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2026
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| Online-Zugang: | https://doi.org/10.5281/zenodo.19200405 |
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| _version_ | 1866901211574173696 |
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| author | Lim, Han-Jun |
| author_facet | Lim, Han-Jun |
| contents | <p>Abstract</p> <p><br>We prove that n = 12 is the unique positive integer n ≥ 2 satisfying five conditions from five independent branches of mathematics: (C1) highly composite, (C2) {2, 3}-smooth, (C3) genus g(X0(n)) = 0, (C4) class number h(Q(ζn)) = 1,<br>and (C5) Fibonacci square Fn = n2 (Cohn 1964). Three extended conditions (C6– C8) are verified, and the double perfection theorem is proved: n = 12 is the unique integer with both d(n) and σ(n) perfect numbers, established unconditionally via<br>discriminant analysis of the quadratic p 2 +p−6 = 0 (∆ = 25 = 52 , the only perfectsquare discriminant). The master identity σ(12) = φ(12) · (11 −2 3 d(12)) = 4 × 7 encoding spacetime dimension times QCD beta-function coefficient—holds among<br>C1–C5-satisfying integers uniquely at n = 12. The string-theoretic landscape of ∼10500 vacua collapses to a single rendering algebra R12 ∼= M3(C) ⊗ M4(C). We further establish three consequences of the R12 framework: (1) UV finiteness: all n-point correlation functions are bounded by Q ∥Ai∥ < ∞, with natural cutoff Λ = 1/Znoise; (2) gauge coupling: αs(MZ) = ζ(3)Znoise = 0.1180 (0.06% error) derived from two independent paths, with b0 = 11 −2 3 d(12) = 7 universal and scheme-independent; (3) non-perturbative confinement: the internal partition function Zint(β) = 1 + 8e −3β is analytic for all β > 0 with Boltzmann suppression e −3/Znoise ≈ 5 × 10−14, proving σ > 0 without perturbation theory.<br>In appendices we present a conditional mass gap theorem for SU(3) Yang–Mills<br>and the observation that the genus-zero property of X0(12) eliminates discrete spectral contamination from the Riemann-zero encoding in the scattering matrix of Γ0(12).</p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19200405 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Uniqueness of Twelve, UV Finiteness, and the Gauge Coupling from First Principles: Array Cosmology Paper XXV Lim, Han-Jun <p>Abstract</p> <p><br>We prove that n = 12 is the unique positive integer n ≥ 2 satisfying five conditions from five independent branches of mathematics: (C1) highly composite, (C2) {2, 3}-smooth, (C3) genus g(X0(n)) = 0, (C4) class number h(Q(ζn)) = 1,<br>and (C5) Fibonacci square Fn = n2 (Cohn 1964). Three extended conditions (C6– C8) are verified, and the double perfection theorem is proved: n = 12 is the unique integer with both d(n) and σ(n) perfect numbers, established unconditionally via<br>discriminant analysis of the quadratic p 2 +p−6 = 0 (∆ = 25 = 52 , the only perfectsquare discriminant). The master identity σ(12) = φ(12) · (11 −2 3 d(12)) = 4 × 7 encoding spacetime dimension times QCD beta-function coefficient—holds among<br>C1–C5-satisfying integers uniquely at n = 12. The string-theoretic landscape of ∼10500 vacua collapses to a single rendering algebra R12 ∼= M3(C) ⊗ M4(C). We further establish three consequences of the R12 framework: (1) UV finiteness: all n-point correlation functions are bounded by Q ∥Ai∥ < ∞, with natural cutoff Λ = 1/Znoise; (2) gauge coupling: αs(MZ) = ζ(3)Znoise = 0.1180 (0.06% error) derived from two independent paths, with b0 = 11 −2 3 d(12) = 7 universal and scheme-independent; (3) non-perturbative confinement: the internal partition function Zint(β) = 1 + 8e −3β is analytic for all β > 0 with Boltzmann suppression e −3/Znoise ≈ 5 × 10−14, proving σ > 0 without perturbation theory.<br>In appendices we present a conditional mass gap theorem for SU(3) Yang–Mills<br>and the observation that the genus-zero property of X0(12) eliminates discrete spectral contamination from the Riemann-zero encoding in the scattering matrix of Γ0(12).</p> <p> </p> |
| title | The Uniqueness of Twelve, UV Finiteness, and the Gauge Coupling from First Principles: Array Cosmology Paper XXV |
| url | https://doi.org/10.5281/zenodo.19200405 |