1. What the declaration says in plain English
The theorem ricci_scalar_equiv establishes the algebraic equivalence of two different formulas for the same curvature quantity. Specifically, it proves that an expression using hyperbolic functions (ricciQ) evaluates to the exact same real number as an expression using exponential functions (ricciZexp), given any real parameters $a$ and $b$, and any coordinate $q$, provided $q$ is not zero and a specific denominator does not vanish.
2. Why it matters in Recognition Science
In Recognition Science, the cost function $J$ induces a "Hessian metric" defining the geometry of multidimensional cost spaces (MODEL). The curvature of this space—measured by the Ricci scalar—governs underlying structural properties. This theorem guarantees that calculating the Ricci scalar in the transformed hyperbolic $q$-coordinates is perfectly consistent with calculating it in the rational $Z$-coordinates. It provides a formal algebraic consistency check for the formulas used in the Multidimensional Cost Geometry calculations.
3. How to read the formal statement
theorem ricci_scalar_equiv (a b q : ℝ)
(hq : q ≠ 0)
(hLC : (a + b) * Real.cosh q - Real.sinh q ≠ 0) :
ricciQ a b q = ricciZexp a b q
(a b q : ℝ): The variables $a$, $b$, and $q$ are real numbers.(hq : q ≠ 0): A required premise that $q$ is non-zero.(hLC : ... ≠ 0): A structural premise ensuring that the denominator from the Levi-Civita connection does not evaluate to zero, preventing undefined division.ricciQ a b q = ricciZexp a b q: The conclusion, stating that the $q$-form definition equals the $Z$-form definition.
4. Visible dependencies in the source
The proof operates by triangulating both expressions through a third canonical rational form, ricciW, formulated in terms of $w = e^q$.
- It relies on ricciZexp_eq_ricciW, proving the $Z$-form equals the $w$-form.
- It relies on ricciQ_eq_ricciW, proving the $q$-form also equals the $w$-form (utilizing the non-vanishing conditions).
By showing both equal
ricciW a b (Real.exp q), the theorem bridges them by transitivity.
5. What this declaration does not prove
This theorem strictly proves an algebraic identity between two explicitly written formulas. It does not prove the differential geometry steps that generated these formulas. The computation of the metric tensor from the Hessian, the derivation of the Christoffel symbols, and the tensor contractions that yield the Ricci scalar are mathematically upstream of this equivalence and are not mechanically certified by this specific Lean theorem.