The 12-gon Area Identity, Defect Reciprocity, and the A Priori Derivation of the Fine-Structure Constant from the R12 Rendering Algebra
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2026
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| _version_ | 1866901142581018624 |
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| author | Lim, Han-Jun |
| author_facet | Lim, Han-Jun |
| contents | <p>Abstract</p> <p>We derive the inverse fine-structure constant α −1 = 137.035 999 083 from the rendering algebra R12 ∼= M12(C) through NNLO with zero free parameters.</p> <p>Part I (LO + NLO). The regular 12-gon inscribed in the unit circle has area exactly Nc = 3 (since sin(π/6) = 1 2 ), making the geometric defect δ = π − Nc; Niven’s theorem proves N=12 unique. The Generalised Landauer Allocation gives the defect<br>leakage 1/δ with normalisation c=1. The Dimensional Allocation Theorem classifies the self-referential series Z/Nk by sector (k=1 excluded at 30σ as M3; k≥4 truncated by D=3), yielding Kknot = Z(N+1)/N3 with KMS normalisation 1/Tw. Result: α−1 NLO = 137.036 000 (43σ residual). </p> <p>Part II (NNLO). The defect operator V on M12 has M4-sector matrix D =v1v † 1 − v2v†2 (rank-1 difference). Five structural identities: rank(D) = 2 = Tw; eigenvalues of D2 are {N+1, N+1, 0, 0}; |⟨v1|v2⟩|2 = Nc; ⟨v1|D|v2⟩ = 0; dim(stabsu(4)(D)) =<br>5 = Tw C2(∧2). This stabiliser–Casimir identity holds only for N/Nc = 4 (cubic k3−6k 2+7k+4=0, unique integer root k=4), providing a new N=12 uniqueness proof. Three exact cancellations–(N+1), b0, (N/Nc)/8=1/Tw–reduce the perturbation to NNLO =−Z 2/(TwN2· 5 2).</p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19402385 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | The 12-gon Area Identity, Defect Reciprocity, and the A Priori Derivation of the Fine-Structure Constant from the R12 Rendering Algebra Lim, Han-Jun Array Cosmology 배열우주론 R12 Rendering Algebra Rendering Defect Zero-Parameter Derivation Z_noise Topo-Thermodynamic Framework 5 Axioms Array Cosmology Hanjun Lim 임한준 <p>Abstract</p> <p>We derive the inverse fine-structure constant α −1 = 137.035 999 083 from the rendering algebra R12 ∼= M12(C) through NNLO with zero free parameters.</p> <p>Part I (LO + NLO). The regular 12-gon inscribed in the unit circle has area exactly Nc = 3 (since sin(π/6) = 1 2 ), making the geometric defect δ = π − Nc; Niven’s theorem proves N=12 unique. The Generalised Landauer Allocation gives the defect<br>leakage 1/δ with normalisation c=1. The Dimensional Allocation Theorem classifies the self-referential series Z/Nk by sector (k=1 excluded at 30σ as M3; k≥4 truncated by D=3), yielding Kknot = Z(N+1)/N3 with KMS normalisation 1/Tw. Result: α−1 NLO = 137.036 000 (43σ residual). </p> <p>Part II (NNLO). The defect operator V on M12 has M4-sector matrix D =v1v † 1 − v2v†2 (rank-1 difference). Five structural identities: rank(D) = 2 = Tw; eigenvalues of D2 are {N+1, N+1, 0, 0}; |⟨v1|v2⟩|2 = Nc; ⟨v1|D|v2⟩ = 0; dim(stabsu(4)(D)) =<br>5 = Tw C2(∧2). This stabiliser–Casimir identity holds only for N/Nc = 4 (cubic k3−6k 2+7k+4=0, unique integer root k=4), providing a new N=12 uniqueness proof. Three exact cancellations–(N+1), b0, (N/Nc)/8=1/Tw–reduce the perturbation to NNLO =−Z 2/(TwN2· 5 2).</p> <p> </p> |
| title | The 12-gon Area Identity, Defect Reciprocity, and the A Priori Derivation of the Fine-Structure Constant from the R12 Rendering Algebra |
| topic | Array Cosmology 배열우주론 R12 Rendering Algebra Rendering Defect Zero-Parameter Derivation Z_noise Topo-Thermodynamic Framework 5 Axioms Array Cosmology Hanjun Lim 임한준 |
| url | https://doi.org/10.5281/zenodo.19402385 |