Chronotopic Metric Theory (CTMT)

CTMT is a conservative geometric framework for local observation systems. It does not begin with spacetime, particles, fields, ontology, or a predefined physical metric. Instead, it begins with observations, observable responses, and covariance. The central question is:

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

Given an observation space O, an observable-response Jacobian

J = DθF,

and a covariance structure Σ, CTMT constructs the resolution geometry

𝒢 = (O ; R , N , Σ),

where

R = Im(J),
N = Ker(Jᵀ).

The resolved sector R contains directions visible to admissible perturbations. The null sector N contains directions that are observationally present but invisible at first order.

This object is the primitive geometric structure of modern CTMT.

Core Claim

Two regimes belong to the same local CTMT class if and only if they preserve the same resolution geometry.

φ ∈ Mor_CTMT ⇔ φ*𝒢 = 𝒢.

Equivalently, admissible morphisms preserve:

• resolved structure R,
• null structure N,
• observational covariance Σ.

When only relative covariance scale is observable, CTMT admits the conformal variant

(O ; R , N , [Σ]),

where covariance is preserved up to one global positive scale.

Relation to Fisher Geometry

Fisher geometry remains an important local diagnostic. Given observational covariance Σ,

G_F = Jᵀ Σ⁻¹ J.

However, Fisher geometry is not the primitive invariant. It is induced by the pair (J, Σ).

Two systems may exhibit similar Fisher geometry while possessing different null structure or different resolved–null covariance coupling. For that reason, Fisher similarity is necessary but not sufficient for transport-class identification.

Covariance Geometry

Relative to the decomposition

O = R ⊕ N,

covariance admits the block structure

Σ = [ ΣR CRN CᵀRN ΣN ].

CTMT distinguishes:

• ΣR — resolved covariance,
• ΣN — null covariance,
• CRN — resolved–null covariance coupling.

The coupling term is structurally important. When CRN vanishes, automorphism groups factor into independent resolved and null components. When CRN is present, the admissible group shrinks to the stabilizer of that coupling.

Similarity

Similarity in CTMT is not defined by observable correlation alone, not by parameter fit alone, and not by Fisher geometry alone.

Similarity is determined by proximity of resolution geometries:

• resolved sector similarity,
• null sector similarity,
• resolved covariance similarity,
• null covariance similarity,
• resolved–null coupling similarity.

This allows CTMT to distinguish:

• global scale changes,
• null-sector modifications,
• resolved-sector modifications,
• coupling distortions,
• fitted impostors,
• genuine physical response changes.

Transport and Holonomy

Local charts may be connected by admissible transports. A transport is admissible when every transition preserves the relevant resolution geometry.

Composed transports define holonomy operators. Current synthetic batteries demonstrate that compatible coil and Navier-like chart families can be assembled into transport loops with numerically stable closure, while injected covariance-coupling defects produce transport obstructions.

This establishes a practical route from local geometry toward global transport geometry, although a general holonomy classification remains an open problem.

What CTMT Does Not Claim

• No new physical ontology.
• No replacement for established statistics.
• No universal physical law.
• No proof that nature must realize CTMT geometry.
• No automatic globalization from local charts.

CTMT proposes a falsifiable geometric structure for local observation systems and studies which transport relations preserve that structure.

Current Status

The modern CTMT program is organized around three connected results:

1. Resolution Geometry: 𝒢 = (O ; R , N , Σ)
2. Covariance Geometry: (ΣR , ΣN , CRN) relative to R ⊕ N
3. Transport Geometry: admissible morphisms, similarity classes, and holonomy tests

The strongest current conclusion is:

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.