The claim
For a chain U_0 → U_1 → … → U_n, two quantities are easy to conflate:
- Retention, I(U_0; U_n), or I(G(U_0); U_n) for a feature G: information about a particular starting point.
- Drift, D[P(U_n), P(U_0)]: how far the population’s distribution has moved from where it began.
They come apart in both directions. A contractive chain started from different states becomes indistinguishable across starts, so retention falls toward zero, while it converges to a stationary distribution that may be far from the initial one. CASE-017 is a constructed instance: chains started in different states converged on each other while staying “substantially different from the original”. A chain can also stay close in distribution while scrambling which individual started where.
The data-processing inequality says retention cannot increase along a Markov chain. It is “not a theorem that messages get less comprehensible”: it says nothing about drift, legibility or structure. A state can become more conventional, and easier to read, while carrying less information about its origin.
Where the record applies it
CLAIM-081 applies the distinction to serial reconstruction of utterances. CLAIM-189 and CLAIM-191 apply it to an argument restated across sessions. All three now rest on this claim for the distinction.
What it does not say
- Which quantity matters for a given question. That depends on whether the question is about the original or about the convention.
- That a real transmission chain is Markov. Adding independent historical evidence to later states changes the graph and invalidates a naive use of the inequality.
Note of 2026-10-19: the four combinations, constructed
Take two states {a, b}, drift measured in total variation, and one step.
| start P(U_0) | kernel | retention I(U_0; U_1) | drift TV(P(U_1), P(U_0)) |
|---|---|---|---|
| (1/2, 1/2) | both states go to a | 0 | 1/2 |
| (1/2, 1/2) | resample uniformly | 0 | 0 |
| (0.9, 0.1) | swap a and b | H(U_0), the maximum | 0.8 |
| (0.9, 0.1) | identity | H(U_0), the maximum | 0 |
The first two rows have the same retention and different drift. The last two do as well, so neither quantity is a function of the other. That meets the defeat condition’s terms and fails it: no monotone relation holds for Markov chains in general.