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\[K_X^p := H²ᵖ(X,Q)∩ Hp,p(X),\]the algebraic cycle span\[A^p(X):=spanQ\{\,cl(Z): Z⊂ X algebraic of codimension p\,\},\]and the residual obstruction\[R_X^p:=K_X^p/A^p(X).\] The central detector no-loss target is\[Ann(K_X^p)^A^p(X)=0.\]Equivalently, for every nonzero rational Hodge detector\[0≠ λ∈ (K_X^p)^,\]one must construct an algebraic cycle \(Z\) such that\[λ(cl(Z))≠ 0.\] The module introduces monodromy-invariant windows, invariant Hodge carriers, invariant algebraic cycle spans, and invariant residual quotients\[KX,inv^p,X,inv^p,X,inv^p.\]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:\[∀\,0≠λ∈ (K_X^p)^,∃\,Z∈ Z^p(X)Q thatλ(cl(Z))≠0.\]