Minimal Reconstruction and the Canonical Six-Channel Operator in Constrained Null Geometry

This paper completes a previously open step in the Constrained Null Geometry reconstruction program by deriving the canonical quadratic operator of the common six-channel domain. Earlier work established protected topology, first-loss boundaries, bridge-supported reconstruction, and reduced geometric certificates, but left the complete microscopic Hessian of the common domain undetermined. The present work introduces the principle of minimal reconstruction sufficiency. According to this principle, a completed common domain contains exactly the geometric structure required for its present admissibility and no redundant historical labels without a current geometric carrier. Under this principle, the completed six-channel domain becomes invariant under the full permutation group S6. This uniquely determines the common-domain Hessian, fixes the previously undetermined stiffness ratio h parallel divided by h perpendicular to one half, and yields the explicit induced five-channel operator after rank-one first loss. The paper clearly distinguishes the exact mathematical theorem from the physical hypothesis of partition erasure and formulates explicit falsification conditions. No phenomenological fitting to nuclear binding energies is used. The result fixes the normalized operator structure while leaving the overall physical scale to the complete Constrained Null Geometry closure map. A separate Supplement contains the full mathematical derivations, permutation-symmetry arguments, pullback calculations, spectral analysis, and the conditions under which the result must be weakened or rejected.