Orbitwise Uniqueness of the Path-Integral Measure from Observer Equivariance

This paper isolates a symmetry-first structural point about path-integral quantization within the Observer-Equivariance (OE) programme. The central question is not how to construct a new regularization, but what symmetry alone can and cannot fix about a quantization scheme once one works with equivariant half-densities on finite-stage cylindrical approximations of path space. The main result is an orbitwise uniqueness statement: for any two strictly equivariant schemes, their ratio is constant along each symmetry orbit wherever that ratio is defined. This separates three issues that are often run together in informal discussions of path integrals: (i) orbitwise rigidity forced by symmetry, (ii) possible projective/anomalous mismatch encoded by character data, and (iii) additional physical input needed to determine orbit-space or sector weights. The paper is deliberately modest in its claims. The finite-stage statement is the robust core result. The projective-limit extension is treated as conditional on the analytic compatibility built into a chosen scheme, and determinant-line data are used only as a geometric bookkeeping device for global phase coherence within an appropriate regularized Fredholm package. The note also discusses how zeros of the relevant half-densities affect the domain on which ratio statements are defined. Examples include the free particle, the harmonic oscillator, and 2D Yang–Mills theory. The first two illustrate how symmetry constrains variation along orbits while leaving orbit-space dependence open; the Yang–Mills example shows the same structural separation in an infinite-dimensional gauge-theoretic analogue, where symmetry fixes the within-sector form but not the sector weights themselves. The broader OE motivation is methodological: changes of observer are treated through symmetry actions, and objectivity is tied to what survives those changes. In that setting, the present paper extracts one narrow but useful lesson. Symmetry can enforce strong orbitwise rigidity in the measure-theoretic core of quantization, but it does not by itself select a unique global weighting across orbit space. Any such further selection must come from additional principles beyond bare equivariance.