registry_predictions_cert_exists
plain-language theorem explainer
The theorem shows that a certificate structure collecting explicit bounds on the cosmological constant and golden-ratio powers for fermion hierarchies is inhabited. Researchers verifying Recognition Science registry items C-010 and P-002 would cite it when confirming Omega_Lambda bounds and phi-ladder mass ratios. The proof is a direct term construction that assembles the six fields from upstream lemmas and applies trivial to witness existence.
Claim. There exists a certificate structure asserting that the cosmological constant parameter satisfies $0 < 11/16 - (alpha / pi) < 11/16$, that $17 < phi^6 < 18$, that $phi^{11} > 180$, and that for every positive integer rung difference $Delta r$ there exists a mass ratio equal to $phi^{Delta r}$ which exceeds 1.
background
The module supplies calculated proofs for registry items C-010 (cosmological constant Lambda with Omega_Lambda bounds) and P-002 (fermion mass hierarchy via phi^6 and phi^11 structure). RegistryPredictionsCert is the structure whose fields record these bounds: omega_lambda_upper and omega_lambda_lower for the expression 11/16 minus alpha over pi, phi_6_lower and phi_6_upper for the sixth power of phi, phi_11_lower for the eleventh power, and hierarchy_exists for the forall Delta r > 0 claim that phi to the Delta r exceeds 1. Upstream lemmas supply the concrete inequalities: omega_lambda_lt_11_16 proves the strict upper bound using alpha > 0 and alphaInv > 2; omega_lambda_positive proves positivity of the same expression; phi_6_hierarchy_bounds and phi_11_hierarchy_lower establish the numerical intervals via phi_cubed_eq and phi_eleventh_eq; hierarchy_phi_power_structure witnesses the ratio existence.
proof idea
The term proof constructs the certificate by supplying its six fields directly from the upstream results omega_lambda_lt_11_16, omega_lambda_positive, phi_6_hierarchy_bounds.1, phi_6_hierarchy_bounds.2, phi_11_hierarchy_lower, and the lambda abstraction that applies hierarchy_phi_power_structure, then closes the existential with trivial.
why it matters
This theorem completes the RegistryPredictionsProved module by exhibiting an explicit certificate for the cosmological constant bounds and the phi-power structure in fermion masses. It supports the Recognition Science phi-ladder mass formula and the alpha band inside (137.030, 137.039). No open questions remain in this module; all listed registry items are discharged.
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