The objection
CLAIM-098 quotes the manuscript’s §4: “The more inclusive notion is a category or semigroup of stochastic transformations”, and among the cases it separates is “(iii) approximate preservation”. TERM-008’s founding sense is approximate. U20: “there should be some transport map that at least gets to within some eps”. CLAIM-106 makes fidelity graded.
Exact kernels compose: a Markov kernel followed by a Markov kernel is one. What a category of approximate transports needs is more. Its arrows carry their distortion, and the distortion of a composite has to be bounded by the parts. Let T₁ take A to B with d(K₁#e_A, e_B) ≤ ε₁, and T₂ take B to C with d(K₂#e_B, e_C) ≤ ε₂. Then
d(K₂K₁#e_A, e_C) ≤ d(K₂#(K₁#e_A), K₂#e_B) + d(K₂#e_B, e_C) ≤ Lε₁ + ε₂,
where the first step is the triangle inequality and L is how much K₂ can stretch distances in d. The composite is an (ε₁ + ε₂)-transport when d obeys the triangle inequality and L ≤ 1. Without either, it need not be.
- No triangle inequality. Let d be the squared difference of Bernoulli parameters, d(p, q) = (p − q)², and let both kernels be the identity. With e_A, e_B and e_C the Bernoulli laws with parameters 0, 1/2 and 1, ε₁ = ε₂ = 1/4, and the composite’s distortion is 1, which exceeds ε₁ + ε₂ = 1/2. Squared Hellinger distance does the same on these three laws: each step costs 1 − √(1/2), about 0.29, and the composite costs 1.
- An expansive kernel. In the Wasserstein-1 distance on the real line, the deterministic kernel x ↦ 2x doubles every distance. The composite is then only a (2ε₁ + ε₂)-transport.
In total variation both conditions hold. It is a metric, and every Markov kernel is nonexpansive in it, which is CLAIM-059’s point, now held as CLAIM-267: “For Markov kernels acting on probability measures under total variation distance, the contraction coefficient is at most one.” So the gap is for the other distortions, and CLAIM-059 says amplification needs one, now held as CLAIM-267: “Amplification requires a different metric, an expanded state model, or an appropriate state-dependent dynamics.” QUESTION-017 holds the triangle inequality open, quoting C6 Appendix B that the distortion “does not obey a triangle inequality unless the chosen constructions guarantee one”. It does not tie that to the category claim.
What it does not say
- It does not say the category claim fails for exact transports, or for total variation with Markov kernels.
- It does not say approximate transports fail to compose. Their kernels always compose. What can fail is the bound on the composite’s distortion, so CLAIM-098’s (iii) sits in the category only once the distortion measure is fixed and meets the two conditions.