Cost Ndim Ricci Scalar Exp Three Mul

A small algebraic step

The declaration exp_three_mul is a small algebraic lemma inside a machine-checked library of formal theorems. It states that for any real number q, the exponential of 3 times q equals the cube of the exponential of q: exp(3q) = (exp q)^3. This is a standard property of the real exponential function, proved here by splitting 3q into q + q + q and applying the addition rule for exponentials. The proof is a short chain of rewrites, closed by the ring tactic, and it carries no additional hypotheses.

Why does this matter? The page's main result compares two formulas for the Ricci scalar curvature of a Hessian metric: one written in coordinates (q, r), the other in coordinates (x, y). The two formulas look different, but they are supposed to describe the same geometric object. To show they agree, the proof rewrites both into a common rational form in the variable w = exp q. The lemma exp_three_mul is exactly the bridge that converts a term like exp(3q) in one formula into w^3 in the common form. Without it, the algebraic comparison would stall.

The theorem is proved in the framework's library, a machine-checked collection of formal theorems. It is a private helper lemma, not a standalone result about curvature. Its role is purely algebraic: it does not assert anything about the geometry of the metric, the meaning of the Ricci scalar, or the physical interpretation of the coordinates. It only provides the exponential identity that the larger proof needs.

What the declaration does not claim: it does not say that the two Ricci formulas are equal, nor that the curvature has any particular value. Those claims belong to the surrounding theorems ricci_scalar_equiv and its supporting lemmas, which carry additional conditions (q ≠ 0 and a nonzero denominator). The lemma also does not claim that the exponential function is injective or that q is real versus complex; it is stated only for real numbers. In short, exp_three_mul is a workhorse, not a headline.

private theorem exp_three_mul (q : ℝ) :
    Real.exp (3 * q) = (Real.exp q) ^ 3 := by
  rw [show (3 : ℝ) * q = q + (q + q) from by ring, Real.exp_add, Real.exp_add]; ring

What this page does not claim

The lemma does not prove equality of the two Ricci scalar formulas. It does not assign any numerical value to the curvature. It is stated only for real numbers, not complex or other fields.

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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