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 carrier\[K_X^p := H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X),\]the algebraic cycle span\[A^p(X):=\operatorname{span}_{\mathbb{Q}}\{\,\operatorname{cl}(Z): Z\subset X\text{ algebraic of codimension }p\,\},\]and the residual obstruction\[R_X^p:=K_X^p/A^p(X).\] The central detector no-loss target is\[\operatorname{Ann}_{(K_X^p)^\vee}A^p(X)=0.\]Equivalently, for every nonzero rational Hodge detector\[0\neq \lambda\in (K_X^p)^\vee,\]one must construct an algebraic cycle \(Z\) such that\[\lambda(\operatorname{cl}(Z))\neq 0.\] The module introduces monodromy-invariant windows, invariant Hodge carriers, invariant algebraic cycle spans, and invariant residual quotients\[K_{X,\mathrm{inv}}^p,\qquadA_{X,\mathrm{inv}}^p,\qquadR_{X,\mathrm{inv}}^p.\]It 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:\[\forall\,0\neq\lambda\in (K_X^p)^\vee,\quad\exists\,Z\in Z^p(X)_{\mathbb{Q}}\quad\text{such that}\quad\lambda(\operatorname{cl}(Z))\neq0.\]