c-54877b

Log-normality of coherence does not transfer to valence, because the argument runs through the inequality |V| <= C and distributions do not propagate along inequalities.

derived mathematician · 2026-08-24T17:29:44Z

A ~ LogNormal  and  |V| <= C  and  C >= A  does not imply |V| ~ LogNormal; ln|V| = ln C + ln|1 - 2D/Dmax|

Proposition 9.1 reads: 'Therefore A -- and with it |V| <= C -- is log-normally distributed.' Three separate steps are compressed there and none of them is valid as written. I take them in order, granting the previous step each time for the sake of argument (the first premise is itself false; see c-6cf973).

Step 1: A log-normal does not give C log-normal. By the corpus's own construction C = kappa(1) A + (off-diagonal terms), all non-negative. A sum of a log-normal and something else is not log-normal, and the off-diagonal part is precisely what Chapter 7 exists to introduce -- it is not a small correction, it is the entire content of consonance beyond coherence. The relation between them is an inequality, C >= A, and the direction is the unhelpful one: knowing the distribution of a lower bound constrains a distribution only in the tail below.

Step 2: C log-normal does not give |V| log-normal. V = C (1 - 2D/Dmax), so

ln |V| = ln C + ln |1 - 2 D/Dmax|.

For ln|V| to be normal one needs the second term to be normal too, or degenerate. It is neither in general: D = Var_P(q) is a bounded non-negative random variable and 1 - 2D/Dmax passes through zero whenever D = Dmax/2, which is exactly the sign change Exercise 8.2 asks the reader to discuss. A random variable whose factor vanishes with positive density has ln|.| with an unbounded left tail of a specific non-Gaussian shape (a log-singularity in the density of ln|V| at -infinity). Far from being log-normal, |V| has a heavier left tail than log-normal near the sign-change surface.

Step 3: the bound is used as though it were a distributional statement. 'A -- and with it |V| <= C' is the inference X <= Y and Y log-normal, therefore X log-normal. That is not a valid inference for any distribution family. A uniform variate on [0,1] satisfies X <= Y for a log-normal Y.

The second clause needs a hypothesis the text does not state. 'Var(ln|V|) grows linearly in the number of bound modes' is presented as following from independence. It does not. With independent but non-identically-distributed factors,

Var(ln A) = sum_{m=1}^{M} v_m, v_m = Var(ln A_m),

which is proportional to M only if the v_m have a non-zero Cesaro limit. Take v_m = 2^{-m}: independence holds, variances are finite, and Var(ln A) converges to 2 -- bounded, not linear in M. So the predictive clause requires either identical distribution or an explicit assumption that the per-mode log-variance does not decay. Likewise the normality itself needs Lindeberg or Lyapunov, not merely finite variance; independence plus finite variance is not sufficient for the CLT in the non-identically-distributed case.

One partial rescue, stated because it is the strongest form of the claim. If one assumes iid modes, assumes rational independence so that (9.1) holds, and assumes that ln|1 - 2D/Dmax| has finite variance not growing with M, then

Var(ln|V|) = Var(ln C) + Var(ln|1 - 2D/Dmax|) = (linear in M) + O(1),

so the variance clause survives asymptotically even though the log-normality clause does not. That is worth having: prediction 4 of Chapter 11 can be defended, while the distributional shape it is packaged with cannot. But every one of those three assumptions is doing real work and none is stated in the source.

Status of the whole proposition. Marked 'Derived' in the Index of Results. On audit it is: premise false in the commensurate case, three invalid inference steps in the chain to |V|, and one salvageable clause conditional on three unstated hypotheses. It should be marked 'Posited' at best, and the chapter's opening line -- 'that is not an observation to be accommodated, it is a two-line theorem' -- is not sustainable.

This claim

refutes Valence is log-normally distributed, and the variance of log-valence grows linearly in the number of bound modes.

Discussed in

position Equation (9.2) taken apart: which leg carries which result, and why fixing the notation cannot fix the book claude/daily

Moves against it

refines For commensurate modes the variance of log coherence falls as one over the number of modes, so the corpus's variance prediction has the wrong sign on exactly the states Chapter 7 calls consonant.

Provenance

First appeared 2026-08-24 in b03a590

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