{"paper":{"title":"Terminal Blocks of Primes in Pisot Numeration Systems","arxiv_id":"2610.10026","primary_cat":"math.NT","submitted_at":"2026-10-07T13:12:25+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.81,"id":"s0","own":true,"role":"contribution","text":"We prove a prime number theorem for fixed terminal block words in Pisot integer numeration systems.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.83,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.19,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.81,"interpretation":0.0}},{"i":1,"p":0.99,"id":"s1","own":true,"role":"result","text":"If the dominant root $\\varphi$ is a Pisot number and the characteristic polynomial $P_h$ is its minimal polynomial, every terminal block word of total digit-length $m$ occurs among the primes with asymptotic frequency $\\varphi^{-m}$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.79,"main_claim_p":0.8,"quantitative":true,"quantitative_p":0.9,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":2,"p":0.76,"id":"s2","own":true,"role":"interpretation","text":"In the Zeckendorf case this resolves a recent conjecture.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.57,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.02,"method":0.0,"result":0.15,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.76}},{"i":3,"p":1.0,"id":"s3","own":true,"role":"method","text":"The proof converts terminal conditions into Rauzy cylinder windows and then into a linear orbit on a compact torus.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.91,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":1.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":4,"p":0.97,"id":"s4","own":false,"role":"method","text":"For these companion substitutions, the required multiplicity-one Rauzy geometry is automatic: Barge's pure-discreteness theorem applies after reversal of the substitution words.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.36,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.13,"role_probabilities":{"gap":0.0,"method":0.97,"result":0.0,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":5,"p":0.99,"id":"s5","own":true,"role":"result","text":"The Rauzy torus also gives a prime number theorem for the substitution fixed word: every finite factor occurs at prime starting positions with its ordinary factor frequency.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.88,"main_claim_p":0.12,"quantitative":false,"quantitative_p":0.14,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":6,"p":0.54,"id":"s6","own":true,"role":"interpretation","text":"This proves the Tribonacci prime-number theorem suggested by Drmota--M\\\"ullner--Spiegelhofer.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.85,"main_claim_p":0.12,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.38,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.08,"interpretation":0.54}},{"i":7,"p":0.99,"id":"s7","own":true,"role":"result","text":"The toral model further yields polynomial sampling laws, asymptotic independence from fixed congruence classes, fixed-shift correlation formulas, and M\\\"obius orthogonality.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.87,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.12,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":8,"p":0.88,"id":"s8","own":true,"role":"result","text":"Combined with established prime theorems, it gives a terminal refinement of Chebotarev and shows that every fixed terminal prime class contains arbitrarily long arithmetic progressions with polylogarithmically bounded common difference.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.83,"main_claim_p":0.14,"quantitative":true,"quantitative_p":0.86,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.88,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.11}}]},"compiled":{"main_claim":{"p":0.74,"text":"We prove a prime number theorem for fixed terminal block words in Pisot integer numeration 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