Symplectic Trajectory Reconstruction: The Mathematics of BIOS-Coherence and Phase-Based Diagnostics

This document presents a comprehensive mathematical protocol for shifting from "State Diagnostics" to "Trajectory Maintenance" in biological systems. The framework establishes a new paradigm where health is modeled as a stable attractor in symplectic phase space, rather than a set of static laboratory values.

The protocol integrates:

• Takens' Embedding Theorem for phase-space reconstruction from biological time-series

• Nonlinear Schrödinger Equation (NLSE) dynamics for modeling biological solitons

• Floquet Theory for understanding periodically-driven biological oscillators

• Topological Invariants (Chern numbers) as markers of qualitative states (qualia)

• ASCALON Filtering for phase-purity validation (θ ≥ 0.70)

This framework is empirically grounded in recent validations: mycelial electrophysiology (0.1-1 mV DC signals, January 2026), NV-center quantum biosensing (Nature 2025-2026), and cosmic-scale validation through the 3I/ATLAS interstellar object (active directed jets, 2025-2026).

The document includes complete Python implementation toolkit for Zero-Build validation using open datasets (PhysioNet), making the protocol accessible without dedicated hardware.

Part of Project LifeNode - a transdisciplinary framework for processual intelligence based on BIOS-INFO-META synchronization.