Is a number that recurs across unrelated systems a count or a well-measured magnitude?

Independence means independent in method, not in publication. Two Monte Carlo papers are one family; MC, high-temperature expansion, field-theoretic expansion and the conformal bootstrap are four.

The 3D Ising row is the real test. Six determinations across four method families give chi2/dof = 0.62, worst pairwise tension 1.6 sigma. Values scattering at the scale of their stated uncertainties is what a well-measured quantity looks like, not what a count looks like. Eta and omega behave the same (1.52 and 2.29).

The category is mixed, which confirms CASE-19 branch 4. The 2D Ising exponents are exact rationals because Onsager solved the model - nu = 1, eta = 1/4, zero spread - and they sit with the kissing number, not with their own three-dimensional counterparts. So "critical exponent" is not a kind. An exactly solved model's exponents are counts; an unsolved model's are computed magnitudes. The discriminator is whether an exact solution exists, and the signature detects it without being told. The same physical quantity lands on opposite sides of the founding split in two dimensions and in three.

The discrepant case agrees with its own field. 3D XY alpha: -0.01507(21) from improved-clock MC against -0.0127(3) from Lipa's microgravity measurement, flown on Columbia to remove the gravitational rounding that smears the transition on Earth. chi2/dof = 41.9, tension 6.5 sigma, and the condensed-matter literature states the incompatibility directly. The diagnostic agreeing with people who already know is the right outcome.

And an edge case found a limit that was not stated when the tool was proposed. Feigenbaum's delta returns chi2/dof = 0.00 and is not a count - a real number computed to arbitrary precision with no experimental side, so there is nothing to disagree. The test discriminates only when a quantity has both a computed and a measured determination. Fix: report the number of method families and an experimental-row flag alongside chi2/dof. Zero spread with no experimental row is unresolved, not exact.