Part IV · Valence
# Symmetry as Almost-Periodicity
The Symmetry Theory of Valence says how good an experience feels depends on the symmetry of the object describing it. This chapter turns "symmetry" into a number in [0,1] that can be computed from a spectral measure.
> Assumes
> Chapter 5. Familiarity with the spectral theorem for self-adjoint operators.
> Delivers
> Definition 6.1 (the coherence index $\mathcal{A}$); Theorems 6.2 and 6.3; Proposition 6.4.
6.1 The spectral measure of a state
An experience is symmetric under its own dynamics to the extent that the flow returns it to itself. To make this precise, let $|\Psi\rangle$ be the subject's state on the split factor $\mathcal{N}$, and let $H$ generate the modular flow of Chapter 5. By the spectral theorem $H=\int\lambda\,dP(\lambda)$, and the state defines a probability measure on the spectrum,
$$ \mu_\Psi(d\lambda)=\langle\Psi|\,dP(\lambda)\,|\Psi\rangle, \qquad \int d\mu_\Psi=1 $$
Every measure on $\mathbb{R}$ decomposes uniquely into three pieces — a pure point part carried by countably many atoms, an absolutely continuous part with a density, and a singular continuous part carried by an uncountable set of measure zero. This decomposition is the technical spine of the rest of the book, so it is worth attaching physical meaning to each piece now.
The return amplitude is the Fourier transform of the measure,
$$ A(s)=\langle\Psi|e^{-iHs}|\Psi\rangle=\hat\mu_\Psi(s), \qquad P(s)=|A(s)|^2 $$
and $P(s)$ is the probability that the state, carried along by its own intrinsic time, is found to have come back to where it started.
6.2 Wiener's theorem and the coherence index
> Theorem 6.2 · Wiener
>
> Established
>
> For any finite measure $\mu$ on $\mathbb{R}$,
>
> $$ \lim_{S\to\infty}\frac{1}{S}\int_0^{S}\bigl|\hat\mu(s)\bigr|^2\,ds \;=\; \sum_{\lambda}\mu(\{\lambda\})^2 $$ (6.1)
>
> The long-run average of the squared Fourier transform recovers exactly the squared masses of the atoms, and is blind to the continuous part.
> Definition 6.1 · Coherence index
>
> Posited
>
> The coherence of a state is the atomic mass of its modular spectral measure,
>
> $$ \mathcal{A}[\Psi]\;:=\;\sum_{\lambda}\mu_\Psi(\{\lambda\})^2\;\in\;[0,1] $$
>
> For a purely discrete spectrum with $|\Psi\rangle=\sum_k c_k|k\rangle$ this is $\mathcal{A}=\sum_k|c_k|^4$, the inverse participation ratio — equivalently the purity $\mathrm{Tr}\rho_{\mathrm{diag}}^2$ of the diagonal ensemble.
> Three familiar quantities turn out to be the same object: the long-run return probability, the inverse participation ratio, and the purity of the time-averaged state.
$\mathcal{A}=1$ exactly when $|\Psi\rangle$ is an eigenstate of $H$ — invariant under the flow, hence maximally symmetric. $\mathcal{A}\to 0$ when the measure is continuous, and $\mathcal{A}=1/N$ for a state spread evenly over $N$ atoms. So $\mathcal{A}$ simultaneously measures concentration, recurrence, and invariance, which is a strong hint that it is the right object.
6.3 What the continuous spectrum means
The interpretation is not a matter of taste; there is a theorem.
> Theorem 6.3 · RAGE
>
> Established
>
> (Ruelle; Amrein–Georgescu; Enss.) The Hilbert space splits as $\mathcal{H}=\mathcal{H}_{\mathrm{pp}}\oplus\mathcal{H}_{\mathrm{c}}$, and for any compact operator $C$,
>
> - states in $\mathcal{H}_{\mathrm{c}}$ satisfy $\lim_{S\to\infty}\frac1S\int_0^S\|C e^{-iHs}\Psi\|^2ds=0$ — they escape every compact region and never return;
> - states in $\mathcal{H}_{\mathrm{pp}}$ have almost-periodic, bounded, recurrent orbits.
So the dichotomy is exact. Point spectrum means the orbit is confined and keeps coming back. Continuous spectrum means irreversible dispersal, mixing, no return. And these are the two limiting phenomenological cases we are trying to distinguish.
