Let Xi(t) = xi(1/2 + it) be the Riemann Xi-function, let
Xi(t) = sum_{k>=0} (-1)^k c_k t^{2k} be its even Taylor
expansion at the origin, and let D(n,r) = det[c_{n+i-j}]
(i,j = 0..r-1) be the Toeplitz minors whose positivity for
all n >= 0, r >= 1 is equivalent to the Riemann Hypothesis
(RH). We prove a sharp and quantitative obstruction in the
resulting hierarchy of criteria.
Parametrising the zeros of G(z) = sum c_k z^k by
beta_j = -1/z_j, we establish by the dual Jacobi-Trudi
identity the exact identity D(n,r) = s_{(r^n)}(beta), the
Schur function of the rectangular partition (r^n) in the
variables beta. A zero of zeta off the critical line
contributes to beta a conjugate pair rho e^{+-i theta} whose
angle theta is exactly the vertical angle of the violating
pair of zeros, while zeros on the critical line contribute
positive real variables. For two variables we prove the
closed form
s_{(a,b)}(rho e^{i theta}, rho e^{-i theta})
= rho^{a+b} sin((a+1-b) theta) / sin theta
= rho^{a+b} U_{a-b}(cos theta),
U_k being the Chebyshev polynomial of the second kind, so
that the sign is governed by the classical criterion
U_k(cos theta) > 0 if and only if (k+1) theta < pi.
Since a Schur function in two variables can be expanded
through the coproduct s_lambda(X sqcup Y) = sum_mu s_mu(X)
s_{lambda/mu}(Y), and since s_mu(X) > 0 for positive real X,
positivity reduces to the two-variable factors, whose index
k = a - b is bounded by the number of columns r, not by the
shift n. This yields:
Blind Zone Theorem. If a violating pair subtends a vertical
angle theta < pi/(r+1), then every Toeplitz minor D(n,r')
with r' <= r is strictly positive, notwithstanding the
presence of non-real zeros.
The critical strip together with gamma_1 = 14.134725141...
forces theta <= theta_max = 2 arctan(1/(2 gamma_1)) =
0.07071826... rad, whence pi/theta_max = 44.424... and
therefore every order r <= 43 is completely
non-discriminating.
We emphasise that the exponential bound is on the order r
and not on the shift n; we exhibit the mechanism and show
the contrary n-based bound is false. We also record the
refutation of a "universal covering law" previously
conjectured by the author, and a floating-point artefact
that nearly produced a false refutation of the main theorem.
No claim of a proof of RH is made.