RegistryPredictionsCert
plain-language theorem explainer
RegistryPredictionsCert is a structure that packages explicit bounds on the cosmological constant fraction derived from alpha and pi, together with inequalities on phi to the sixth and eleventh powers and a general statement that positive integer steps yield ratios exceeding one. Researchers verifying unification predictions against the COMPLETE_PROBLEM_REGISTRY would cite this certificate when checking C-010 and P-002 items. The declaration is a plain structure definition that directly encodes the listed inequalities with no additional proof.
Claim. A structure certifying the bounds $0 < 11/16 - (alpha / pi) < 11/16$, the inequalities $17 < phi^6 < 18$ and $180 < phi^{11}$, and the statement that for every positive integer step $Delta r$ there exists a ratio equal to $phi^{Delta r}$ that exceeds 1.
background
The module RegistryPredictionsProved supplies calculated proofs for selected items from the COMPLETE_PROBLEM_REGISTRY. It addresses the cosmological constant prediction Omega_Lambda less than 11/16 and greater than zero (C-010) and the fermion mass hierarchy expressed through phi to the sixth and eleventh powers (P-002). All bounds are obtained via norm_num, nlinarith and positivity tactics with explicit numerical constants.
proof idea
The declaration is a structure definition with an empty proof body. Its six fields directly record the upper and lower bounds on 11/16 minus alpha over pi, the two-sided bounds on phi to the sixth power, the lower bound on phi to the eleventh power, and the universal quantifier asserting that every positive integer rung produces a ratio strictly larger than one.
why it matters
RegistryPredictionsCert is the data type consumed by the downstream theorem registry_predictions_cert_exists, which asserts that an instance exists. It supplies the concrete numerical checks required for registry items C-010 and P-002. Within Recognition Science it connects the alpha definition to the cosmological constant bound and the phi-ladder to the mass hierarchy, consistent with the eight-tick octave and the phi-forcing chain.
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