Exact certificate representation

Use JSON for the exact numerical and polynomial data. Rational numbers must be rational strings. A real algebraic number may be specified by an integer polynomial and a rational isolating interval selecting its real embedding; arithmetic in a finite real algebraic extension may be represented by coefficient vectors in a specified power basis. An equivalent explicit exact algebraic representation is acceptable if the verifier checks its interpretation.

Represent each polynomial by a finite list of exponent vectors of length twelve and exact coefficients. Every exponent must be a nonnegative integer. Gram matrices must be square, exactly symmetric, and accompanied by an exact positivity argument or exact verification procedure.

The primal certificate may list all thirteen SOS Gram representations directly. It may instead exploit permutations of the twelve variables, provided the group action, orbit multiplicities, and normalization factors are explicitly specified and checked. No particular basis, support, block count, matrix size, symmetry decomposition, or coefficient field is prescribed.

The dual certificate may list its values on every monomial through total degree twelve, or give a finite exact rule specifying all of those values. Its verification must establish positivity on the complete degree-six moment space and every complete degree-five localizing space, rather than a chosen subset of polynomials.

All certificate data must be supplied. The verifier may use general-purpose exact arithmetic libraries, but may not recover missing numerical values from an optimizer or rely on tolerances. Include precise commands for running the verifier.
