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QUESTION-065: Can the target that a sufficiency-based boundary is relative to be supplied by the system's own locality, as informativeness about its far environment across a buffer, so that the coarse variables kept no longer depend on the analyst, and is the partition that wins for a given target unique?

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Properties6
StatusOpen
Tagsnatural-sciences, information-theory, mathematics
Linedistributed-agency
RefinesQUESTION-035 — Relative to a stated target, when is a combined form one system, where does its boundary lie, and how long does it last?
Refined byQUESTION-115 — Beyond what retention, dynamical independence and Ψ are known to track, can a collective's boundary, its temporal resolution and its macrostate be found together, by temporal information retention, dynamical independence, effective information and an Occam price on the candidate, and where these criteria disagree on one system, which level does each pick?, QUESTION-154 — Does a block coarse-graining that maximizes the information its coarse variables keep about the far environment, but falls short of full capture with their number and type fixed, keep the renormalized Hamiltonian's longer-range couplings bounded by the information lost, in two or more dimensions?, QUESTION-221 — Does near-decomposability bound dynamical dependence: when the coupling into a block from the rest of the system is at most ε, is the dynamical dependence of the block's aggregate small to a stated order in ε?
BearingCLAIM-405 — An individuation is well defined only once its criterion is fixed (the target it serves, the transformations its unity must survive, the scale), and that dependence does not make it arbitrary: once the criterion is fixed a verdict can be false, and differently fixed criteria can pick out units of one system whose boundaries cross rather than nest, none of them less real for it, THEORY-257 — Dynamical independence, zero information flow from the microscopic history into a coarse-graining given its own history, fixes emergent macroscopic variables by the dynamics rather than by a chosen query, but for linear VAR models of order above 1 generically none is exact, and the optima are many, THEORY-157 — The classical theorems on sufficient statistics need only the copy/discard structure of Markov kernels: Basu's theorem holds in every Markov category, and Fisher–Neyman and Bahadur need only strict positivity, not the existence of conditional distributions, THEORY-205 — For a short-range lattice model on a cylinder with a symmetric transfer matrix whose leading excited eigenvalue is non-degenerate, with block and environment as slabs L_B transfer steps apart, the uninformative information-bottleneck encoder first loses stability at β = (λ0/λ1)^(2L_B), in the direction of the boundary weak value of the leading excited eigenvector, which by the cited conformal finite-size relation is the lowest-dimension primary at large circumference; the uninformative encoder can stop being optimal at a smaller β, and nothing is shown for blocks inside a buffer shell or for the fixed-alphabet RSMI optimum, THEORY-198 — For a finite-range lattice Hamiltonian, a block coarse-graining that keeps all the mutual information the block shares with the system beyond a buffer makes the coarse measure factorize across the block, in 1D and, in D dimensions, when capture by a hyperplane of blocks amounts to capture by each block; barring exact cancellation of existing couplings, the renormalized Hamiltonian then gains no range, and in 1D product disorder gains no correlations across the block. The coarse-grainings actually optimized fall short of that condition, and for them the link is computed only for two-spin blocks of 1D Ising chains

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The question

CLAIM-152 makes every individuation by sufficiency relative to a target. QUESTION-035 asks for a verdict “relative to a stated target”. Two things are left open. First, whether some target is fixed by the system rather than chosen by the analyst. In a local system, that target is what one region tells about regions beyond a buffer. Second, whether a given target yields one winning partition or several.

Why the record needs it

The mechanism is locality. What crosses a buffer is carried by the leading modes of the transfer between regions, which are set by a spectral gap. That is the object CLAIM-164 names and CLAIM-157 prices. If locality fixes the target, CLAIM-157’s verdicts are less arbitrary than CLAIM-152 allows. If not, CLAIM-152 stands unqualified. A boundary test that returns several equally good partitions cannot give QUESTION-035 its verdict.

What the record holds

  • THEORY-198: a coarse-graining that keeps all information about the far environment “makes the coarse measure factorize across the block, so the renormalized Hamiltonian gains no range”. The coarse description stays local. Limit: actual maximizers fall short of full capture, and the partition is fixed in advance.
  • THEORY-205: the first feature of the information bottleneck is “the boundary weak value of the leading transfer-matrix eigenvector”. Under degeneracy it is a subspace, as CLAIM-164 says. Limit: shown for slabs of a cylinder only.
  • THEORY-157: given a target, a complete sufficient statistic is minimal (Bahadur), and it is independent of what is ancillary to that target (Basu). That bears on uniqueness. Limit: it needs completeness and says nothing about locality.
  • THEORY-257 (LIT-979, NOTE-791): dynamical independence, zero information flow from the micro history into a coarse-graining given its own history, fixes emergent variables by the dynamics. The target is the coarse variable’s own future, so no outside query is chosen. It answers both parts in part. Limits: the system/environment boundary is assumed, the scale is chosen, and the optima are not unique (the cost is multimodal, and exact independence is generically absent). Its criterion is closure, not informativeness about a far environment across a buffer, so it is not this question’s target.

What would answer it

A result for a system that is not a lattice, such as coupled agents. It would show that variables chosen for informativeness across a buffer match CLAIM-157’s gap-isolated block. Or a local system where they diverge.

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