Master Equation Revisited: Reflected Semantic Gradient Flow and the Viability Body. A single-action derivation of the two-sided memory law, the Weak Energy Condition, and finite-acuity causal structure

AbstractThe Canonical Semantic Framework (CSF) has, until now, carried a master equation that read as a slogan with attachments: a least–action principle decorated with a memory floor, a Fisher ceiling, a holonomic transport term, a gravity slot, and a complexity firewall, each justified in a separate paper. This work replaces the slogan with a single derived object. We show that one joint two–sector semantic action, profiled over its unregistered sector, generates the entire apparatus: its profiled Hessian is the minimum–memory floor; its profiled kinetic dressing is the no–ghost condition; and the same positive–semidefinite back–reaction matrix appears as a debit in the floor and a credit in the propagator, the information–geometric Gauss equation. We prove that the floor condition is exactly –strong convexity of the profiled action, making the realized dynamics a reflected gradient flow on a convex viability body, a well–posed object via maximal–monotone–operator theory. We then derive that the Weak Energy Condition (WEC) of the effective stress–energy does not reduce to one inequality but splits into three independent positivity conditions, which are precisely CSF’s reserve wall, floor wall, and kinetic leg; the back–reaction is asymmetric, manufacturing tachyons but never ghosts. The KKT multiplier of the floor is shown to be the field-theoretic analogue of the energy density, , identifying the Fifth Interaction with the non–negativity of the energy condition and recasting chronology protection as ceiling saturation. Throughout, we correct a methodological drift: the floor an observer reconstructs is not a pointwise Hessian (an infinite–acuity object) but the best band–limited approximation built from the 2001 sinc/Paley–Wiener kernel; finite acuity smooths the viability corners and regulates the kinetic energy. We further show that the framework’s causal structure is a bicone, not a single cone: the two Schur complements of the Fisher matrix define the forward viability cone and the inverse reconstructibility cone, their inverses are the two diagonal blocks of the inverse Fisher, sharing one singular surface, with the anti–Hessian diagnosing precisely the hidden–sector–sourced channel of their joint collapse. And we generate, rather than name, the holonomic kinetic operator : it is the mixed fiber/network transport, the curvature of the combined connection, which twists the local bicone as it is carried along the causal spine, with loop–reversal antisymmetry verified to machine precision. The boundary leg of the operator stack is likewise generated, not named: the retained Paley–Wiener band carries a conserved spectral Noether current in its interior, and the boundary term is the edge charge, the band derivative of the Fisher budget whose bulk integral is the firewall ceiling, with a spectral defect that vanishes for a reciprocal observer and pairs the two bicone legs. Every structural claim is verified by an explicit, reproducible computation on a minimal field–dependent action, and every claim is tiered: machine–exact, derived, anchored, or frontier. The single remaining frontier, upgrading the Lorentzian mass–matrix signature to a genuine causal lightcone via an eikonal/homogeneity program, is isolated cleanly.