Physics from Finite Validation: The Bridge — How a Validation Network Becomes a Field

Across the preceding four works of this series, one debt was named load-bearing and left open: the two model regimes used throughout — a Kuramoto-type phase network (vacuum energy, relaxation, entropy) and a sine-Gordon field (tunnelling, charge, the force trough) — were never shown to be the same object. Without that bridge, the programme could be read as two metaphors stitched by a principle rather than one theory. This work builds the bridge in the linear and solitonic regime, with numerical support at every step, and names the one remaining analytic debt precisely. The chain: (1) the finiteness of the tick — the founding requirement of the series — means a node sees its neighbours through a delay, and a first-order delayed network generates an effective second-order (inertial) dynamics; the oscillation period scales with the delay, confirmed in-model. (2) The linearised phase network with inertia is identical to the linearised discrete sine-Gordon chain — the dispersion relations coincide to machine precision within the model, recovering continuum sine-Gordon as ka tends to zero (the classical Frenkel-Kontorova limit). (3) The network's soliton behaves as a sine-Gordon kink: it carries the topological charge of Paper 3 (Q = 1, conserved), is stable on the lattice, and Lorentz-contracts as sqrt(1 minus v squared). (4) An amorphous 3D network suppresses the directional anisotropy of a regular lattice — a modest in-model trend, stronger at short wavelengths. (5) The exact delay-equation spectrum (Lambert-W roots) shows the two-mode sine-Gordon reduction is rigorous for Stau up to about 3 and exact at the Hopf threshold Stau = pi/2 — the regime where stable oscillating modes are born. The remaining debt is analytic, not numerical: a full nonlinear reduction of the delayed network to sine-Gordon at arbitrary S*tau. The record contains two files: the main article and the Numerical Companion (full protocols, parameters, per-result status labels). Code: https://github.com/ivan-denysov/finite-validation-bridge