Order-Sensitive Sequential Interventions on Ideal Lattices

• URL: http://arxiv.org/abs/2604.26472v1
• Date: Wed, 29 Apr 2026 09:29:10 GMT
• Title: Order-Sensitive Sequential Interventions on Ideal Lattices
• Abstract summary: We study sequential interventions under prerequisite constraints.<n>In this setting, admissible intervention sequences are paths in the ideal lattice of a finite prerequisite poset.<n>We give an exact local-to-global theory of order sensitivity on this state space.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study sequential interventions under prerequisite constraints. In this setting, admissible intervention sequences are paths in the ideal lattice of a finite prerequisite poset rather than unconstrained action strings. We give an exact local-to-global theory of order sensitivity on this state space. First, we prove that any two admissible paths with the same endpoints differ by a finite sequence of elementary diamond swaps. Second, for edge-additive path valuations, we show that path-independence is equivalent to vanishing diamond curvature, yielding an endpoint potential with a canonical Möbius parameterization on the ideal lattice. Third, we prove that a local diamond field is induced by an edge-based path model if and only if it satisfies cube consistency, with uniqueness after fixing a reference-tree gauge. Under reduced-state longitudinal assumptions, supported reference paths identify reference-path scores, whereas local order effects require two-sided support of both orders on each diamond. These results yield exact planning consequences, including an order-insensitivity bound and dynamic programming on the truncated ideal lattice.

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