Deformation Induced Transition Between Algebraic and Transcendental Structures on K3 Surfaces: A Period Based Reconstruction Framework

This paper propose and analyze a conceptual experiment on K3 surfaces aimed at understanding theinterplay between algebraic cycles and transcendental structures under deformation. Starting from an explicitalgebraic K3 surface with nontrivial Néron–Severi group, this paper introduces a controlled complexdeformation and track its impact on cohomology, algebraic cycles, and periods. This work show that algebraiccycles do not disappear in a literal sense but lose their algebraic realizability when their cohomology classes exitthe Hodge subspace 𝐻1,1. Their information persists in the transcendental lattice and becomes accessible onlythrough period integrals. This provides a precise mathematical description of a transition from discrete geometricdata to continuous analytic data, analogous to a loss of observability rather than loss of structure. The experimenthighlights a fundamental obstruction underlying Grothendieck’s standard conjectures, especially Conjecture D.