alphaLock_numerical_bounds
The locked fine-structure constant α_lock = (1 − 1/φ)/2 satisfies 0.18 < α_lock < 0.21 when φ lies in (1.61, 1.62). Researchers deriving coupling constants from the Recognition Science ledger would cite this bound to confirm the numerical range of the parameter-free coupling. The proof unfolds the definition, invokes the two phi bounds, derives the reciprocal inequalities via division properties, and closes both sides with linear arithmetic.
claim$0.18 < α_{lock} < 0.21$, where $α_{lock} = (1 - φ^{-1})/2$ and $φ$ satisfies $1.61 < φ < 1.62$.
background
In the FineStructureConstant module the RS derivation of the fine-structure constant is formalized as α_lock = (1 − 1/φ)/2. This expression follows directly from the reciprocal symmetry J(x) = J(1/x) of the ledger and the self-similar fixed point φ forced by the J-uniqueness condition T5. The module records the conversion to the observed α ≈ 1/137 as a separate ledger-to-lab calibration step.
proof idea
The tactic proof first unfolds alphaLock. It then applies phi_gt_onePointSixOne to obtain 1/φ < 1/1.61 via div_lt_div_iff_of_pos_left and closes the lower bound with linarith. The symmetric upper bound is obtained from phi_lt_onePointSixTwo by showing 1/1.62 < 1/φ and again applying linarith.
why it matters in Recognition Science
This supplies the coarse numerical interval for the RS-native coupling α_lock, directly discharging the C-001 resolution that the constant is determined by φ with no free parameters. It anchors the fine-structure derivation inside the forcing chain (T5–T8) and the reciprocal symmetry of the ledger. No downstream theorems are recorded yet.
scope and limits
- Does not establish the exact numerical value of α_lock.
- Does not perform the ledger-to-SI unit conversion to match the experimental 1/137.
- Does not derive or tighten the bounds on φ.
- Does not address higher-precision refinements inside the alpha band.
formal statement (Lean)
42theorem alphaLock_numerical_bounds :
43 (0.18 : ℝ) < alphaLock ∧ alphaLock < (0.21 : ℝ) := by
proof body
Tactic-mode proof.
44 unfold alphaLock
45 have h_phi := phi_gt_onePointSixOne
46 have h_phi' := phi_lt_onePointSixTwo
47 constructor
48 · have h_inv : 1 / phi < 1 / 1.61 := by
49 rw [div_lt_div_iff_of_pos_left (by norm_num) phi_pos (by norm_num)]
50 exact h_phi
51 linarith
52 · have h_inv : 1 / 1.62 < 1 / phi := by
53 rw [div_lt_div_iff_of_pos_left (by norm_num) (by norm_num) phi_pos]
54 exact h_phi'
55 linarith
56
57/-! ## C-001 Resolution Statement -/
58
59/-- **C-001 Resolution**: The fine structure constant is determined by φ.
60
61 α_lock = (1 − 1/φ)/2 has no free parameters.
62 It is the unique coupling compatible with the ledger's
63 reciprocal symmetry and self-similar closure.
64
65 Relationship to measured α ≈ 1/137 requires the conversion
66 from RS-native to SI units (λ_rec calibration). -/