Chronotopic Metric Theory (CTMT)

The central question is:

When do two observation regimes represent the same transport-identifiable structure?

Observation Structure

Let

\[
O
\]

denote an observation space equipped with an observation pairing (or equivalently a precision form) allowing observable responses to be compared.

Given

\[
J=D_\theta F,
\]

the observable-response Jacobian, together with observational covariance (or equivalently precision), CTMT defines the local resolution geometry

\[
\boxed{
\mathcal G
=
(O,g,R,N,\Sigma),
}
\]

where

\[
R=\operatorname{Im}(J),
\]

and

\[
N=\ker(J^\ast),
\]

with \(J^\ast\) denoting the adjoint induced by the observation pairing \(g\).

The resolved sector \(R\) contains perturbation directions observable to first order.

The null sector \(N\) contains directions that remain locally invisible to first-order observations while remaining part of the observation structure.

This local resolution geometry is the primitive object of CTMT.

Admissible Morphisms

CTMT studies transformations that preserve observable structure.

A local morphism is admissible whenever it preserves the underlying resolution geometry,

\[
\boxed{
\phi\in\operatorname{Mor}_{\mathrm{CTMT}}
\iff
\phi^\ast\mathcal G
\cong
\mathcal G.
}
\]

Depending on the observational protocol, covariance may be preserved

• exactly,
• or only up to one global positive scale.

The latter yields the conformal resolution geometry

\[
(O,g,R,N,[\Sigma]).
\]

Derived Local Geometry

The primitive object is the observable resolution geometry.

Several familiar geometric objects appear as derived constructions.

Most notably, Fisher geometry is recovered locally through

\[
G_F
=
J^\ast
\Sigma^{-1}
J.
\]

Within CTMT, Fisher geometry is therefore a derived local metric induced by the observable response operator and observational uncertainty rather than the primitive invariant itself.

Covariance Geometry

Relative to the decomposition

\[
O=R\oplus N,
\]

the covariance tensor decomposes into

\[
\Sigma=
\begin{pmatrix}
\Sigma_R & C_{RN}\\
C_{RN}^{\!\top} & \Sigma_N
\end{pmatrix}.
\]

CTMT distinguishes

• resolved covariance \(\Sigma_R\),
• null covariance \(\Sigma_N\),
• resolved--null covariance coupling \(C_{RN}\).

The coupling term governs admissible automorphisms.

When

\[
C_{RN}=0,
\]

resolved and null sectors decouple.

Non-zero coupling constrains admissible transformations to the stabilizer of the coupling structure.

Similarity

Similarity is determined by proximity of observable resolution geometries, rather than by parameter values or Fisher geometry alone.

The comparison involves

• resolved structure,
• null structure,
• resolved covariance,
• null covariance,
• resolved--null covariance coupling.

This distinguishes genuine transport similarity from fitted, correlational, or scale-equivalent models.

Transport Geometry

Compatible local charts define a transport atlas.

Admissible transports preserve local resolution geometry, while composed transports generate holonomy operators.

Holonomy measures the obstruction to assembling locally compatible charts into a globally consistent transport geometry.

Current synthetic benchmark studies demonstrate numerically stable transport closure for compatible chart families and controlled transport obstructions produced by covariance-coupling perturbations.

A complete mathematical classification of transport holonomy remains an open problem.

Relationship to Existing Mathematics

CTMT does not replace differential geometry, information geometry, inverse-problem theory, or statistical estimation.

Instead, it proposes a common observable resolution structure from which these established mathematical tools naturally arise as derived or compatible constructions.

Current Mathematical Status

The current local theory establishes

• local resolution geometry;
• admissible automorphism classification;
• covariance geometry;
• similarity of observable structures;
• local transport geometry;
• computational invariant reconstruction.

Open mathematical problems include

• globalization of compatible local charts;
• classification of transport holonomy;
• physical interpretation of resolved--null covariance coupling;
• robustness under observational uncertainty;
• applications to experimentally observed transport systems.

Scope

CTMT makes the following explicit non-claims.

• It does not propose a new physical ontology.
• It does not replace General Relativity or Quantum Mechanics.
• It does not replace information geometry.
• It does not derive the observation pairing from first principles.
• It does not claim that nature must realize CTMT geometry.
• It does not claim that every local geometry admits globalization.

Instead, CTMT proposes a falsifiable mathematical framework describing observable resolution geometries, admissible transport structures, and their associated invariants.

Local resolution geometry, covariance geometry, similarity classification, and synthetic transport batteries are mathematically computable, experimentally testable, and have survived the current falsification program. The principal remaining open problem is global transport and holonomy structure.