Why it is a question
A86 §10 named three results the theory should aim at: (a) compression that preserves a chosen family of decision problems; (b) when iterated reconstruction preserves or destroys sufficiency; (c) how much decoder adaptation compensates for lower rate (QUESTION-020). This is (b).
Where it stands
The manuscript §8 has I(Z;U_(n+1)) ≤ I(Z;U_n) for a fixed Markov chain, with the caveat that added evidence changes the graph, and a bound on distributional error (CLAIM-056). Neither says when a chain keeps the Blackwell sufficiency for a family Q of communicative decisions (CLAIM-050). A garbling of a garbling is a garbling, so sufficiency for Q can only decline along a pure chain. The open part is the case the manuscript flags, interpreters with side information, where a retelling “can then improve an audience’s task performance” (A86 §3).
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.
- THEORY-157: The classical theorems on sufficient statistics need only the copy/discard structure of Markov kernels:…
- CLAIM-081: Retaining information about the ancestor and drifting in the population’s distribution are different…
- THEORY-155: Iterated learning by Bayesian agents who sample from the posterior converges to the shared prior
- CLAIM-182: Storage is not anchoring, addressability is: a structure that is kept but cannot be pointed at part by…
- THEORY-194: For the 1D and 2D Ising and 2D dimer models, the block coarse-graining that maximizes mutual information…
- THEORY-198: For a finite-range lattice Hamiltonian, a block coarse- graining that keeps all the mutual information the…
CLAIM-182 bears from the other end: side information that is stored is not thereby used unless it is addressable.
Note of 2026-10-12: entries on the side-information case
The note of 2026-10-11 lists material as unassessed. PR #121’s round of
2026-10-10 filed three entries on the part this question holds open,
interpreters with side information. CLAIM-199, a proposal, separates
what a rendering makes possible from what a given receiver can use, and
holds that a garbling can raise a receiver’s usable information “but never
above what the source makes possible”. CLAIM-258, an open counter,
says a translator with context knowledge yields a rendering that is not a
garbling of the source, and suggests handling the side-information case by
a measure such as deficiency rather than by Blackwell’s order.
CLAIM-237, granted, says that for a contextual model what is open is
the converse direction or a deficiency bound. None is linked by answers.
Note of 2026-10-14: a bounded-degradation instance
THEORY-289, filed from LIT-971, is an instance of a chain anchored to
the source in which degradation is bounded. When each round of a linear
regression is refit to labels from the previous fit and every earlier
round’s data is kept, test error is bounded at π²/6 times the first fit’s;
when the data are replaced, it grows linearly. The loss is bounded, not
stopped: every round adds error. What is retained is the real sample, not
an interpreter’s side information, and every model is still a garbling of
the real data, for one squared-error task rather than a family Q. It
carries no answers edge.