== A. Theorem (main): multi-affinity, degrees, certificate (exact) ==
  m= 1: n=3, deg(P,Q)=(2,3), #terms Q=3, certificate OK
  m= 2: n=5, deg(P,Q)=(2,4), #terms Q=6, certificate OK
  m= 3: n=7, deg(P,Q)=(2,4), #terms Q=10, certificate OK
  m= 4: n=9, deg(P,Q)=(2,4), #terms Q=15, certificate OK
  m= 5: n=11, deg(P,Q)=(2,4), #terms Q=21, certificate OK
  m= 6: n=13, deg(P,Q)=(2,4), #terms Q=28, certificate OK
  m= 7: n=15, deg(P,Q)=(2,4), #terms Q=36, certificate OK
  m= 8: n=17, deg(P,Q)=(2,4), #terms Q=45, certificate OK
  m= 9: n=19, deg(P,Q)=(2,4), #terms Q=55, certificate OK
  m=10: n=21, deg(P,Q)=(2,4), #terms Q=66, certificate OK
  m=11: n=23, deg(P,Q)=(2,4), #terms Q=78, certificate OK
  m=12: n=25, deg(P,Q)=(2,4), #terms Q=91, certificate OK
  m=13: n=27, deg(P,Q)=(2,4), #terms Q=105, certificate OK
  m=14: n=29, deg(P,Q)=(2,4), #terms Q=120, certificate OK
  m=15: n=31, deg(P,Q)=(2,4), #terms Q=136, certificate OK
  m=16: n=33, deg(P,Q)=(2,4), #terms Q=153, certificate OK
== B. zero set: exact rational points in every sign class, m<=10 ==
  m= 1: 2 sign classes = 2^1, each contains an exact common zero
  m= 2: 4 sign classes = 2^2, each contains an exact common zero
  m= 3: 8 sign classes = 2^3, each contains an exact common zero
  m= 4: 16 sign classes = 2^4, each contains an exact common zero
  m= 5: 32 sign classes = 2^5, each contains an exact common zero
  m= 6: 64 sign classes = 2^6, each contains an exact common zero
  m= 7: 128 sign classes = 2^7, each contains an exact common zero
  m= 8: 256 sign classes = 2^8, each contains an exact common zero
  m= 9: 512 sign classes = 2^9, each contains an exact common zero
  m=10: 1024 sign classes = 2^10, each contains an exact common zero
== C. Lemma 3.1 (block lemma): random multi-affine blocks (exact) ==
  250 random instances (k<=5, d<=4, blocks up to 5 vars, extra vars): (a),(b) OK
== D. Corollary 3.3: f_j = z_j1...z_jd - 1 (exact degrees; sign classes) ==
  d=2 k=2: n=5, degrees (2,4), 4 sign classes = 2^2, exact zero in each
  d=2 k=3: n=7, degrees (2,4), 8 sign classes = 2^3, exact zero in each
  d=2 k=4: n=9, degrees (2,4), 16 sign classes = 2^4, exact zero in each
  d=2 k=5: n=11, degrees (2,4), 32 sign classes = 2^5, exact zero in each
  d=3 k=2: n=7, degrees (3,6), 16 sign classes = 2^4, exact zero in each
  d=3 k=3: n=10, degrees (3,6), 64 sign classes = 2^6, exact zero in each
  d=3 k=4: n=13, degrees (3,6), 256 sign classes = 2^8, exact zero in each
  d=3 k=5: n=16, degrees (3,6), 1024 sign classes = 2^10, exact zero in each
  d=4 k=2: n=9, degrees (4,8), 64 sign classes = 2^6, exact zero in each
  d=4 k=3: n=13, degrees (4,8), 512 sign classes = 2^9, exact zero in each
  d=4 k=4: n=17, degrees (4,8), 4096 sign classes = 2^12, exact zero in each
  d=4 k=5: n=21, degrees (4,8) exact; classes 2^15 (not enumerated)
  d=5 k=2: n=11, degrees (5,10), 256 sign classes = 2^8, exact zero in each
  d=5 k=3: n=16, degrees (5,10), 4096 sign classes = 2^12, exact zero in each
  d=5 k=4: n=21, degrees (5,10) exact; classes 2^16 (not enumerated)
== E. Remark 3.2: shift x0 -> x0 - m (exact) ==
  m=1..12: P'(x0-m) = P_m and Q'(x0-m) = Q_m - m P_m exactly
== F. Proposition 4.1 (bounded versions) ==
  (a) membership criterion 1/R <= a_i <= R confirmed on random rational points, m<=4
  (b) m<=5: Ptilde,Qtilde multi-affine of degrees (2,4)/(2,3); phi(p_s) in c+[-r,r]^n and common zero, all s
== G. Remark 3.4: BP Example 2.2 via Newton identities (exact in sigma's) ==
  for k=1..11: N4-2N3+N2 at s1=k, s2=C(k,2) equals (4k-6)s3-4s4-k(k-1)^2(k-2)/2
  sum x^2(x-1)^2 = N4 - 2N3 + N2 confirmed
== H. Lemma 5.2 (Newton polytope / support) random exact tests ==
  300 random instances: 2M = max ell over supp(G)
== I. Proof of Prop 5.5(b): nonzero multi-affine quadratic forms are indefinite ==
  250 random nonzero square-free quadratic forms: T2(e_a+e_b)=c, T2(e_a-e_b)=-c
ALL EXACT CHECKS PASSED
