Event-Deletion Reconstruction of Two-Dimensional Causal Orders: Pin-Seed Localization, Exceptional-Card Closure, and Cyclic Switch Towers

This research release studies reconstruction of finite partially ordered sets of Dushnik-Miller dimension at most two from the ordinary multiplicity-retaining multiset of their unlabeled one-point-deleted subposets. These structures can also be interpreted as finite generic causal orders in a spacetime with one spatial and one temporal dimension. The central question is whether every such order is uniquely determined, up to isomorphism, by its event-deletion deck. The release develops a combined structural, algebraic, spectral, permutation-theoretic, and computational program for this problem. It proves several all-order reconstruction theorems, closes the entire decomposable branch, reconstructs important critical and exceptional families, reduces the remaining prime case to sharply defined occurrence-alignment problems, and supplies exact finite classifications through parent order twelve. The unrestricted reconstruction conjecture for all two-dimensional orders remains open. A major all-order result closes the decomposable case. Every connected and coconnected decomposable two-dimensional poset is reconstructible from its ordinary one-point deck. The proof permits nonrigid prime quotients and repeated isomorphic factors. It combines a modular reconstruction theorem for parents with at least two surplus factor points with a complete treatment of the formerly unresolved one-bump case, in which exactly one vertex of a prime quotient is inflated to a two-point factor. The one-bump proof uses canonically recoverable safe-prime subprofiles, safe-site cancellation, critical-root analysis, simple-permutation insertion rigidity, occurrence-cycle cancellation, and exact deck formulas. This removes every decomposable parent from the unresolved class. The release also reconstructs every finite minus-one-critical graph and every finite minus-one-critical poset. Consequently, every unresolved prime parent must be noncritical and must possess at least two prime one-point-deletion cards. Additional all-order results show that the series and parallel nodes appearing in a one-point card of a simple parent have only tightly restricted binary or ternary forms. General branching ambiguities can therefore be reduced by exact leaf-message elimination to a residual cyclic core constraint problem. For prime parents, the work develops an occurrence-atlas formulation of reconstruction. An occurrence atlas records the possible ways anonymous deletion-card occurrences can be aligned across their common double deletions. Reconstruction becomes the assertion that all globally coherent atlases yield the same parent isomorphism type. The release proves degree-class attachment, fixed-pair attachment, and degree-separated occurrence-incidence theorems. It also shows that if the deck-derived ambiguity graph is a forest, then the parent type is forced. Any genuine residual ambiguity must therefore survive in a cyclic two-core. The spectral and linear-algebraic part of the program proves a regular-orthogonal attachment theorem for prime posets. Rooted generalized cospectrality confines every rival missing neighborhood to an exact binary walk-moment cell. Controllable cards make that cell unique, while almost-controllable cards leave one explicit orthogonal branch that oriented-card information can eliminate. Prime-oriented moment-cell criteria, centered-star graph forcing, paired walk invariants, automorphism reductions, and deletion-visibility lemmas provide additional exact mechanisms. Together these methods close all 15,821 stored prime two-dimensional types of orders four through nine by explicit certificates, with no residual cases. These computations verify theorem-driven mechanisms and are not being used as substitutes for all-order proofs. A further all-order theorem proves that a fixed prime poset-card type can occur at most twice in a prime two-dimensional parent. When it occurs twice, the common card is rigid and the inherited two-realizer frames have opposite orientations. A compatible root-preserving marked external-card map is proved to extend to a parent isomorphism. The release also replaces an earlier higher-order alignment fallback with a sharp aligned-three criterion: one global bijection preserving all induced types on subsets of at most three points must be an isomorphism. The ordinary deck determines the ordered multiset of strict predecessor and successor counts. Combining this parent bidegree profile with the multiplicity-retaining prime-card subdeck gives a particularly strong residual signature. An exhaustive screen through parent order twelve covers 23,540,019 qualifying prime two-dimensional parent types. The prime-card subdeck alone leaves 64 nonsingleton groups containing 131 parents and 71 cross-parent candidate pairs. Exact complete-deck comparison rejects all 71 pairs. The reconstructed parent bidegree profile independently rejects