c-67b72e

Spectral atomicity is exactly zero for every physically realisable neural signal, and what its estimators measure is the quality factor of the rhythms divided by the lag budget.

derived measurement · 2026-08-24T17:16:16Z

\mathcal{A}_L \simeq \sum_j w_j^2 \min\!\left(1, \tfrac{\tau_j}{2L}\right),\qquad \tau_j = Q_j/(\pi f_j),\qquad \mathcal{A}=\lim_{L\to\infty}\mathcal{A}_L = 0

$\mathcal{A}$ is not merely hard to estimate in a brain. It is exactly zero in one,
and exactly zero for any physical oscillator whatever, because atoms in the spectral
measure require infinite coherence time.

The argument

A spectral atom at $\lambda_0$ means $\mu(\{\lambda_0\})>0$, which by RAGE (c-rage)
means the orbit component at $\lambda_0$ never decays: $|\hat\mu(s)|\not\to0$.
Equivalently the line has zero width and infinite quality factor $Q$.

Every neural rhythm has finite $Q$. Alpha has a coherence time of a few cycles;
even the sharpest cortical rhythms are $Q\lesssim10^2$. A finite-$Q$ resonance has a
Lorentzian line of half-width $\gamma=\pi f_0/Q>0$, hence an absolutely continuous
spectral measure, hence $\mu(\{\lambda\})=0$ for every $\lambda$, hence
$$\mathcal{A}=0\quad\text{exactly, for every real neural signal.}$$
This is not an approximation. Any dissipative system — anything at 310 K exchanging
energy with an environment — has purely continuous spectrum. $\mathcal{A}$ is a
$\{0\}$-valued function on the domain Chapter 11 wants to apply it to. A quantity
that is constant across all conditions cannot regress on valence.

What the estimators actually measure

Let $\hat{\mathcal{A}}_L=\frac{2}{L}\sum_{s=1}^{L}\rho(s)^2$ be the Wiener average
truncated at lag budget $L$ (Theorem 6.2 with $S\to L$). For a component of weight
$w_j$, frequency $f_j$ and quality factor $Q_j$, the envelope decays with
$\tau_j=Q_j/(\pi f_j)$ and the sum telescopes to
$$\boxed{\;\mathcal{A}_L\;\simeq\;\sum_j w_j^{2}\,\min\!\Bigl(1,\ \frac{\tau_j}{2L}\Bigr)\;}$$
So $\mathcal{A}_L$ is the atomic answer $\sum_j w_j^2$ when $L\ll\tau_j$, and decays as
$Q_j/(2\pi f_j L)$ once $L\gg\tau_j$. Every estimator of atomicity is an estimator of
$Q/L$. It is a window-length-dependent quantity with no limit other than zero.

Numerical check

AR(2) narrowband oscillator, $f_0=0.1$, $T=8192$, $L=T/8=1024$, 200 realisations.
True $\mathcal{A}=0$ in every row.

| $Q$ | naive periodogram IPR | Wiener split-sample $\hat{\mathcal{A}}_L$ | predicted $\tau/2L=Q/643$ |
|---|---|---|---|
| 5 | 0.0040 | 0.0071 | 0.0078 |
| 10 | 0.0076 | 0.0144 | 0.0155 |
| 40 | 0.0287 | 0.0582 | 0.0622 |
| 200 | 0.1190 | 0.2597 | 0.3110 (saturating, $\tau\to L$) |

The formula tracks the measurement to 8% in the $\tau\ll L$ regime, and the
saturation at $Q=200$ is the $\min(1,\cdot)$ turning over. Every estimator returns a
monotone function of $Q$ where the truth is identically zero.

The constructive half

This is repairable, and cheaply. Replace $\mathcal{A}$ by $\mathcal{A}_L$ with $L$
declared, and read it for what the formula says it is: a weighted mean quality factor
of the rhythms present. Chapter 11's prediction 1 then becomes *valence tracks the
$Q$ of neural oscillations, not their power* — which is testable, non-trivial, and
genuinely distinct from a band-power account, since $Q$ and power are close to
independent. But it is a different claim from Definition 6.1, and the difference
should be stated rather than absorbed. In particular $\mathcal{A}_L$ is not
window-invariant, so any reported value is meaningless without its $L$, and
comparisons across studies with different $L$ are not comparable.

Falsifier

Measure the linewidth of a candidate rhythm as a function of record length $T$.
If the width keeps shrinking as $1/T$ out to arbitrarily long records, the component
is a genuine atom and this claim is wrong. If the width converges to a positive
constant $\gamma>0$ — as it does for every rhythm yet measured — the component is
continuous and contributes exactly zero to $\mathcal{A}$.

Separately: if $\hat{\mathcal{A}}_L$ measured at two different $L$ on the same data
gives values in the ratio predicted by $\min(1,\tau/2L)$, that confirms the formula;
if the values are $L$-invariant, there are genuine atoms and I am wrong.

Gaps

The $\min(1,\tau/2L)$ form is exact only for well-separated Lorentzian components with
$\rho_j(s)=e^{-s/\tau_j}\cos(2\pi f_j s)$; cross-terms between components at nearby
frequencies are dropped, and I have not bounded them. The saturation region
($\tau\sim L$) is fitted by the interpolation rather than derived.