Physical distinctions between phases must be detectable by measurements on finite-energy physical states

The verdict rests on three observations, re-examined after a change in the standing of one supporting premise.

First, the two camps in the primary literature explicitly divide over this criterion. Hayashi's continuity argument (arxiv.org/abs/2303.02129, published as PRL 132, 221901 (2024)) is built on it: the claim that the smoothly interpolating dressed Aharonov-Bohm phase supports Higgs-confinement continuity has force only if quantities inaccessible to measurements on physical states cannot mark phase distinctions. Cherman, Jacobson, Sen and Yaffe (arxiv.org/abs/2401.17489, JHEP 06 (2024) 200) reject the criterion as a necessary condition by counterexample, proposing the phase of a gauge-invariant vortex-line correlator as a diagnostic of Higgs versus confining dynamics even though it is not the expectation of a local observable in a finite-energy state. Both papers are peer-reviewed, technically careful, and mutually engaged; neither disputes the other's calculations. That is the signature of a live criterion dispute, not an empirical disagreement one side is winning.

Second, the subclaims split by side and none is decisive. Two supporting premises remain settled physics: thermodynamic transitions produce nonanalyticities in measurable bulk observables↗︎, and local gauge-variant order parameters cannot distinguish gauge-theory phases↗︎ (Elitzur's theorem and its consequences). The third, that same-symmetry regimes can always be connected without a transition↗︎, now stands contradicted as a universal principle: topological order (fractional quantum Hall liquids, the toric code versus a trivial paramagnet) shows that matching symmetry-breaking patterns do not guarantee connectability. This weakens the premise as stated but not the use made of it here, which is the Fradkin-Shenker instance: Higgs and confining regimes of a lattice gauge-Higgs system with fundamental matter are in fact connected, so a formally suggestive local quantity misidentified phase structure in exactly the setting at issue. Moreover, the topological counterexamples do not count against this claim itself: topological distinctions are detectable by measurements on finite-energy states (ground-state degeneracy on nontrivial manifolds, anyon braiding statistics), so they are cases where the operational criterion succeeds, not fails. Even with the premises at full strength, the supporting argument established only that measurement-backed nonanalyticity is sufficient for a physical distinction and that unmeasurable local quantities have misled before; the step to necessity is an inductive generalization the opposing camp simply declines. On the other side, the lattice result that the vortex-line correlator phase changes non-analytically between regimes↗︎ is accepted by both camps, but it refutes the criterion only granting that nonanalytic gauge-invariant correlator behavior marks a physical distinction on its own↗︎, which is the contested criterion question restated. Neither argument can compel the other side without begging the question, which is why the claim bottoms out as a criterion choice.

Third, the earlier literature search found no development that resolves the choice: follow-up lattice work on nonlocal order parameters (Shimada and Yamamoto, PTEP 2025, 043B05) develops the against-side toolkit but does not tie the disputed phase to a finite-energy measurement, and no operational-side result rules such a tie out. The present change is internal to the graph and conceptual in character; no new external evidence bears on the verdict, so no fresh search was made.

Credence is omitted: the claim is a methodological criterion, and a single probability that it is "true" would be false precision. What would change the verdict: a derivation connecting the vortex-line correlator phase to a measurable response of finite-energy states (which would tend to vindicate a broadened operational criterion while dissolving this particular dispute), or a community consolidation around one classificatory practice. Confidence 0.85 that contested is the right status: the only rival reading is that this is a definitional stipulation not worth a verdict at all, but the two camps treat it as substantive and argue it, so contested fits.

[Status relabeled from contested to inconclusive by migration 0066 (#524), when the contested status was retired.]