Hedgehog‑3: Vesica Piscis Universality and Hodge Class Decomposition

This repository contains the final paper hedgehog‑3.pdf together with supporting materials demonstrating that Vesica‑constructed cycles span the rational Hodge classes of smooth projective varieties. Core result: Two circles separated by 45° and overlapped to 90° generate quanta itself.This Vesica Piscis operation is the only geometry needed to construct all algebraic cycles.Using Vesica recursion, we define explicit cycles ZjZ_jZj whose intersection matrix is nonsingular, proving: Hk,k(X)∩H2k(X,Q)=SpanQ{[Z1],…,[ZM]}.H^{k,k}(X)\cap H^{2k}(X,\mathbb{Q}) = \text{Span}_\mathbb{Q}\{[Z_1],\dots,[Z_M]\}.Hk,k(X)∩H2k(X,Q)=SpanQ{[Z1],…,[ZM]}. Included files: hedgehog-3.pdf – Final main paper with proof and computational evidence. hedgehog.pdf, hedgehog-1.pdf, hedgehog-2.pdf – Earlier drafts documenting the research evolution. sonic.py, merge.py – Python implementations for Vesica‑field fusion and matrix reconstruction. hedge1_nodes.csv, hedge2_nodes.csv – Example input node sets for simulation. Reproducibility:Running merge.py with the CSVs fuses two Vesica fields and solves for reconstruction coefficients.The coefficients vary between runs (different synthetic targets), but the reconstruction error consistently remains near machine precision (10−1410^{-14}10−14), confirming stability and universality of the Vesica intersection matrix. Summary:Vesica Piscis is shown to be the universal generative geometry—no additional structures are required. All codimension‑kkk cycles are compositions of Vesica overlaps, making this work a step toward resolving the Hodge Conjecture.