Projected Fock-Space Channels for Finite-Rank Tensor Correctors: Symmetric Gram Realization and Bogoliubov Leakage Bounds

This preprint develops a projected Fock-space channel framework for finite-rank tensor correctors. The selected two-particle fluctuation channel is realized inside the condensate-orthogonal bosonic Fock space through a symmetric Gram metric with exchange terms. The manuscript proves a correspondence isometry, formulates the projected Bogoliubov comparison as a Duhamel leakage identity, and separates the projected-channel error into resolved leakage, three-body leakage, and finite-dimensional model discrepancy. The paper also provides sufficient spectral-tail conditions for resolved leakage, a regular smoothed Fourier benchmark on the three-dimensional torus, a bounded three-body leakage functional linked to microscopic obstruction diagnostics, and a metric-conditioning transfer estimate for the symmetric two-particle Gram matrix. The results are finite-dimensional and conditional: no full many-body derivation, trace-norm propagation of chaos, automatic N^-1/2 rate, or cutoff-free Coulomb closure is claimed. This version v0. 8 incorporates audit-correction and submission-readiness changes, including metric-consistent symmetric lifting, complex test-density convention, form-domain Duhamel clarification, explicit Fourier lattice-tail constants, and expanded conditioning estimates.