The Fisher metric is unique only on the classical side – a census of information geometry.
The space of probability distributions carries a metric, and so does spacetime (Paper 108). This paper writes two independent reasons why the two are not isomorphic, and then counts how far uniqueness reaches. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Fisher (1925), Rao (1945), Chentsov (1972), Amari (1985), Petz (1996), Stein (1956) and James-Stein (1961) are all standard.
First, the same word metric covers different signatures and different curvatures. Computing the Fisher metric of the Gaussian family gives a diagonal matrix whose entry for the mean is the reciprocal of the variance and whose entry for the standard deviation is twice that, with both eigenvalues positive: the signature is (2,0), positive definite. The Gaussian curvature is computed here to be the constant minus one half, that is constant negative curvature, the same type as the hyperbolic half-plane. The Minkowski metric, by contrast, has signature (1,3), is indefinite, and is flat. There is not one difference but two independent ones, of signature and of curvature, and saying that a statistical manifold carries a metric just as spacetime does overlooks both. And indefiniteness is not decoration: null directions, nonzero vectors of zero length, exist precisely because the metric is indefinite (Paper 111). A positive definite metric has no light cone, which is why the phrase causal structure carries no meaning on the Fisher metric.
Second, the phrase Fisher information names both a number and a tensor. Rescaling the parameter by a constant multiplies the components of the metric by its square, so as a number it moves with the parametrisation. What does not move is the metric as a tensor, and geometric quantities such as curvature and geodesics do not depend on the parametrisation. The first carries a convention and the second does not (Paper 95). Until it is said which is meant, no comparison has begun.
Third, there is a uniqueness theorem, and it carries conditions. Chentsov’s theorem (1972) states that on statistical manifolds over a finite sample space, the Riemannian metric invariant under maps induced by sufficient statistics is the Fisher metric and no other, up to a constant multiple. Information geometry is often introduced by saying that the space of distributions carries a natural metric, but natural is an evaluation and not a theorem. What the theorem says is that exactly one exists. The core of the subject is uniqueness, not existence (Papers 126, 127). And counting the fence gives three conditions: a finite sample space, invariance under Markov morphisms, and being a Riemannian metric. Drop one and uniqueness is gone.
Fourth, and this is the core. That uniqueness lives only inside a fence. On the space of quantum states the metrics monotone under completely positive maps are not unique but form a family parametrised by operator monotone functions (Petz 1996). For a single qubit this paper computes the SLD information and the Kubo-Mori information. On a rotating, non-commuting family they come out as 0.360000 against 0.415888, disagreeing; on a commuting family along the z axis they come out as 0.167002 against 0.167002, exactly equal. The SLD value for the rotating family equals the square of the Bloch radius exactly, which confirms the computation. So on a commuting family every quantum monotone metric collapses onto the classical Fisher metric, and the family opens out only where things fail to commute. The separator is commutativity. This has the shape of Paper 127: uniqueness lives in the middle row and vanishes outside the fence. There the fence was the Weyl relation and the number of degrees of freedom; here it is commutativity.
Fifth, the Cramer-Rao fence is unbiasedness. What the inequality says is that the variance of an unbiased estimator is at least the reciprocal of the Fisher information, and unbiased is a hypothesis rather than an ornament. For a d-dimensional normal mean this paper compares the mean squared errors of the maximum-likelihood and James-Stein estimators over two hundred thousand draws. At dimension two both give 2.005; at dimension three the figures are 2.993 against 1.597, at five 4.996 against 1.399, at ten 9.985 against 1.250, and at thirty 29.968 against 1.137. So from dimension three the James-Stein estimator beats maximum likelihood. This is not a contradiction: the James-Stein estimator is biased and therefore lies outside the range in which the Cramer-Rao statement applies. The theorem is not violated, and the counterexample stands where the theorem does not reach, the same shape as Galapon’s counterexample in Paper 127. And the condition of dimension three or more cannot be relaxed, since at dimensions one and two the maximum-likelihood estimator is admissible. This is a case of a dimension genuinely selecting (Paper 114), of a different standing from the corpus’s n equals d minus one, which does not select.
IMPORTANT (scope): a census of established standard material (Layer 2). Every work cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that information geometry and spacetime geometry are isomorphic (they are not; the signature and the curvature both differ) / that the Cramer-Rao inequality is wrong (it is not; the counterexample stands outside its fence). What is added: making the double difference of signature and curvature explicit, counting the three conditions in the fence around uniqueness, exhibiting numerically where that fence breaks on the quantum side, and giving an instance that stands outside the Cramer-Rao fence. What remains: one word, metric, is being applied to two objects of different signature; and a strong claim of uniqueness always arrives with a fence. Do not accept a uniqueness without counting its fence.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Fisher Metric Is Unique Only on the Classical Side [Paper 133]. Zenodo.
DOI: 10.5281/zenodo.22080550
ORCID: 0009-0006-8587-3450