The claim
The monograph, Part I, “7. just the math”: blow-up “introduces new observables or latent variables that allow one to disentangle previously indistinguishable states”, and its final theorem lists, among what resolution achieves, “Discriminability is locally restored”. “Definitions and Theorems”, Theorem 3.2: “There exists a blow-up π such that in the new coordinates u, the Fisher information matrix I(u) is non-degenerate.” Chapter 9: “Blow-Up as Restoration of Cramér–Rao”. The LEGO reading, throughout: splitting the “slope” bin into finer bins is the blow-up.
Nothing observed changes. A blow-up is a proper map π from a new parameter space onto the old. The model on the new space is p(x | π(u)). It depends on u only through π(u), so if π(u) = π(u′) the two points give one distribution. The exceptional divisor is the set of points mapped to the singular point, so all of it gives the singular point’s distribution. States that no data could tell apart before are still not told apart, and there are now more of them.
The Fisher information can stay degenerate. Take x ~ N(ab, 1), with the true distribution N(0, 1). The divergence is K(a, b) = (ab)²/2, singular along the axes. In the chart a = u, b = uv, the mean is u²v and K = u⁴v²/2, a normal crossing. The gradient of the mean is (2uv, u²), zero at every point with u = 0. So the Fisher matrix is zero along the exceptional divisor, and Theorem 3.2 fails for this model. It was checked here by finite differences at v = −2, 0.5 and 3. In general, a normal crossing K = Π u_j^{2k_j} has a zero Hessian at the origin whenever any k_j ≥ 2, or whenever two or more coordinates appear.
What resolution does buy. The normal form lets the divergence’s behaviour near the true parameter be read off. It gives the real log canonical threshold. On THEORY-087’s reading of Watanabe, held as cited and not checked here, that threshold then sets the Bayesian complexity penalty λ log n in place of (d/2) log n. That is a statement about how hard learning is near the singularity. It is not one about separating the states there.
What splitting a LEGO bin is. Replacing a sorting rule f by a finer f′ (f a function of f′) is refining the observation. It is a more informative experiment, and it can separate brick types that f put together. That is the comparison of experiments the record holds in its Blackwell claims (for example CLAIM-126), not resolution of singularities. The monograph’s analogy joins the two because both are described as “separating what was collapsed”. MOM-CLM-099 records the same word used for three different operations. This claim adds that none of them is a way of gaining observations.
Why it is Active
The first point is the definition of a pulled-back model. The second is a two-line computation, checked numerically. The third reads off the normal form, and is worked for this model in CASE-152 (Active), where the real log canonical threshold is ½. What that threshold then does to the Bayesian penalty is stated only on THEORY-087’s reading, which is Proposed.
What it does not say
- Not that blow-up is useless for epistemology. The learning coefficient is a real measure of how much the data can say near a singularity.
- Not that the monograph’s identifiability material is wrong. MOM-CLM-135 grades its quotient construction correct. The error is in the step from resolution to discriminability.
Note of 2026-10-27: the penalty stated conditionally
The status review of this date (ADR-057) found this Active claim grounded in THEORY-087, which is Proposed, for one clause: that the real log canonical threshold sets the Bayesian penalty λ log n. That clause is Watanabe’s singular result, which the record holds only as cited. The clause is now conditional on THEORY-087’s reading. The claim is grounded in CASE-152 for the worked threshold, and THEORY-087 is cited and no longer a ground.