The objection
CLAIM-078 makes half of the theory’s success that the measures “follow its compositional predictions across reconstructions”. It is defeated if “their behaviour along reconstruction chains departs from the composition bounds”. CLAIM-056 is defeated if “end-to-end drift is not predicted by per-step discrepancies combined with the measured sensitivity of later steps”, and CLAIM-059 if “Measured per-generation sensitivity of real reconstruction chains shows no difference between conditions that the theory assigns to different regimes”.
A bound, not a prediction. The recursion is CLAIM-056’s. Let u_i be the chain’s state after step i and r_i the reference transport’s, with e_i = d(u_i, r_i). Let the chain’s step be K̃_i and the reference’s K_i, with local discrepancy ε_i = d(K̃_i u_i, K_i u_i) and κ_i a Lipschitz constant of K_i. By the triangle inequality,
e_(i+1) = d(K̃_i u_i, K_i r_i) ≤ d(K̃_i u_i, K_i u_i) + d(K_i u_i, K_i r_i) ≤ ε_i + κ_i e_i,
and unrolling from e_0 gives e_n ≤ (∏(j<n) κ_j) e_0 + Σ(i<n) ε_i ∏_(i<j<n) κ_j. If the κ_i are true Lipschitz constants and the ε_i the true discrepancies, this holds for every chain: it is a theorem, and no data can contradict it. A measured e_n above the bound shows only that some κ_i or ε_i was estimated too low. A measured e_n below it is consistent with it however far below. So “departs from the composition bounds” has no outcome that counts against the theory. Read as CLAIM-056’s “predicted”, the recursion gives no point prediction to compare with.
The constants need kernels. κ_i is a property of the reteller’s kernel, the distribution of retellings it produces for each input. To estimate it needs the kernel at several inputs, with many retellings of each, at every generation. Two errors then enter. A maximum over the input pairs sampled can only fall short of the supremum, which biases the bound low. And each distance between output distributions is itself estimated from finite retellings, which adds a bias of its own (upward for a plug-in total variation). Either can produce or hide a “violation”. A serial reproduction chain gives one retelling of one input at each node. The chain designs (CASE-009; manuscript §8; Case II’s “heterogeneous-agent chains”, as CLAIM-085 reports them) do not provide several independent retellings of one input per generation.
The metric. No design fixes the metric d in which κ_i is computed. CLAIM-059 itself says amplification “needs a metric other than total variation” (now held as CLAIM-267): for a Markov kernel acting on distributions, the contraction coefficient in total variation is at most one, so in that metric no chain is amplifying. A chain counted as amplifying is so only in some other metric, chosen afterwards. Which regime a chain is in therefore depends on a choice the theory does not make, and CLAIM-059’s defeat condition, about conditions “the theory assigns to different regimes”, has no assignment to test until that choice is made.
QUESTION-017 asks what bound holds when the distortion is directed and the triangle inequality fails. That is a question about the theorem. This objection is about measuring its constants.
What it does not say
- It does not say error propagation along chains is unimportant, or that the three regimes are not real.
- It does not say the recursion is wrong. It is correct, and for that reason cannot be tested as stated.
What would answer it
- A design that estimates each step’s kernel from several independent retellers on held-out inputs and perturbed inputs at each generation.
- A metric fixed before the data, with the reason it is the right one.
- A prediction stated as a point estimate of end-to-end drift with an interval, computed from kernels estimated on one set of chains and tested on another, so that it can fail.