First Fix. Radial Decomposition of the FTLE Field

2. First Fix: Radial Decomposition of the FTLE Field

The original scalar diagnostic was

G_{λ} = {Var}_{x} [λ_{T} (x)]

This quantity is mathematically well-defined, but conceptually under-specified.

Why?

Because the FTLE field $λ_{T} (x)$ is:

A global variance collapses all of the following distinctions:

Thus, a decrease in $G_{λ}$ is ambiguous. It could mean:

The metric cannot distinguish these cases.

Why radial structure is the minimal correct refinement?

Define radius:

r (x) = ∥ x ∥, r_{*} = median radius .

Let $r_{*}$ be the median radius of the input grid.

The task geometry is radially symmetric:

Then defined (schematically):

λ_{bdry} = E [λ_{T} (x) ∣ r (x) \approx r_{*}]

λ_{center} = E [λ_{T} (x) ∣ r (x) < α r_{*}]

Therefore, the FTLE field naturally decomposes along radial coordinates.

Any diagnostic that ignores $r (x)$ is guaranteed to mix:

Radial decomposition is the weakest possible refinement that:

Importantly, this is not an ad hoc trick — it is imposed by symmetry.

The radial decomposition is introduced to answer a very specific question that $G_{λ}$ cannot:

Is the observed reduction in FTLE variance caused by a collapse of geometric structure, or by spatial homogenization across task-aligned regions?

More precisely:

Only a conditioned statistic can tell the difference.

Radial decomposition addresses where variation lives.
Anisotropy later addresses how stretching is oriented.

This ordering is essential:

Skipping step (1) would make anisotropy uninterpretable.

Pipeline:

Global G_{λ} \to Radial FTLE \to Directional alignment

Verdict:

Radial decomposition is a necessary methodological correction.
No assumptions are violated; no conclusions are prematurely drawn.

Progress:

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