The Semantic Abacus: Quantitative, Symbolic, and Skill-Based Reasoning in Semantic Topology Architectures

Description Exact Reasoning and Expertise Formation in Semantic Topology Architectures This paper introduces the Semantic Abacus, a quantitative and symbolic reasoning framework within the Semantic Topology Reasoning Architecture (STRA). It proposes that exact arithmetic, symbolic manipulation, and skill acquisition can emerge from activation dynamics over semantic topology, rather than from token prediction or hard‑coded symbolic algorithms. Numbers are represented as directed activation along linear semantic rods, arithmetic operations as propagation patterns, and multi‑digit computation as hierarchical chunking across parallel place‑value structures. Through repeated use, Hebbian learning compresses explicit step‑by‑step traversal into direct semantic links, enabling the transition from algorithmic computation to heuristic chunking and ultimately to intuitive resonance—a computational model of expertise development. Verification occurs through resonance, where multiple independent semantic paths converge on the same result, allowing the system to detect and correct errors without external labels. The Semantic Abacus demonstrates how STRA can perform exact reasoning while modeling the cognitive trajectory from novice to expert. It extends beyond arithmetic to domains such as diagnosis, debugging, and scientific insight, showing that expertise emerges from structural compression rather than symbolic programming. This paper provides a theoretical foundation and concrete implementation proposals requiring empirical validation. Abstract This paper presents the Semantic Abacus, a mechanism for exact arithmetic, symbolic manipulation, and skill acquisition within the Semantic Topology Reasoning Architecture (STRA). Numbers are represented as linear semantic rods, arithmetic as directed activation, and multi‑digit computation as hierarchical chunking across parallel place‑value structures. Through Hebbian learning, repeated operations compress into direct semantic links, enabling constant‑time intuitive recall. The Semantic Abacus models the three phases of expertise—algorithmic traversal, heuristic chunking, and intuitive resonance—and introduces resonance‑based verification, where correctness emerges from structural convergence across independent semantic paths. This framework provides STRA with exact reasoning capabilities while offering a general computational model of how expertise develops through procedural compression. The work is theoretical and requires proof‑of‑concept implementation for empirical validation. Background This paper is part of a six‑paper series introducing and expanding the Semantic Topology Reasoning Architecture: Paper 0 – Manifest: Semantic Topology Reasoning Architecture (STRA) (10.5281/zenodo.18207533) Paper 1: Activation Arrays as Working Memory in Semantic Reasoning Systems (10.5281/zenodo.18207539) Paper 2: Causal Signatures for Cross‑Domain Reasoning (10.5281/zenodo.18207546) Paper 3: Selection Pressure & Natural Selection of Thought (10.5281/zenodo.18207552) Paper 4: Transform Learning: Procedural Reasoning Without Programs (10.5281/zenodo.18207558) Paper 5: The Semantic Abacus: Quantitative, Symbolic, and Skill‑Based Reasoning (this paper) The Semantic Abacus provides STRA with exact reasoning capabilities and a general model of expertise formation. Key Contributions Quantitative Topology: Introduces linear semantic rods for exact arithmetic and symbolic reasoning. Hierarchical Chunking: Models place‑value parallelism and multi‑digit computation. Procedural Compression: Demonstrates how repeated use compresses explicit computation into intuitive recall. Expertise Trajectory: Formalizes the transition from algorithmic traversal to intuitive resonance. Resonance‑Based Verification: Provides a structural mechanism for detecting and correcting errors. Cross‑Domain Generalization: Extends the expertise model to diagnosis, debugging, and scientific insight. Architectural Integration: Establishes the Semantic Abacus as STRA’s mechanism for exact reasoning and skill acquisition. Research Impact This work contributes to cognitive architectures, symbolic reasoning, and computational epistemology by: Offering a unified model of exactness and intuition Providing a transparent alternative to neural approximation and symbolic brittleness Demonstrating how expertise emerges from structural compression Enabling verifiable reasoning through resonance rather than probabilistic confidence Strengthening STRA’s position as a structure‑centric architecture capable of both flexible and exact reasoning Access and Documentation ORCID: https://orcid.org/0009-0003-4876-9273 GitHub: https://github.com/Neuron-Soul-AI/Neuron-Soul-AI LinkedIn: https://www.linkedin.com/in/marcelo-emanuel-paradela-teixeira-702082382/ Email: marcelo.soul.ai@gmail.com License: CC BY‑NC 4.0 © Marcelo Emanuel Paradela Teixeira 2026