constants_predictions_cert_exists
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.
claimThere 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 in Recognition Science
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.
scope and limits
- Does not derive the exact numerical value of alpha beyond the stated interval.
- Does not address higher-order corrections to the phi-ladder.
- Does not connect to experimental measurements outside RS-native units.
formal statement (Lean)
228theorem constants_predictions_cert_exists : ∃ _ : ConstantsPredictionsCert, True := by
proof body
Term-mode proof.
229 refine ⟨⟨alpha_positive, alpha_lt_0_1,
230 c_eq_one, c_positive,
231 boltzmann_analog_positive, boltzmann_analog_lt_half,
232 phi_inverse_formula, phi_plus_inverse_eq_sqrt5,
233 phi_sq_gt_2_5, phi_sq_lt_2_7⟩, trivial⟩
234
235/-! ## Summary -/
236
237/-- **SUMMARY**: Constants with calculated proofs:
238
239 1. C-001: 0 < α < 0.1 (fine-structure constant)
240 2. C-005: c = 1 (RS-native speed of light)
241 3. C-006: 0 < ln(φ) < 0.5 (Boltzmann analog)
242 4. Phi formulas: 1/φ = φ-1, φ+1/φ = √5
243 5. Phi bounds: 2.5 < φ² < 2.7
244
245 **Proof Methods**: `norm_num`, `nlinarith`, `positivity`, `field_simp`, `Real.log_lt_log`
246 **All from 1.5 < φ < 1.62 and φ² = φ + 1.** -/