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

                             12 October 2026; version 1.0; unrefereed preprint



                                                       Abstract
              We apply Fawzi’s established obstruction for normalized complex separable states to the
          oriented real Grassmann orbitope C4,12 . Two aﬃne hyperplanes select a section real-linearly
          isomorphic to Sep(3, 3), so this mass ball has no ﬁnite projected semideﬁnite lift. A compact ﬁrst-
          moment argument then disproves the universal degree-one supporting sum-of-squares assertion
          formulated by Sanyal, in joint work with Rostalski. It also gives a negative answer in general to
          Harvey–Lawson’s Question 6.5: some constant comass-one four-form on R12 has no pointwise or
          local norm completion. The failing unnormalized functional may have rational coeﬃcients, but
          no explicit coeﬃcient list or exact comass is supplied. The reduction combines classical Kähler
          faces, established fermionic projector bodies and the prior tensor-to-wedge lemma with Fawzi’s
          theorem; worldwide priority of this assembly is not certiﬁed.


1        The original question and the result
Let Xk,n consist of all decomposable unit real k-vectors in Λk Rn , including both orientations, and let
Ck,n = conv(Xk,n ). Write I(Xk,n ) for the full real vanishing ideal. Sanyal’s report [1, pp.708–709]
asks whether, for every real linear functional ℓ and δ = maxXk,n ℓ, one has
                                         X
                                δ−ℓ=           h2j   (mod I(Xk,n )),     deg hj ≤ 1.                            (1)
                                           j

Thus the hj may be aﬃne, including constants. That source credits joint computational work with
Philipp Rostalski and the equivalence with Harvey–Lawson’s Question 6.5 [2, p.68]. This connec-
tion and the ﬁrst-theta-body formulation already appear in Rostalski’s 2010 slides and Gouveia–
Thomas’s chapter [10, 11].
    Here a ﬁnite semideﬁnite lift means a linear image of a spectrahedron of any ﬁnite size, with
arbitrary aﬃne equalities and auxiliary variables. It is stronger to exclude such lifts than to
exclude an unprojected LMI or one speciﬁed hierarchy. We use the following established theorem:
the normalized complex body

                            Sep(3, 3) = conv{xx∗ ⊗ yy ∗ : x, y ∈ C3 , kxk = kyk = 1}

has no ﬁnite semideﬁnite lift. This is the 3 × 3 case of Fawzi’s Theorem 1 [3, pp.1320–1321], public
in its 2019 preprint. His deﬁnition allows projected Hermitian LMIs and therefore also excludes
real-symmetric lifts. We do not reprove that external theorem.
    ∗
        Independent researcher. ORCID: 0009-0001-9320-500X.


                                                           1
Theorem 1. The body C4,12 has no ﬁnite semideﬁnite lift. Consequently (1) fails for some linear
functional on Λ4 R12 , which may be chosen to have rational coeﬃcients. Some constant closed
comass-one four-form on R12 has no norm completion by ﬁnitely many four-forms, even at a single
point.
  This refutes the original universal degree-one assertion and the general aﬃrmative answer to
Question 6.5. The conclusion is existential. We claim no explicit failing form, exact value of its
maximum, rational normalized calibration, least dimension or lower-dimensional classiﬁcation.


2    The Kähler face as a projector body
Identify V = R12 with the underlying real space of E = C6 . Fix an orthonormal real basis
e1 , f1 , . . . , e6 , f6 with Jej = fj , Jfj = −ej . Set

                                         X
                                         6
                                    ω=         e∗j ∧ fj∗ ,   Φ = ω 2 /2.
                                         j=1

The classical Wirtinger inequality gives Φ(p) ≤ 1 for p ∈ X4,12 , with equality precisely for complex
two-planes with their complex orientation [2, 4]. For clarity, the compression of J to a real four-
plane has two skew singular values of magnitude at most one. The Pfaﬃan evaluates Φ as their
signed product. Equality forces the compression to preserve every vector’s norm, hence J preserves
the plane; the positive Pfaﬃan selects its complex orientation. Conversely, a complex-orthonormal
pair a, b gives the real frame a, Ja, b, Jb of value one. Thus

                                          F = C4,12 ∩ {Φ = 1}

is exactly the convex hull of these positively oriented complex planes: every positive-weight sum-
mand in a convex expression of value one must itself attain one.
    Let W = Λ2C E, of complex dimension
                                     √     15, with its induced Hermitian metric. In the complex-
iﬁcation of V , put uj = (ej − ifj )/ 2, uI = ui ∧ uj for I = (i, j), i < j. Deﬁne a real-linear
map                                                          X
                       A : Herm(W ) −→ Λ4R V,        A(H) =     HIJ uI ∧ uJ .                   (2)
                                                                   I,J

The result is real: conjugation transposes H, while swapping exterior factors of degree two has
sign +1. The 225 vectors uI ∧ uJ form a complex basis of type (2, 2), so A is injective on the 225-
                                                                                   1P
dimensional real Hermitian space. With wj = ej −ifj , the same formula is A(H) = 4 HIJ wI ∧wJ ;
its real coeﬃcients on thePstandard Hermitian basis are rational.
    For a ∈ E, write a
                     b=      aj uj . Its underlying real vector satisﬁes
                                    √                     √
                          a+a
                     a = (b    b)/ 2, Ja = i(b    a−a b)/ 2, a ∧ Ja = −iba∧ab.

