Relational Lysis: The Primitive Architecture of Structure, Constants, and Physical Law

Relational Lysis (RL) is presented as a primitive mathematical theory: a finite, deterministic architecture in which logic, arithmetic, geometry, dimensionless constants, and the structural skeleton of physical law are derived from a single generative act of distinction governed by nine axioms. The development is stratified. The primitive tier derives the emergence of a unique forced second-order structure on the solved relational state --- which renders, downstream of the gate, as a self-adjoint operator; the binding constant $\rho=(\sqrt5-1)/2$ from the containment identity $s^{2}+s=1$, with incommensurability established by exact unit-defect brackets; a complete exact combinatorial geometry (triangle census $4N{+}1$, circulation rank $3N{+}1$ with integral triangle generation, bending density six, metric balls $4r{+}2$, shell growth with a sharp bulk-to-boundary transition); a parity involution commuting with the operator; and the constants $e$ and $\sqrt2$ as census limits with exact defect laws. The rendering tier, entered through exactly two declared postulates (continuation and isotropy), determines the Pythagorean separation rule uniquely; constructs $\pi$ as the closure measure of the derived direction group; yields pair-symmetric dynamics with an exact massless/massive sector split (substrate speed $\sqrt{2\rho}$; spectral gap $\sqrt5+1$); derives the unique inter-sector vertex with running effective coupling $\rho\sqrt2/\sqrt M$; and produces the structural layers of gauge theory, quantum mechanics, statistical mechanics, and hydrodynamics, including unconditional a priori bounds on density, transport speed, and vorticity, and an exact second-order closure of the transport moment hierarchy with the complete relaxation spectrum in closed form. The dependency structure of the development is itself verified: $286$ nodes and $799$ edges, acyclic, terminating exclusively in the nine axioms, the two postulates, and self-contained identities, with longest chain $26$. Companion papers: https://doi.org/10.5281/zenodo.19116249, https://doi.org/10.5281/zenodo.19144445 derive the Yang--Mills mass gap and the gravitational regime on this architecture; the present paper is their foundation.