No single polynomial inequality defines a real binary Gram locus

For a real tensor with n binary factors, the Gram locus is the tuple of determinants of its one-mode Gram matrices as the tensor ranges over the closed Frobenius unit ball. For every n>=3, no single real polynomial nonnegativity condition in those determinant coordinates defines this locus inside the cube [0,1/4]^n, or inside its known convex hull. This gives a negative answer to Seigal's single-polynomial question for n>=4. The self-contained proof uses established marginal polygon inequalities, an explicit real three-factor realization, exact polynomial multiplicities and zero-determinant slices. A strictly positive rational four-factor tuple also passes the proposed polynomial condition but is unrealizable at any allowed norm. The marginal criteria and real constructions of Higuchi-Sudbery-Szulc, Bravyi and Domanov-Stegeman-De Lathauwer are credited antecedents. Sinn's earlier general boundary/interior obstruction is also credited. Alper Ferudun's unrefereed October 2, 2026 preprint, DOI10.5281/zenodo.23111048, contains an overlapping stronger theorem, including an all-positive domain. Our original theorem-bearing proof is present in GitHub PR170, created September 30, 2026 at13:15:23UTC, and is unchanged. We describe these as contemporaneous separate disclosures. Commit timestamps are author-controlled; current public repository visibility was verified, but historical visibility was not independently archived. This October12 formal preprint is not backdated. Mutual independence of authorship, literal simultaneity, exclusive firstness and complete historical novelty are not certified. The accompanying bounded priority audit records exact source versions, read coverage and access limits. The deposit contains the manuscript PDF and a support archive with standalone LaTeX, exact standard-library controls, ordinary and optimized outputs, deliberately false variants, chronology evidence and priority reviews. Finite controls supplement the universal written proof; they cannot prove an all-polynomial impossibility or certify historical priority. AI tools were used extensively in solving, drafting, source comparison, reproduction and adversarial verification. This work is unrefereed and has not undergone conventional human peer review. The author is responsible for its contents.