Mandelbrot Fractal Neural Synthesis: Zero-Storage Procedural Weight Derivation and Non-Linear Decision Boundaries

Abstract

Mandelbrot Fractal Neural Synthesis: Zero-Storage Procedural Weight Derivation and Non-Linear Decision Boundaries What's New in Version 2.0 (Major Revision & Q1 Readiness) Systems Trade-Off Framing: Reframed the central research question from raw parameter fitting to fundamental systems architecture: trading deterministic local compute (O(M) escape iterations) for extreme memory footprint elimination (>99.99999998% reduction). Formal Algorithmic Specification: Added formal pseudocode (Algorithm 1: Coordinate-Space Evolutionary Synthesis) defining deterministic spatial exploration, quadrant area integration, and synaptic mapping. Quadtree Tensor Scalability (Section IV-F): Introduced multi-scale 2p × 2p recursive spatial partitioning and seed-offset generation to scale synthesis from single neurons to dense weight matrices (W ∈ ℝd_out × d_in). Non-Linear Topological Context: Grounded the non-linear XOR solution within Minsky & Papert's (1969) perceptron limitations. Bilingual Standalone Labs (TR / EN): Both interactive labs now feature client-side Turkish/English bilingual toggle switches with localStorage persistence. English Abstract Modern deep learning architectures store billions or trillions of parameters as independent floating-point scalars across dense tensor arrays, trained via backpropagation. While exceptionally capable, this paradigm incurs severe storage footprints, memory bandwidth bottlenecks (“the memory wall”), and excessive energy consumption. In this work, we propose and empirically validate an alternative weight generation paradigm: deriving synaptic weights and threshold biases procedurally from the non-linear visual morphology of the Mandelbrot fractal set. By specifying a 3-parameter coordinate tuple (cx, cy, zoom) in the complex plane, a 128×128 pixel patch is sampled via the quadratic escape recurrence z = z2 + c. Using our 4-Quadrant Partitioning method, non-escaping dark areas directly determine synaptic weights and biases. Key Results: 128×128 resolution eliminates discretization noise within 0.08% of 256×256 while executing 15× faster (~15.6 ms). All fundamental linear logic gates (AND, OR, NAND, NOR) achieved 100% accuracy. The historically non-linear XOR problem was resolved with 100% accuracy using a 2-layer composite fractal network. An entire functional decision cell requires ZERO persistent weight tensors, retaining only 24 bytes of coordinate metadata, yielding >99.99999998% memory reduction compared to conventional weight arrays. The “Escape Horizon Principle” links biological motor error boundaries (e.g., mastering hammering or balance via catastrophic edge feedback) to the topological boundary of the Mandelbrot set. Genişletilmiş Türkçe Özet (Extended Turkish Abstract) Bu çalışma, derin öğrenme modellerinde milyonlarca parametreyi bellek çiplerinde statik sayılar olarak saklamak yerine; deterministik kaosun ve fraktal geometrinin en bilinen örneği olan Mandelbrot kümesinden (z = z2 + c) ihtiyaç anında canlı türeten yeni bir yapay zeka paradigmasını teorik ve deneysel olarak kanıtlamaktadır. Temel Bulgular ve Yenilikler: 24-Bayt Sıfır-Bellek Modeli: Bir karar hücresi için hiçbir tensör matrisi saklanmaz. Yalnızca 24 baytlık (üç Float64 koordinat: cx, cy, zoom) bir konum tutularak %99.99999998+ bellek tasarrufu sağlanır. Hata Kümesi ve Kaçış Ufku İlkesi: İnsanın çekiç vururken eline 1-2 kez vurup canı yandığında hata sınırını kilitleyerek 1000 yerine 300 adımda öğrenmesi gibi; Mandelbrot'un matematiksel kaçış sınırı da yapay nörona en keskin karar eşiğini doğal olarak sunar. %100 Doğruluk: OR, AND, NAND, NOR kapıları tek nöronla; yapay zeka tarihinde tek nöronun çözemediği ünlü XOR problemi ise 2 katmanlı kompozit fraktal ağımızla %100 doğrulukla çözülmüştür. Müstakil Açık Kaynak Laboratuvarı: Çalışma, hiçbir sunucuya veya harici kütüphaneye ihtiyaç duymadan internetsiz çalışan, Türkçe/İngilizce çift dilli interaktif web simülatörleriyle doğrulanmıştır. Official GitHub Repository: https://github.com/pCwOrM/mandelbrot-fractal-neural-synthesisLive Interactive Labs: https://pcworm.github.io/mandelbrot-fractal-neural-synthesis/License: Creative Commons Attribution 4.0 International (CC-BY-4.0)