Foundations of a Quantum Number Theory - Cyclotomic Levels, Halving Lattices and Class Numbers

Description (v4) This preprint develops a purely arithmetic study of discrete, level-controlled quantities attached to cyclotomic fields: the two-adic contribution of individual Galois orbits to the relative class number, its relation to circular units and their signatures, and the narrow class groups of real cyclic fields of odd degree. The word “quantum” is used only as a heuristic label for this step-like behaviour; the paper makes no physical claim. The central results identify the layer size of an orbit with its contribution to the minus class number, reduce it to a parity condition over the field with two elements, and connect the vanishing of individual blocks of the distinguished circular unit to the two-class group and to jumps of the narrow class group. For the canonical twist by the character of conductor four, a two-adic excess of the generalised Bernoulli number at a block is shown to be equivalent to narrow two-class group at that block, together with an exact order formula; the second two-adic digit is identified as the length of the corresponding module of the minus class group. Several of these theorems are proved relative to published deep results (Greither’s main conjecture at two, Atsuta’s theorem on finite submodules, and the classification of Breen, Varma and Voight); this dependence is stated explicitly. New in version 4: • Index theorem for the halving lattice (proved relative to the class number formula). The lattice has full rank exactly when every prime q | N generates the group (Z/N_q)^×/{±1}; in that case the index in the full unit group is h⁺ n^{ω−2} 2^{n−2}. The formula of version 3 is corrected and kept on record. • Moduli with ω(N)=5, computed up to 5000. They include the first exception of odd index (9, at N=2730). • Cyclicity theorem (proved). If h_k is odd, every 2-free block of the minus 2-class group of k(i) is cyclic of length v(T). This determines Cl(k(i))₂ ≅ (Z/4)⁴ for a field of degree 30, given a GRH computation. • Test in two prime-conductor families, 31,000 fields of degree 7 and 48,142 fields of degree 15. A block rate 2^{-d} is refuted as a law, and a model combining Breen–Varma–Voight with Cohen–Lenstra fits. • A length excess that did not replicate. It was checked with a test fixed before the second sample was computed and is recorded as withdrawn. The paper is written as a transparent research record. Every statement carries a label: proved, proved relative to cited results, framework-internal, numerical, conditional, open, refuted, negative, or withdrawn. Refuted candidates and withdrawn predictions are kept in the text with later-status notes instead of being removed. Class group computations with PARI/GP are conditional on the generalised Riemann hypothesis where indicated. The work has not been peer reviewed. Subjects (MSC 2020): 11R18 (Cyclotomic extensions), 11R29 (Class numbers, class groups), 11R27 (Units and factorization), 11R23 (Iwasawa theory), 11Y40 (Algebraic number theory computations) Keywords cyclotomic fields; relative class number; circular units; cyclotomic units; narrow class group; signature rank; generalized Bernoulli numbers; Stickelberger element; 2-adic valuation; main conjecture at p=2; real cyclic fields; Cohen–Lenstra heuristics; computational number theory; PARI/GP; research record