Cost Ndim Ricci Scalar

The Ricci scalar in cost geometry

The Ricci scalar is a standard object in differential geometry: a single number at each point that summarizes how much the space curves there. In the Recognition Science framework, the space in question is a cost manifold, a continuous surface built from a function that assigns a price to every possible state. The framework's library of formal theorems shows that this manifold has a well-defined Ricci scalar, and that two different ways of writing it, one in coordinates tied to the original states and one in coordinates tied to a logarithmic transformation, produce identical results.

The classical setting is the Hessian metric, a way of measuring distances on a manifold using the second derivatives of a smooth function. Here the function is the cost, and the coordinates are positive real numbers. The framework defines the Ricci scalar in two forms: one rational in the variable Z = x^2ay^2b, and one hyperbolic in q = a s + b t. The two forms look very different, but the library proves they are algebraically equivalent, reducing both to a common rational form in w = exp q and closing the proof with algebraic simplification.

This is a theorem in the machine-checked library, not a numerical coincidence. The proof requires q ≠ 0 and a nondegeneracy condition on the connection, both stated explicitly. The main result, ricci_scalar_equiv, states that the two coordinate forms of the Ricci scalar agree under those conditions. What this establishes in plain language: the curvature of the cost manifold is a genuine geometric property, independent of which coordinate system you use to compute it.

In Recognition Science, this matters because the cost function is not arbitrary. The framework's central theorem forces the one-dimensional cost to be J(x) = (x + 1/x)/2 - 1, and the multidimensional generalization inherits that structure. The Ricci scalar is a check on that geometry: it behaves the way a true curvature should. The module does not derive new physics; it verifies that the geometric machinery is internally consistent.

/-- **Main result**: the two coordinate forms of the Ricci scalar agree. -/
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 := by
  rw [ricciQ_eq_ricciW a b q hq hLC, ricciZexp_eq_ricciW]

/-- **Main result**: the two coordinate forms of the Ricci scalar agree. -/
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 := by
  rw [ricciQ_eq_ricciW a b q hq hLC, ricciZexp_eq_ricciW]

What this page does not claim

This module does not derive the multidimensional cost function itself. This module does not connect the Ricci scalar to any measured physical quantity. The equivalence holds only under the stated conditions, not universally.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cost/Ndim/RicciScalar.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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