Developmental Harmonic-Topological Entrainment in Growing Neural Tissue

This repository contains the manuscript, simulation code, and structured data artifacts for "Developmental Harmonic-Topological Entrainment in Growing Neural Tissue: A Coupled In-Silico Atlas of Geometry, Folding, Electrical/Calcium Dynamics, Metabolism, and Hopf-Compatible Phase Structure." Overview This computational working paper establishes a stage-resolved framework to test whether self-organizing neural matter is shaped by the coupled interactions of time-evolving domain geometry, differential growth mechanics, excitable electrical/calcium dynamics, mitochondrial-metabolic feedback, and complex phase topology. Rather than treating brain harmonics as a static, adult-only phenomenon or assigning sole causality to mechanics, the project synthesizes these layers across four numerical investigations and an integrated developmental atlas: Cranial Proxy Modal Mixing: Analyzes scalar Laplace-Beltrami eigenmodes within skull-shaped geometries, demonstrating territory formation and coherent web-like nodal interference. Coupled Developmental Surface Model: Demonstrates how metabolic and electrical/calcium activity fields modulate the timing and spatial placement of cortical folding instabilities without replacing mechanical differential growth. Topological Fibration Testing: Distinguishes visual nodal loops from true Hopf topology, demonstrating that while unseeded harmonic fields possess near-zero Hopf charge, an explicitly phased Hopf state is efficiently captured by a low-order cranial basis (99.95% charge retention and pairwise Gauss linking $\approx 1$). Hyperspherical Sweep & Atlas Synthesis: Evaluates harmonic spectra across unit spheres $S^2$–$S^8$ (including classical Hopf dimensions 2, 4, 8) and introduces a scored, gate-controlled Developmental Entrainment Atlas ranking stage-eligible spatial sequences, nested carriers (e.g., 13–20 Hz metabolic windows), and ultradian-to-network envelopes across developmental time.