registry_predictions_cert_exists
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.
claimThere 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 in Recognition Science
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.
scope and limits
- Does not compute the precise numerical value of Omega_Lambda.
- Does not derive the phi forcing chain or J-uniqueness from first principles.
- Does not address registry items outside C-010 and P-002.
- Does not supply error estimates tighter than the stated inequalities.
formal statement (Lean)
179theorem registry_predictions_cert_exists : ∃ _ : RegistryPredictionsCert, True :=
proof body
Term-mode proof.
180 ⟨⟨omega_lambda_lt_11_16,
181 omega_lambda_positive,
182 phi_6_hierarchy_bounds.1,
183 phi_6_hierarchy_bounds.2,
184 phi_11_hierarchy_lower,
185 fun Δr hΔr => hierarchy_phi_power_structure Δr hΔr⟩,
186 trivial⟩
187
188end RegistryPredictionsProved
189end Unification
190end IndisputableMonolith