Smoothing Operators and Algebraic Invariance in Axiomatic Second–Variation Geometry

The paper shows that in Axiomatic Second–Variation Geometry (ASVG), every symmetric second variation decomposes as a Laplacian polynomial (P(Δ_ω)) plus a smoothing operator (R). It proves that the smoothing term is completely harmless: (R) cannot change Hodge type, cannot create or destroy ((p,p))-classes, and vanishes on all ((p,p))-blocks except for a scalar action on harmonic forms. As a result, all geometric and algebraic behavior of ASVG is governed solely by (P(Δ_ω)). The minimizers of any ASVG-admissible functional coincide precisely with harmonic representatives of their Hodge classes, which are real-analytic and possess analytic (and for rational classes, algebraic) zero loci. This yields a canonical analytic mechanism that turns positive integral ((p,p))-classes into algebraic cycles. The analysis further shows that ASVG induces a rigid, positively curved geometry whose flows suppress all non-algebraic components. Consequently, ASVG provides an analytic pathway toward key parts of the Hodge and Grothendieck standard conjectures by ensuring algebraic rigidity of all relevant minimizers.