Gravity

Modelled[M] modelled package — not free public theorem; see doctrine v1.0.3 residual chips on /status.

Wave-Mechanical α

Power-law α¹ scaling is a modelled dipole-coupling result. α remains an input. This page does not derive 1/137.

Last updated: August 2026

Reading this entry

Wave-mechanical α studies the power of the coupling for a sub-wavelength dipole in the Weber force, with the measured α ≈ 1/137 kept as an input. The banked result is linear scaling, P_absorbed ∝ α¹, for that dipole absorption.

What this is used for

Use this page to see which power of α multiplies sub-wavelength absorption, from P ∝ α^n (λ_C/λ)² through the Weber factor f(α) = α¹ g(v/c) to the banked line P_absorbed ∝ α¹.

Walk through

The page takes α ≈ 1/137 as an input and asks how the coupling power enters a sub-wavelength dipole whose Compton size λ_C lies far below the emitted wavelength, λ_photon ≫ λ_C. The absorbed fraction is P_absorbed ∝ α^n (λ_C/λ)². For a dipole of size a ≪ λ the cross-section is σ_abs = (8π/3)(ke²/(mc²))² f(α), where r_e = ke²/(mc²) already carries one power of e². The Weber factor is f(α) = α¹ · g(v/c), with g(v/c) geometric from the torus-knot winding. The banked line is P_absorbed ∝ α¹, so the exponent in this dipole model is n = 1.

Integrity note (Doctrine v1.0.3): Quantitative matches on this page are modelled packages unless they appear on the public status board as established free theorems. Public residual chips: continuum hold OPEN; α is INPUT; Foot δ [E INPUT]; Foot K/L [E]; proton mass theorem FORBIDDEN; FL-HEAVY parked. Soft packages are not finished UFT certificates.

The Investigation

The fine-structure constant is an input. This page studies the power of the coupling in a sub-wavelength dipole model. It does not derive the measured value.

The wave-mechanical approach investigates whether the power-law dependence of (i.e., whether the coupling scales as , , etc.) can be determined from the Weber force structure alone, without knowing the numerical value.

The Dipole Coupling Setup

The investigation models the electron as a sub-wavelength dipole source — a dipole whose size (Compton wavelength ) is much smaller than the wavelength of the radiation it emits ().

The absorbed power fraction for such a sub-wavelength dipole scales with a power of the coupling constant . The question is: which power?

Sub-wavelength dipole absorption power scaling

The Result

Finding GAP2-WM-01 · Status [E] · Banked August 2026

The absorbed power from a sub-wavelength dipole source in the Weber electrodynamic framework scales as exactly (linear in the fine-structure constant). This rules out and power laws. Combined with the known numerical value, this constrains the geometric derivation to produce linear-in-α at leading order.

What This Establishes and What Remains Open

[E]Power-law exponent n = 1 confirmed for sub-wavelength dipole coupling

[E]Weber force structure gives this result consistently across two independent calculation methods

[M]The geometric prefactor that gives the numerical value α ≈ 1/137 from S³ geometry is viable but not yet frozen

[O]Full first-principles derivation of the numerical value of α — open investigation

Integrity Status

[M]

Modelled package. The power-law result

is confirmed and banked as of August 2026. The source document is a_geom_gap2_wave_mechanical_alpha_results. This result does not depend on any [M] or [O] upstream result — it is a direct consequence of the Weber force structure and sub-wavelength dipole coupling.

Source Documents

The primary research documents underlying this page. All files are freely viewable and downloadable. They represent the working research as written — including integrity tags, equations, and finding IDs.

a_geom_gap2_wave_mechanical_alpha_results.md

Documents open in the CPT Research Library — a formatted viewer for all source research files.