Event-Deletion Decks of Two-Dimensional Causal Orders: Safe-Prime Cancellation and Complete Modular Reconstruction
Year: 2026
Type: article
Abstract: Can a finite causal order be recovered from all the smaller orders obtained by deleting one event at a time, when the events and deletion locations are anonymous? This paper investigates that question for finite partially ordered sets of dimension at most two. The reconstruction data consist of the complete multiset of unlabeled one-point deletions, with repeated card types counted separately. Through the correspondence between two-dimensional orders and finite generic causal orders in one spatial and one temporal dimension, the problem also has a direct interpretation in discrete spacetime reconstruction. The paper develops a unified framework connecting order theory, simple permutations, permutation graphs, modular decomposition, automorphism groups, and exact computation. An occurrence-atlas construction records how individual card occurrences can be identified and assembled into a common parent. Its components represent compatible parent isomorphism types, while its stabilizers capture parent symmetries. This formulation distinguishes agreement of anonymous card types from the stronger requirement that their occurrences and orientations fit together globally. Structural reconstruction results and established reductions dispose of decomposable and critical cases and concentrate the unresolved problem on prime parents with multiple prime deletion cards. Such parents can be compared as one-point extensions of a common prime root. Permutation insertion gaps give an explicit finite candidate space, and punctured-deck profiles encode the remaining deletion information after a selected root occurrence is removed. The paper proves restrictions on prime-card multiplicities, sharp control of prime-card counts under extension, and exact relationships between deletion behavior and the modular structure exposed inside a card. Several independent sufficient reconstruction criteria are developed using incidence information, centered configurations, spectral attachment constraints, feasible moment cells, and compatible small-suborder data. The paper also establishes reconstruction for specific infinite classes, including parents with a full exceptional parallel-alternation card, parents with a full prime wedge card, and a hierarchy associated with spanning-cycle repeated-card symmetries. These results require the stated full-card or structural hypotheses; the occurrence of a smaller related pattern alone does not provide the necessary alignment. Further reconstruction theorems use either a specified pair of dual spine cards or a separated common prime root. For the dual spine pair, the number of prime deletion cards and the number of comparable pairs distinguish arbitrary competing prime parents. A general completion theorem bounds the number of possible prime parents from two full prime cards by a quantity linear in card size. For separated roots, a geometric condition and an explicit bipartite matching obstruction make the complete prime-card subdeck sufficient. An infinite family of affine permutation roots satisfies this criterion. The local analysis explains both the strength and the limitations of these approaches. Proper pin sequences, modular-decomposition scars, marked boundaries, and safe deletions reveal structure that can survive the loss of a point. Ordered lower and upper degree profiles provide additional deck-reconstructible information, while exact identities separate persistent sparse candidate families. Point-image bounds sharply restrict individual card isomorphisms, but explicit rigid safe cards still admit exponentially many realizer frames. Counterexamples to proposed shortcuts show why local rigidity and compatible-looking matches need not supply a single consistent alignment across the deck. Triple literal-card coincidences receive an exact anchored-component reduction and a conditional exclusion of free points, while their complete classification remains open. The theoretical development is accompanied by reproducible exact enumeration, explicit witnesses, symbolic classifications, and verification code. A combined prime-card and degree-profile audit covers more than twenty-three million qualifying parent types through order twelve without a surviving collision. Computer-assisted arguments also establish statements valid in every order: the full-wedge classification and the exclusion of the minimum monotone degree-profile collision use arithmetic formulations with an unbounded size parameter. Solver inputs, proof traces, replay procedures, and separate encoding checks document these arguments. Finite computational evidence, analytic proofs, and computer-assisted theorems are distinguished throughout. The general reconstruction conjecture remains open. Its principal unresolved step is to coordinate anonymous matches that may move the retained point or change modular alignment. Under the paper’s reductions, bounding the common safe-card overlap of the remaining nonisomorphic candidates by four occurrences would contradict the overlap forced by equal decks and complete the argument. The work therefore provides a collection of reconstruction theorems, structural reductions, exact obstructions, and reproducible tests directed toward this remaining problem. Its causal interpretation concerns recovery of the finite order; it does not determine embedding coordinates, metric distances, or continuum spacetime geometry.
Author: Fathi, Kevin
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204929054
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