The symmetry group of an experience is then not a metaphor but a set. Define the $\delta$-almost-periods
$$ G_\Psi^{(\delta)}=\bigl\{\,s\in\mathbb{R}\;:\;\|e^{-iHs}\Psi-\Psi\|\lt\delta\,\bigr\} $$ (6.2)
> Proposition 6.4 · Symmetry is almost-periodicity
>
> Derived
>
> $G_\Psi^{(\delta)}$ is relatively dense in $\mathbb{R}$ for every $\delta\gt 0$ — that is, $s\mapsto e^{-iHs}\Psi$ is a Bohr almost-periodic function — if and only if $\mu_\Psi$ is purely atomic. Symmetry, recurrence and positive coherence are one fact stated three ways.
6.4 What this looks like
> [figure] Three spectral measures, three fates. Each trace is $P(s)=|\langle\Psi|e^{-iHs}|\Psi\rangle|^2$; the dashed line is its running Cesàro mean, converging by Theorem 6.2 to $\mathcal{A}$. The top two traces converge to the same $\mathcal{A}=\sum_k w_k^2=0.222$ yet look nothing alike — which is exactly why Chapter 7 is necessary. The bottom trace disperses and never returns: $\mathcal{A}\to 0$, the limiting case of suffering.
6.5 Why this is the Symmetry Theory of Valence
QRI's claim is that valence depends on the symmetry of the mathematical object describing a moment of consciousness. The object is fixed by Axiom 2.2; the relevant symmetry is invariance under the object's own intrinsic dynamics, which is the only dynamics available without importing external structure; and Proposition 6.4 says that invariance is measured by $\mathcal{A}$.
This is a genuine sharpening rather than a restatement, in three respects. It says which symmetry group is meant — the time-translation group of the modular flow, not an unspecified geometric symmetry. It supplies a scalar in $[0,1]$ instead of a qualitative comparison. And it makes the notion estimable from data: the atomic mass of a power spectrum is something one can compute from a magnetoencephalogram, which is Chapter 11, prediction 1.
It is not yet a theory of valence. Figure 6.1 shows why: two states can agree on $\mathcal{A}$ and differ structurally, and — as Chapter 8 argues at length — a state can be highly recurrent and still be terrible. Two ingredients are missing, and the next two chapters supply them.
> ### What Chapter 6 established
>
> - The modular spectral measure $\mu_\Psi$ decomposes into pure point, absolutely continuous, and singular continuous parts.
> - Theorem 6.2 (Wiener): the long-run mean return probability equals the squared atomic mass, defining the coherence index $\mathcal{A}$.
> - Theorem 6.3 (RAGE): point spectrum ⟺ recurrent bounded orbits; continuous spectrum ⟺ irreversible dispersal.
> - Proposition 6.4: the symmetry group is the set of Bohr almost-periods, relatively dense iff the measure is atomic.
> - $\mathcal{A}$ is the Symmetry Theory of Valence made precise — but it is not yet sufficient.
> ### Exercises
>
> 1. Compute $\mathcal{A}$ for a state spread uniformly over $N$ atoms, and for a state with weights $w_k\propto 2^{-k}$. Which has the larger effective participation number? [1]
> 2. Verify Theorem 6.2 directly for a two-atom measure $\mu=w\delta_{\lambda_1}+(1-w)\delta_{\lambda_2}$ by computing $|\hat\mu(s)|^2$ and averaging. [2]
> 3. Show $\mathcal{A}$ is invariant under $H\mapsto H+c$ and under relabelling of eigenvalues, but not under $H\mapsto\lambda H$ combined with a finite observation window. What does this imply for estimating $\mathcal{A}$ from finite data? [2]
> 4. The singular continuous case was set aside above. Show that such a state has $\mathcal{A}=0$ yet is not mixing in the RAGE sense, and speculate on what phenomenology this third category might correspond to. [2]
> 5. A finite sum of 400 closely spaced atoms mimics a continuous measure over short times. Estimate the timescale at which its recurrence becomes visible, and confirm it lies outside the window of Figure 6.1. [2]
> 6. Open. Give an estimator of $\mathcal{A}$ from a finite, noisy time series that is unbiased under $1/f$ backgrounds. The naive periodogram estimator is badly biased, and this is the main obstacle to testing prediction 1. [3]
For agents
GET /api/library/spectral-panpsychism/ch6.md