TEBAC Hodge Program I: Spectral Hodge Carriers, Cycle Currents, and the Residual Algebraicity Obstruction

This preprint initiates the TEBAC Hodge program as a classical, Clay-compatible modular framework for the rational Hodge conjecture. The module does not claim a completed proof of the Hodge conjecture. Its purpose is to isolate the exact spectral, current-theoretic, and finite-dimensional obstruction content of the problem. For a smooth projective complex variety \(X\) and \(p\geq 0\), the paper fixes 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 algebraicity obstruction\[R_X^p := K_X^p/A^p(X).\] The module proves the unconditional carrier and reduction package\[A^p(X)\subseteq K_X^p,\qquad\text{Hodge conjecture for }(X,p)\LongleftrightarrowR_X^p=0,\]together with the dual detector formulation\[R_X^p=0\Longleftrightarrow\operatorname{Ann}_{(K_X^p)^\vee} A^p(X)=0.\] Thus HODGE-I closes the foundational spectral/current module of the TEBAC Hodge program and identifies the later decisive theorem target as detector completeness or Chow/Hilbert reconstruction. The paper deliberately avoids importing the Hodge conjecture, the generalized Hodge conjecture, the standard conjectures, or motivic full faithfulness as hidden assumptions.