Part IV · Valence

# Intensity and the Logarithmic Scales

QRI's empirical finding is that human valence is long-tailed — that it plots linearly on a log scale. In this model that is not an observation to be accommodated. It is a two-line theorem.

> Assumes
> Chapters 6 and 8.
> Delivers
> Proposition 9.1 (log-normality); the identification of felt intensity with a Rényi entropy and a replica free energy.

9.1 Coherence is multiplicative

Suppose the subject's state factorises over $M$ quasi-independent collective modes within the split factor $\mathcal{N}$. Since the inverse participation ratio of a product state is the product of the factors' ratios,

$$ \mathcal{A}=\prod_{m=1}^{M}\mathcal{A}_m \qquad\Longrightarrow\qquad \ln\mathcal{A}=\sum_{m=1}^{M}\ln\mathcal{A}_m $$ (9.1)

This is a small observation with a large consequence, because it converts a product of many bounded quantities into a sum, and sums of many independent quantities have a universal distribution.

> Proposition 9.1 · Log-normal valence
>
> Derived
>
> Since $\ln\mathcal{A}$ is a sum of $M$ independent contributions, the central limit theorem gives $\ln\mathcal{A}\sim\mathcal{N}(M\mu,\,M\sigma^2)$. Therefore $\mathcal{A}$ — and with it $|\mathfrak{V}|\le\mathcal{C}$ — is log-normally distributed. Reported valence is linear on a logarithmic scale, with a heavy tail, and the variance of log-valence grows linearly in the number of bound modes.

The second clause is the part that is genuinely predictive. Log-normality by itself is cheap: many mechanisms produce it. But $\mathrm{Var}(\ln|\mathfrak{V}|)\propto M$ ties the spread of reported intensities to a structural quantity — the number of modes bound into the moment — and that is a relation one can go and check. It is Chapter 11, prediction 4.

It also explains something QRI has emphasised on ethical grounds: that the extremes of the distribution are far more extreme than a linear intuition suggests. If $\ln\mathcal{A}$ is normal with variance growing in $M$, then the ratio between a typical bad day and the worst states accessible to a highly integrated nervous system is exponentially large. Whether one accepts the suffering-focused ethics QRI draws from this, the mathematical point stands independently: log-normal tails are not a rhetorical exaggeration.

9.2 Intensity as a Rényi entropy

Writing $N_{\mathrm{eff}}=1/\mathcal{A}$ for the effective participation number, the natural scale on which to report intensity is already an entropy:

$$ S_2=-\ln\mathrm{Tr}\rho^2=-\ln\mathcal{A}=\ln N_{\mathrm{eff}}, \qquad \mathcal{A}=\mathrm{Tr}\rho^2=\frac{Z_2}{Z_1^2}=e^{-\beta\,\Delta F_{\text{replica}}} $$ (9.2)

So the logarithm is not a psychophysical convention imposed on the data after the fact. Valence is an exponentiated free-energy difference — between the two-replica system and two copies of the one-replica system — and the log scale is simply that free energy, measured in its natural units.

9.3 A bridge to variational free energy

The second equality in (9.2) has a consequence worth drawing out, because it connects two research programmes that rarely speak. The Fristonian account holds that nervous systems minimise variational free energy. This model holds that they maximise symmetry. Equation (9.2) says these are, up to the replica structure, the same optimisation: maximising $\mathcal{A}$ is minimising $\Delta F_{\text{replica}}$.

The bridge is not free of charge, and the discrepancy is informative. Free-energy minimisation is defined on a single replica; $\mathcal{A}$ is a two-replica quantity. The gap between them is exactly the object of Chapter 8 — whether the two replicas agree, which is $\mathcal{D}$. So the relationship is:

| Quantity | Replicas | Reads out |
| --- | --- | --- |
| Variational free energy $F_1$ | One | Prediction error; how surprised the system is |
| Coherence $\mathcal{A}=Z_2/Z_1^2$ | Two | Intensity; how concentrated the state is |
| Frustration $\mathcal{D}=\mathrm{Var}_P(q)$ | Two, compared | Sign of valence; whether the system is one thing |

A system can therefore be minimising prediction error perfectly well and still be in the glass, because $F_1$ is blind to whether the minimum it has found is one basin or a hierarchy of them. That, in this framework, is what chronic suffering is: a well-fitted model of the world that is nonetheless replica-symmetry-broken. It is also why merely getting better at prediction is not a route out.

> ### What Chapter 9 established
>
> - $\mathcal{A}$ is multiplicative over independent modes, so $\ln\mathcal{A}$ is a sum.
> - Proposition 9.1: valence is therefore log-normal, and $\mathrm{Var}(\ln|\mathfrak{V}|)$ grows linearly in the number of bound modes.
> - Felt intensity is the Rényi-2 entropy $S_2=\ln N_{\mathrm{eff}}$; the log scale is a replica free energy.
> - Free-energy minimisation and symmetry maximisation coincide at one replica and diverge at two — and the divergence is exactly $\mathcal{D}$.

> ### Exercises
>
> 1. Verify that $\mathcal{A}$ is multiplicative for a product state, and show the claim fails for entangled states. What does the failure imply for (9.1) in a strongly bound moment? [2]
> 2. Derive $S_2=-\ln\mathcal{A}$ from the definition of the Rényi entropy $S_n=\frac{1}{1-n}\ln\mathrm{Tr}\rho^n$, and state why $n=2$ rather than $n=1$ is the relevant index here. [1]
> 3. A log-normal with parameters $(M\mu,M\sigma^2)$ looks like a power law over a finite range. Compute the apparent Pareto exponent over three decades for $M\sigma^2=4$. [2]
> 4. Design an experiment distinguishing $\mathrm{Var}(\ln|\mathfrak{V}|)\propto M$ from $\mathrm{Var}(\ln|\mathfrak{V}|)=$ const, given that $M$ is not directly observable. What proxy would you use, and what confound would it introduce? [2]
> 5. Open. The replica trick requires analytic continuation in $n$, which is not justified in general. Give conditions on the modular spectrum under which (9.2) is rigorous. [3]

For agents

GET /api/library/spectral-panpsychism/ch9.md

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