Constants Alpha Higher Order Delta 1 Denominator Nat

The first correction term

The fine-structure constant α is one of physics' most precisely measured numbers, and its origin remains a deep open question. Within the Recognition Science framework, a construction proposes to derive its inverse from a geometric seed, a gap weight, and a series of correction terms. The first correction term, δ₁, is defined as a negative rational number: minus 103 divided by (102 times π⁵). The declaration delta_1_denominator_nat is a machine-checked proof that the natural number 102, which appears in that denominator, is exactly the product of the six faces of a cube and the 17 wallpaper groups that can tile each face.

This is a theorem about arithmetic, not about physics. The declaration proves that 6 times 17 equals 102. It does not prove that this number is the correct correction to α⁻¹, nor that the series converges to the measured CODATA value. The framework's own documentation marks the exact value of α as a free boundary datum, and the convergence of the full series to the CODATA target is an explicit hypothesis, not a proved result.

In Recognition Science, the framework models the fine-structure constant's inverse as a seed of 4π times 11 (about 138.230), minus a gap weight, plus the infinite sum of corrections δₙ. The first-order additive formula lands at about 137.035, which is roughly 8 parts per million below the CODATA 2022 value of 137.035999206(11). The exponential variant overshoots by about 6 ppm. These are near misses, not derivations; the framework itself lists the second-order correction as the key open deliverable.

What the declaration delta_1_denominator_nat establishes, in plain language, is a counting fact: the denominator 102 arises from a specific combinatorial structure on a three-dimensional cube. It is a small, exact, and verified step in a larger construction whose physical validity remains unproven.

theorem delta_1_denominator_nat : (face_wallpaper_pairs : ℕ) = 102 := face_wallpaper_pairs_eq

theorem face_wallpaper_pairs_eq : face_wallpaper_pairs = 102 := rfl

theorem Q3_faces_eq : Q3_faces = 6 := rfl

/-- First-order curvature correction. -/
def delta_1 : ℝ := -(curvature_numerator : ℝ) / ((face_wallpaper_pairs : ℝ) * π ^ measure_dimension)

/-- Geometric seed: 4π × passive_edges. -/
def alpha_seed : ℝ := 4 * π * passive_edges

/-- CODATA 2022 value of α⁻¹. -/
def CODATA_alpha_inv : ℝ := 137.035999206

What this page does not claim

The declaration does not prove that 102 is the physically correct denominator for the fine-structure constant. The declaration does not establish that the series converges to the CODATA value. The declaration does not derive the fine-structure constant itself; the seed identification remains open.

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)

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Derived articles

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