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This research investigates the Hodge Conjecture, focusing on the relationship between algebraic cycles and cohomological classes in smooth projective varieties.

May 22, 2026Open Access

The Hodge Conjecture as Abstract-to-Geometric Cycle Closure in IE-016 Identity Architecture — Approach Geometry of Cohomology Substrate Applied at the Algebraic-Cycle-Resolution-Class

Key Points

This research investigates the Hodge Conjecture, focusing on the relationship between algebraic cycles and cohomological classes in smooth projective varieties.
Application of HLRP #1 Approach Geometry of Amplitude Zeros methodology at the algebraic-cohomology-substrate-resolution-class.
Exploration of the IE-016 Identity Persistence architectural mechanism and how it operates across projections.
Integration of various substrate-resolution classes to understand the cross-substrate manifestation.
Identified connections between abstract cohomological classes and geometric cycles indicating a structural coherence in the proposed architecture.
Proposed substrate-coupling mechanism that aligns Hodge classes with algebraic cycles, expanding understanding of the Hodge Conjecture.
Demonstrated that internal consistency in the architectural framework is essential for closure in various resolution classes.

Abstract

The Hodge Conjecture proposes that, on a smooth projective complex algebraic variety X, every rational Hodge class is a Q-linear combination of cohomology classes of algebraic subvarieties. Formally: for X smooth projective over C, H^p, p (X) H^2p (X, Q) = Z ₐ where Z ranges over algebraic cycles of codimension p. This companion deposit proposes that the Hodge Conjecture is the formalism-substrate-resolution-class readout of an underlying substrate-coupling-architecture in which abstract cohomological classes and geometric algebraic cycles are substrate-coupling-architecture-identity-persistence-readouts at two substrate-resolution-classes. The architectural-mechanism: IE-016 (Identity / Persistence) operates across the projection from geometric-substrate (algebraic cycles) to cohomological-substrate (Hodge classes). The Hodge Conjecture asserts that this projection admits identity-persistence-substrate — i. e. , every cohomological readout (Hodge class with rational coefficients) corresponds to a geometric-substrate-coupling-architecture realization (algebraic cycle). Architectural translation: the HLRP #1 Approach Geometry of Amplitude Zeros methodology (Zenodo DOI 10. 5281/zenodo. 18727570 (https: //doi. org/10. 5281/zenodo. 18727570), originally developed at gauge-field amplitude-substrate, then extended to Riemann zeta-zeros in Companion II) applies at the algebraic-cohomology-substrate-resolution-class. Same architectural-substance: amplitude-zero / cohomology-class distribution as substrate-coupling-architecture-substance with axial closure-coordinate constraints. Cross-substrate manifestation: abstract-to-geometric identity-persistence renders at: algebraic-cohomology-substrate (Hodge, this paper) ; zeta-function-substrate (Riemann, Companion II) ; gauge-field-substrate (Yang-Mills, Companion III) ; elliptic-curve-substrate (BSD, Companion VI) ; fluid-substrate (Navier-Stokes, Companion IV) ; cardiac-substrate (HLRP #198 v3 empirical). Same architectural-mechanism rendered at six substrate-resolution-classes. Internal-grammar consistency is the load-bearing closure. Keywords: Hodge Conjecture; algebraic cycles; cohomology; complex algebraic varieties; smooth projective; Millennium Prize Problems; substrate-coupling architecture; Hydrogen Lifecycle Research Programme; HLRP; approach geometry of cohomology substrate; IE-016 Identity Persistence; IE-019 Residual; IE-007 Perceptual Closure; abstract-to-geometric closure; identity-persistence-substrate; cross-substrate architectural mechanism; architectural formalization

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