TEBAC Hodge Program III: Detector No-Loss, Monodromy-Invariant Windows, and Algebraic Cycle Separation

This preprint is the third module of the TEBAC Hodge program. It develops the detector no-loss battlefront for a Clay-compatible modular attack on the rational Hodge conjecture. The module is a hardened theorem-target and obstruction-separation paper; it does not claim a completed proof of the full Hodge conjecture. Starting from the HODGE-I and HODGE-II framework, the paper works with the rational Hodge carrierXᵖ: = H^2p (X, Q) H^p, p (X), algebraic cycle spanᵖ (X): =spanₐ\\, cl (Z): Z X algebraic of codimension p\, \, the residual obstructionXᵖ: =KXᵖ/Aᵖ (X). \ The central detector no-loss target is (₊ₗ㵵) ^Aᵖ (X) =0. , for every nonzero rational Hodge detector\0 (KXᵖ) ^, must construct an algebraic cycle \ (Z\) such that\ (cl (Z) ) 0. \ The module introduces monodromy-invariant windows, invariant Hodge carriers, invariant algebraic cycle spans, and invariant residual quotientsₗ, ₈₍ₕᵖ, ₗ, ₈₍ₕᵖ, ₗ, ₈₍ₕᵖ. separates the cohomological control supplied by monodromy invariance and Hodge-locus methods from the stronger cycle-algebraicity output required to prove the Hodge conjecture. The hardened conclusion is that monodromy invariance alone gives cohomological control, not automatically an algebraic cycle separator. A genuine proof must add a Hilbert--Chow cycle output or a detector-matrix rank theorem producing actual algebraic cycles whose classes separate all nonzero detectors. Thus HODGE-III is Zenodo-ready as a detector no-loss and algebraic cycle separation module. The remaining terminal target is the Algebraic Separator Theorem: \\, 0 (KXᵖ) ^, \, Z Zᵖ (X) ₐ that (cl (Z) ) 0. \