Topological Semantics for Scoped Computational Paths

[2608.04228] Topological Semantics for Scoped Computational Paths Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Logic in Computer Science arXiv:2608.04228 (cs) [Submitted on 4 Aug 2026 ( v1 ), last revised 14 Sep 2026 (this version, v3)] Title: Topological Semantics for Scoped Computational Paths Authors: Arthur Freitas Ramos , Ruy J. G. B. de Queiroz , Anjolina Grisi de Oliveira , Tiago M. L. de Veras View a PDF of the paper titled Topological Semantics for Scoped Computational Paths, by Arthur Freitas Ramos and 3 other authors View PDF HTML (experimental) Abstract: Computational paths record the steps of an equality derivation. We give them a topological semantics that distinguishes derivable rewrites from arbitrary homotopies. Coherent representatives pair traces with paths homotopic to their realizations. We compare a topology retaining the entire trace with one observing only endpoints, length, and paths. Quotienting by the declared rewrites gives a groupoid. Multiplication is continuous when composable pairs carry the quotient topology inherited from composable representatives. This topology can differ from the usual subspace topology on pairs of quotient arrows. We characterize when they agree, give compact-Hausdorff and discrete sufficient conditions, and use the Hawaiian earring to exhibit a failure of agreement. The comparison with geometric homotopy classes is injective exactly when the presentation is geometrically complete. Normal-form certificates give a criterion for completeness. In the universal presentation, all paths are primitive steps and all endpoint-fixed homotopies are allowed rewrites; its quotient recovers the quotient-topologized fundamental groupoid. Circle and torus examples recover the classical based-loop classifications by $\mathbb Z$ and $\mathbb Z^2$. A Lean development supports the construction. A focused Lean 4.32.0 result registered in Palomar covers the topology comparison, additive circle and torus classifications, and a conditional Hawaiian-earring obstruction transfer. We distinguish that result from the earlier Lean 4.24.0 development and from the mathematical exposition. Comments: 35 pages, 8 figures, 2 tables. Revised exposition and expanded diagrams; added a precise Palomar registration boundary. Lean artifact: this https URL (tag topological-paper-v12); Zenodo: this https URL Subjects: Logic in Computer Science (cs.LO) ; Algebraic Topology (math.AT) Cite as: arXiv:2608.04228 [cs.LO] (or arXiv:2608.04228v3 [cs.LO] for this version) https://doi.org/10.48550/arXiv.2608.04228 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Arthur Ramos [ view email ] [v1] Tue, 4 Aug 2026 21:06:00 UTC (26 KB) [v2] Fri, 14 Aug 2026 20:15:32 UTC (37 KB) [v3] Mon, 14 Sep 2026 00:18:15 UTC (41 KB) Full-text links: Access Paper: View a PDF of the paper titled Topological Semantics for Scoped Computational Paths, by Arthur Freitas Ramos and 3 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.LO < prev | next > new | recent | 2026-08 Change to browse by: cs math math.AT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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