Cost Ndim Ricci Scalar Ricci Q Eq Ricci W

Two forms, one curvature

In differential geometry, the Ricci scalar curvature summarizes how a surface bends at a point, a single number that does not depend on the coordinates you use to describe the surface. For a two-dimensional surface built from a cost function, the Recognition Science framework writes that number in two ways. One form uses the original positive coordinates (x, y) and looks like a rational function; the other uses logarithmic coordinates q and looks like a hyperbolic expression with sinh and cosh. The theorem ricciQ_eq_ricciW proves that, under two explicit conditions, these two expressions are equal: it establishes that the q-form equals the w-form, where w is simply the exponential of q.

The proof is algebraic, not geometric. The framework's machine-checked library of formal theorems reduces both expressions to a common rational form in w = exp q, then closes the equality using algebraic simplification. The two conditions are that q is not zero and that a certain denominator term, (a+b) cosh q minus sinh q, is not zero. These are not hidden assumptions; they are the places where the formulas would otherwise divide by zero. The theorem also chains with a companion result to show that the q-form equals the original Z-form, so all three coordinate styles agree.

What this theorem does not claim is just as important. It does not say that the Ricci scalar has any particular value, nor that it is constant, nor that it encodes physical gravity. It only says that two algebraic expressions, derived from the same geometric object, are equal. The parameters a and b remain free real numbers; the result holds for all of them. The theorem is a coordinate-consistency check, not a physical law. It confirms that the framework's two bookkeeping styles for curvature do not secretly describe different surfaces.

For a reader, the practical consequence is that you may compute the curvature in whichever coordinate system is more convenient, and the answer will not change. That is the property that makes the curvature a genuine geometric invariant rather than an artifact of your coordinate choice. The proof is a small but necessary link in the framework's larger program of deriving geometry from cost.

/-- The q-form is also `ricciW` at w = exp q. -/
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) := by
  set w := Real.exp q with hw_def
  have hw_pos : 0 < w := Real.exp_pos q
  have hw_ne : w ≠ 0 := hw_pos.ne'
  have hsinh_ne : Real.sinh q ≠ 0 := Real.sinh_ne_zero.mpr hq
  have hcosh_w : Real.cosh q = (w ^ 2 + 1) / (2 * w) := by
    rw [Real.cosh_eq, Real.exp_neg]; field_simp; ring
  have hsinh_w : Real.sinh q = (w ^ 2 - 1) / (2 * w) := by
    rw [Real.sinh_eq, Real.exp_neg]; field_simp; ring
  have hw2m1 : w ^ 2 - 1 ≠ 0 := by
    intro h; exact hsinh_ne (by rw [hsinh_w, h, zero_div])
  have hLCw : (a + b - 1) * w ^ 2 + (a + b + 1) ≠ 0 := by
    intro h; apply hLC; rw [hcosh_w, hsinh_w]; field_simp; linarith
  show ricciQ a b q = ricciW a b w
  unfold ricciQ ricciW
  rw [hcosh_w, hsinh_w]
  field_simp [hw_ne, hw2m1, hLCw]
  ring

/-- **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]

/-- The q-form is also `ricciW` at w = exp q. -/
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) := by
  set w := Real.exp q with hw_def
  have hw_pos : 0 < w := Real.exp_pos q
  have hw_ne : w ≠ 0 := hw_pos.ne'
  have hsinh_ne : Real.sinh q ≠ 0 := Real.sinh_ne_zero.mpr hq
  have hcosh_w : Real.cosh q = (w ^ 2 + 1) / (2 * w) := by
    rw [Real.cosh_eq, Real.exp_neg]; field_simp; ring
  have hsinh_w : Real.sinh q = (w ^ 2 - 1) / (2 * w) := by
    rw [Real.sinh_eq, Real.exp_neg]; field_simp; ring
  have hw2m1 : w ^ 2 - 1 ≠ 0 := by
    intro h; exact hsinh_ne (by rw [hsinh_w, h, zero_div])
  have hLCw : (a + b - 1) * w ^ 2 + (a + b + 1) ≠ 0 := by
    intro h; apply hLC; rw [hcosh_w, hsinh_w]; field_simp; linarith
  show ricciQ a b q = ricciW a b w
  unfold ricciQ ricciW
  rw [hcosh_w, hsinh_w]
  field_simp [hw_ne, hw2m1, hLCw]
  ring

What this page does not claim

The theorem does not assign a numerical value to the Ricci scalar. The theorem does not state that the Ricci scalar is constant or has any particular sign. The theorem does not claim any physical gravitational content for the curvature.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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