Observational Distinguishability of Reparameterized Field Theories: No-Go, Codimension, and Local-Power Results with a Compact-Phase Case Study

What problem does this paper solve?

New physical frameworks often introduce different fields, coordinates, topologies, or response functions while leaving open whether the resulting observations differ from those of an established theory. This paper gives a rigorous procedure for answering that question. It separates a change of variables from a new prediction, determines how many observable restrictions survive nuisance fitting, identifies which of those restrictions distinguish a named competitor, and converts the surviving directions into asymptotic power and a sample-size requirement.

Main result

For a closed model with observation Jacobian J_C in an m-dimensional observation space, the predictive codimension is m - rank(J_C). Against a comparator with Jacobian J_S, the number of exposed comparator directions is rank([J_C J_S]) - rank(J_C). Covariance whitening and a declared perturbation metric then determine the noncentrality, local power, and required sample size. The theorem therefore distinguishes three questions that are often conflated: whether a model is falsifiable, whether it differs from a specified competitor, and whether the difference is measurable at an attainable precision.

What is calculated?

• A regular CHC scalar branch and its matched canonical-scalar representation have the same local observation map. Greater local precision cannot distinguish them.
• Compact and noncompact scalars are locally indistinguishable on a zero-winding chart, but a measured nonzero winding separates their global image sets.
• A symmetric three-mode compact-holonomy model is exactly the standard symmetric magnetic graph. A generic unequal trimer shares its first-order tangent at the symmetry point, while explicit unequal-site and unequal-hopping perturbations appear at quadratic order. Their detection boundary scales as N^-1/4.

One prospective test

The paper fixes one future-data test for a symmetric three-mode ring: twelve nondegenerate phase settings, independently measured spectra and generalized currents, full covariance propagation, one minimum-distance statistic, explicit validity gates, and one rejection threshold. The unchanged machine-readable specification is included in PROSPECTIVE_PREDICTION.json. The public timestamp counts as a preregistration only if it precedes acquisition of every primary datum.

Scope of the conclusion

This is a methods-and-theory paper with a compact-phase case study. It presents no new experimental or observational data. Agreement with the trimer identity would support the finite symmetric closure at the achieved precision, but it would not distinguish CHC from standard magnetic-graph theory and would not identify a cosmological compact field. A valid rejection would reject the stated finite symmetry assumptions, not establish an alternative cosmology.

The deposit contains the 35-page paper, complete TeX source, deterministic validation program and report, prospective specification, citation metadata, revision map, manifest, and SHA-256 checksums. It is derived from the CHC Framework Series v2.0. The representative paper has DOI 10.5281/zenodo.22636043.