cLagLock_pos
The positivity of the locked lag constant defined as the golden ratio raised to the power of negative five is established. Researchers deriving RS-native constants such as the reduced Planck constant cite this result to confirm that derived quantities remain positive. The proof is a direct term-mode reduction that invokes the positivity of the golden ratio and the preservation of positivity under real exponentiation.
claimThe locked lag constant satisfies $0 < C_{lock}$ where $C_{lock} = phi^{-5}$ and $phi$ is the golden ratio.
background
The Constants module works in RS-native units where the fundamental time quantum tau_0 equals one tick. The locked lag constant is introduced as the canonical value phi to the power of negative five. Upstream results include the definition of cost for recognition events as the J-cost and the derived cost for multiplicative recognizers, both of which presuppose non-negative quantities in the recognition framework.
proof idea
The proof asserts positivity of phi from the phi_pos lemma, unfolds the definition of the locked lag constant, and applies the real-analysis fact that a positive base to any real power remains positive.
why it matters in Recognition Science
This result is invoked directly by E_coh_pos and by hbar_pos, the latter stating that hbar equals phi^{-5} in RS-native units and corresponding to theorem C-004.1. It anchors the framework where hbar = phi^{-5}, c = 1, and G = phi^5 / pi, supporting the forcing chain from T5 J-uniqueness through the eight-tick octave to D = 3 spatial dimensions.
scope and limits
- Does not establish the numerical value or irrationality of phi.
- Does not address positivity for negative bases or non-real exponents.
- Does not derive the locked lag constant from first principles.
- Does not extend beyond RS-native units.
Lean usage
lemma hbar_pos : 0 < hbar := mul_pos cLagLock_pos tau0_posformal statement (Lean)
243lemma cLagLock_pos : 0 < cLagLock := by
proof body
Term-mode proof.
244 have hphi : 0 < phi := phi_pos
245 unfold cLagLock
246 exact Real.rpow_pos_of_pos hphi (-(5 : ℝ))
247
248/-- The elementary ledger bit cost J_bit = ln φ. -/