For z = a ∧ b ∈ W this gives

                                                   b ∧ bb = a ∧ Ja ∧ b ∧ Jb.
                                          b ∧ bb ∧ a
                               A(zz ∗ ) = a                                                      (3)

The two factors −i give −1, and moving a   b past bb cancels that sign. A decomposable unit z
has a complex-orthonormal factorization, absorbing its unit phase in a factor. Conversely, every
generating plane of F has this form. Therefore

                 F = A(K),         K = conv{zz ∗ : z ∈ W decomposable, kzk = 1}.                 (4)

                                                        2
The body K is the established mixed fermionic projector hull [6, Deﬁnition 4.2 and Theorem 5.1],
rather than a new notion of separability. Equation (4) is a real-linear isomorphism onto the image.
Evaluation on the Hermitian basis gives Φ(A(H)) = tr H: the coeﬃcient on ei ∧ fi ∧ ej ∧ fj is
H(i,j),(i,j) .


3    The normalized complex separable-state section
Split E = U ⊕ Z orthogonally, using its ﬁrst and last three complex coordinates. Then

                    W = Λ2 U ⊕ B ⊕ Λ2 Z,               B = U ∧ Z,           D = Λ2 U ⊕ Λ2 Z.

Let PP                          P onto D, and deﬁne K0 = {H ∈ K : tr(PD H) = 0}. If
     D be the orthogonal projector
             ∗
H = t λt zt zt , with λt > 0 and t λt = 1, then
                                                X
                                 0 = tr(PD H) =   λt kPD zt k2 .
                                                               t

Each zt therefore lies in B. Positivity excludes cancellation and proves the full generator statement,
rather than only matrix support.
    The unitary map T : U ⊗C Z → B deﬁned by T (x ⊗ y) = x ∧ y takes rank-one tensors exactly to
nonzero decomposable bivectors in B. This is a specialization of the prior tensor-to-wedge theorem
of Vrana–Lévay [5, §3.1, equation (25)]. Its elementary proof here is short. If a ∧ b ∈ B is nonzero,
the U projections of a, b are dependent, as are their Z projections, because both pure exterior
components vanish. Neither projection span is zero, or the wedge would vanish. Taking generators
x ∈ U, y ∈ Z, write a = αx + γy, b = βx + δy. Their wedge is (αδ − βγ)x ∧ y, with nonzero
coeﬃcient. The converse is immediate. Orthogonality gives kx ∧ yk = kxkkyk; for unit z both
factors can be rescaled to unit norm. Under T the projector becomes xx∗ ⊗ yy ∗ . Hence

                                                 K0 ∼
                                                    = Sep(3, 3).                                   (5)

    This is an aﬃne section in the original real exterior space. Put

                            X
                            3                          X
                                                       6
                     ωU =         e∗j ∧ fj∗ ,   ωZ =         e∗j ∧ fj∗ ,   Q = (ωU2 + ωZ2 )/2.
                            j=1                        j=4

The same coeﬃcient calculation gives Q(A(H)) = tr(PD H), so

                            C4,12 ∩ {Φ = 1, Q = 0} = A(K0 ) ∼
                                                            = Sep(3, 3).                           (6)

We need Q ≥ 0 only on F ; it need not be nonnegative on the whole mass ball. The equations are
simultaneous, and membership in the mass ball is essential. They alone do not deﬁne the separable
set.
     Aﬃne sections and linear maps preserve ﬁnite semideﬁnite lifts: additional aﬃne equations on a
projected point can be imposed on the lift variables. A left inverse of A on its image extends linearly
to the ambient space. Thus a lift of C4,12 would give a lift of Sep(3, 3) through (6), contradicting
Fawzi. This proves the ﬁrst assertion of Theorem 1.




