KM Energy Decay, Calibrated Return, and a Conditional Proof of the Rational Hodge Conjecture Subtitle: Finite Obstruction Matrices, Algebraic Cone Shadows, and Density Rigidity

Abstract

This paper identifies a precise analytic-to-algebraic bridge under which the KM compression and return formalism yields the rational Hodge conjecture for a fixed smooth complex projective variety. The bridge is formulated in terms of integral-current realizations of effective KM compression cones, uniform mass bounds, absence of mass loss, and asymptotic saturation of the Kähler calibration inequality. We first compare the calibration defect with a pointwise complex-plane defect through explicit Kähler-angle estimates. Elliptic estimates give a spectral gap after removal of genuine zero modes, and a forced energy identity records exactly which return errors must be integrable. Smooth geometric realizations preserve integral homology by a homotopy-current formula; a minimizing-movement construction then gives a conditional weak continuation principle across singular times. The finite-basis theorem requires neither a globally smooth flow nor exponential convergence. It is enough that, for a rational basis of the Hodge space, signed pairs of closed integral currents have uniformly bounded mass and calibration defect tending to zero along a sequence. Compactness, lower semicontinuity, the structure theorem for calibrated integral currents, and Chow algebraicity then produce classical algebraic cycles representing the basis. We assemble the successive KM return obstructions into a finite total obstruction map and show that the basis construction annihilates this map. A separate density-rigidity theorem proves that positive lattice density of algebraic Hodge classes already forces all rational Hodge classes to be algebraic. Consequently, exceptional loci obtained from a particular KM certificate are algorithmic degeneracy loci, not a possible positive-codimension collection of genuinely nonalgebraic exceptions once positive density has been established. The result is conditional, not an unconditional proof of the Hodge conjecture. Its unresolved input is the calibrated KM cone-shadow hypothesis: each effective compression cone must admit a closed integral realization with no loss of class or mass, zero limiting calibration defect, proper projectivized support, and a cycle-faithful comparison between its Segre shadow and the corresponding compression layer. Under this hypothesis, the finite return construction proves the rational Hodge conjecture for the fixed pair \((X,p)\). A finite-dimensional countermodel shows why internal KM algebraicity, Hodge-surjectivity, and termination of compression do not by themselves supply the missing classical cycle. Keywords KM theory; Hodge conjecture; algebraic cycles; calibrated currents; Kähler calibration; energy decay; spectral gap; integral currents; compression obstruction; cone shadows; Segre classes; return–slice algebraization; density rigidity; conditional algebraization.