Chronotopic Metric Theory (CTMT) is developed as a non-ontological, information-geometric framework for assessing when kernel-generated forward maps admit stable prediction, parameter transport, and structural identification across operating regimes. No spacetime, background metric, gauge group, or dynamical ontology is postulated. The primitive object is an admissible transport kernel whose induced forward map yields observable data together with a local Fisher geometry.
The mature CTMT viewpoint is narrower and sharper than earlier foundational formulations. Fisher geometry remains the unique local monotone metric structure on the observable layer, but it is not by itself sufficient to determine transport-class identity. Observable similarity, Fisher coherence, and bounded conditioning can survive under adversarial perturbations while structural equivalence fails. The decisive objects are therefore transport-structural invariants: null-sector stability, quotient-class persistence, winding consistency, and self-reference contraction.
CTMT is thus not presented as a metric-first theory of reality. It is a tested-class framework for when local information can be extended into a coherent structural geometry under admissible transport. All quantities are computable from Jacobians, covariances, and observable comparisons, and all claims are falsifiable through rank loss, instability of the Fisher spectrum, loss of differentiability, or structural transport failure. The practical consequence is a plug-and-play criterion for deciding when coarse grained kernel models support stable calibration, single-tuning transport, and nontrivial gauge-like identification across regimes.
\end{abstract}
CTMT began from a simple operational problem rather than from a physical ontology: under repeated interaction, the same measurable content can often be extracted from different physical carriers, yet the timing, uncertainty, and structural identity of that content are not transparently attributable to any one carrier alone. This suggests that data should not be treated as a property of the observer, the instrument, or the source in isolation. Instead, the correct primitive is the transport-consistent overlap through which observables become jointly reconstructible.
The purpose of mature CTMT is therefore not to postulate a new substance, dimension, or metaphysical layer. Its purpose is more restricted and more operational:
Given a kernel-generated forward map, when does local observable information extend to a stable, composable, and transport-identifiable structure across regimes?
The early CTMT program established a Fisher-admissibility viewpoint: kernel-generated observables induce Jacobians and hence Fisher information geometry; rank stability, bounded conditioning, coherence density, and coherence proper time become operational diagnostics of stability. The present mature formulation keeps that foundation but adds a crucial refinement. Local Fisher structure is necessary, but it does not by itself determine whether two regimes belong to the same transport class. Observable similarity and Fisher coherence may survive while transport identity fails.
Accordingly, CTMT now distinguishes two layers:
This distinction is the central mature result. Information is local, but geometry is not merely local information. Geometry appears only when local information closes consistently under transport.
CTMT is intentionally non-ontological. It does not assume:
The framework is also intentionally conservative. It does not claim that every coherence quantity survives every perturbation, nor that synthetic batteries establish universal laws of nature. What it claims is narrower and testable: on a given tested class, one can determine which local and structural invariants remain transport-admissible and which do not.
The strongest tested-class claim supported by mature CTMT is this:
Local observable similarity and Fisher coherence are necessary but not sufficient for transport-class identification. Structural transport invariants are required to decide whether two regimes belong to the same admissible class.
This matters because it moves CTMT beyond a merely metric reading. The Fisher matrix remains a uniquely admissible local information metric, but the identity of a regime is decided only after transport structure is checked. In this sense, CTMT is best read as a coherence-constrained structural geometry rather than as a Fisher-only metric theory.
To keep the paper disciplined, we separate four levels of claim:
Imported mathematical background
Classical Fisher information geometry, rank/null structure, monotonicity under coarse-graining, and related invariance ideas are used as imported mathematical tools.
Operational diagnostics
Jacobians, Fisher spectra, coherence density, admissibility indicators, structural transport signature, and scenario-level acceptance or refusal are computed directly from forward maps and observable comparisons.
Tested-class results
All theorem-level conclusions in this note are restricted to the explicit kernel families and attacked suites studied here.
Explicit non-claims.
No universal ontology, no laboratory closure claim, and no assertion that local metric similarity alone determines physical or transport identity in general.
Why this mature reformulation was necessary
Earlier CTMT drafts were useful as foundational program statements, but they still allowed an overly clean reading in which Fisher rigidity and coherence could appear to decide too much. The mature work shows that this is not the strongest defensible formulation. If observable similarity and Fisher admissibility survive under compensated attack while structural identity fails, then the theory must say so plainly.
That refinement does not weaken CTMT. It strengthens it. A serious framework must survive adversarial perturbation without collapsing into either trivial acceptance or universal rejection. Mature CTMT passes that test by revealing a nontrivial split between the informational layer and the structural transport layer.
The present note should be read as a tested-class refinement of the foundational CTMT preprint on coherence geometry and Fisher-rigid kernel transport.
The foundational formulation introduced the primitive transport kernel, the induced Jacobian and Fisher geometry, coherence density, and coherence proper time. It established the central admissibility idea: a forward map is acceptable only when differentiability is preserved, the Fisher rank remains stable, and coherence does not diverge under extension.