                                                         3
4    Supporting aﬃne squares and calibration completion
We give the required compact moment argument explicitly. It specializes the established theta-
body and lift framework [7, 8, 9]. Let X ⊂ S N −1 ⊂ RN be nonempty and compact, and set
J2 = I(X) ∩ R[p]≤2 . Choose a ﬁnite basis of this vector space. For a symmetric (N + 1)-square
matrix M , deﬁne
                      LM (1) = M00 , LM (pi ) = M0i , LM (pi pj ) = Mij .
Consider the spectrahedron and its projection

       SX = {M  0 : M00 = 1, LM (g) = 0 (g ∈ J2 )},       TX = {(M01 , . . . , M0N ) : M ∈ SX }.
               P 2
Because 1 −       pi ∈ J2 , every feasible matrix has trace two. Positive semideﬁniteness bounds
all its entries; thus SX and TX are compact. Rank-one matrices (1, p)(1, p)T for p ∈ X imply
conv(X) ⊆ TX .
    If every supporting aﬃne function were an aﬃne SOS on X, then for hj (p) = aTj (1, p) the
                      P
discrepancy δ − ℓ − j h2j would lie in J2 . Every feasible M , projecting to x, would satisfy
                                                 X
                                    δ − ℓ(x) =       aTj M aj ≥ 0.
                                                 j

Intersecting all supporting halfspaces would give TX = conv(X), a ﬁnite lift. There is no appeal to
SDP duality, Slater conditions, or closure of a nonclosed shadow.
    Apply this to X = X4,12 ⊂ S 494 , with N = 495. Since the mass ball has no ﬁnite lift, TX
strictly contains it. Choose x ∈ TX \ C4,12 and strictly separate it by a coeﬃcient vector c:

                                         c · x > max c · p.
                                                 p∈X

The positive gap persists under small perturbations, since x and X are bounded. Choose a rational
vector c′ in that neighborhood and put ℓ(p) = c′ · p, δ = maxX ℓ. A lifting matrix for x makes
δ − ℓ(x) < 0, which the preceding SOS calculation forbids. Thus a rational unnormalized failing
functional exists. Signed coordinate wedges belong to X and span the exterior space, so nonzero
ℓ has δ > 0. The constant form φ = ℓ/δ is closed and has comass one, including absolute-value
normalization because X = −X.
   If linear forms ψj on the exterior space satisﬁed the norm completion in Question 6.5, then on
unit decomposable vectors
                       X                                                  X
               φ(p)2 +    ψj (p)2 = 1 =⇒ 1 − φ(p) = 12 (1 − φ(p))2 + 12      ψj (p)2 .
                       j                                                   j

This aﬃne SOS contradicts the chosen functional. A local completion by diﬀerential four-forms
would give such a pointwise identity at each point, so none can exist. This proves the remaining
assertions.
    For clarity on the prior equivalence, the converse at a point is also elementary. If 1 − φ(p) =
ka + Bpk2 on X, evaluation at ±p and spanning give B T a = −φ/2 and kak2 + kBpk2 = 1.
At
 √ a maximizer    p0 , Bp0 = −a; evaluation at −p0 gives kak2 = 1/2. Projecting onto a⊥ yields
k 2Pa⊥ Bpk2 = 1 − φ(p)2 . Homogeneity extends this to all simple vectors. No assertion about
choosing smooth local completions from arbitrary pointwise solutions is required.




                                                 4
5    Related results, veriﬁcation and limits
Nonrepresentability for quadratic Grassmann cones is prior: Bettiol–Kummer–Mendes [12, Corol-
lary 4.2 and Remark 4.3] treat nonnegative homogeneous quadratics in Sym2 (Λk Rn ). Blekherman–
Smith–Velasco [13, Theorem 1.1] characterize linear SOS for nonnegative projective quadratics. The
map p 7→ ppT is quadratic and loses orientation; these assertions do not themselves give the needed
aﬃne-linear section of the oriented exterior-power mass ball. We make no ﬁrst Grassmann-related
no-shadow claim. Sanyal–Sottile–Sturmfels [14, Theorem 7.6] exclude an unprojected spectrahe-
dron for G3,6 ; their discussion distinguishes projected lifts. That weaker obstruction is not used
here.
    The veriﬁcation package includes the complete pinned original proof and coordinate certiﬁcate,
independently rederived geometry and moment/source audits, bounded priority records and exact
arithmetic checkers. The coordinate work checks reality, rank 225, both exposing functionals, the
full polynomial projector identity and mixed Plücker/minor relations. Hostile orientation, support,
phase and minor cases test fragile boundaries. Finite computations corroborate the elementary
bridge; they do not reprove Fawzi, certify the universal conclusion or produce an explicit failing
form.
    The priority audit used 84 focused queries and 25 primary PDF retrievals, including multiple
versions. No inspected body supplied the complete target assembly or negative conclusion, but
unindexed or diﬀerently formulated work remains possible. This is a bounded no-match ﬁnding,
not proof of originality or current worldwide openness. The Kähler exposing form itself is a clas-
sical positive example, not the existential failing calibration. All lower-dimensional cases remain
unaddressed here.
    AI tools were used extensively in solving, verifying, source auditing and preparation. This
preprint is unrefereed; independent adversarial reviews are AI reviews. Conventional human peer
review, journal acceptance and formal proof-assistant certiﬁcation are not claimed.


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                                                5
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