Add certified polynomial enthalpy bounds and feasible direction repair

experiments/POLYNOMIAL_ENTHALPY_ADMISSIBILITY_2026-09-17.md

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+# Polynomial enthalpy admissibility and feasible direction repair
+
+This branch adds a sufficient polynomial nonnegativity certificate and an interchangeable enthalpy direction repair. It does not alter the EOS or declare a constrained residual minimum an equilibrium.
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+`TensorBernsteinPolynomial<Dimension>` encodes spatial dimension in its type and accepts polynomial order through the mesh's runtime nodal points. A cached one-dimensional Lagrange-to-Bernstein transformation is applied by tensor contractions. Every nontrivial arithmetic operation in the transformation and de Casteljau subdivision encloses its result using `nextafter`; a polynomial is certified only when all coefficient lower bounds in every leaf are nonnegative. The implementation assumes ordinary IEEE binary64 arithmetic, gradual underflow, and no unsafe fast-math transformations. The underlying mathematical FE is the Lagrange polynomial at the supplied floating-point nodal coordinates.
+
+A negative control coefficient is not a negative polynomial witness. Such cells are refined. A negative upper interval bound at a tensor corner proves a negative polynomial value. Cells that remain ambiguous at the work budget return `unresolved`, including some truly nonnegative polynomials with tangential interior zeros. No negative tolerance is accepted as nonnegative. Exact endpoint interpolation preserves identically zero tensor faces without manufacturing a small negative error interval.
+
+The production adapter supports conforming scalar nodal H1 hexahedra with closed Gauss-Lobatto, closed uniform, or closed Gauss-Legendre points. Unsupported element types are rejected explicitly. Its reduced field map and MFEM prolongation preserve shared true DOFs; no elementwise coefficient clipping occurs. The prepared object assumes the finite-element space and topology remain unchanged. Mesh adaptation requires a new object.
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+Given a certified base `h`, a certified anchor `a`, and proposed direction `p`, the repaired full candidate is `(1-beta)*(h+p)+beta*a`. Thus the altered enthalpy direction is `(1-beta)*p+beta*(a-h)`. Zero-trace reduced rows are fixed to zero. The adapter constructs a sufficient convex cone from a subdivision on which the anchor has nonnegative Bernstein bounds, and computes the largest required fraction across all coefficients and MPI ranks. This is the minimum sufficient fraction for those conservative coefficient inequalities, not the least fraction for the complete nonnegative polynomial set. A caller can choose any larger fraction through one, then verify the actual rounded full candidate. The original direction is left unchanged when verification fails.
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+This changes the enthalpy direction while allowing the stellar strategy to retain the other field directions. It is a feasible direction repair, not a constrained Newton optimizer. Full residual acceptance, actual descent checks, and the original physical convergence requirements remain necessary. Invalid or unresolved bases are rejected explicitly; this mechanism is not a feasibility-restoration solver. A certified checkpoint or initial seed is required as anchor. A frozen repaired full candidate and certified base define a nonnegative exact-arithmetic segment for backtracking; every floating-point candidate must nevertheless pass the same check.
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+Collective adapter calls must occur in the same order on all ranks. Sign status, polynomial bounds, and repair fraction are globally combined. Witness element indices, coordinates, and work counters remain rank-local. Unequal repair fractions and work budgets are rejected collectively.
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+Tests cover nodal-positive/interior-negative cubics, a positive polynomial with negative coarse Bernstein coefficients, exact zero faces, very small negative pockets, unresolved tangential zeros, arbitrary runtime degree, nonfinite input, finite-element ordering, true-DOF sharing, invalid bases, immutable failure output, and prescribed trace handling. Integration tests are intended for both one and two MPI ranks.