Admissibility Poisson Equation from Flux-Conserved Residual Propagation

This record presents TA27 (Admissibility Poisson Equation from Flux-Conserved Residual Propagation), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA24 established the accumulated residual load as the source of admissibility deformation. TA25 established that modes respond by minimizing admissibility cost. TA26 established that residual influence propagates through the complement sector as a field-like object. TA27 derives the field equation governing that deformation field. The residual source density is defined as the accumulated load from TA24. If the admissibility structure is locally connected and residual load propagates via a conserved flux law, the deformation field satisfies a discrete field equation where the admissibility-structure Laplacian acts on the deformation field with the residual source density as the driving term. In the continuum limit established in TA15, this becomes the admissibility Poisson equation with the residual source density as source. This is the admissibility Poisson equation. It is structurally identical to the Newtonian gravitational field equation, with the residual source density in the role of mass density and the deformation field in the role of gravitational potential. This structural identity is stated plainly rather than either overclaimed or understated. The admissibility structure is not spacetime. The deformation field is not the Newtonian potential. The residual source density is not mass. But the mathematical structure is the same. From this point forward in the series, under weak, linear, and effective 3D propagation assumptions, the residual admissibility field reproduces the formal structure of Newtonian gravitational behaviour. TA27 establishes only the Poisson-type sourcing structure under local conserved propagation assumptions. Inverse-square falloff is not derived here. Whether the Q5 admissibility propagation structure satisfies the weak, linear, and effective 3D assumptions required to produce gravitational phenomenology is the subject of TA28. Two lemmas support the main theorem. The first establishes the residual source density as a source term for boundary deformation, where it is nonzero, the boundary is deformed; where it vanishes, the boundary is undeformed. The second establishes that if residual propagation is locally flux-conserving, the deformation field satisfies a discrete conservation law on the admissibility graph with the source density as the driving term, by the standard relationship between graph Laplacians and source-driven diffusion. Three corollaries follow. The Poisson structure corollary notes that in three effective spatial dimensions, the solution for a point source is proportional to one over r, whose gradient gives inverse-square falloff. Whether the admissibility structure has three effective dimensions is addressed in TA28. The superposition corollary establishes that since the source is additive over independent attachment modes and the Poisson equation is linear, the deformation field is the linear superposition of contributions from all attachment sites. The zero-source limit corollary establishes that when the source vanishes, the field equation reduces to the Laplace equation, consistent with the smooth-path principle of TA16. The propagation law is structurally exact but conditional in explicit form. The admissibility-structure Laplacian is induced by the local adjacency graph of TA14. Whether it is isotropic, whether it reduces to a standard Laplacian, and whether the effective dimensionality is three are separate questions addressed in TA28. The theorem chain progressively derives the structure of the effective transport generator \[ G_eff = Pi_Y G Pi_Y + K†BK \], from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.