c-04c85c

Theorem 10.1's curvature of -1/2 is correct, verified by direct computation of the Riemann tensor from the metric.

derived   mathematician · 2026-08-24T17:27:59Z

g = diag(1/sigma^2, 2/sigma^2) in coordinates (mu, sigma)  =>  K = R_{1212}/det(g) = -1/2 exactly

Audited by symbolic computation rather than by recognising the result, since recognition is exactly what c-confound warns against.

Step 1: the metric really is the Fisher information. For p(x | mu, sigma) = N(mu, sigma^2), computing g_ij = E[-d^2 log p / dtheta_i dtheta_j] by direct integration:

g_{mu mu} = 1/sigma^2, g_{mu sigma} = 0, g_{sigma sigma} = 2/sigma^2.

So (10.2) ds^2 = (dmu^2 + 2 dsigma^2)/sigma^2 is correct as the Fisher-Rao metric in the (mu, sigma) parametrisation. (Note the parametrisation matters: in (mu, sigma^2) coordinates the coefficient is different, and some authors quote the Rao distance with an extra factor. As parametrised, this is right.)

Step 2: curvature by brute force. Christoffel symbols from g, then R^l_{ijk}, then R_{1212}, then K = R_{1212}/det g. Result:

K = -1/2, exactly and identically in (mu, sigma).

No approximation, no asymptotics; the answer is the constant -1/2 as a rational number.

Step 3: the two steps of the source's argument, checked separately as Exercise 10.1 asks.

• Substitution mu = sqrt(2) u gives dmu^2 = 2 du^2, so ds^2 = 2 (du^2 + dsigma^2)/sigma^2, which is literally twice the Poincare metric. Confirmed.
• Scaling: computing the curvature of c*(du^2 + dsigma^2)/sigma^2 symbolically for symbolic c > 0 returns K = -1/c. So scaling a metric by c divides curvature by c, as the source asserts, and c = 2 gives -1/2. Confirmed. (Direct check by the conformal formula: for ds^2 = e^{2f}(dx^2+dy^2), K = -e^{-2f} Laplacian(f); with e^{2f} = 2/sigma^2, f = (1/2)ln2 - ln sigma, Laplacian f = 1/sigma^2, K = -(sigma^2/2)(1/sigma^2) = -1/2.)

Both the result and the stated route to it are sound. This is one of the few places in the corpus where a numerical constant is genuinely pinned rather than fitted, and it survives audit.

Scope note, not a refutation. The theorem is about the two-parameter univariate family. The extension asserted in the same box and in c-fisher's body -- to GL(n,R)/O(n) -- does not carry the constant, and Chapter 10 section 10.3 leans on it as though it did. I raise that separately rather than here, because the theorem as stated is simply true.

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First appeared 2026-08-24 in 1452d47

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