alphaLock_lt_one
alphaLock_lt_one shows that the locked fine-structure constant α_lock = (1 - 1/φ)/2 is strictly less than one. Cosmologists and gravity modelers cite it when bounding dark-energy density and amplitude parameters. The proof is a short term reduction that unfolds the definition, invokes positivity of φ, and closes with linarith.
claimThe locked fine-structure constant satisfies $α_{lock} < 1$, where $α_{lock} := (1 - φ^{-1})/2$.
background
Recognition Science fixes φ as the self-similar fixed point of the J-cost function J(x) = (x + x^{-1})/2 - 1. The constant α_lock is introduced as the canonical RS-native fine-structure coupling (1 - 1/φ)/2. Upstream lemmas supply φ > 1 and the explicit definition of α_lock. The module treats τ₀ = 1 tick as the fundamental RS time quantum.
proof idea
The term proof unfolds alphaLock, applies phi_pos to obtain 1/φ > 0 via one_div_pos, and finishes with linarith.
why it matters in Recognition Science
It feeds fine_structure_derived, which states that α_lock has no free parameters and resolves C-001. Downstream uses appear in omega_lambda_pos for cosmology and in A_amplitude_bounds together with p_steepness_pos for gravity parameters. The lemma closes the algebraic step from the eight-tick octave to observable couplings inside the Recognition framework.
scope and limits
- Does not establish numerical agreement with the measured 1/137.
- Does not derive the definition of alphaLock.
- Does not perform conversion from RS-native units to SI.
- Does not prove uniqueness of the coupling beyond the given algebraic form.
formal statement (Lean)
234lemma alphaLock_lt_one : alphaLock < 1 := by
proof body
Term-mode proof.
235 have hpos : 0 < phi := phi_pos
236 unfold alphaLock
237 have : 1 / phi > 0 := one_div_pos.mpr hpos
238 linarith
239
240/-- Canonical locked C_lag constant: C_lock = φ^{−5}. -/