GeoFormer: Non-Equilibrium Metric Manifolds and Quantized Thermodynamic Attractor Convergence for Multi-Hop Transitive Reasoning

GeoFormer is a neuromorphic foundation architecture grounded in Endogenous Propagation Geometry (Q-PG) designed to solve catastrophic forgetting, topological blindness, and compositional collapse in deep neural networks without requiring floating-point matrix multiplications. Abstract Standard Transformer architectures treat latent sequence space as an immutable, flat Euclidean coordinate system. In dot-product attention, dense synaptic weights in ℝD are forced to simultaneously perform semantic feature transformation and topological signal routing, leading to compositional collapse on unseen multi-hop transitive reasoning tasks. GeoFormer structurally decouples frozen 1.58-bit ternary transformation weights (θ ∈ {-1, 0, +1}) from an episodic 4-bit metric conductance lattice W. Relational associations deform this lattice, contracting effective resistances dij = (Wij + ε)-1 → 0 into persistent geodesic highways. Multi-hop transitive deductions are resolved in a single forward pass via continuous Quantized Thermodynamic Attractor Convergence (Q-TAC) graph Laplacian heat diffusion. Core Mathematical & Empirical Highlights Formal Deductive Machine Verification: All 6 governing algebraic invariants (discrete geodesic contraction, Fiedler spectral gap preservation λ2(L) > 0, precision-topology duality, S5 Cayley projection stability, Lyapunov energy monotonicity, and bounded plasticity) are certified UNSAT (counterexample-free) using the Microsoft Z3 SMT solver. Empirical Multi-Hop Generalization: Evaluated across 5 independent random seeds on unseen 2-hop compositions, directed Q-TAC achieves 40.00% ± 7.30% Hits@1 and 89.33% Hits@3, matching non-neural pure graph random walks (45.33%) within 1-to-N candidate branching limits (4.01 ± 2.11 targets/head). Standard Euclidean Transformers collapse to 1.33%. Autonomous Self-Regulation (Irrational Geometry Engine): Integrates golden-ratio quasi-crystalline positional encodings (0.000% collisions, 0 trainable parameters), Diophantine continued-fraction multi-scale relaxation schedules (1.84× latency speedup), and ergodic Weyl rotation drivers for bounded background memory consolidation (zero invariant violations over 1,000 steps). Sub-Nanojoule Memristor Compilation: A first-principles device physics simulation confirms isomorphic mapping to analog 1T1R ReRAM crossbars, dissipating 443.9 picajoules per inference pass (9,462× lower dynamic energy than FP32 GPU baselines) with analog noise absorption up to 20% programming variance. Open Source Code & Repositories GitHub Repository: https://github.com/bhosaleone/geoformer Hugging Face Weights: https://huggingface.co/SetMD/geoformer Author Research Dispatches: https://praman.blog Licensing & Commercial Rights GeoFormer is released under a Dual-Licensing Model: Open-Source Research: Software and algorithms are licensed under the GNU Affero General Public License v3.0 (AGPLv3). Any network service or cloud API utilizing this architecture must provide complete corresponding source code. Commercial Closed-Source Licensing: Commercial entities seeking proprietary exemptions from AGPLv3 copyleft terms must obtain a commercial license directly from the author (ishrikantbhosale@gmail.com / bhosaleone@gmail.com). Preprint & Documentation: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0).