We show that a generative causal process equipped with compact edge phase transport has a layered consequence: the primitive \( \mathrm{U}(1) \) phase first contains a pure arithmetic cover branch, while placing that same phase on a directed causal network with Wilson curvature admits a unique minimal complete simplicial causal-gauge cell at the Hodge–Bianchi resonance dimension \( d = 4 \). In that dimension curvature and response are both middle-degree two-forms; the Bianchi and response equations live in the same equation-degree; \( F \wedge F \) is a top form; and the curvature sector admits a self-dual/anti-self-dual split. The minimal complete simplicial realisation of this \( d = 4 \) closure is the 4-simplex, whose one-skeleton is the complete graph \( K_5 \). The familiar equality \( \binom{5}{2} = \binom{5}{3} = 10 \) is therefore the simplicial ledger shadow of this middle-degree closure, not the primary selection principle.

\( K_5 \) is not a spacetime brick; it is a sliding causal window. Time is continuation; space is synchronisation. Particles are stable causal motifs: charged leptons are phase defects, baryons are \( k = 3 \) closures, pions are \( k = 2 \) bridges, photons are coherent transport of \( \mathrm{U}(1) \) phase non-equivalence, the Higgs is the radial stiffness of causal ordering, and the \( k = 4 \) overclosure is the minimal local seed of a future-sealed black-hole cluster.

From this Hodge-closed causal cell we construct a one-scale web of theorem-level, derived-law, and candidate results, with no additional dimensionless fit parameters in the theorem-level sectors: the SM gauge group \( \mathrm{SU}(3)\times \mathrm{SU}(2)\times \mathrm{U}(1) \); the minimal fermion carrier spectrum \( \bar{5} \oplus 10 \) (15 Weyl states, no sterile singlet) in three generations; the theorem-level bare orientation coupling \( \alpha_{\text{bare}}^{-1} = |A_5| = 60 \); the electromagnetic conductor weight \( W_{\mathrm{EM}} = \varphi(60)/2 = 8 \) for charged channels, with the observed low-energy \( \alpha_{\mathrm{em}}^{-1} \simeq 137 \) obtained as an A−/LO dressed-response structure after lepton dressing and open-channel completion; \( \sin^2 \theta_W = 0.232 \); the causal depth \( L^* = 26 \); the Higgs ratio \( \frac{m_H}{v} = \frac{3}{\sqrt{35}} \ (124.9\,\text{GeV},\,0.2\%) \); the proton radius law \( \frac{m_p r_p}{\hbar c} = 4 \ (0.04\%) \); \( \frac{m_\pi}{m_p} = \frac{1}{3\sqrt{5}} \ (0.2\%) \); the neutrino law \( \frac{\Delta m^2_{21}}{\Delta m^2_{32}} = \frac{1}{35} \); the gravitational response \( V_0 = \frac{1}{10\pi} \); the absolute scale bridge \( M_{\mathrm{Pl}} = m_e \exp(51 + 19/36) \) (about \( 10^{-4} \) agreement, A−/near-theorem, with denominator \( \operatorname{Tr}'(d_2^\top d_2) = 30 \)); the operator-level vacuum split \( \Omega_\Lambda = \frac{2}{3} \); and the dark-matter frontier relic candidate ratio \( \frac{\Omega_{\mathrm{DM}}}{\Omega_b} \simeq 5.48 \) (2.2% relative to the Planck 2018 ratio, with the number closed and halo phenomenology open).

Scattering is a boundary-to-boundary response kernel on the thermodynamically selected causal skeleton; unitarity is reconstructed in the infrared stable-motif sector; Lorentz invariance emerges as the kinematics of the surviving gapless response sector. Throughout the construction we use generated completeness as a working discipline: a quotient or readout may lose information, but it may not add an ungenerated degree of freedom. The companion arithmetic note treats ordinary arithmetic as the pure cover-shadow of the same compact phase and records RH as the corresponding cover-shadow completeness theorem; RH is not used by any finite physical derivation. All claims are classified as theorem, derived law, candidate, benchmark, or open structural target.