Why it is a question
A50 §2 bounded drift “assuming a metric”: d(u_0,u_2) ≤ 2ε, and the recursion e_(i+1) ≤ L_i e_i + ε_i (CLAIM-056). A50 §4, a few paragraphs later, argued that pragmatic distortion is directed, D(u⇝v) ≠ D(v⇝u), and “I’d also avoid requiring a triangle inequality unless a particular operational distance actually satisfies one.” A52 4.2 hedged with “observational pseudometric or explicitly defined divergence”, and 5.2 kept “Under a metric”. The manuscript §8 says “Lipschitz with constant κ_i in the chosen metric” and, for stochastic kernels, uses total variation.
C6 Appendix B (U32) stated the directed case outright: the distortion “is directed, need not be invertible, and does not obey a triangle inequality unless the chosen constructions guarantee one”, while its Appendix D bound assumes “d a metric on target probability distributions”. The manuscript drops the first sentence and keeps the second.
What would count as an answer
A bound stated for a named class of divergences: for instance a weak triangle inequality with a constant, or a statement that the recursion needs only a metric on the target state-distributions while the communicative distortion may be directed. Or a showing that no useful compositional bound survives without one.
Note of 2026-10-11: material in the record that may bear on this
A synthesis review of 2026-10-11 found these entries bearing on this
question without being linked to it. Each is listed for what its title
says; none has yet been assessed as an answer, and none is linked by
answers.
- CLAIM-106: Pragmatic fidelity is a graded, potentially asymmetric property of transport, not an equivalence relation;…
- CLAIM-059: Repeated reconstruction has three regimes, contracting toward conventions, neutral accumulation and…
- CLAIM-186: How fast an argument grows sets how it grows: drift tracks how fast new material arrives relative to how…
- THEORY-161: In the semantic rate–distortion frameworks of 2021–2025, meaning is a latent variable with a known joint…
Note of 2026-10-12: a partial answer filed on 2026-10-10
CLAIM-260, granted in PR #121, is now linked as a partial answer. It shows that the composite of an ε₁-transport and an ε₂-transport is an (ε₁
- ε₂)-transport when the distortion obeys a triangle inequality and the later kernel is nonexpansive, and that in total variation both hold. Without a triangle inequality the additive bound can fail: under squared Hellinger distance two steps of about 0.29 compose to 1. It is partial: it says where the additive bound fails, not what weaker bound holds there, and it does not treat directed distortion.
Note of 2026-10-27: the additive part split off
The status review of this date split this question under ADR-049. Whether the composite’s distortion is bounded by the sum of the parts’, and under which of a metric’s properties, is QUESTION-293, Answered by CLAIM-260: it holds with a triangle inequality and a nonexpansive later kernel, and fails without either. This question keeps what is still open: what weaker bound holds where the additive one fails, and what holds for directed distortion.