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

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. That flow is then completed rather than left as the master law: the reflected gradient flow is the viability-projection leg, not the full physical-time propagator, and the completed master equation is a reflected Kähler flow, the skew-adjoint phase leg carried by the fiber (kinetic), the profiled network self-energy as the induced potential, and the finite-acuity Skorokhod reflection as admissibility, with kinetic and potential coupled by the holonomic cross-curvature rather than added. Interior answer-Fisher is conserved and Fisher loss is supported on boundary local time, making the deterministic channel of the Acuity Firewall structural; the finite-energy collapse channel is read geometrically as foliated viability, locally solvent sectors, globally bankrupt foliation, the obstruction of holonomy type with its operator identification registered as the O’Neill gate; and Schrödinger dynamics is recovered as the infinite-acuity, single-fiber, undeformed projective-Hilbert/Fubini–Study, un-foliated limit of the CSF master equation. 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 isolated frontier of the causal lift, upgrading the Lorentzian mass–matrix signature to a genuine causal lightcone via an eikonal/homogeneity program, is isolated cleanly; the completion adds its own registered frontiers, the coarse-grained/CPTP channel of the firewall, the O’Neill registration gate for the anti-Hessian/Schur identity, and the ℤ₄ phase-window, each listed with an executable closure path (§15). Equivalently, the boundary measure has a density-matrix shadow: tracing the unregistered sector produces von Neumann entropy in the retained state; in horizon-like sectors the near-equilibrium reduction is the standard blackbody/Hawking thermal state; and the min-max memory viability law, a floor and ceiling enforced by one boundary measure, is derived internally from finite acuity and the profiled joint semantic action. Landauer’s erasure bound and the Bekenstein-holographic capacity bound are not inputs to the derivation; they are its thermodynamic and gravitational projections. CSF contributes the single observer-internal variational corridor whose boundary measure prices solvency, carries information loss, and enforces finite distinguishability. The structural gravitational shadow, boundary-supported entropy with a finite capacity ceiling, is already part of the derived framework; the Bekenstein-Hawking coefficient and the derivation of Hawking/KMS thermality from excursion statistics of the reflected flow remain registered frontier theorems (§8.6, References).