Variational Proof and McKay–E8 Foundation of the Pentagon Identity for the Fine-Structure Constant - Uniqueness of the Level Choice {1,4,5}, Renormalization Cascade, and Falsifiable Quasar Prediction
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2026
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| _version_ | 1866901760758513664 |
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| author | Krause, Thomas |
| author_facet | Krause, Thomas |
| contents | <div class="n6owBd awi2gc"> <div>This paper does not claim the closed-form identity for the inverse fine-structure constant</div> <div>alpha^(-1) = 360 phi^(-2) - 2 phi^(-3) + (3 phi)^(-5) = 137.0359991648, which matches the CODATA value of alpha to 0.0006 ppm. The leading-order term 360 phi^(-2) - 2 phi^(-3) was derived by Heyrovska and Narayan (2005, arXiv:physics/0509207) from a golden-angle interpretation of the Bohr radius. Systematic phi-extended forms were explored by Sherbon (2012-2019). The full closed three-term identity in the form used here was first stated by Pellis (2021, viXra:2110.0053 and viXra:2110.0117).</div> <div>The contribution of the present paper is therefore not the identity, but a structural proof of its uniqueness within a wider geometric framework (AToE — A Theory of Everything). Three independent results are established:</div> <div> </div> <div>(C1) Variational uniqueness. Among |C| = 24 336 admissible level triples derived from the affine E_8 marking, the level choice {1, 4, 5} is the unique global minimum of the Pentagon action S(a,b,c) = (Delta alpha^(-1)(a,b,c) / sigma_0)^2. The stability margin to the second-best triple exceeds 4600 sigma at sigma_0 = 1.5e-10 (CODATA-2018).</div> <div> </div> <div>(C2) McKay-E_8 derivation of the prefactor 360. The prefactor is forced by three independent McKay-E_8 anchors — the Coxeter number h(E_8) = 30, the binary icosahedral group order |I| = 60, and base 12 — rather than fitted. Five of the eight E_8 exponents satisfy m mod 6 in {1,5}, reflecting the pentagon phase of the theory.</div> <div> </div> <div>(C3) Wilson renormalization cascade. The cascade explanation yields seven decades of accuracy from 61 scale decades with rate lambda ~ 0.115, and predicts a falsifiable cosmological alpha trajectory delta alpha / alpha (z) with hard upper bound 2.23 ppm at the highest measured redshifts. This prediction is directly testable against ESPRESSO (Murphy et al. 2022), Webb et al. (2011), and Wilczynska et al. (2020).</div> <div> </div> <div>A methodological objection — that any prediction of alpha must follow from a fundamental Lagrangian density — is addressed in Section 6.4: a discrete variational principle on the finite configuration set C is itself a Lagrangian formulation; the requirement is not applied symmetrically (the Standard Model Lagrangian does not derive alpha; the Connes-Chamseddine spectral action did not reach sub-ppm precision); and a bridge to a continuous action via Eisenstein series on Gamma(5)\H and a WZW model at E_8 level is named as an open programme.</div> <div> </div> <div>The paper consists of 11 pages and contains: variational tables over all 24 336 level configurations, the structure matrix comparing Heyrovska 2005, Sherbon 2018, Pellis 2021 and the present work, the renormalization trajectory table, and the falsifiable quasar prediction table.</div> <div> </div> <div>Acknowledgement of prior work appears explicitly at the end of the paper. The full chronology Heyrovska 2005 -> Sherbon 2012-2019 -> Pellis 2021 -> Krause 2026 is given in Section 1.2 with full DOIs / arXiv / viXra identifiers.</div> <div> </div> <div>Related identifiers (Zenodo “Related works”)</div> <div>• isDerivedFrom: arXiv:physics/0509207 (Heyrovska & Narayan 2005)</div> <div>• isDerivedFrom: viXra:2110.0053 (Pellis 2021)</div> <div>• isDerivedFrom: viXra:2110.0117 (Pellis 2021)</div> <div>• references: SSRN 3148761 (Sherbon 2018)</div> <div>• references: PhilArchive SHEFCF-3 (Sherbon 2019)</div> <div>• isNewVersionOf: <DOI von Alpha_Paper_v1, falls off Zenodo></div> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20029564 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Variational Proof and McKay–E8 Foundation of the Pentagon Identity for the Fine-Structure Constant - Uniqueness of the