The Adelic Structure of the R12 Rendering Algebra: Ramification, L-Functions, and the Standard Model Gauge Hierarchy
Abstract
We establish that the R12 rendering algebra of Array Cosmology carries a natural adelic structure dictated by the prime factorisation 12 = 2 2 × 3. Eleven A-layer results are proved (using only standard number theory): ten theorems and one corollary.
(1) The structure constant ω = e 2πi/12 = ζ12 defines the cyclotomic field Q(ζ12) whose Galois group Gal(Q(ζ12)/Q) ∼= (Z/12Z)
∗ ∼= Z2 × Z2 has four elements corresponding to the four Volma types of R12.
(2) The four Dirichlet characters mod 12 define four L-functions; the special value L(1, χ1) = π/(2√3) (where χ1 is the lift of the primitive character χ−3 to mod 12) provides a number-theoretic invariant of the electroweak sector, while the
Weinberg angle sin2 θW = ln 2/3 is independently derived from the Landauer principle (Paper V).
(3) The tensor factorisation M3(C) ⊗ M4(C) = M12(C) is the algebraic realisation of Rankin-Selberg functoriality GL(3) × GL(4) → GL(12).
(4) The ramification structure of Q(ζ12) determines the gauge hierarchy: p = 3 ramified → SU(3)C, p = 2 ramified → SU(2)L × U(1)Y, p ≥ 5 unramified → energy desert.
(5) The Z3 centre obstruction (ω3 residual phase) is the Artin conductor at p = 3, providing the number-theoretic origin of colour confinement.
Related
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BibTeX
@misc{lim2026adelic,
title = {{The Adelic Structure of the R12 Rendering Algebra: Ramification, L-Functions, and the Standard Model Gauge Hierarchy}},
author = {Han-Jun Lim},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23603559},
url = {https://zenodo.org/records/23603559},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Adelic Structure of the R12 Rendering Algebra: Ramification, L-Functions, and the Standard Model Gauge Hierarchy
AU - Han-Jun Lim
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23603559
UR - https://zenodo.org/records/23603559
AB - We establish that the R12 rendering algebra of Array Cosmology carries a natural adelic structure dictated by the prime factorisation 12 = 2 2 × 3. Eleven A-layer results are proved (using only standard number theory): ten theorems and one corollary. (1) The structure constant ω = e 2πi/12 = ζ12 defines the cyclotomic field Q(ζ12) whose Galois group Gal(Q(ζ12)/Q) ∼= (Z/12Z) ∗ ∼= Z2 × Z2 has four elements corresponding to the four Volma types of R12. (2) The four Dirichlet characters mod 12 define four L-functions; the special value L(1, χ1) = π/(2√3) (where χ1 is the lift of the primitive character χ−3 to mod 12) provides a number-theoretic invariant of the electroweak sector, while the Weinberg angle sin2 θW = ln 2/3 is independently derived from the Landauer principle (Paper V). (3) The tensor factorisation M3(C) ⊗ M4(C) = M12(C) is the algebraic realisation of Rankin-Selberg functoriality GL(3) × GL(4) → GL(12). (4) The ramification structure of Q(ζ12) determines the gauge hierarchy: p = 3 ramified → SU(3)C, p = 2 ramified → SU(2)L × U(1)Y, p ≥ 5 unramified → energy desert. (5) The Z3 centre obstruction (ω3 residual phase) is the Artin conductor at p = 3, providing the number-theoretic origin of colour confinement.
ER -