Orbital Error Dynamics: Self-Organized Criticality, Ephemeral Parameter Resonance, and Non-Linear Biological Ontologies in Zero-Storage Neural Synthesis

Volkan Dağlı; Zerrin Dağlı; Dağhan Dağlı

cs.NESep 24, 2026

Orbital Error Dynamics: Self-Organized Criticality, Ephemeral Parameter Resonance, and Non-Linear Biological Ontologies in Zero-Storage Neural Synthesis

Authors: Volkan Dağlı, Zerrin Dağlı, Dağhan Dağlı

Organizations: Anadolu University, Eski¸sehir, Turkey · ITouch Systems, Mersin, Turkey · Mersin University, Mersin, Turkey · Toros Science College, Mersin, Turkey

Abstract

Modern deep neural networks treat parameters as static floating-point matrices stored in physical memory, incurring Von Neumann memory bottlenecks and representation collapse. We formulate Orbital Error Dynamics (OED), an analytical framework wherein synaptic weights are not stored masses (O(W)), but transient topological resonances (O(1)) derived procedurally from the complex quadratic polynomial map z_{n+1} = z_n^2 + c. We introduce the Bent Sine Wave Hypothesis, demonstrating that non-equilibrium living systems emerge when harmonic waves curl inward through environmental drag toward the cardioid cusp (c = 1/4). We define the Observer Horizon Geometry in parameter space, identifying interior resonance shoulder loci X_upper = (0.25, +0.18) and X_lower = (0.25, -0.18) between the fixed-point basin and the true boundary at c = 0.25 +/- 0.50i. To escape non-convex stagnation without loss zeroing, we introduce a heavy-tailed Biomimetic Perturbed Jump Operator (Omega_tunneling) inspired by mammalian fertilization zinc sparks. We further couple an enteric-cranial Dual-Brain architecture shielded by adaptive CD4+ regulatory immune gating (M_CD4), and project the 4-nucleotide genetic basis (A, T, C, G) across quadrants in C. Multi-seed empirical validation on the Two-Moons manifold (5 seeds, 80/20 train/test split, 32x32 grid, zero test-time updates, zero label leakage) demonstrates that procedural parameterization from a 24-byte coordinate seed achieves 77.67% +/- 5.35% clean test accuracy (within an 8.00-point paired difference of an unconstrained gradient baseline at 85.67% +/- 5.35%, 95% CI: [-1.07%, 17.07%]) and 71.33% +/- 3.80% under distribution shift (N(1.2, 0.4)), alongside conceptual equivalence with an analog optical co-processor.

Figures & tables

Figure 1: Parameter Space Geometry & Critical Loci (Schematic Topological Illustration): Main cardioid boundary $\partial\mathcal{M}$ , analytic cusp singularity at $c=1/4$ , true boundary loci at $(0.25,\pm 0.50i)$ , sub-boundary interior resonance shoulder loci $\mathbf{X}_{upper}=(0.25,+0.18)$ and $\mathbf{X}_{lower}=(0.25,-0.18)$ , radial escape fluxes $\mathbf{v}_{escape}$ , and homeostatic surfing corridor ( $\lambda_{z}\approx 0$ ).

Figure 2: Morphological Comparison of Cybernetic Dynamics Across Regimes (Schematic Illustration): (a) Static equilibrium: $\frac{dx}{dt}=0$ , complete entropic death; (b) Conservative oscillation: linear reversible Hamiltonian cycle; (c) Non-equilibrium bent cycle: non-linear dissipative cardioid cusp ( $c=1/4$ ) where asymmetric phase-folding sustains continuous information generation.

Figure 3: Biomimetic Perturbed Jump Operator Escaping Non-Convex Gradient Traps (Schematic Illustration): Energy landscape $V(\Theta)$ with local stagnation. When $\|\nabla\mathcal{L}\|<\epsilon_{\mathrm{tol}}$ , heavy-tailed jump $\Omega_{\mathrm{tunneling}}\sim\text{Cauchy}(0,\gamma_{\mathrm{spark}})$ traverses the potential barrier into the functional basin.

Figure 4: Conceptual Phase Space $(z_{n},z_{n+1})$ Trajectory Regimes (Schematic Illustration): (1) Hyperbolic sink collapse ( $\lambda_{z}<0$ ); (2) Unbounded chaotic divergence ( $\lambda_{z}>0$ ); and (3) Active OED homeostatic horizon surfing ( $\lambda_{z}\approx 0$ ).

