Derivation of Fundamental Constants from Geometric First Principles
We present a geometric framework in which the fine structure constant α emerges from the topology of a helicoidal manifold with zero free parameters. Beginning with a single axiom—the dimensional coupling ratio γ = 2/3 derived from holographic boundary considerations—we derive the spinor period T = 18 and channel count N = 11, yielding 1/α = 137.0359477 with 0.376 ppm precision against the experimental value. The same framework predicts ionization energies across 118 elements (R² = 0.944) and explains the Koide formula for lepton mass ratios.
The framework rests on a single geometric axiom:
This ratio emerges from holographic boundary considerations: in a system where information is encoded on a 2D boundary surrounding a 3D bulk, the coupling between surface and volume degrees of freedom is 2/3. This is not fitted—it is the only value consistent with holographic entropy scaling.
Fermions require 720° rotation to return to their initial state (spinor double-cover). Combined with the 40° angular step of the 9-fold boundary lattice:
The number of information channels through the veil, accounting for Period 1 exclusion:
The "−1" reflects Period 1 (H, He), which exists in a pre-axial state—present before Carbon provided the organizing center. This is geometrically observable: H and He sit at ±120° (γ × 180°), encoding the framework ratio without passing through the axis.
The fine structure constant emerges as the total phase-space coupling:
Expanding with T = 18, N = 11:
| Term | Calculation | Physical Meaning |
|---|---|---|
| T² | 324 | Full spinor-squared lattice |
| −N(T−1) | −187 | Channels × wrap-around (bulk contribution) |
| +N/(T(T−1)) | +11/306 = 0.03595... | Phase-slip correction (veil leakage) |
The integer part (137) arises from lattice alignment. The fractional part (11/306) represents residual phase-slip—information leaking through the boundary at non-nodal points.
The complete derivation requires zero fitted parameters:
Each step follows necessarily from the previous. No parameter is adjusted to match experiment.
The helicoid geometry predicts first ionization energies across the periodic table:
| Block | Elements | R² |
|---|---|---|
| Noble gases | He, Ne, Ar, Kr, Xe, Rn | 0.99 |
| d-block | Sc–Zn, Y–Cd, etc. | 0.88 |
| Main group | Groups 1, 2, 13–17 | 0.86 |
| f-block | Lanthanides, Actinides | 0.83 |
| Overall | 118 elements | 0.944 |
The ~6% residual variance is attributed to dynamic effects (electron correlation, relativistic contraction) that require the full quantum treatment of the dual-sheet manifold.
The Koide formula for charged lepton masses (1981) gives:
This matches γ = 2/3 to 0.0009%. The Koide relation, unexplained for 44 years, is a direct expression of the dimensional coupling constant in the lepton sector.
The periodic table maps onto a helicoidal manifold with the following properties:
The ~6% variance in ionization predictions suggests the "geometric skeleton" requires a "quantum flesh" treatment. Preliminary analysis indicates this residual maps to the phase-slip term 11/306, implying quantum mechanical effects emerge as interference between dual stable states on the helicoid.
Key open problems:
From a single axiom (γ = 2/3), the EiG framework derives:
The periodic table is not flat. It is a helicoid, and its geometry encodes the coupling constants of physics.
Contact: contact@discernibility.org
Interactive visualization: EiG Periodic Helicoid
Full framework: 9 Laws of Emergent Information Geometry (PDF)