How Inflation Ends Is How Matter Generation Begins: An N-Body Embedding of the Universal Interaction with No Timing Rule, and an End-to-End Genesis Run

Two preceding papers each supplied half of a story: the onset-mode paper gave the way a universe begins (only unstable self-consistent closures undergo latent exponential amplification, ending in a three-direction metastable structure) but no matter; the two-body paper gave the generation mechanism (a universal inelastic vertex acting on waveforms alone) but no universe - its collisions were loop counters, with when and where unanswered. This paper closes the gap without introducing any timing rule. Embedding the two-body vertex into the N-body relational wave system (M=N(N-1)/2 relational waves on the complete graph x a multi-slice register axis) with the interaction always on, we measure: (1) a two-factor law - generation rate = condensation gate x C f^2 (gate ratio 2.74) - where the condensation is precisely the endpoint of the onset paper: the ending of the previous paper is, by measurement rather than assumption, the opening condition of this one; (2) transplantation of the two-body laws unchanged (ignition power p=2.0011 with the coefficient constant to 0.4%; the pair-structure census holds up to the hair-complete version, sum-rule exclusivity ratio 1e289, hair lock 0.9997); (3) an end-to-end genesis run (vacuum -> latent exponential amplification -> three-direction condensation -> matter precipitation) under a single dynamics with no switches, discovering seed unification: the perturbation that ignites inflation and the seed that becomes matter are the same (epsilon-law agreement to 2%); (4) locality does not live on the graph (concentration ratios 1.28, 1.02) but on the register axis (response x19.6), products are pinned at their birth position, and harmonic-closure dispersion provides shape-invariant translation (1D deviation below 3e-4 cells, 3D deviation 0.0000); (5) the generation direction is a phase lottery (P(+)=0.52 over 50 random phases) rectified by the C f^2 autocatalysis (80% of the ensemble grows net). As conditional derivations we prove the mediated-vertex uniqueness theorem (closure conservation forces g2=-g1 and pair symmetry of the mediated strengths - the two-body uniqueness is its M=2 special case) and the identity of the computational structure (the adjacency sum folds exactly into two edge-node-edge passes; the operation count is linear in the state size, guaranteed by anonymity itself). An E8 special-value search is null on all instruments and recorded as such. Assumptions, conditional derivations, and measurements are strictly separated. Japanese and English full texts included.