Constants Alpha Higher Order Measure Dimension Eq

The dimension of the correction

The fine-structure constant α measures the strength of electromagnetic interaction between elementary particles. Its inverse, 1/α, is approximately 137.035999206, a value known experimentally to about eleven decimal places. Within the Recognition Science framework, a construction aims to produce this number from geometric ingredients. The declaration measure_dimension_eq is a small but precise piece of that larger effort: it states that a certain counting number, called measure_dimension, equals five.

In the framework's model, this dimension arises from the cube Q₃, which has 8 vertices, 12 edges, and 6 faces. The measure_dimension is defined as 3 + 1 + 1, a sum that combines the three spatial dimensions of the cube with two additional counting terms. The theorem measure_dimension_eq proves this sum equals 5 by direct computation. This number then feeds into the half-period dimension, which counts the sectors used in the correction series. The theorem half_period_dim_eq establishes that this half-period dimension is also 5, meaning the integration measure splits into 2⁵ = 32 sectors.

The number five matters because it appears in the first correction term δ₁ = -103/(102π⁵), a formula that uses the dimension as the exponent of π. This correction is part of a series designed to close the gap between the framework's seed value and the measured constant. The seed identification itself remains open, not derived. The first correction brings the additive formula to about 137.035, which sits roughly 8 parts per million below the measured value. The exponential formula lands about 6 parts per million above it.

What measure_dimension_eq does not claim is just as important as what it proves. It does not derive the fine-structure constant, nor does it establish that the correction series converges to the measured value. The second-order correction δ₂, the key deliverable for improving the approximation, remains an open computation. The convergence of the full series to the CODATA value is a hypothesis, not a theorem. The declaration is a counting fact, rigorously proved, inside a larger construction whose physical success is still being tested.

theorem measure_dimension_eq : measure_dimension = 5 := rfl

theorem half_period_dim_eq : half_period_dim = 5 := rfl

theorem Z2_sectors_eq : Z2_sectors = 32 := by native_decide

/-- The residual between additive formula and CODATA.
    This is the amount the remaining δ_n terms must sum to. -/
def additive_residual (w8_val : ℝ) : ℝ :=
  CODATA_alpha_inv - (alpha_seed - f_gap w8_val + delta_1)

What this page does not claim

The declaration does not derive the fine-structure constant from first principles. The declaration does not prove that the correction series converges to the measured CODATA value. The declaration does not identify the geometric seed 4π × 11 as a derived quantity.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Constants/AlphaHigherOrder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: