Complete physical complex G-closure and proof-carrying finite-data compilation for two-dimensional two-phase conductivity

Abstract

We give a standalone, constructive theory of the complete quasistatic conductivity-function closure for two scalar isotropic phases in two dimensions. The physical class is identified with reflection-complement positive matrix measures and with a precisely symmetry-constrained matrix Schur class on the unit disk. For nonreal reciprocal samples, one explicitly computable positive-semidefinite matrix decides feasibility; its rank is the exact shortest pure-phase-host lamination length. Rational inner degree is exactly twice that physical length, with endpoint ranks counted explicitly. The main quantitative advance is a sharp minimax information theorem. At prescribed completed contrast nodes, the smallest possible worst-case error for recovery of the normalized weak-contrast coefficient is exactly one half times the product of the squared moduli of the associated disk coordinates. Two shortest physically realizable rational inner interpolants attain the lower bound. Requiring the estimator itself to have shortest host length doubles its conditional minimax error; one additional host layer restores the optimum. A Blaschke-product envelope controls all compatible responses, self-dual sampling yields O(log(1/epsilon)) host-step approximation on fixed compact contrast sets, and an adversarial construction matches the exponential information rate. Joint unmeasured responses form an exact spectrahedron; linear prediction extrema have finite physical realizations and admit rational primal-dual certificates in the strictly feasible case. Classical Schur, Stieltjes and conic-duality ingredients are identified as such; the physical-information synthesis is not presented as a historically certified first discovery. Full releases with files: Zenodo: https://zenodo.org/records/22721915 Hugging Face: https://huggingface.co/datasets/PureOne/phase-orbit-complex-g-closure-v3.5.0

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