What Must Survive? Exact Task-Information--State Frontiers Under Partial Task Revelation
Abstract
A state may need to be compressed before its exact downstream task is known. For a finite family of linear task operators {Tu}u∈U, we characterize exactly how much state must survive when only one of K coarse task messages is available at compression time: p*(K)= minsubstackPcal partition of U |Pcal|≤ K maxC∈Pcalrank(TC), where TC stacks the tasks that remain unresolved within cell C. Thus task information reduces retained state precisely by separating tasks whose joint task-visible subspace is expensive. We derive the dual bit frontier, an irreducible common-core floor, and a sharp singular-value characterization for nonzero tolerance. For overlapping task families, a cumulative overlap deficit places the exact frontier within an additive Δ of weighted load balancing. Optimal partitioning is strongly NP-hard, yet direct-sum families admit classical approximation guarantees and a PTAS. We stress-test the theory on learned task operators from CIFAR-10, Fashion-MNIST, SVHN, and Burgers observables. For Burgers with K=2 and eps=0.35, a constructive partition attains p=4, while balanced random partitions require median p=6, with held-out worst standardized RMSE $0.0368$. The results distinguish uniform task sufficiency from distributional and decision-level performance.
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BibTeX
@misc{katende2026what,
title = {{What Must Survive? Exact Task-Information--State Frontiers Under Partial Task Revelation}},
author = {Ronald Katende},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21523},
eprint = {2609.21523},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21523},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - What Must Survive? Exact Task-Information--State Frontiers Under Partial Task Revelation
AU - Ronald Katende
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.21523
UR - https://arxiv.org/abs/2609.21523
AB - A state may need to be compressed before its exact downstream task is known. For a finite family of linear task operators {T_u}_u∈U, we characterize exactly how much state must survive when only one of K coarse task messages is available at compression time: p^*(K)= min_substackPcal partition of U |Pcal|≤ K max_C∈Pcalrank(T_C), where T_C stacks the tasks that remain unresolved within cell C. Thus task information reduces retained state precisely by separating tasks whose joint task-visible subspace is expensive. We derive the dual bit frontier, an irreducible common-core floor, and a sharp singular-value characterization for nonzero tolerance. For overlapping task families, a cumulative overlap deficit places the exact frontier within an additive Δ of weighted load balancing. Optimal partitioning is strongly NP-hard, yet direct-sum families admit classical approximation guarantees and a PTAS. We stress-test the theory on learned task operators from CIFAR-10, Fashion-MNIST, SVHN, and Burgers observables. For Burgers with K=2 and eps=0.35, a constructive partition attains p=4, while balanced random partitions require median p=6, with held-out worst standardized RMSE $0.0368$. The results distinguish uniform task sufficiency from distributional and decision-level performance.
ER -