α-π-Helix: Geometric Unification of Fundamental Constants

In the paper's words

Geometric unification of fundamental constants using α (fine-structure) and π (geometric) helix structures, classifying Standard Model particles as topological vortex/helicon states. v2.1 — Phase 10 complete with 6-domain synthesis including experimental roadmap, cosmology, mathematical rigor, QFT correspondence, alternative theory comparison, and philosophy of science.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Lepton Knot Type $(p,q)$ Crossing # $n$ Prime Factor Predicted $m/m_e$ Observed (PDG 2024) Error $e$ Unknot $(1,2)$ 0 1 1 (input) 1 — $\mu$ Trefoil $(3,2)$ 3 $69 = 3 \times 23$ 207 206.77 0.1% $\tau$ Cinquefoil $(5,2)$ 5 $695 = 5 \times 139$ 3477 3477.2 <0.01% Average accuracy: 0.08% — competitive with the Koide formula's precision but derived from topological integers rather than an ad-hoc algebraic relation.

   from 15.2 Lepton Mass Spectrum

2. Sector Particles Free Parameters Mass Range Quarks 18 (×3 colors) 6 masses + 4 CKM angles + 1 phase 2 MeV → 173 GeV Charged Leptons 6 (+antiparticles) 3 masses 0.511 MeV → 1.777 GeV Neutrinos 6 (+antiparticles) 3 masses + 4 PMNS angles + 2 phases ≲ 0.1 eV Gauge Bosons $12 (8g+W^{+}W^{-}Z\gamma )$ 3 couplings ($g_{s}, g, g$') 0 → 91 GeV Higgs 1 1 mass + 1 vev 125 GeV Total ~61 26 free params 12 orders of magnitude

   from 13.1 Standard Model Census

3. What explains the Koide formula? $Q = (m_{e} + m_{\mu } + m_{\tau }) / (\sqrt{m_{e}} + \sqrt{m}_{\mu } + \sqrt{m}_{\tau })^{2} \approx 2/3$, to within 0.04% — coincidence or principle?

   from 13.2 Unexplained Facts

4. Helicity ($\mathcal{H}$): $\mathcal{H} = \int \mathbf{v} \cdot (\nabla \times \mathbf{v})\ d^3x$, a conserved quantity in ideal hydrodynamics measuring the linkage of vortex lines.

   from 14.1 Core Premise

5. Status: Current $|V_{ub}|$ measurements show ~20% tension between inclusive and exclusive determinations.

   from Prediction 3: CKM Ratios

6. Our model differs in crucial respects: Feature Bilson-Thompson (2005) This Work Topological object Braided ribbons Knotted/linked vortex rings Preons Helons (charged twists) None (vortex is fundamental) Mass predictions Qualitative Quantitative (0.08% accuracy) Generation mechanism Braid crossings Writhe mod 3 Base object Preon substructure Single helical Compton vortex

   from 20.1 Bilson-Thompson Helon Model (2005)

How it is done

1. Both $\pi$ and $\alpha$ are dimensionless — they are pure numbers.

   from 3.6 Dimensional Analysis

2. David Hestenes, over two decades of work, developed the most complete geometric model of the electron based on the Dirac equation and Spacetime Algebra (STA) — a Clifford algebra formulation of relativistic physics that eliminates the need for Dirac matrices.

   from 6. Hestenes' Helical Zitterbewegung Model

3. In Zitterbewegung in Quantum Mechanics — a research program (2008), Hestenes proposed a "self-contained dynamical model of the electron as a lightlike particle with helical zitterbewegung".

   from 6.1 The 2008 Model: Lightlike Helical Particle

4. In Zitterbewegung structure in electrons and photons (2019), Hestenes extended the model dramatically.

   from 6.2 The 2019 Model: Toroidal Vortex

What would falsify it

1. A scientific model must be falsifiable.

   from 11.4 What Would Falsify the Model?

2. The Helical Compton Vortex model would be falsified by: Experimental disconfirmation of Hestenes' channeling resonance prediction — if dedicated experiments find no ZB-related resonance after exhaustive search.

   from 11.4 What Would Falsify the Model?

3. The topological particle zoo model makes five concrete, falsifiable predictions that distinguish it from the Standard Model and its extensions:

   from 19. Falsifiable Predictions

Where the claim stops

1. We note that this interpretation is currently limited to the electron; extension to other particles remains an open problem.

   from Abstract

2. Physics rests on a set of numbers we call "fundamental constants." Among them, three stand out for their ubiquity and elusiveness: $\pi \approx 3.14159\ldots$ — appears in formulas from Coulomb's law to the Heisenberg uncertainty principle, from Gaussian integrals to Fourier transforms.

   from 1. Introduction — The Three Constants Problem

3. In the Attic acrophonic numeral system, used in Athens from approximately the 7th to the 3rd century BCE, numbers were represented by the first letters of their Greek names. $\Pi$ (pi) stood for πέντε (pente), meaning "five." The Attic numeral for 5 was written as $\Pi$ or, in its decorative form, as 𐅃 (a pi with a decorative right stroke).

   from 2.1 The Dual Identity of $\pi$

Where to read

1. Abstract189 words · 1 min
2. 1. Introduction — The Three Constants Problem360 words · 2 min
3. 2. Historical Foundations665 words · 3 min
4. 3. Mathematical Foundations — Exact Identities531 words · 2 min
5. 4. Mass-Frequency Identity306 words · 1 min
6. 5. The Dirac Electron and Zitterbewegung358 words · 2 min
7. 6. Hestenes' Helical Zitterbewegung Model579 words · 3 min
8. 7. Extended Electron Models370 words · 2 min
9. 8. The Unified Geometric Object463 words · 2 min
10. 9. Reification of Constants — A Critique740 words · 3 min
11. 10. Catalog of $\pi$ Appearances in Physics379 words · 2 min
12. 11. Implications and Predictions399 words · 2 min