Explain the Lean theorem `ricciQ_eq_ricciW` in module `IndisputableMonolith.Cost.Ndim.RicciScalar`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

1. What the Declaration Says in Plain English

The theorem ricciQ_eq_ricciW states that the Ricci scalar curvature of the 2D cost Hessian metric, when written in hyperbolic coordinates ($q$) using hyperbolic sines and cosines, is algebraically identical to a purely rational expression evaluated at $w = e^q$. The proof holds provided $q \neq 0$ and a specific denominator term relating to the Levi-Civita connection does not vanish.

2. Why it Matters in Recognition Science

In the RS multidimensional cost framework, the cost function defines a Hessian Riemannian metric. The Ricci scalar describes the curvature of this cost manifold. Because this curvature can be calculated in multiple coordinate systems—such as the $Z$-form (rational powers) and the $q$-form (hyperbolic functions)—it is necessary to formally prove they describe the same geometry. This theorem acts as the crucial translation step. By reducing the hyperbolic $q$-form to a canonical rational $w$-form, it enables the capstone theorem ricci_scalar_equiv, which confirms that the two main coordinate representations are geometrically identical.

3. How to Read the Formal Statement

theorem ricciQ_eq_ricciW (a b q : ℝ)
    (hq : q ≠ 0)
    (hLC : (a + b) * Real.cosh q - Real.sinh q ≠ 0) :
    ricciQ a b q = ricciW a b (Real.exp q)

• (a b q : ℝ): Declares three real-valued parameters. $a$ and $b$ are coordinate constants, and $q$ is the coordinate variable.
• (hq : q ≠ 0): A hypothesis requiring $q$ to be non-zero. This prevents $\sinh(q) = 0$, avoiding division by zero in the ricciQ denominator.
• (hLC : ... ≠ 0): A hypothesis ensuring that the Levi-Civita denominator term (a + b) * cosh q - sinh q does not vanish.
• Conclusion: ricciQ a b q = ricciW a b (Real.exp q) asserts that the hyperbolic representation ricciQ is exactly equal to the rational representation ricciW when evaluated at $w = e^q$.

4. Visible Dependencies

Within the supplied source, this is a THEOREM-grade algebraic proof that relies strictly on:

• The RS definitions ricciQ and ricciW from the same module.
• Mathlib real analysis primitives (Real.exp, Real.cosh, Real.sinh) and the identities relating hyperbolic functions to exponentials (e.g., cosh q = (w^2 + 1)/(2w)).
• Lean's algebraic simplifiers (field_simp and ring) to verify the rational polynomial equality. There are zero external assumptions, unproved axioms, or sorry gaps.

5. What this Declaration Does Not Prove

• It does not prove the final equivalence between the $Z$-form and $q$-form (that is done by ricci_scalar_equiv, which uses this theorem as a stepping stone).
• It does not calculate specific bounds or numerical values for the curvature.
• It does not map the cost manifold's Ricci scalar to physical General Relativity or spacetime curvature.