Reconstruction of Finite Posets: Anonymous Decks and Structural Alignment

How much of a finite partially ordered set survives the loss of one point at a time? Its ordinary deletion deck records the isomorphism types of all one-point-deleted subposets, with multiplicities, but does not identify the missing points. The finite-poset reconstruction conjecture asks whether this information determines every poset with at least four points. This two-paper work develops structural reductions, reconstruction theorems for infinite families, and exact finite certificates. The central reductions isolate two obstacles: aligning attachments to a uniquely occurring induced subposet, and locating a doubled point in a common base. The general conjecture remains open, but the results substantially restrict the forms a counterexample could take. The main paper, Reconstruction of Finite Posets: Structural Reductions and Global Alignment (68 pages), develops these reductions and the general alignment framework. The companion, Reconstruction of Finite Posets: Families, Certificates, and Further Methods (414 pages), presents the family theorems, specialized methods, and finite classifications. The main paper’s central proofs do not depend on the companion’s family classifications. Both papers are by Kevin Fathi, [email protected]. Throughout, a deletion card retains its partial order, not merely its comparability graph. Reconstruction theorems concern arbitrary competing finite posets unless a different setting is explicitly stated. Colored, rooted, and group-action formulations serve as intermediate problems; their transfer to ordinary decks requires the recognition and compatibility conditions given in the papers. Structural reductions and global alignment Every nonisomorphic equal-deck pair of order at least five falls into one of two branches. Either both parents have prime comparability graphs and prime deletion cards, or they arise by placing the same autonomous two-point fiber at different points of a common ordinary card. The fiber is a chain or an antichain, and the common base may contain other, already matched fibers. Thus the decomposable case becomes a placement problem on one known base. Anchor-free posets of order at least four reconstruct. Here an anchor is an induced isomorphism type occurring on exactly one vertex subset. The proof combines rank-uniform principal-neighborhood data, ordered-degree restrictions, and degree-preserving card comparisons; the four-point endpoint is checked by the complete classification of the 16 four-point posets. Every unresolved parent therefore contains a proper anchor. Occurrence hulls of longest chains and suitable principal neighborhoods provide concrete candidates. In a prime parent, an anchor maximal among all proper anchors has an antichain exterior whose distinct attachment rows lie in one automorphism orbit. With at least three exterior points, the anchor-containing cards recover the corresponding one-deletion subset deck. The remaining issue is simultaneous alignment under a single anchor automorphism. The paper distinguishes this full maximality hypothesis from restricted notions of maximality, which yield weaker information. The alignment framework uses comparisons between separate occurrences of cards, common-neighbor matrices, degree constraints, and proper walk moments. For a selected card with a ≥ 3 points, comparability matrix M, and unknown binary attachment vector b, the deck determines bᵀMᵏ1 and bᵀMᵏb for 0 ≤ k ≤ a − 2. Reconstruction follows when the compatible binary vectors form one orbit under the card’s order-automorphism group. An explicit kernel condition gives a sufficient test. A separate orientation argument then recovers the ordered attachment. A whole-card compatibility construction gives an exact orbit formula for the number of parent types realizing a deck of order at least five. This formula organizes the global consistency problem, while matrix and rooted-moment constraints provide further separators. It does not prove that the parent count is always one. Its state space may be factorial, even when the compatibility constraints have a width-three presentation. What the deck reveals about clone placement The common-base problem is studied through internal fiber deletions and boundary deletions. A polynomial model and normalized merger operators describe how boundary information is lost when marks are forgotten. The reconstruction question asks whether a genuine difference of two parent types can survive both deletion operators. Explicit multi-parent cancellation examples show why injectivity on the entire linear space would be a stronger, generally unavailable conclusion. For a doubled point u in a known base R, the ordinary deck recovers the contracted deletion data of R − u at every depth. The proof sorts deletions according to whether they retain both fiber points, retain one, or remove both, and inverts the resulting triangular recurrence. A concrete example shows