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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19158670 |
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| _version_ | 1866901210882113536 |
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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 drawn from five independent branches of mathematics: (C1) n is highly composite, (C2) n = 2a· 3b, (C3) g(X0(n)) = 0, (C4) h(Q(ζn)) = 1, and (C5) Fn = n2<br>(Cohn 1964). Three additional conditions -(C6) ψ(n) = π(n) (Dedekind psi equals Pisano period), (C7) σ(n) is a perfect number, (C8) d(n) is a perfect number -are verified for n = 12 and shown to hold simultaneously for no other n ≤ 10,000. Ten<br>structural identities at n = 12 are catalogued, culminating in the master identity σ(12) = φ(12) ·11 −23d(12),<br>which encodes the factorisation 28 = 4 × 7 of the divisor sum into the Euler totient (spacetime dimension in AC) and the QCD one-loop beta-function coefficient. This identity holds for no other positive integer. Within Array Cosmology, the uniqueness<br>theorem reduces the string-theoretic landscape of ∼10500 vacua to a single rendering algebra R12 ∼= M3(C) ⊗ M4(C). In appendices, we present (A) a conditional mass gap theorem for SU(3) Yang–Mills theory: if the theory exists on R4 satisfying<br>OS axioms, then ∆ > 0; and (B) the observation that the genus-zero property of X0(12) eliminates all discrete spectral contamination, so that the Riemann zeros are encoded purely in the scattering matrix of Γ0(12)—a structure singled out uniquely<br>by the conditions of Theorem 1.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19158670 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | The Uniqueness of Twelve: Eight Independent Number-Theoretic Conditions, the Master Identity σ(n) = φ(n)·(11 − 2 3 d(n)), and the Collapse of the Landscape to a Single Point Lim, Han-Jun <p>Abstract</p> <p><br>We prove that n = 12 is the unique positive integer n ≥ 2 satisfying five conditions drawn from five independent branches of mathematics: (C1) n is highly composite, (C2) n = 2a· 3b, (C3) g(X0(n)) = 0, (C4) h(Q(ζn)) = 1, and (C5) Fn = n2<br>(Cohn 1964). Three additional conditions -(C6) ψ(n) = π(n) (Dedekind psi equals Pisano period), (C7) σ(n) is a perfect number, (C8) d(n) is a perfect number -are verified for n = 12 and shown to hold simultaneously for no other n ≤ 10,000. Ten<br>structural identities at n = 12 are catalogued, culminating in the master identity σ(12) = φ(12) ·11 −23d(12),<br>which encodes the factorisation 28 = 4 × 7 of the divisor sum into the Euler totient (spacetime dimension in AC) and the QCD one-loop beta-function coefficient. This identity holds for no other positive integer. Within Array Cosmology, the uniqueness<br>theorem reduces the string-theoretic landscape of ∼10500 vacua to a single rendering algebra R12 ∼= M3(C) ⊗ M4(C). In appendices, we present (A) a conditional mass gap theorem for SU(3) Yang–Mills theory: if the theory exists on R4 satisfying<br>OS axioms, then ∆ > 0; and (B) the observation that the genus-zero property of X0(12) eliminates all discrete spectral contamination, so that the Riemann zeros are encoded purely in the scattering matrix of Γ0(12)—a structure singled out uniquely<br>by the conditions of Theorem 1.</p> |
| title | The Uniqueness of Twelve: Eight Independent Number-Theoretic Conditions, the Master Identity σ(n) = φ(n)·(11 − 2 3 d(n)), and the Collapse of the Landscape to a Single Point |
| url | https://doi.org/10.5281/zenodo.19158670 |