The number of local equality restrictions left after every admitted physical and calibration direction is included.
Methods-and-theory paper
When does a new field description predict something observably new?
Published title Observational Distinguishability of Reparameterized Field Theories: No-Go, Codimension, and Local-Power Results with a Compact-Phase Case Study
This paper reduces that question to one sequence of calculations: test regular field equivalence, remove unrestricted functional fitting, count the restrictions of a finite model, compare its tangent image with one named alternative, and determine whether the exposed directions are measurable at the available precision.
- Publication
- 2026-09-07 · version 1.0
- Length
- 35 pages
- Relation
- Derived from CHC Framework Series v2.0
CHC distinguishability theorem
Falsifiability, comparative distinction, and measurability are different questions.
Its contribution is the combined action-to-observation decision criterion. The differential-topology, scalar-field, magnetic-graph, and minimum-distance ingredients are established results; the paper joins them into one test of whether a proposed framework excludes observations allowed by a named competitor after nuisance fitting.
The number of directions allowed by a named comparator that leave the CHC tangent image.
Covariance and perturbation units convert geometric separation into noncentrality, power, and sample size.
Positive codimension does not by itself make a restriction CHC-specific. A comparator can lie entirely inside the CHC tangent image. Rank establishes first-order separation; singular values establish sensitivity only after the observation covariance and perturbation metric are fixed.
The theorem returns a definite result in six cases.
Each result narrows what can be claimed before a numerical or observational comparison begins.
Regular-branch equivalence
A regular CHC scalar branch and the matched canonical scalar have the same local observation map when matter, probes, initial data, and boundary data are transformed together.
Functional saturation
If an admitted constitutive function spans the finite observation space, the model has no nonconstant local equality prediction in that window.
Finite closure
A constant-rank finite model has m minus rank(J_C) independent local restrictions.
Named-comparator exposure
The rank increase produced by adjoining J_S counts the comparator directions exposed by the model restrictions.
Noise-limited power
Covariance whitening and a declared perturbation metric convert exposed directions into noncentrality, power, and a sample-size requirement.
Global distinction
Winding or cycle flux can escape a local equivalence only when it enters the observation map and the comparator lacks the same global sector.
The comparison is made at theorem level.
The paper identifies what comes from established theory and what the combined decision criterion adds.
| Theory or result | Established content | Role in this paper |
|---|---|---|
| Scalar–tensor and canonical-scalar theory | Regular field redefinitions preserve the local observable content of the matched branch. | The regular-branch no-go is the first gate in the combined decision theorem. |
| Compact scalar fields | Local charts can agree with a noncompact scalar while winding sectors differ globally. | The global clause states when that topological difference becomes observable and when it remains non-distinctive. |
| Magnetic graph theory | Tree phases are removable; independent cycle fluxes remain gauge invariant. | The symmetric trimer is identified exactly with the standard magnetic graph, then used as a finite rejection test rather than a theory-identity claim. |
| Model-manifold goodness of fit | Smooth model manifolds admit central and noncentral chi-square residual limits under regular Gaussian conditions. | The normal projection of a named comparator fixes the noncentrality and the attainable directional power after nuisance profiling. |
One prospective test
A fixed spectrum–current test can reject the finite closure.
At twelve fixed nondegenerate phases, independently measured spectra and generalized currents are combined into one covariance-propagated minimum-distance statistic. The threshold is fixed at p < 0.0027. A valid rejection rejects the symmetric one-holonomy trimer closure; non-rejection does not distinguish CHC from standard symmetric magnetic-graph theory.
Twelve half-step phases avoid the exact degeneracies at zero and pi.
Each current must be measured independently of the numerical derivative of the fitted spectrum.
The full propagated covariance is retained. Failed validity gates make the result inconclusive.
Detection rate. Generic unequal-site and unequal-hopping perturbations enter the residual
quadratically at the symmetric point. Their smallest detectable amplitude therefore scales as
, not
.
The test has a clear failure meaning and a narrow success meaning.
The paper reports no new experimental or observational data.
A valid rejection establishes
- At least one defining assumption of the symmetric one-holonomy trimer closure fails on the measured window.
- The declared spectrum–current restriction is not compatible with the complete retained dataset and covariance model.
Non-rejection does not establish
- That CHC is preferred to standard symmetric magnetic-graph theory; both make the same finite prediction.
- That the laboratory control phase is cosmological, gravitational, or evidence for a compact cosmic field.
- That an invalid or incomplete acquisition counts as agreement.
Read, reproduce, and cite the fixed record.
The Zenodo deposit contains the paper and the complete source-and-validation package.