CosmologicalPredictionsCert
plain-language theorem explainer
This structure bundles nine explicit numerical inequalities that certify cosmological predictions derived from the golden ratio φ. A unification theorist or cosmologist would cite it to anchor the spectral index n_s = 1 - 2/φ³ inside (0.5, 1), to confirm H₀ > 0 via ln(φ)/8, and to record tight intervals for φ², φ⁴ and φ⁵. The definition simply collects these bounds as a single record type so that an existence theorem can witness them all at once.
Claim. A structure certifying the inequalities $0.5 < 1 - 2/φ^3 < 1$, $ (ln φ)/8 > 0 $, $2.59 < φ^2 < 2.62$, $6.7 < φ^4 < 6.9$ and $10.7 < φ^5 < 11.3$, where φ is the golden ratio.
background
In Recognition Science the golden ratio φ is the unique positive fixed point of the J-map J(x) = (x + x^{-1})/2 - 1 that appears in the T5–T6 steps of the forcing chain. This module supplies calculated proofs for selected entries in the COMPLETE_PROBLEM_REGISTRY, notably EU-003 (primordial spectral index bounds from φ³) and T-001 (Hubble positivity from ln φ). The structure simply aggregates the nine real inequalities that have already been verified by norm_num and positivity so they can be referenced as a single object.
proof idea
This is a structure definition with no proof body. Each field is an independent proposition whose truth is established by direct numerical evaluation once φ > 1 is known; the downstream existence theorem discharges the fields with Real.log_pos and positivity.
why it matters
The certificate is the immediate parent of the existence theorem cosmological_predictions_cert_exists, which constructs an instance to witness that the listed predictions hold. It closes the calculated-proofs section for EU-003 and T-001 inside the module, supplying the numerical anchor for the eight-tick octave and D = 3 predictions that follow from the T7–T8 forcing steps.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.