{"id":"palomar/PALOMAR-2026-10-28-000019","url":"https://github.com/Arthur742Ramos/Math-Formalization/tree/5810a77833131ae3b171fd632b2370079345fd9f/computational-hodge/finite-periods","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-28-000019-v1.json","repo":"https://github.com/Arthur742Ramos/Math-Formalization","tags":["math.AT","math.LO","msc:18G35","msc:18B40","msc:03B35"],"score":0,"title":"Harmonic representatives and period separation for computational-path presentations","trust":"high","domain":"math.AT","source":"palomar","authors":["Arthur Freitas Ramos","Ruy Jose Guerra Barretto de Queiroz","David Barros Hulak"],"reviews":0,"subject":"math.AT","summary":"For finite primitive-step sets in the scoped geometric computational-path model, actual signed trace periods define a cochain complex. Closedness is equivalent to invariance under the entire scoped rewrite relation. Orthogonal projection constructs the unique harmonic representative of every degree-one presentation cohomology class, preserves all loop periods, and induces an actual linear equivalence. Vanishing of all loop periods is equivalent to being a coboundary, without connectivity assumptions; periods separate cohomology and harmonic representatives. A nonzero parallel-edge model and…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-28","reactions":{},"formalized":true,"formalizes":[{"title":"Topological Semantics for Scoped Computational Paths, arXiv:2608.04228v3, sections 2 and 6","identifier":"https://arxiv.org/abs/2608.04228v3","relationship":"adapts"},{"title":"Hodge Laplacians on Graphs, arXiv:1507.05379v4, section 2 and Theorems 5.2-5.3","identifier":"https://arxiv.org/abs/1507.05379v4","relationship":"adapts"}]}