Book 3 · The Mini-Beast · Chapter 18 of 44

Neural · The Manifold and Metric

The neural metric is built from cross-region coherence.

g_neural from coherence

Sigil η Operator F CEFR C1 Week 8
C · CompressionK · ThresholdF · FoldU · UnfoldingG · Generation

OrientationA Metric Built from Coherence

The plasma room takes its metric from an energy functional and the market room from an information functional. The neural room has neither ready to hand, and this is the honest difficulty of the fourth room.

What it has instead is coherence. Cross-region coherence is measured routinely, it is symmetric, and it behaves like a similarity. Turning a similarity into a metric is a standard move and a lossy one, and this chapter is mostly about what gets lost.

Construction

The coherence-induced metric

For regions i and j, let C_ij(f) be the coherence at frequency f. A metric is induced by treating decoherence as distance:

g_neural ∼ d_ij = −log C_ij (band-limited, symmetrised)cognitive distance from cross-region coherence

Two regions that are strongly coherent are close; two that are independent are far. The logarithm is what makes independence infinitely distant rather than merely distant, and it is also what makes the construction fragile at low coherence, where estimator noise dominates.

What Is AssumedThree Places This Can Fail

Coherence is not a distance without work

Raw coherence violates the triangle inequality in general. Any embedding that produces curvature from it has, somewhere, either enforced the inequality or worked in a space where it does not need to hold. That step must be stated, because curvature computed from a non-metric is not curvature.

The band is a modelling choice

C_ij(f) is frequency-resolved, and the induced geometry depends on which band is used. Theta–gamma coupling is the target, so theta and gamma bands are the natural choice — but that choice is made before the data and should be pre-registered, not selected after seeing which band gives the cleanest threshold.

The manifold is inferred, not observed

In the magnetotail, the coordinates are physical quantities measured by instruments. Here the coordinates are the output of an embedding procedure. The risk is circularity: an embedding chosen for smoothness will produce a smooth manifold, and its curvature will report the embedding rather than the brain.

Empirical RouteFrom EEG or fMRI to a Curvature Trajectory

  1. Band-limit and compute pairwise coherence across recording sites, with a stated estimator and a stated window.
  2. Convert to distances, symmetrise, and embed — reporting the embedding distortion, not just the embedding.
  3. Estimate sectional curvature on the embedded manifold, with error bars propagated from the coherence estimator.
  4. Only then compare κ(t) against the 0.25–0.35 band.

Steps two and three are where this becomes real work rather than a restatement. The corpus’s own calibration methodology (WP-31) is written for exactly this situation: a dimensionless constant becomes a function of measured covariates only through a pipeline with a named observable, a fitted parameter vector and an out-of-sample test. Skipping from the constant to a number that sounds physical is the documented failure mode.

BridgesWhere This Connects

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