alpha_s_value
The theorem establishes that the Recognition Science prediction for the strong gauge coupling satisfies 0.117 < α_s < 0.118. Researchers comparing gauge unification models to PDG data on α_s(M_Z) would cite this bound when checking consistency with the wallpaper-derived value. The proof is a direct numerical verification that 2/17 lies inside the interval, performed by splitting the conjunction and applying simplification plus normalization.
claim$0.117 < α_s < 0.118$, where $α_s = 2/17$ is the strong coupling constant obtained from the wallpaper group count in the Recognition Science framework.
background
The module derives the three Standard Model gauge couplings from Recognition Science ledger geometry. For the strong sector the key definitions are the wallpaper fraction 2/17 (a rational constant) and its embedding as the real-valued prediction. These rest on the upstream theorem that the seven axioms imply four structural conditions plus three definitional facts, which here fix the wallpaper group count W = 17. The local setting is registry item C-014, which asks what determines α_s and answers it by α_s = 2/W ≈ 0.1176, stated to match the PDG 2022 interval 0.1179 ± 0.0009.
proof idea
The term-mode proof opens the conjunction with constructor. Each conjunct is discharged by simp only on the definitions of the predicted coupling and the geometric fraction, reducing the claim to a numerical comparison that norm_num then confirms.
why it matters in Recognition Science
This result supplies the concrete numerical anchor for the strong coupling entry in the C-014 gauge couplings derivation. It supports the unification hint that α, α_s and α_w converge near the GUT scale, drawing on the crystallographic origin of the strong force within the ledger structure. The bound is consistent with the eight-tick octave and D = 3 spatial dimensions fixed earlier in the forcing chain.
scope and limits
- Does not derive the wallpaper group count W = 17 from more primitive axioms.
- Does not address the running of α_s with energy scale.
- Does not incorporate experimental uncertainties beyond the stated open interval.
- Does not connect the strong coupling to the electromagnetic or weak sectors inside this statement.
formal statement (Lean)
137theorem alpha_s_value : (0.117 : ℝ) < (alpha_s_pred : ℝ) ∧ (alpha_s_pred : ℝ) < (0.118 : ℝ) := by
proof body
Term-mode proof.
138 constructor
139 · -- Lower bound: 2/17 > 0.117
140 simp only [alpha_s_pred, alpha_s_geom]
141 norm_num
142 · -- Upper bound: 2/17 < 0.118
143 simp only [alpha_s_pred, alpha_s_geom]
144 norm_num
145
146/-- **CALCULATED**: sin²θ_w from φ ≈ 0.230 (matches observed 0.2229 within ~3%) -/