Figure 5: Cybernetic Dual-Brain Architecture & Adaptive Immune Filter Flow: Coupling Central/Cranial Brain ( $\Theta\in\mathbb{C}\times\mathbb{R}^{+}$ , $O(1)=24$ B) with Enteric/Visceral Brain ( $X_{visc}$ ), shielded by Adaptive CD4+ Immune Gating Mask ( $M_{CD4}$ ).

Figure 6: Complex 4-Quadrant Genetic Base Decomposition: Mapping quadrant mass ratios $R_{1},R_{2},R_{3},R_{4}\in[0,1]$ to canonical biological bases: Adenine ( $Q_{1}\to w_{1}$ ), Thymine ( $Q_{2}\to w_{2}$ ), Cytosine ( $Q_{3}\to w_{3}$ ), and Guanine ( $Q_{4}\to b$ ), synthesizing weights without persistent tensor storage.

Figure 7: Empirical Multi-Seed Benchmark Evaluation Across 5 Independent Runs ( $T=50$ Epochs): (a) Training Loss Convergence curve (Mean $\pm$ 1 Std Dev); (b) Generalization comparison: clean test vs distribution shift (noise perturbation $\mathcal{N}(1.2,0.4)$ ) evaluated with zero test-time updates and zero label leakage; (c) Theoretical storage scaling: Standard $O(W)$ linear matrix allocation vs OED constant $O(1)=24$ Bytes (extrapolated across multi-layer scaling).

Paired Difference (GLM $-$ OED): Clean = 8.00% $\pm$ 7.30% [95% CI: -1.07%, 17.07%] — Noisy = 9.00% $\pm$ 6.52% [95% CI: 0.91%, 17.09%]
Architecture / Model	Clean Test Accuracy	Distribution Shift ( $\mathcal{N}(1.2,0.4)$ )	Memory Footprint	Optimization Scheme
Standard Logistic Regression (GLM)	85.67% $\pm$ 5.35% [79.03%, 92.31%]	80.33% $\pm$ 7.21% [71.38%, 89.28%]	16 B (Float32) / 32 B (Float64) [ $O(W)$ ]	Direct unconstrained gradient descent
OED (Zero-Storage Synthesis)	77.67% $\pm$ 5.35% [71.03%, 84.31%]	71.33% $\pm$ 3.80% [66.61%, 76.05%]	24 Bytes (3 Float64 coords) [ $O(1)$ ]	Boundary coordinate surfing ( $\delta=0.02$ )

Table 1: Rigorous Multi-Seed Empirical Benchmark Summary (5 Independent Runs, 80/20 Train/Test Split, 95% CI,

t_{4}=2.776

)

Hyperparameter	Symbol	Value / Setting
Coordinate Learning Rate	$\eta$	0.06
Finite Difference Step	$\delta$	0.02 ( $32\times 32$ pixel grid)
Initial Coordinate Seed	$\Theta_{0}$	$(-0.72,0.28,2.8)$
Total Training Epochs	$T$	50
Sampling Resolution	$N_{res}$	$32\times 32$ (Train & Test)
Zinc Spark Jump Scale	$\gamma_{\mathrm{spark}}$	0.10

Table 2: Algorithmic Hyperparameters & Experimental Protocol

Natural & Physical Phenomenon	Ontological Significance	Cybernetic & Mathematical Formalism
Multi-Agent Boundary Exploration	Decentralized non-linear phase exploration	Multi-polar parameter search with dynamic coordinate shifts
Homogeneous Enforcers	Rigid compliance forcing equilibrium	Isotropic gradient descent causing over-smoothing ( $\mathcal{L}\to 0$ )
Edge of Chaos Balance	Equilibrium between flexibility and structure	Critical state balancing exploration/exploitation ( $\lambda_{z}\approx 0$ )
Scale-Free Neural Hubs	Hierarchical network coordination	Central coordinate routing hub governing parameter projection
Protective Damping / Sacrifice	Boundary preservation under stress	Early-stopping dissipative regularization absorbing divergence
Spurious Signal Rejection	Disregarding secondary harmonic reflections	Filter isolating primary non-linear attractors from high-frequency noise

Table 3: Ontological Concept Mapping: Natural Systems to Cybernetic & Physical Formalisms

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Oscillations and synchronization are widely believed to play a fundamental role in representation and computation. However, existing machine learning approaches based on synchronization dynamics have largely been confined to specialized settings such as object discovery, with limited evidence of scalability to standard vision benchmarks or logic reasoning tasks. We propose the Winfree Oscillatory Neural Network (WONN), a dynamical neural architecture based on generalized Winfree dynamics. WONN evolves representations on the torus

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Abstract

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