Orbital Error Dynamics: Self-Organized Criticality, Ephemeral Parameter Resonance, and Non-Linear Biological Ontologies in Zero-Storage Neural Synthesis
Volkan Dağlı, Zerrin Dağlı, Daghan Dagli · arXiv (Cornell University) · 2026
Abstract — Modern deep artificial neural networks treat parameters as static, unconstrained floating-point scalar matrices stored permanently in physical memory and optimized through empirical loss minimization (L → 0). While computationally powerful, this classical paradigm incurs severe Von Neumann memory bandwidth bottlenecks, thermodynamic inefficiencies, and representation collapse. In this work, we propose a foundational theoretical paradigm shift: Existence is an active set of non-vanishing errors (ℰ), and biological intelligence is an ongoing, non-equilibrium resistance against dynamic attractor collapse rather than passive convergence to zero loss. Building upon our initial procedural framework, we formulate Orbital Error Dynamics (OED), an analytical framework wherein synaptic weights are not stored masses (O(W)), but transient topological resonances (O(1)) sampled dynamically from the complex quadratic polynomial map zn+1 = zn2 + c. We introduce the Bent Sine Wave Hypothesis, demonstrating that while unperturbed linear waves are static and harmonic waves are repetitively conservative, living non-equilibrium systems emerge when waves curl inward through environmental drag toward the main cardioid cusp (c = 1/4). We analyze the Observer Horizon Geometry in parameter space, identifying sub-boundary interior resonance shoulder loci Xupper = (0.25, +0.18) and Xlower = (0.25, -0.18) positioned between the central fixed-point basin and the true analytic boundary at c = 0.25 ± 0.50i, surveying radiant uncommitted potential fields while anchored to somatic dissipation axes. To overcome non-convex stagnation without loss zeroing, we formalize a Biomimetic Perturbed Jump Operator (Ωtunneling) inspired by mammalian fertilization zinc sparks. We further formulate an enteric-cranial Dual-Brain Cybernetic Architecture incorporating adaptive CD4+ regulatory immune gating masks (MCD4) for streaming noise stabilization, and decompose the 4-nucleotide genetic basis (A, T, C, G) directly across the quadrants of the complex plane (ℂ). Finally, multi-seed empirical validation on the non-linear Two-Moons manifold (5 seeds, 80/20 train/test split, consistent 32 × 32 sampling grid, zero test-time updates and zero label leakage) demonstrates that procedural parameterization from a 24-byte seed achieves 77.67% ± 5.35% clean test accuracy (within an 8.00-point paired difference of an unconstrained gradient-descent baseline at 85.67% ± 5.35%, 95% CI of paired difference: [-1.07%, 17.07%]), while under distribution shift (𝒩(1.2, 0.4)), OED preserves 71.33% ± 3.80% test accuracy compared to 80.33% ± 7.21% for the baseline, alongside conceptual equivalence with a proposed analog optical co-processor theoretically modeled for sub-nanosecond evaluation. Version 3.0 Revision Notes (Methodological Rigor & Prior Art Integration): This revised preprint incorporates comprehensive peer and community review feedback: (1) Zero Label Leakage Protocol: Test evaluations are conducted in 100% pure feedforward inference with zero test-time gradient steps and zero label access; (2) Consistent Grid Resolution: Standardized sampling resolution to a uniform 32×32 grid across training and evaluation; (3) Finite Difference Step & Gradient Dynamics: Verified δ = 0.02 traversing discrete pixel boundaries, achieving 0.0% zero-gradient epochs (0/250) and 0.0% Zinc Spark invocations with T = 50 epochs; (4) Rigorous Statistical Reporting: Exact Student-t distributions for N=5 (t_4 = 2.776), paired differences and 95% Confidence Intervals reported for both clean and distribution shift regimes; (5) Literature & Prior Art Integration: Contextualized against intrinsic dimension projections (Li et al., 2018), HyperNetworks (Ha et al., 2017), and HashedNets (Chen et al., 2015), substantiating the distinct O(1) non-linear dynamical formulation and patent claims (TR 2026/016285); and (6) Clarified CD4+ Immune Gating: Formulated as an active cybernetic defense mechanism for online streaming data against anomalous sensory shocks. 🏛️ Official Patent Priority Notice:The procedural weight derivation architectures, the adaptive CD4+ immune gating mask mechanism, and the zero-storage hardware co-processor implementations disclosed in this record and associated preprint files are subject to an official national priority patent application:• Patent Office: Turkish Patent and Trademark Office (TÜRKPATENT)• Application Number: 2026/016285• Official Priority Timestamp: September 22, 2026 (14:42:15 UTC+3)• Applicant: Volkan Dağlı• Inventors: Volkan Dağlı, Zerrin Dağlı, Dağhan Dağlı• Official Repository: https://github.com/pCwOrM/mandelbrot-fractal-neural-synthesis