all 71. Thus the combined prime-card and bidegree signature has no collision anywhere in the exhaustive range through order twelve. This is a complete finite theorem, not an all-order reconstruction proof. The punctured-deck program gives an even finer analysis. A complete census through parent order eleven examines 1,728,081 qualifying prime parent types at order eleven, more than eleven million prime-card occurrences, and more than eleven million designated-card profiles. It finds no cross-parent collision for the complete punctured profile. Weakening the signature to a designated prime card and its prime-card subdeck leaves exactly seventeen collision groups supported on seven parent pairs. These consist of one proper-pin pair, four parallel-alternation pairs, and two type-one wedge-simple pairs. In every case, decomposable cards in the complete punctured deck separate the parents, and the reconstructed parent bidegrees provide an independent separation. A companion occurrence census exhausts all 1,893,399 prime oriented two-dimensional types through order eleven. Among 7,142 repeated prime-card groups, exactly ten occurrence pairs cross parent-automorphism orbits. Every common card in these cases is rigid, every missing-point extension produces a spanning cyclic disagreement with full-star relation defect, and every case remains a same-parent rerooting rather than a collision between nonisomorphic parents. These examples reveal why locally valid card identifications can create global cyclic identity swaps without producing reconstruction counterexamples. A targeted order-twelve continuation exhausts all simple one-point extensions of sixteen order-eleven roots chosen deterministically from the complete prime-only residue and rerooting certificates. It produces 1,536 rooted gap records and 1,446 qualifying unrooted prime parents, with no complete punctured-profile collision. This result is exact over the declared root set but is not presented as an exhaustive order-twelve punctured-profile census. Separate targeted searches through orders twelve to twenty and fixed-root structural-family audits provide additional evidence without being promoted to universal theorems. The latest attack is organized around the unavoidable-substructure framework for simple permutations described by brignall, Ruskuc, and Vatter. That framework directs attention to proper-pin sequences, parallel alternations, and wedge-simple permutations. For proper-pin structures, the release proves an all-order geometric prefix-localization theorem after an occurrence has been selected. Deleting an interior pin places every later chosen pin in an open quadrant outside the earlier prefix rectangle. Ambient points already inside the rectangle can obstruct this localization only through coordinate side strips, and each fixed side slicer affects one interval of deletion indices. The resulting pin-prefix and slicer incidence graph is convex and chordal bipartite. If it contains a cycle, every shortest cycle is therefore a four-cycle. The latest version sharpens this result to an intrinsic marked localization theorem. Once a two-point active seed has been selected, deletion of an interior pin from a spanning proper-pin sequence leaves a least seed-containing interval whose size recovers the deleted pin’s position. For a simple one-point extension and an index outside the new point’s side-slicer interval, the analogous set is the unique least oriented-poset module containing the marked seed. Its size increases strictly along the pin sequence. Exact audits through root order nine cover 20,596 simple marked extensions and 53,536 nonsliced cases, with no violations. The same audits provide explicit counterexamples to several stronger proposals based on unmarked cards, strong modules, unique interval sizes, or naive active-knight transport. The release also proves strong restrictions on prime deletions of spanning proper-pin sequences. Only the first two pins and the terminal pin can yield prime one-point cards. A prime double deletion must meet one of the first two pins or be the terminal pair, leaving only a linear number of candidate pairs. Exact geometric and deletion audits through the stated finite ranges find no violations. For same-parent repeated-card switches whose card isomorphism extends to one cycle, a cyclic star-word theorem determines every missing-point relation and gives exact conditions for parent symmetry and full-star disagreement. The theorem produces explicit infinite families of prime two-dimensional parents having two equal prime cards in different parent-automorphism orbits. Their cyclic defects have unbounded depth. The order-thirteen witness is the third level of one such tower. This establishes that cyclic disagreement is not a bounded exceptional phenomenon. It does not claim that the displayed families exhaust every possible cyclic word. The exceptional parallel-alternation branch is closed in all orders. Every prime two-dimensional parent of order at least seven with a codimensio