The mature formulation preserves those primitives but clarifies their scope. Fisher geometry remains the unique local monotone metric on the observable layer, and informational coherence remains a necessary admissibility condition. What is new is that tested adversarial suites show this layer to be incomplete for structural identification. Transport-class identity requires additional invariant structure.
This does not contradict the foundational formulation. Rather, it resolves an ambiguity left open there. The original preprint asked when a coarse-grained kernel admits stable prediction and parameter transport. The mature theory answers more sharply:
Stable Fisher geometry is necessary for admissibility, but transport equivalence is decided only after structural invariants are.
Thus the role of the present paper is not to replace the foundational CTMT program, but to stress-test and sharpen it.
CTMT is intended to be operational from the beginning. The framework is therefore best introduced not only through definitions, but also through compact plug-and-play demonstrations that show how the admissibility logic appears in practice.
Demonstration A: Oscillatory transport kernel
The minimal entry point is an oscillatory transport kernel
\begin{equation}
T(t;\theta)
=
\int_{t_0}^{t}
\Xi(t')\,
\exp\!\left(
\frac{i}{\mathcal S_\ast}\,\Phi(t,t')
-
\epsilon(t')
\right)\,dt',
\qquad
\theta \in \mathbb R^d,
\end{equation}
with observable vector
\begin{equation}
Y(\theta)=\mathcal F_\theta[T].
\end{equation}
From the forward map one computes the Jacobian
\begin{equation}
J(\theta)=\frac{\partial Y}{\partial \theta},
\end{equation}
and, with observational covariance \(C\), the local Fisher object
\begin{equation}
F(\theta)=J(\theta)^\top C^{-1}J(\theta).
\end{equation}
This first demonstration is intentionally minimal. It exhibits:
Demonstration B: Fixed-point calibration and single-tuning transport
A second operational entry point is the fixed-point calibration logic. A parameter \(\theta^\ast\) is accepted only if it lies in a stable coherence class:
\begin{equation}
\theta^\ast
=
\arg\min_{\theta}
\Bigl[
\kappa\!\bigl(F(\theta)\bigr)
+
\Gamma(\theta)
\Bigr]
\quad
\text{subject to}
\quad
\operatorname{rank}F(\theta)=\text{const},
\end{equation}
where \(\kappa(F)\) penalizes ill-conditioning and \(\Gamma(\theta)\)
is a hazard or decoherence surrogate.
When this fixed-point holds, a single calibration constant can often be transported meaningfully across the class. Loss of transportability is not diagnosed first by empirical drift but by structural degradation of the Fisher layer: rank collapse, instability of conditioning, or the appearance of unresolved null structure.
This plug-and-play demonstration is particularly useful for real instrumentation, because it shows that CTMT calibration is not a collection of ad hoc heuristics. It is a geometric admissibility test.
Demonstration C: Partial or malformed data
CTMT is naturally compatible with masked and relative observables. Suppose some channels are missing, malformed, or uncertain. Then a mask operator \(M\) and relative normalization can be introduced at the observable level without changing the admissibility logic:
\begin{equation}
Y_{\mathrm{rel}}(\theta)
=
\frac{M\,Y(\theta)}{\mathcal N(M\,Y(\theta))}.
\end{equation}
In this setting, redundant directions are not removed by hand. They are pushed into near-null curvature of the Fisher spectrum. This is one of the most practical features of the framework: missing or uncertain channels do not force a metaphysical argument about what "really" exists; they simply alter the observable layer and hence the induced geometry.
Demonstration D: Transport-identifiability chaos battery
The most mature entry point is the transport-identifiability battery. Here a single synthetic platform is attacked by perturbations that can preserve coarse observable appearance while altering resolved structure. This is the decisive test because it distinguishes:
This battery shows that the mature CTMT core is not the claim that "every invariant survives chaos." The genuine result is sharper: coarse observable similarity and local Fisher coherence may survive, but gauge-like transport identity is decided only after null-sector, quotient-class, winding, and self-reference structure are checked.
Demonstration E: Coherence extraction on scalable magnetostatic pipelines
A final operational demonstration treats magnetostatic or coil-like pipelines with explicit sensor arrays, finite-difference Jacobians, windowing, coarse-graining, and structural diagnostics. In this form, CTMT becomes directly deployable:
define the forward map and observable layer,
This is the intended use of CTMT in practice: not as a metaphysical replacement for physics, but as a strict criterion for when observables and instruments jointly support a stable transport geometry.
The remainder of the paper develops this program in increasing order of commitment.
A reader interested only in deployment may treat the demonstrations above as a practical checklist: define observables, compute the Jacobian, inspect the Fisher layer, and refuse any class that fails transport-structural persistence. A reader interested in foundations may instead read the paper as a minimal route from local information to transport-constrained geometry.
Publication Date: 2026-06-18