Global Hessian realizations of statistical manifolds

Every smooth Hausdorff second-countable statistical manifold of fixed finite dimension n at least 1, without boundary and with positive-definite metric, admits a proper global statistical embedding into an open subset of R^N with one smooth positive Hessian potential, where N=binomial(2n+4,3). Orthogonal projection of the standard flat connection and its metric dual induces the prescribed ordered pair of torsion-free dual connections. For a general mutually dual pair, both torsions vanishing is necessary and sufficient for realization by a torsion-free dually flat ambient manifold. Countable zero-dimensional inputs are treated separately. The ambient neighborhood may be nonconvex and incomplete; dual affine coordinates are local. No finite-probability-model, global-gradient-injectivity or optimal-dimension conclusion is asserted. The construction answers the historical realization question in item1(a) of Furuhata, Matsuzoe and Urakawa, Open Problems in Affine Differential Geometry and Related Topics (1998), in the stated smooth positive boundaryless scope. A proper Whitney embedding followed by all polynomial features through degree three gives a fixed finite scalar-jet spanning family. A smooth fiberwise right inverse produces an exact global Lauritzen pair, including the prescribed cubic and connection signs. A normal-neighborhood extension and a nonuniform Schur-complement penalty give one positive Hessian potential on a neighborhood of the entire embedded manifold, without compactness or uniform geometric bounds. This is a fully credited self-contained construction note. Lauritzen, Matumoto, Le, the Nash/Gromov free-map lineage, Marugame, Whitney and tubular-neighborhood theory provide substantial ancestry; Osipov gives genuine global special-curvature predecessors. Early unrestricted finite-simplex claims are disclosed. Actual Ay-Jost-Le-Schwachhofer2017 Theorems4.10 and4.11 distinguish finite models for compact inputs from countable models for noncompact inputs. Bounded independent primary-source priority audits did not establish a verified unrestricted predecessor, but material final-body access gaps remain. Historical novelty, first priority and continued openness of the question are not certified. The deposit includes the editable LaTeX source and an explicit-inventory verification package preserving all33 original PR946 files byte for byte, curated mathematical and priority reviews, exact symbolic controls and separately bounded numerical replay. The original strict platform-dependent floating-output mismatch is preserved; current transport compares twelve named finite sample floats under documented tolerances and every other mathematical field exactly. Those tolerances are diagnostics, not proved roundoff bounds, and finite controls do not replace the universal proof. The original PR declares4/5 proof responses, but native historical invocation events were unavailable; the audit adds zero proof-search responses and fabricates no history. Raw third-party source bodies are excluded. AI tools were used extensively in research, proof development, source checking, computation, writing and adversarial review. This preprint is unrefereed; no independent human peer review, journal acceptance or formal proof-assistant certification is claimed.