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

Description (v6) 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 since version 3 (versions 4 to 6; versions 4 and 5 were not deposited separately): **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 (in version 5 without GRH, see below). **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 deficit in the family of composite conductor q₃q₅ did not replicate.** A pre-fixed test on 21,674 further fields of degree 15 with conductor up to 10⁶ finds the single-block rate consistent with Breen–Varma–Voight (z = −0.97); the earlier value 1/16 is recorded as negative. **Two geometric readings of the block choice tested.** Antipodes and levels on the wheel modulo 30 do not predict which of two dual blocks a field selects (pre-fixed replication 15:11); recorded as negative. **Jacobi sums and root number tested as deciders.** Neither the norm datum of the Jacobi sums modulo P² (a Barrucand–Cohn analogue) nor the root number decides the block choice (negative); a small degeneracy lemma explains why two pre-fixed features were uninformative by construction. **Consolidation and deposited code (version 5).** Abstract, introduction, overview table, methodology, state of results and road map are brought up to date; superseded statements carry later-status notes; the labels refuted, negative and withdrawn are distinguished; scripts and raw data are deposited as QNT_v5_code_data.zip, one folder per subsection 13.21–13.32. **Quarter formula for the block choice (proved).** In the family q3q5 the layer counts halve exactly, and B_{1,χ} = −2εψ(4)R(ζ) (Washington normalisation) with R the colour count of ψ on (0, f′/4); the block choice is divisibility of a 15-bit parity word by x⁴+x+1 or x⁴+x³+1. Checked on all 29082 fields with f′ ≤ 10⁶. **CRT form of the quarter formula (proved).** By the Chinese remainder theorem, R(ζ), hence B_{1,χ} and the block pattern, is computed in O(q3+q5) operations via prefix counts of the quintic classes; 29082/29082 fields agree, and conductors up to 5·10¹² take about 2 s **Block statistics up to f′ ≤ 10⁷ (numerical, pre-registered).** 249235 further fields: single-block rate 0.11754 against Breen–Varma–Voight 2/17 (z = −0.16), block rate 0.0660 against Cohen–Lenstra 0.0664, length distribution and block choice as in the model. The earlier block rate 1/16 is refuted as a law (z = +10.2). A closed non-vanishing criterion and an algorithm below O(√f′) remain open. **GRH removed at conductor 8401 (proved relative to Greither and Atsuta).** Quadratic residue symbols at the 15 primes above q = 84011 show that the circular units, including those of the cubic and quintic subfields, have odd index in the unit group of the degree-15 field. This gives signature rank 11 unconditionally and, relative to Greither, odd class number, hence Cl(k(i))₂ ≅ (Z/4)⁴ without GRH. A direct bnfcertify is out of reach (Minkowski bound about 3.9·10¹⁵). A failure is documented: the top-level circular units alone have rank 12 because 31 is a cube modulo 271. **One sign convention (version 6).** Washington's normalisation B_{1,χ} = (1/f)Σ aχ(a) is used throughout; the half-sum identity reads S_χ = (χ̄(2)−2)B_{1,χ}. For conductor 4f′ the relation T = −B_{1,χ} is proved separately (it does not follow from the odd-conductor case). No valuation, block or vanishing statement changes; 29 conductors are checked numerically. The conductor hypothesis of the block class group theorem and the probability of the selection test (two-sided 0.0022) are stated precisely, and at splitting level 3 both lower bounds for the coefficients are shown to be attained (56 triples). **Assessment of the open points.** Each of the eight remaining steps is classified (core: parity word, rates; completion: quarter formula in other families, the case g>1, the halving lattice beyond the index; optional: faster algorithm, independent proofs, higher degrees), with the grounds for deferring it to later work. None is a gap in a proof. 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. 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) Related works: as „References“ (Resource type: Publication): • https://doi.org/10.5281/zenodo.22833875 (Prime-Hyperoctahedron Witt Tower) • https://doi.org/10.5281/zenodo.22857700 (The Geometry of Unity) • https://doi.org/10.5281/zenodo.22883976 (Prime Information Conservation)