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 0\), the paper fixes 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 algebraicity obstructionXᵖ: = KXᵖ/Aᵖ (X). \ The module proves the unconditional carrier and reduction packageᵖ (X) KXᵖ, conjecture for (X, p) Xᵖ=0, with the dual detector formulationXᵖ=0 (₊ₗ㵵) ^ Aᵖ (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.