Level Choice {1,4,5}, Renormalization Cascade, and Falsifiable Quasar Prediction Krause, Thomas fine-structure constant Pentagon Path McKay correspondence E8 Lie algebra, icosahedron enormalization group quasar spectroscopy AToE deterministic natural constants <div class="n6owBd awi2gc"> <div>This paper does not claim the closed-form identity for the inverse fine-structure constant</div> <div>alpha^(-1) = 360 phi^(-2) - 2 phi^(-3) + (3 phi)^(-5) = 137.0359991648, which matches the CODATA value of alpha to 0.0006 ppm. The leading-order term 360 phi^(-2) - 2 phi^(-3) was derived by Heyrovska and Narayan (2005, arXiv:physics/0509207) from a golden-angle interpretation of the Bohr radius. Systematic phi-extended forms were explored by Sherbon (2012-2019). The full closed three-term identity in the form used here was first stated by Pellis (2021, viXra:2110.0053 and viXra:2110.0117).</div> <div>The contribution of the present paper is therefore not the identity, but a structural proof of its uniqueness within a wider geometric framework (AToE — A Theory of Everything). Three independent results are established:</div> <div> </div> <div>(C1) Variational uniqueness. Among |C| = 24 336 admissible level triples derived from the affine E_8 marking, the level choice {1, 4, 5} is the unique global minimum of the Pentagon action S(a,b,c) = (Delta alpha^(-1)(a,b,c) / sigma_0)^2. The stability margin to the second-best triple exceeds 4600 sigma at sigma_0 = 1.5e-10 (CODATA-2018).</div> <div> </div> <div>(C2) McKay-E_8 derivation of the prefactor 360. The prefactor is forced by three independent McKay-E_8 anchors — the Coxeter number h(E_8) = 30, the binary icosahedral group order |I| = 60, and base 12 — rather than fitted. Five of the eight E_8 exponents satisfy m mod 6 in {1,5}, reflecting the pentagon phase of the theory.</div> <div> </div> <div>(C3) Wilson renormalization cascade. The cascade explanation yields seven decades of accuracy from 61 scale decades with rate lambda ~ 0.115, and predicts a falsifiable cosmological alpha trajectory delta alpha / alpha (z) with hard upper bound 2.23 ppm at the highest measured redshifts. This prediction is directly testable against ESPRESSO (Murphy et al. 2022), Webb et al. (2011), and Wilczynska et al. (2020).</div> <div> </div> <div>A methodological objection — that any prediction of alpha must follow from a fundamental Lagrangian density — is addressed in Section 6.4: a discrete variational principle on the finite configuration set C is itself a Lagrangian formulation; the requirement is not applied symmetrically (the Standard Model Lagrangian does not derive alpha; the Connes-Chamseddine spectral action did not reach sub-ppm precision); and a bridge to a continuous action via Eisenstein series on Gamma(5)\H and a WZW model at E_8 level is named as an open programme.</div> <div> </div> <div>The paper consists of 11 pages and contains: variational tables over all 24 336 level configurations, the structure matrix comparing Heyrovska 2005, Sherbon 2018, Pellis 2021 and the present work, the renormalization trajectory table, and the falsifiable quasar prediction table.</div> <div> </div> <div>Acknowledgement of prior work appears explicitly at the end of the paper. The full chronology Heyrovska 2005 -> Sherbon 2012-2019 -> Pellis 2021 -> Krause 2026 is given in Section 1.2 with full DOIs / arXiv / viXra identifiers.</div> <div> </div> <div>Related identifiers (Zenodo “Related works”)</div> <div>• isDerivedFrom: arXiv:physics/0509207 (Heyrovska & Narayan 2005)</div> <div>• isDerivedFrom: viXra:2110.0053 (Pellis 2021)</div> <div>• isDerivedFrom: viXra:2110.0117 (Pellis 2021)</div> <div>• references: SSRN 3148761 (Sherbon 2018)</div> <div>• references: PhilArchive SHEFCF-3 (Sherbon 2019)</div> <div>• isNewVersionOf: <DOI von Alpha_Paper_v1, falls off Zenodo></div> </div> |
| title | Variational Proof and McKay–E8 Foundation of the Pentagon Identity for the Fine-Structure Constant - Uniqueness of the Level Choice {1,4,5}, Renormalization Cascade, and Falsifiable Quasar Prediction |
| topic | fine-structure constant Pentagon Path McKay correspondence E8 Lie algebra, icosahedron enormalization group quasar spectroscopy AToE deterministic natural constants |
| url | https://doi.org/10.5281/zenodo.20029564 |