{"id":"palomar/PALOMAR-2026-08-31-000004","url":"https://github.com/stalex444/mahler-measure-minima/tree/ca3df22ba91ce60e5199aee4a0c35a6ca8ee559d","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-08-31-000004-v1.json","repo":"https://github.com/stalex444/mahler-measure-minima","tags":["math.NT","msc:11R06","msc:11R09","msc:12D10","number-theory","mahler-measure","pisot-numbers"],"score":0,"title":"stalex444/mahler-measure-minima","trust":"high","domain":"math.NT","source":"palomar","authors":["Stephanie Alexander"],"reviews":0,"subject":"math.NT","summary":"For each degree d in {2, 3, 4}, the polynomial x^d - x - 1 attains the minimal Mahler measure among monic irreducible integer polynomials of degree d with Mahler measure above one, and the three minimal values are exactly the golden ratio (M^2 = M + 1), the plastic ratio (M^3 = M + 1), and the root of x^4 - x^3 - 1 above one (M^4 = M^3 + 1); by Siegel's classical theorem (1944, cited as background, not formalized here) the latter two are the smallest and second-smallest Pisot numbers. Classically the degree-4 minimizer is not itself a Pisot polynomial, yet its measure lands back on the Pisot…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-08-31","reactions":{},"formalized":true,"formalizes":[{"title":"Quantum Mechanics as the Arithmetic of a Complex Place (v8B)","identifier":"","relationship":"formalizes"},{"title":"D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), 461-479","identifier":"","relationship":"formalizes"}]}