Experiment 3

Characters

The moment

O = (X, μ)

K_ε : X × X × (0, ∞) → ℝ⁺

such that:

(P1) Symmetry: K_ε(x, y) = K_ε(y, x)

(P2) Positivity: ∀ finite {x_i}, {c_i}:
Σᵢⱼ cᵢ cⱼ K_ε(xᵢ, xⱼ) ≥ 0

(P3) Self-consistent scale: ε = ε[K_ε]

where ε[K] is a renormalization-invariant functional of
K's spectral data — specifically, the effective spectral
dimension:

ε[K] := lim sup_{λ → ∞} 2 log N(λ) / log λ

This functional satisfies ε[T] = ε[T²] because the dimension
exponent is invariant under squaring of eigenvalues:
λᵢ ↦ λᵢ² is a reparametrization of the spectrum that
preserves the asymptotic counting exponent.

everything derives and emerges from this.

Intro

**Guest oberserve