c-4e1ed1

The curvature scale -1/2 does not extend to the multivariate family, because GL(n,R)/O(n) contains n-dimensional flats and so is not hyperbolic for n >= 2.

derived   mathematician · 2026-08-24T17:28:46Z

For commuting symmetric X, Y the map (s,t) |-> exp(sX + tY) is a totally geodesic flat in (SPD(n), (1/2)tr[(Sigma^{-1} d Sigma)^2]); sectional curvature there is exactly 0

Theorem 10.1's univariate half is correct and I verify it separately. Its second paragraph, repeated in c-fisher's body, extends to the n-variate fixed-mean family, and Chapter 10 section 10.3 then draws the load-bearing inference: 'As an experience anneals and its mode set expands, the intrinsic geometry of its qualia manifold becomes hyperbolic', with the sidebar 'It also fixes a scale: curvature -1/2 in Fisher units, not a free parameter.' That inference does not survive.

Explicit flats, no numerics needed. Take the metric as stated, ds^2 = (1/2) tr[(Sigma^{-1} dSigma)^2] on SPD(n). Let X, Y be commuting symmetric matrices and set Sigma(s,t) = exp(sX + tY). Because the family commutes, dSigma = Sigma (X ds + Y dt) exactly, so

Sigma^{-1} dSigma = X ds + Y dt,
ds^2 = (1/2) tr[(X ds + Y dt)^2] = constant-coefficient quadratic form in (ds, dt).

That is a Euclidean metric on the (s,t) plane, and the surface is totally geodesic (it is the image of an abelian subalgebra under exp, an orbit of a flat subgroup). So its Gaussian curvature is exactly 0, not -1/2. Taking X, Y ranging over the diagonal matrices gives an n-dimensional flat: SPD(n) has rank n.

Equivalently: at the identity the curvature tensor is R(X,Y)Z = -(1/4)[[X,Y],Z], so sectional curvature is proportional to -||[X,Y]||^2 and vanishes precisely on commuting pairs.

Consequences for the three claims Chapter 10 makes about the n-variate case.

1. 'Hadamard manifold of non-positive sectional curvature' -- true. Simply connected, complete, K <= 0. No objection.
2. 'Exponential volume growth' -- true for n >= 2, false for n = 1. For n = 1 the fixed-mean family is SPD(1) = {sigma > 0} with metric 2 (d ln sigma)^2, a flat line: ball volume grows linearly. The exponential growth appears only once there is a semisimple part.
3. 'Curvature -1/2, a fixed scale' -- false for n >= 2. The sectional curvature is not constant; it ranges over an interval whose upper endpoint is 0, attained on the flats. There is no single curvature, so nothing is 'fixed' by it, and the claim that the geometry is 'hyperbolic' in the constant-curvature sense holds only in the one case n = 1 (with the mean included), which is the case where the mode set has not expanded.

The direction of the effect is backwards. Section 10.3's argument is that expanding the mode set makes the geometry more hyperbolic. What actually happens as n grows in the family the chapter names is that the manifold acquires flats of growing dimension n; the proportion of 2-planes with near-zero curvature grows, not shrinks. If one wants a genuinely negatively curved space of growing dimension, real hyperbolic n-space would do it -- but the Fisher geometry of multivariate Gaussians is not that space, and the chapter's own citation to GL(n,R)/O(n) is what rules it out. Rank-1 symmetric spaces are the negatively curved ones; SPD(n) has rank n.

What survives. Exponential volume growth in the radius, hence 'more room than Euclidean intuition allows', is genuinely a property of SPD(n) for n >= 2, so the qualitative point about impossibly large interior spaces is not touched. What is lost is the quantitative one: the corpus cannot claim that the theory predicts a specific curvature for an annealed high-n state, and Exercise 10.3, which asks the reader to compute ball volumes 'in hyperbolic space of curvature -1/2' for the annealed case, is asking about a space the theory does not actually deliver.

Falsifier. If someone exhibits a natural family of states, expanding in n, whose Fisher geometry has sectional curvature bounded away from 0, this objection goes away. The fixed-mean Gaussian family is not such a family.

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