Specular Field Framework: Mathematical Structure of the Two-Sheet Informational Geometry

This work develops the mathematical foundations of the Specular Field Framework, an informational two‑sheet geometry governed by a global involutive symmetry. Each sheet carries its own metric and kinetic signature, with the specular sheet endowed with a negative‑sign kinetic term required by informational complementarity. Together with a global informational ledger, this structure induces a natural symmetric/antisymmetric decomposition of the fields, dynamically suppressing the antisymmetric sector and leaving the symmetric mode as the only effective propagating degree of freedom. The paper constructs the full mathematical sequence, including informational axioms, dual-sheet geometry, Lagrangian and Hamiltonian formulations, gravitational sources, field equations, flavour-space Hamiltonian, and the effective path-integral structure. Integrating out the antisymmetric mode yields an effective symmetric action, from which a minisuperspace Schrödinger equation follows as a purely mathematical consequence of the framework. No cosmological model or phenomenological scenario is introduced. A minimal numerical experiment illustrates the structural predictions of the theory: exponential suppression of the antisymmetric mode, stability of the symmetric sector, and coherent reorganization under the specular involution. These results confirm the internal consistency of the Specular Field Framework and establish its mathematical foundations as a basis for future developments in informational and geometric field theory.