MOTION THE OBJECT INDUCES IN THE MEDIUM AROUND IT, and the identification is exact rather than approximate in the solvable case. AND PUH'S VERSION IS THE STRONGER ONE: for a real sphere in water the added mass is a correction to a mass the sphere already possessed, and only part of the kinetic energy lives in the fluid. In this framework a particle IS a localised fold in the lattice — there is no sphere underneath — so the added mass is NOT AN ADDITION BUT THE WHOLE OF THE MASS, and ALL the kinetic energy is lattice motion with nothing left over. This is also why T185 can say inertial mass is not intrinsic: on this picture it could not be, since there is nothing for it to be intrinsic to except the lattice configuration itself. A STRUCTURAL CHECK AT HIGH SPEED. T186 establishes the lattice's wave speed as c, identifying ε₀ as electrostatic stiffness and μ₀ as inertial density, so that c = 1/√(ε₀μ₀) is the elastic wave speed √(stiffness/density) of a continuous medium. A body can only be gotten out of the way of at the medium's own signal speed, so as v → c the lattice cannot reconfigure ahead of it and the induced-flow energy must diverge — which is exactly what (γ−1)mc² requires. THE PICTURE HAS THE RIGHT BEHAVIOUR AT HIGH SPEED, NOT MERELY THE RIGHT VALUE AT LOW SPEED. CAVEAT STATED RATHER THAN GLOSSED: obtaining the exact relativistic form from an induced-field calculation is not routine, and the nearest classical analogue — the energy and momentum of a moving charged sphere's own field — is known not to come out consistently without careful treatment. The low-speed result is exact, the high-speed structure is right, and the exact relativistic form is NOT established here. A SEPARATE ARITHMETIC CORRECTION. T185's coefficient carries a factor c². Substituted into its own force equation, the product of coefficient, ground-state tension and vortex volume gives mc², so the force comes out as mc²a rather than ma; the paper divides by c² in its final step, but that division is not in the product as written. Defining the coefficient without the c² makes the equation exact with nothing left over. Typographical rather than physical, and no conclusion depends on it. WHAT THIS DOES NOT TOUCH. Cosmological velocity decay is a different regime. In an expanding universe the geodesic equation contains a term proportional to Hv, so peculiar velocity falls as 1/a — matter's momentum redshifts exactly as a photon's does. That IS a genuine velocity-dependent effect, it is standard general relativity, and it costs no new parameter. It does not conflict with Result 316.2 because H is utterly negligible on laboratory timescales: the first law holds locally to any testable precision while momentum still decays cosmologically. T198's lattice-friction mechanism is in that second category, and nothing here bears on it. THE DISTINCTION TO KEEP IS BETWEEN A LOCAL DRAG, WHICH IS EXCLUDED, AND A COSMOLOGICAL ONE, WHICH IS REQUIRED. KILL-CONDITIONS: (i) if any velocity-dependent force on a uniformly moving body is detected in the laboratory, the added-mass reading fails and the lattice is an ether after all; (ii) if the exact relativistic kinetic energy cannot be recovered from an induced-field calculation, Section 5's structural agreement is insufficient and the picture holds only at low speed; (iii) if the vortex volume in T185's coefficient is velocity-dependent, the acceleration-only reading fails and the drag objection returns; (iv) if a particle is shown to possess mass independent of its lattice configuration, Theorem 316.3's identification fails. NOT CLAIMED: that the exact relativistic form is derived; that the vortex volume has been computed for any particle, which remains open; that the added-mass coefficient for a lattice fold equals the one half obtained for a sphere in an incompressible fluid, since the lattice is neither; that T185's conclusions require revision, since only its terminology and one factor do; or that this bears on cosmological velocity decay.