alphaLock_gt

In the ILG a priori prediction certificate, the Recognition Composition Law forces the unique cost $J(x) = \tfrac12(x+x^{-1})-1$ (T5), which in turn forces the self-similar scale $\varphi$ with $\varphi^2 = \varphi+1$ (T6). The memory kernel of the discrete ledger then inherits a $\varphi$-structure whose fractional exponent is locked at

$$\alpha_{\mathrm{lock}} = \frac{1-1/\varphi}{2}.$$

The module’s stated purpose is to convert that algebraic lock into a genuine prediction made before any SPARC fit, closing the paper’s objection that “parameter agreement is post-hoc, not predictive.” The companion lemma already supplies the numerical upper bound $1/\varphi < 0.622$ (from $\varphi > 1.61$). The present lower bound on $\alpha_{\mathrm{lock}}$ is the matching half of the decimal sandwich used downstream.

Unfold $\alpha_{\mathrm{lock}}$ to $(1-1/\varphi)/2$. Invoke the upstream bound $1/\varphi < 0.622$. Subtract from 1 (linarith) to obtain $1-1/\varphi > 1-0.622$. Divide both sides by the positive constant 2. A one-line norm_num identification $0.189 = (1-0.622)/2$ then yields the strict inequality by transitivity.

The parent theorem alpha_prediction_validated needs the two-sided decimal window $0.189 < \alpha_{\mathrm{lock}} < 0.192$ to prove

$$|\alpha_{\mathrm{lock}} - 0.19| < 0.002 < 0.02,$$

i.e. that the a priori prediction lies inside the SPARC $1\sigma$ band $\alpha_{\mathrm{obs}} = 0.19\pm 0.02$. Together with the matching upper bound, this lemma therefore discharges the numerical half of the certificate that $\alpha$ is derived from self-similarity (the forcing chain RCL → T5 → φ → kernel exponent) rather than fitted after the fact. It sits inside the broader Recognition claim that the same φ-ladder also produces the fine-structure band and the eight-tick octave.