R1 — Recursive Geometry and Non-Markovian Invariants in Field Dynamics

This paper derives a geometric invariant for non-Markovian dynamical systems based on the survivability function S(t) = τ(t) cos(∇Φ(t)) Q(t), where τ represents memory depth, Q measures coherence-energy efficiency, and ∇Φ captures phase gradient. Three principal results are established: (1) Persistence Criterion (Theorem 2.1): trajectories with S(t) > 0 remain globally admissible even when smoothness fails, establishing a persistence criterion independent of regularity assumptions. (2) Irreducible Non-Markovian Property (Theorem 3.1): systems with dynamically varying memory depth (dτ/dt ≠ 0) exhibit irreducible path-dependence that cannot be captured by any finite-state Markov process. (3) Recursive Conservation Law (Theorem 4.1): the product Q(t)S(t) is conserved along admissible trajectories, defining a recursive information-geometric invariant. Numerical validation with Taylor-Green vortex simulations gives R² = 0.89 correlation between time-averaged S and inverse cascade timescale. Re-analysis of the Walker, Vanderlinden & Lipfert (2018) single-molecule magnetic tweezers benchmark dataset yields force-dependent coefficient of variation (CV = 1.78 at 3 pN, CV = 3.05 at 4 pN) and dwell-time autocorrelation persisting beyond 10³ events, consistent with dτ/dt ≠ 0 and inconsistent with any finite-state Markov model. A decisive experimental discriminator — glycerol titration under Kramers viscosity scaling — is proposed. CV invariance is algebraically guaranteed by any telegraph model, making CV decrease under glycerol a theorem-level falsification of the Markov class. The full protocol is described in the companion paper (Paper C). Pre-committed classification thresholds were archived 17 February 2026 (DNA-CH-02), approximately two months after the theoretical framework was first deposited (v1.0, 2025-12-17). Version history: v1.0 (2025-12-17) — foundation paper establishing all theorems and numerical validation. v2.0 (2026-03-13) — added Section 5.4 empirical re-analysis. v3.0 (2026-03-18, this version) — consolidated edition; proof of Theorem 2.1 expanded; theorem counter corrected; no theorem statement or empirical value changed.