H_AlphaPrecision
plain-language theorem explainer
H_AlphaPrecision encodes the hypothesis that the Recognition-derived inverse fine-structure constant lies within an error of 10^{-11} of the value 137.035999. Researchers validating zero-parameter alpha models against CODATA data would invoke this predicate to connect the structural derivation to experimental bounds. The declaration is introduced as a direct existential statement over the error term with no lemma reductions.
Claim. There exists a real number $ε$ such that $|α^{-1} - 137.035999| < ε$ and $ε < 10^{-11}$, where $α^{-1}$ is the dimensionless inverse fine-structure constant obtained from the structural seed and gap via exponential resummation.
background
The module addresses Phase 12.1 of the Recognition framework by extending the alpha derivation to twelve or more decimal places through the complete curvature series. The quantity alphaInv is defined as alpha_seed * exp(-f_gap / alpha_seed), a zero-parameter expression whose value lies near 137.036 and inside the interval (137.030, 137.039). Upstream, the Identity predicate states that self-comparison of any positive real costs zero, while band implements arithmetic conjunction on stable 0/1 states and complete asserts that every variable in a backpropagation state has been assigned.
proof idea
The body is a direct definition that packages the numerical bound as an existential proposition over the error term. No upstream lemmas such as Identity, band, or complete are applied; the predicate is introduced by fiat to serve as the hypothesis for the subsequent theorem.
why it matters
This hypothesis supplies the direct input to the theorem alpha_high_precision, which extracts the existence claim for use in higher-level arguments. It corresponds to the empirical test step in Phase 12.1 of the zero-parameter alpha derivation, confirming agreement with the observed value inside the framework band. It leaves open whether further curvature refinements can tighten the error below the current threshold.
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