23. Refinement cannot manufacture missing information
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An atlas change is an invertible recoding of the same represented information. Predictive refinement usually adds information. Treating these as identical operations would undo the central distinction of the foundation.
Proposition 22: The upward-lift obstruction
Let \(\gamma:X\to Z\) be a coarse representation and \(\eta:X\to W\) a proposed refined representation. There is a deterministic map \(J:\gamma(X)\to\eta(X)\) satisfying \(\eta=J\gamma\) if and only if
For linear representations, this is equivalent to \(\ker\gamma\subseteq\ker\eta\). If two states share a coarse representation but their refined representations differ, any predicted refined value \(w\) has worst-case error at least
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Proof
A lift must be constant on each coarse fiber. Conversely, the stated condition defines it unambiguously on every fiber. The linear version is the kernel factorization criterion. The lower bound follows from the triangle inequality between the two refined values through \(w\). \(\square\)
For a strict refinement, the coarse state is recoverable from the refined one but not conversely. In the linear case this typically means \(\ker\eta\subsetneq\ker\gamma\), exactly the opposite of the condition required for an upward lift.
For example, \(\gamma(x_1,x_2)=x_1\) and \(\eta(x_1,x_2)=(x_1,x_2)\). The states \((0,1)\) and \((0,-1)\) have the same coarse state, but any common point initialization has worst-case error at least one in Euclidean norm. Padding a vector with zeros is a choice, not information recovery.
A refinement step therefore needs one of three explicitly distinguished inputs: an archive that already contains the relevant observations, new observations, or a set-valued or probabilistic state retaining uncertainty about the missing details. An exact archive of measured outputs can initialize a longer history when those samples were stored. It cannot recover a latent coordinate that never affected any measurement.
For adaptive prediction, an attempted lift \(J_t\) contributes its own initialization defect
It belongs in Proposition 18's initial error or in the defect of the composed transition. It cannot be set to zero merely because a finer packet tree has been allocated.
A data-dependent representation schedule also requires care. The pathwise bound applies to the realized schedule when all of its hypotheses hold along that schedule. Comparing two different schedules chosen by true and predicted histories introduces a selection discrepancy. No uniform statistical consistency of adaptive selection follows solely from the deterministic propagation law.