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The proof follows Godsil's path tree: unfold G into the tree T(G,u) of its paths (where the matching polynomial equals a characteristic polynomial) and transport back along the divisibility mu_G | mu_{T(G,u)}; the root bound is proved from scratch by a weighted Gershgorin / Collatz-Wielandt argument reduced to two facts about distances in a tree. All headline theorems are sorry-free (axioms: propext, Classical.choice, Quot.sound). It formalizes classical mathematics and claims no new theorem; the novelty is the formalization, supported by a search of the Lean/Mathlib, Isabelle/AFP and Coq/mathcomp ecosystems.  Lean sources: github.com/karlesmarin/godsil-gutman-lean. English and Spanish editions are included, with the full Lean 4 sources."},{"descriptionType":"Other","description":"This is Part II, and a direct continuation of Part I -- 'Random Signs into Matchings: A Godsil-Gutman Identity, Formalized in Lean 4' (doi 10.5281/zenodo.20517350). Part I formalized the matching polynomial and the Godsil-Gutman identity, and proposed, as its explicit 'next stone', proving the Heilmann-Lieb theorem through Godsil's path tree rather than through interlacing-family machinery. Part II carries out exactly that proposal: it builds the path tree, proves the divisibility mu_G | mu_{T(G,u)} and the forest identity, and proves the Ramanujan bound -- completing the path-tree route that Part I had mapped but not formalized. Part II thus continues and completes Part I."}],"geoLocations":[],"fundingReferences":[],"url":"https://zenodo.org/doi/10.5281/zenodo.20561832","contentUrl":null,"metadataVersion":6,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"api","isActive":true,"state":"findable","reason":null,"viewCount":0,"downloadCount":0,"referenceCount":0,"citationCount":8,"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":1,"created":"2026-06-05T19:08:51Z","registered":"2026-06-05T19:08:51Z","published":null,"updated":"2026-08-26T02:07:16Z"},"relationships":{"client":{"data":{"id":"cern.zenodo","type":"clients"}}}},{"id":"10.5281/zenodo.21017525","type":"dois","attributes":{"doi":"10.5281/zenodo.21017525","identifiers":[],"creators":[{"nameType":"Personal","familyName":"叶兴阳双语音标有声读物","name":"叶兴阳双语音标有声读物","nameIdentifiers":[],"affiliation":[]}],"titles":[{"title":"英式书迪克·普雷斯科特在西点军校的第四(Dick Prescott's Fourth Year at West ).pdf"}],"publisher":{"name":"Zenodo"},"container":{},"publicationYear":2026,"subjects":[],"contributors":[],"dates":[{"date":"2026-06-29","dateType":"Issued"}],"language":null,"types":{"schemaOrg":"Book","resourceTypeGeneral":"Book","citeproc":"article","bibtex":"misc","ris":"BOOK","resourceType":""},"relatedIdentifiers":[{"relationType":"HasVersion","relatedIdentifier":"10.5281/zenodo.21017526","relatedIdentifierType":"DOI"}],"relatedItems":[],"sizes":[],"formats":[],"version":null,"rightsList":[{"rightsIdentifierScheme":"SPDX","rightsUri":"https://creativecommons.org/licenses/by/4.0/legalcode","schemeUri":"https://spdx.org/licenses/","rights":"Creative Commons Attribution 4.0 International","rightsIdentifier":"cc-by-4.0"}],"descriptions":[{"descriptionType":"Abstract","description":"Bulk uploaded via High-Speed Tool"}],"geoLocations":[],"fundingReferences":[],"url":"https://zenodo.org/doi/10.5281/zenodo.21017525","contentUrl":null,"metadataVersion":1,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"api","isActive":true,"state":"findable","reason":null,"viewCount":0,"downloadCount":0,"referenceCount":0,"citationCount":0,"partCount":0,"partOfCount":0,"versionCount":2,"versionOfCount":1,"created":"2026-06-29T02:29:48Z","registered":"2026-06-29T02:29:49Z","published":null,"updated":"2026-08-26T02:07:15Z"},"relationships":{"client":{"data":{"id":"cern.zenodo","type":"clients"}}}},{"id":"10.48550/arxiv.2608.23741","type":"dois","attributes":{"doi":"10.48550/arxiv.2608.23741","identifiers":[{"identifier":"2608.23741","identifierType":"arXiv"}],"creators":[{"nameType":"Personal","givenName":"Soumya","familyName":"Ghorai","name":"Ghorai, Soumya","affiliation":[],"nameIdentifiers":[]},{"nameType":"Personal","givenName":"Samir","familyName":"Rom","name":"Rom, Samir","affiliation":[],"nameIdentifiers":[]},{"nameType":"Personal","givenName":"Irina","familyName":"Shamova","name":"Shamova, Irina","affiliation":[],"nameIdentifiers":[]},{"nameType":"Personal","givenName":"N. 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X-ray diffraction analysis reveals that all compounds crystallize in a monoclinic symmetry, with varying degree of structural distortion that increases in moving from La$^{3+}$ to smaller size cation Sm$^{3+}$. Electrical resistivity studies indicate insulating behaviour in all compounds, with variable-range-hopping domination at low temperatures, due to presence of anti-site disorder. AC susceptibility and heat capacity measurements suggest suppression of frustration in Sm-bearing compounds, affecting the magnetic behavior. Our first-principles calculations suggest that the combined effects of lattice distortion and Sm magnetism play a crucial role in weakening magnetic frustration, thereby rationalizing the experimental observations. These findings shed light on the complex interplay of crystal structure and magnetism in Cu-Ru double perovskites, and open up an avenue for tuning of magnetic properties through rare-earth-ion substitution."},{"descriptionType":"Other","description":"21 pages, 16 figures; Accepted for Publication in PRB, in 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We prove that the Toeplitz operator $T_{ω,u}$ is compact on the weighted Bergman space $A_ω^2$ if and only if its Berezin transform satisfies the vanishing condition $\\lim_{|z|\\to 1^-}\\widetilde u_ω(z)=0.$\n Our approach is based on a polynomial frame for $A_ω^2$ and a detailed localization analysis of the resulting infinite matrix representation of $T_{ω,u}$. 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