constants_predictions_cert_exists
plain-language theorem explainer
The declaration establishes existence of a certificate that packages explicit bounds and identities for the fine-structure constant, the speed of light, the Boltzmann analog, and several golden-ratio identities. Researchers working on unification in Recognition Science would cite it to confirm that the predicted intervals align with the phi-ladder and forcing axioms. The proof is a direct term construction that assembles eight sibling lemmas into the certificate record and discharges the existential with trivial.
Claim. There exists a constants predictions certificate such that $0 < α < 0.1$, $c = 1$ with $c > 0$, $0 < ln(φ) < 0.5$, $1/φ = φ - 1$, $φ + 1/φ = √5$, and $2.5 < φ² < 2.7$.
background
The module supplies calculated proofs for registry items C-001 (fine-structure constant), C-005 (speed of light), and C-006 (Boltzmann analog). The certificate structure bundles the assertions alpha > 0, alpha < 0.1, c = 1, c > 0, Real.log phi > 0, Real.log phi < 0.5, 1/phi = phi - 1, phi + 1/phi = sqrt 5, phi² > 2.5, and phi² < 2.7. This sits inside the unification layer that imports PhiForcing and GapWeight. Upstream, PhiForcingDerived supplies the J-cost structure while NucleosynthesisTiers provides the phi-tier scaling for densities.
proof idea
The proof is a term-mode construction. It applies refine to build the certificate record by inserting the eight sibling lemmas alpha_positive, alpha_lt_0_1, c_eq_one, c_positive, boltzmann_analog_positive, boltzmann_analog_lt_half, phi_inverse_formula, phi_plus_inverse_eq_sqrt5, phi_sq_gt_2_5, and phi_sq_lt_2_7, then closes with trivial.
why it matters
This theorem certifies the constant predictions that follow from the phi-forcing chain (T5 J-uniqueness through T8 D=3). It supplies the concrete bounds required by the LawOfExistence.Constants bundle and the alpha band inside (137.03, 137.04). The result closes the calculated-proofs section of the registry with no remaining open questions.
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