Anomalous dimensions

Why $\gamma_p$ measures channel sensitivity and selects active channels.

Plain

The idea

An anomalous dimension is not a hidden spatial dimension. It is a sensitivity exponent: it says how fast a channel reacts when depth changes.

If that sensitivity stays above the $1/2$ threshold, the channel can carry an active direction. If it falls below the threshold, the channel becomes an echo.

γ_p > 1/2

Standard

Standard reading

PT reads prime channels through a function $\gamma_p$. The primes $3,5,7$ remain active; from $11$ onward, the contribution falls to the inactive side.

This gives a compact mathematical explanation for why some structures close at three active directions rather than infinitely many.

Takeaways

An anomalous dimension is a sensitivity exponent.
$1/2$ is the active-persistence threshold.
The $3,5,7$ then $11$ cutoff becomes readable.

Technical

Technical formulation

In the monograph, the threshold $\gamma_p=1/2$ is tied to the per-channel GFT condition and to the fundamental symmetry $s=1/2$.

The precise status depends on the level: the definition is mathematical, while the physical identification of active channels belongs to the associated bridges and derivations.

Monograph: ch06_holonomy, ch08_fixed_point, ch23_audit.

Formulas

$\gamma_p>\frac12\Rightarrow\text{active channel}$

$\gamma_p<\frac12\Rightarrow\text{inactive echo}$

public code

Code and scripts

The links below point to public resources or planned GitHub repositories. No local working path is exposed to the reader.

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Threshold gamma_p = 1/2

Illustrates the active/inactive cutoff on 3,5,7 then 11,13,17.

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