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

Explore similar work

May 20, 2026cs.LG

Winfree Oscillatory Neural Network

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

(S^1)^d

through structured oscillatory interactions, combining phase-based inductive biases with flexible and hierarchical interaction mechanisms instantiated as either fixed trigonometric mappings or learnable neural networks. We evaluate WONN on image recognition and complex reasoning tasks, including CIFAR, ImageNet, Maze-hard, and Sudoku. Across these domains, WONN achieves competitive or superior performance with strong parameter efficiency. In particular, WONN is, to our knowledge, the first synchronization-based oscillatory architecture to scale competitively to ImageNet-1K. Furthermore, on Maze-hard, WONN achieves 80.1% accuracy using only 1% of the parameters of prior state-of-the-art models. These results suggest that structured oscillatory dynamics provide a scalable and parameter-efficient alternative to conventional neural architectures.

Jiawen Dai, Yue Song

3Shanghai Jiao Tong University · 1Shanghai Qi Zhi Institute · College of AI, Tsinghua University

Neural Network Oscillatory Dynamics

Ask PDF

Jun 16, 2026cs.LG

A Link between Shock-wave Theory and Symmetry-reduced Stochastic Gradient Descent for Artificial Neural Networks

We develop a mathematically explicit link between shock-wave theory and the symmetry-quotiented learning dynamics of stochastic gradient descent, drawing on differential geometry, Lie group theory, and fluid mechanics. Specifically, after quotienting parameter symmetries and applying local-entropy coarse-graining, the effective dynamics satisfy a viscous Hamilton--Jacobi equation on the quotient manifold. Moreover, under the assumption that the raw parameter dynamics can be summarized by a gradient field on the quotiented space, the gradient of the coarse-grained loss function obeys a Burgers-type equation, and shock formation can be established rigorously. We apply our theory to multilayer perceptrons, convolutional neural networks, Transformers, and mean-field networks, and show that they obey the Hamilton--Jacobi or Burgers-type equations. We conjecture that this framework also yields practical diagnostics for deep learning. In architectures such as Transformers, raw parameter norms are often distorted by symmetry redundancy and may therefore be misleading, whereas symmetry-corrected quotient observables provide a principled basis for monitoring, forecasting, and controlling training-phase transitions.

Taiki Miyagawa

NEC Corporation

Stochastic Gradient Descent Neural Network Gradient Symmetry Shock Propagation Loss Function

Ask PDF

May 5, 2026cs.LG

Physics-Modeled Neural Networks

We introduce \emph{Dynamical Physics-Modeled Neural Networks} (DynPMNNs), a continuous-time deep learning architecture in which each hidden layer is defined as the solution of an ordinary differential equation. Unlike classical feed-forward networks, this approach replaces static activation functions with time-evolving dynamical systems, providing a biologically inspired interpretation of hidden-layer behavior and enabling the integration of physically meaningful models. The framework is rigorously grounded in Reproducing Kernel Banach Spaces (RKBSs), allowing DynPMNNs to be characterized as finite-dimensional solutions of an abstract training problem and revealing structural connections with standard neural networks. We present a concrete implementation based on the FitzHugh--Nagumo model for neuronal activation, where numerical ODE solvers are embedded into the computational graph via Euler-type schemes. Both network weights and dynamical parameters are trained jointly. Through experiments on the California Housing dataset, we compare DynPMNNs with Neural ODEs (NODEs) and Closed-form Continuous-Time Networks (CfCs). Despite using fewer trainable parameters, DynPMNNs achieve competitive performance. These results position DynPMNNs as a principled bridge between dynamical systems and deep learning, with promising directions for further research in expressivity, stability, and physics-based modeling.

Raul Felipe-Sosa, Angel Martin del Rey, Maria Flores Ceballos

aDepartamento de Matemáticas, Escuela Politénica Superior, Universidad Nebrija, 28015 Madrid, Spain · bDepartment of Applied Mathematics, IUFFyM, Universidad de Salamanca, 37008 Salamanca, Spain · cUnidad de Excelencia en Luz y Materia Estructuradas (LUMES), Universidad de Salamanca, 37008 Salamanca, Spain

Neural Dynamics Neural Ordinary Differential Equations Neural Network Dynamical Systems

Ask PDF