Log-concavity of labelled-poset Eulerian polynomials: Near-chain theorems and uniform representations

Overview This paper studies the descent-count coefficients of labelled-poset Eulerian polynomials. For a finite poset P with an injective numerical labelling, its Eulerian polynomial counts linear extensions according to their number of descents. Brenti's 1989 conjecture asks whether these coefficients are always log-concave and have no internal zeros. The paper establishes log-concavity for several classes with unbounded chain length, develops exact representations valid for every labelling, and specifies the remaining general inequality. Principal log-concavity theorems Arbitrary labels and a chain omitting at most three vertices. Every labelled poset of height at least |P| - 3 has log-concave descent coefficients. Height counts vertices in a longest chain. All induced orders on the exceptional vertices are covered: the antichain, one relation, a total chain, a fork, and its dual. Chain relations, insertion-window endpoints, endpoint ties, and numerical labels are unrestricted. Natural labels and Eulerian degree at most five. Every naturally labelled poset of degree at most five is log-concave. The degree identity deg W = |P| - h(P) makes this equivalent to a longest chain omitting at most five vertices. All exceptional orders, insertion windows, endpoint ties, and chain lengths are covered. The degree-four theorem supplies the lower-degree proof base: exact endpoint-order certificates cover its nonchain exceptional orders, and a known naturally labelled width-two theorem covers the remaining total-chain order. Natural labels and antichain complements. A naturally labelled poset consisting of a chain and at most six mutually incomparable exceptional vertices has strictly log-concave coefficients on its support, whenever the support contains an interior index. All insertion windows and chain lengths are allowed. With arbitrary labels and at most two incomparable exceptions, the polynomial is real-rooted. Explicit five-exception subfamilies. Detailed component proofs treat five exceptional vertices inducing exactly one relation, or two disjoint two-element chains and one singleton (the order 2+2+1). Both component theorems permit arbitrary chain lengths and insertion windows, including coincident endpoints and point windows. Other proof components handle arbitrary induced exceptional orders when all upper chain neighborhoods coincide, when all lower chain neighborhoods coincide, or when the lower endpoints and then the upper endpoints are strictly increasing with common overlap. These families are included in the complete degree-five theorem. Uniform identities and support A set-partition formula enumerates a chain with any number of incomparable exceptions under arbitrary labels. Scalar word formulas and ideal-matrix boundary gauges compress unrestricted chain lengths into finite transfer data. Both edge inequalities have elementary all-degree proofs for naturally labelled chain-antichain families. For natural labels and any number of disjoint exceptional relations, a matching identity expresses the Eulerian polynomial through relation deletion, window clipping, and contraction; two relations give an explicit nine-term formula. For every finite labelled poset, the paper gives signed convex-quotient and reversed-face formulas, an adjacent-label cover-contraction identity, a relative descent-complex representation, and a positive decomposition into compatible decreasing Cartesian-tree orbits. Each tree orbit contributes a polynomial of the form 2^v t^a (1+t)^q. These are exact enumeration identities; their sums require additional correlation inequalities to establish general log-concavity. A descent-marked completion law holds for every injective labelling and retains the internal descents of the exterior together with both numerical boundary labels. Under natural labels, an interface of five active vertices requires at most 112 scalar prefix polynomials. An exact margin identity separates component log-concavity, dispersion between coefficient ratios, and contributions from components with zero lower coefficient. These identities specify the aggregate inequality required of a completion argument. The minimum and maximum descent counts are determined by weighted saturated chains. Connectedness of the linear-extension swap graph proves that the entire interval between these endpoints occurs. A support span of s yields two chains whose union contains at least |P| - s vertices and, consequently, the width bound width(P) <= s + 2. Proofs and reproducibility The arguments combine combinatorial proofs with exact rational, monomial, and integer binomial-basis certificates. The natural degree-five proof covers all 2,017,513 admissible positive-window endpoint/order cells. Combinatorial reductions and component certificates leave 1,689,694 cells, represented by 455,859 classes under actual poset isomorphism and order duality. The complete primary positivity scan checks 1,929,192,024 positive nonzero defect coefficients across these representatives; omitted coefficients are zero. Independent enumeration reconstructs the full endpoint coverage and the complete representative key set. A separate independent global-word audit checks 473 sampled cells, covering all 59 residual exceptional-order types and eight gap dimensions, and matches 253,013 symbolic coefficients. Independent ideal/last-label dynamic programs also agree on 946 finite specializations. The independent polynomial audit is stratified, rather than a complete second polynomial replay. The degree-four result includes complete independent reconstruction of 20,717 endpoint cells and 8,516,827 nonzero positive defect terms. The arbitrary-label fork theorem includes complete independent replay of 32,768 cells and 124,996,265 positive terms. For the natural 2+2+1 theorem, all 81,474 weak endpoint cells are certified by 263,744,844 positive nonzero defect coefficients; omitted coefficients are zero. A complete replay with the original exact transfer engine reconstructs every cell and all four defects. Independent geometric enumeration verifies the full endpoint-cell set, while a separate matching-identity audit checks a 217-cell symbolic sample. The descent-marked completion identity is also checked on all 4,473 labelled posets through five vertices and 85,785 admissible interfaces. The verification records distinguish independent derivations, complete same-engine replay, selected algebra checks, and finite-poset tests. The accompanying source archive contains proof notes, certificate ledgers, coefficient records, independent counting and verification programs, documented reproduction commands, and a SHA-256 manifest. The unrestricted conjecture, arbitrary-label coheight four, and unrestricted higher natural degrees remain unresolved by these results. No general real-rootedness theorem is asserted.