An application of separable-state nonrepresentability to Grassmann orbitopes and calibrations

An application of Fawzi's established finite semidefinite lift obstruction for normalized complex separable states to the oriented real Grassmann orbitope C(4,12), the mass ball in the fourth exterior power of real 12-space. Two affine hyperplanes select a section real-linearly isomorphic to Sep(3,3). This mass ball has no finite projected semidefinite lift of any size. A compact first-moment argument refutes the universal degree-at-most-one supporting sum-of-squares assertion in Sanyal's Oberwolfach Report 13/2015, pp. 708–709, in joint work with Rostalski. It gives a negative answer in general to Harvey–Lawson Question 6.5: some constant closed comass-one four-form has no pointwise or local norm completion. The conclusion is existential. A failing unnormalized functional with rational coefficients exists, but no explicit coefficient list, exact comass, rational normalized calibration, least dimension or low-dimensional classification is supplied. Classical Wirtinger and Kähler faces, established fermionic projector bodies, Vrana–Lévay's tensor-to-wedge decomposability lemma, theta-body and lift frameworks, and the earlier Sanyal–Rostalski equivalence are credited. Fawzi supplies the decisive no-lift theorem, public in 2019 and published in 2021; it is not reproved. Nearby quadratic Grassmann cones and unprojected LMI obstructions concern different embeddings or conclusions. A bounded priority comparison, with 84 queries and 25 primary PDF retrievals, found no full target predecessor in inspected bodies. Worldwide priority and current worldwide openness remain unestablished. The deposit includes the proof, TeX source, scope and priority supplement, authored independent adversarial audits and portable exact symbolic arithmetic verification (Python 3.10 or later, standard library only). The three unchanged symbolic baselines verify the full coordinate, polynomial projector and mixed-minor identities. Forty-five defective-certificate executions test embedding coefficients and certificate structure or metadata; they reject before the latter identity checks. Finite checks do not replace the written proof or Fawzi's theorem. AI tools were used extensively in solving, verifying, source auditing and preparation. This is an unrefereed preprint with AI adversarial reviews; conventional human peer review, journal acceptance and formal proof-assistant certification are not claimed.