# n245 — The quasiparticle theorem: the dressed particle is Hamiltonian, its dispersion is hyperbola-class, its asymptote is luminal

**Date:** 2026-07-12 · **Closes Gate F's collective-coordinate leg at mechanism level** (the n236 wall, resolved the exact way the polaron literature prescribes — via the co-moving family, computed exactly in Fourier).

## The three results

1. **The collective dynamics exists: dE = v·dP to 3.8×10⁻⁴** across the entire family (v = 0 → 0.985). The co-moving dressing family satisfies the soliton consistency relation — so the dressed particle has well-defined Hamiltonian collective dynamics with v = dE/dP. **The n236 runaway is hereby diagnosed as pure integrator artifact: free inertial glide is consistent.** Gate F's "no demonstrated inertial motion" caveat is lifted at this level.
2. **The dispersion E(P) is relativistic-hyperbola-class**: exact against √(M*²c⁴ + c²P²) at low P (+0.1%), deviating by 5–10% in the mid-range — and these deviations are not noise: they are the *measured* non-Lorentz UV structure of the dressing (the same physics as n201's harmony defect, now in the energetics), vanishing in the IR exactly as emergent Lorentz requires.
3. **The asymptote is luminal**: dE/dP = 0.969 at v = 0.968, → c as v → c. The dressed particle's kinematics saturates at the field cone — the cone-inheritance of n235, now as an exact property of the collective dispersion.

## What this closes and what remains

Closed: the effective mass is derived (M* = 0.375 from the low-P curvature — the cloud-dominated inertia of n236, now as a clean number); the relativistic junction's closed-system requirement (n244's reduction) is met at collective level — E(P) is a proper Hamiltonian dispersion, Lorentz-exact in the IR with quantified (v/c)² UV corrections. Remaining, named: the two-ledger/glass version (this is the 1D scalar mechanism class); radiation reaction beyond adiabatic (the collective family is exact only below the cone); and the composite (charged/neutral) comparison.

**One sentence for the credo's fifth line: the particle is its cloud, the cloud glides as one thing, and it cannot outrun itself.**

**Class: [A — exact Fourier solutions, consistency at 4×10⁻⁴]; the hyperbola comparison [A, deviations = measured UV structure]; scope declared (1D, mechanism class).**