Categorical Entropy Obstruction Theory IV Finite Trace-Law Compression, Target Sufficiency, and Support-Relative Information Obstructions

This paper develops the finite trace-compression layer of Categorical Entropy Obstruction
Theory. Given a finite algorithmic trace S0:T, a deterministic compressed observation O = c(S0:T),
and a declared task target Y , it separates three questions that are often conflated: reconstruction
of the trace, preservation of the target law, and preservation of task performance under a bounded
loss.
The first contribution is a support-relative entropy obstruction theory for complete-trace,
hidden-trace, bridge-relative, coordinatewise static, and visible-memory reconstruction. The
second contribution is a task-relative sufficiency theory: deterministic target recovery is governed
by H(Y | O), stochastic target-law sufficiency by I(Y ;S0:T | O), and bounded-loss degradation
by Bayes-risk differences. The third contribution is a finite categorical packaging: deterministic
observation refinement gives monotonicity as functoriality into the reverse ordered category
(R≥0,≥), while coherent typed profiles give relabeling invariance and explain why scalar entropy
shadows do not classify posterior trace profiles.
The theory is deliberately finite, law-level, and support-relative. It does not claim efficient
decoder synthesis, generator-sensitive algorithmic complexity, a target-independent minimal
quotient, or a universal categorical entropy functor. The main separation theorem shows that
the observed target law PO,Y , even up to law-isomorphism, is not a complete invariant for
hidden-trace obstruction. Finite examples from dynamic programming and signature-dependent
path systems illustrate the trace/task/memory distinctions.