that this hierarchy can distinguish placements whose single contracted omitted-root cores agree. The recovered data remain unmarked contractions; the theorem does not restore information discarded by contraction. A complementary spectral invariant assigns separate variables to upward, downward, and incomparable relations. An exact determinant identity, combined with the characteristic-polynomial derivative supplied by the ordinary deck, recovers the full deleted-root relation polynomial over a known base. This strictly strengthens the earlier deleted-base comparability spectrum. An eleven-point example separates the two invariants. Equivalent diagonal moment conditions require only finitely many powers for each fixed base. The induced-pattern transform identifies precisely which rooted information is accessible. Participation counts extend from prime patterns to every pattern without an autonomous pair of the relevant fiber type. Other patterns require rooted correction terms. Equality of all proper unrooted participation counts determines the unmarked root-deleted base, but does not supply root-retaining marked cards. Likewise, commuting spectral variables recover sums of word moments with fixed letter multiplicities, rather than individual noncommutative words. A height-two example has equal diagonal moments for every word in the upward and downward matrices, although directed color refinement separates the two roots. The quadratic two-parent analysis yields an infinite reconstruction class. Under a support-swapping automorphism, a rank-two determinant identity reduces spectral equality to base walk counts. When the swap is an involution, a signed interaction graph controls the placements: different components give isomorphic parents, while a common balanced component gives distinct decks. The stated comparability-tree case also follows. Reflected alternating paths require arbitrarily long separating walks, so no fixed walk-length cutoff covers this family. Explicit thirteen-point clone examples further delimit the available signatures. Their prime eleven-point base produces nonisomorphic parents with equal ordered-degree data, internal deletion subdecks, comparability spectra, and induced-poset counts through four points. Five-point counts distinguish their decks. These examples expose the limits of partial information; they are not counterexamples to reconstruction. The unrestricted placement problem, including the residual unbalanced quadratic case, remains unresolved. Modular families and regular cards The companion proves reconstruction for arbitrary substitutions of nonempty posets into a prime ordered quotient with at most three order-automorphism orbits. No reconstruction assumption is imposed on the individual fibers. Quotient-card lifting gives further families over critical, anchor-free, axis-separated, and specified height-two quotients, including prime regular comparability quotients, incidence orders of connected simple graphs on at least three vertices, and complementary even-path quotients. Every poset of order at least five with at most six singleton maximal autonomous chain factors reconstructs. A corresponding modular theorem allows arbitrary nonsingleton modules. For a fixed prime quotient K of order s with orbit sizes aᵢ, every substitution whose order exceeds max{s + 1, Σᵢ aᵢ(aᵢ + 1)/2 − 1} reconstructs. This bounds the residual problem for each quotient without bounding all possible quotients. Joint fiber inventories and comparison profiles sharpen these restrictions. Independent exchange components of unique fiber types require sufficient capacity in repeated fiber classes; in a nonisomorphic mate, their total size is strictly smaller than the number of repeated-type positions. Exchanges covering at least half the quotient therefore force reconstruction. Factorial divisibility, active-orbit, phase, and coset-Helly criteria provide additional constraints. For the specified four-orbit chain-or-antichain inventories with equal orbit sizes 4 through 8, the refined residual total orders are 28, 39, 53, 67, and 86. Every poset with a seven-regular comparability card reconstructs. Together with the lower-degree theorems, this covers all regular-card degrees through seven for parents of order at least five. The nonextremal degree-seven case follows analytically. The extremal case is settled by a complete boundary classification: 107 admissible types, 2,481 capacity inventories, 1,379 degree-feasible inventories, and 38 completed relation matrices. Each completion has a reconstruction certificate. A separately written verifier regenerates the classification and agrees on the complete canonical multisets, comprising 23 unmarked card types and 27 marked-parent types. Boundary capacities also reduce the higher-degree residual. An unresolved extremal degree-r restoration has parent order n ≤ r(r + 1)/2. In the nonextremal case, the restoring point has p, q ≥ 4 lower and upper neighbors, with p + q = r, and n ≤ 1 + r