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 one object. This work builds that bridge in the linear and solitonic regime, with numerical support and per-result status labels at every step. In one dimension: the finiteness of the tick generates effective inertia (a delayed first-order network behaves as second-order, its period scaling with the delay) ; the linearised network with inertia coincides with the linearised discrete sine-Gordon chain to machine precision — with the on-site term inserted, its microscopic origin named as a debt; the network's soliton is a sine-Gordon kink (topological charge Q = 1, lattice-stable, Lorentz-contracting) ; and the exact delay spectrum (Lambert-W roots) bounds the two-mode reduction to Sτ ≲ 3, exact at the Hopf threshold Sτ = π/2. Rising to three dimensions: an isotropy lemma yields the isotropic Klein–Gordon form as the continuum limit of the amorphous network (mass recovered at 0. 986 against 1. 0, axis isotropy 0. 38%) ; the affine coefficient is a rigorous upper bound, with a measured non-affine disorder renormalisation of the wave speed (c²/c²ₐff ≈ 0. 29–0. 47) ; and the scalar sector's boundary is measured — domain wall and straight vortex line stable, no localised scalar particle. A set of dynamics postulates (locks on links; exclusive one-per-node-per-tick transitions; a derived arc turn-delay; slow link curvature) yields a self-bound flow condensate — steady over ~10³ lock-decay times, confinement by dynamical binding rather than topological winding, "mass = the delayed regime" realised mechanically. New in v3 (§6d): the two branches of the same 3D dispersion are read as two sectors of one network. The massive Klein–Gordon branch is matter (the on-site term is its internal rest mass) ; removing that term leaves a massless, symmetry-protected Goldstone branch — gravity. Four in-model checks: the massless branch is gapless in 1D and 3D with an exact uniform-mode zero on any graph; the masslessness is symmetry-protected (only an absolute-phase term opens a gap) ; matter cannot be scattered pins in the vacuum (those would screen gravity as √ (gn) ), so it couples to gravity as the symmetry-respecting frequency shift Δω of the forces work, leaving gravity unscreened; and the tensor (spin-2) sector lives on the amorphous substrate — bond-bending (the harmonic form of the link-curvature postulate B5) collapses its floppy sector (matching the Maxwell–Calladine count) and opens a shear modulus, with cT/cL = 0. 57–0. 71 measured on the same substrate. The scalar potential sector recovers light deflection via the two-channel reading of Paper 1, so it is not the excluded minimal (Nordström) scalar. All results carry status labels (confirmed-in-model / consistency check / observation / lesson / closed-negative / open). Named open in the ledger: the full nonlinear reduction; the microscopic on-site (mass) term and the Δω-to-m relation; the tensor coefficient and whether cT equals the tick speed c (GW170817) ; the k→0 convergence of the non-affine c²; and the grand map from network elasticity to the Einstein equations. Runs are exploratory instruments within stated models, not measurements of nature. The record contains two files: the main article and the Numerical Companion. Code: https: //github. com/ivan-denysov/finite-validation-bridge