TREP–Hodge Algebraicity by No-Ghost K-Theoretic Selection. A Non-Circular Conditional Endpoint-Exclusion Criterion

We give a compact theorem–proof formulation of the TREP–Hodge endpoint-exclusion criterion for rational Hodge classes on smooth complex projective varieties. The manuscript is written in a deliberately non-circular form. The algebraic component is removed by a rational Gram calculation inside the already realized cycle ledger; any residual class is placed in a guarded branch; Chow/Hilbert/Artin, Noether–Lefschetz/CDK, Abel–Jacobi/normal function, LMHS, boundary, K/Chern/Deligne, derived, and non-LCI alternatives are registered before terminalization. A port-free terminal carrier is then tested only by finite same-parent algebraic de Rham observers, used as charge detectors and not as cycle representatives. The main theorem is conditional on an explicit Hodge–TREP verification package: same-parent primitive branch extraction, port closure or descent, synchronized no-tube witness generation, primitive packing of the de Rham observer frame, and no-late-endpoint closure. Under this package, a nonzero terminal carrier would yield synchronized lower charge on arbitrarily long finite windows, while the fixed primitive ledger gives a uniform upper budget. The resulting finite-budget contradiction excludes the terminal carrier and yields algebraicity. Thus, the paper isolates the exact non-circular proof package needed for a TREP–Hodge argument. It does not use a Hodge-star or Weil projector, CDK as a cycle-production theorem, Abel–Jacobi injectivity, motivic conservativity, standard-conjecture positivity, or any endpoint budget introduced after the branch has been selected.