Cost Ndim Ricci Scalar Ricci Zexp Eq Ricci W

Two names for one curvature

In differential geometry, the Ricci scalar is a single number that summarizes how much a space curves at a point. It is an intrinsic measure: it does not depend on how you choose to label the coordinates. In practice, however, the same curvature can be written in many algebraic forms, and proving that two such forms agree can be a tedious exercise in trigonometric identities. The declaration ricciZexp_eq_ricciW is a machine-checked proof that two specific algebraic expressions for the Ricci scalar, one written using exponentials and the other using a rational function, are equal for every real input.

The two expressions arise from a particular family of curved spaces studied in the Recognition Science framework. These spaces are built from a cost function, a rule that assigns a price to each pair of positive coordinates. The framework's own geometry section defines the Ricci scalar in two coordinate systems. The first, called the Z-form, is a rational expression in the variable Z = e^{2q}. The second, called the q-form, is a hyperbolic expression in q. A third expression, the W-form, is a canonical rational form in w = e^q. The theorem proves that the Z-form equals the W-form when w is set to e^q. A companion theorem proves the q-form also equals the W-form under the same substitution, provided q is nonzero and a certain denominator does not vanish. Together they show that the two coordinate forms describe the same curvature.

The proof itself is a formal algebraic computation. The machine-checked library of formal theorems rewrites the exponential identities e^{2q} = (e^q)^2 and e^{3q} = (e^q)^3, then simplifies both sides to a common rational form and closes the goal with the field_simp and ring tactics. This is not a numerical approximation or a hand-waving argument; it is a symbolic identity verified for all real values of the parameters a, b, and q. The theorem is tagged as a proof in the library's formal system, with no unproven assumptions beyond the standard axioms of the ambient type theory.

What the theorem does not claim is just as important as what it proves. It does not assert that the Ricci scalar itself has any particular physical meaning, nor that the cost function from which the geometry is built is the unique cost function of the framework. The theorem is purely about the equality of two algebraic expressions. It does not say that the q-form and the Z-form are equal for all q; the companion theorem requires q to be nonzero and the denominator condition to hold. The theorem also does not establish that the geometry is well-defined at the points where the denominators vanish, such as q = 0 or the zero of the linear factor in the denominator. Those are separate questions about the domain of the curvature expression.

For a reader, the practical consequence is simple: when working with the Ricci scalar in this framework, you may freely switch between the Z-form and the W-form without changing the value. This is a convenience for computation and for comparing results across different sections of the framework's documentation. It is a small but necessary consistency check, the kind of bookkeeping that a large formal library must verify so that later theorems can rely on a single canonical expression.

/-- The Z-form is `ricciW` evaluated at w = exp q. -/
theorem ricciZexp_eq_ricciW (a b q : ℝ) :
    ricciZexp a b q = ricciW a b (Real.exp q) := by
  unfold ricciZexp ricciW; rw [exp_two_mul, exp_three_mul]

/-- 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]

What this page does not claim

The theorem does not assign physical meaning to the Ricci scalar or to the cost function. The theorem does not prove the q-form and Z-form are equal for all q; it requires q ≠ 0 and a nonzero denominator. The theorem does not address the behavior of the geometry where the denominators vanish.

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.

Derived articles

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