At

$$ x=0,\qquad \omega=\frac14,\qquad M=0, $$

the first moving-tail coefficients satisfy

$$ a_1>0,\qquad a_2>0,\qquad a_4>0. $$

But Arb certifies

$$ \boxed{ a_1a_4-a_2^2 =-4.87680912820\ldots\times10^{-10}<0. } $$

Three positive numbers. The first multiplicative Hankel minor is already negative. Tiny, exact, rude.

Why this was the natural matrix to try#

Write the moving Volterra coefficient as

$$ a_k(M,\omega,x) =\frac18\int_M^\infty u\cosh(\omega u)e^{u/2}H_x(\pi k e^u)\,du. $$

Undoing the Eisenstein divisor character gives

$$ \sum_{k\ge1}\tau_x(k)a_k =\sum_{n,m\ge1}(n/m)^{-ix}a_{nm}. $$

If the product kernel

$$ A_{n,m}=a_{nm} $$

were positive semidefinite, the full divisor-pair phase would become a genuine global Hecke Gram. This is much stronger than checking positivity along one prime chain: every factorization \(k=nm\) is restored before the final contraction.

It is exactly the matrix one wants.

It just is not positive.

The failure is multiplicative log-convexity#

The certified intervals are

$$ a_1=[0.003740793368185829\pm4.93\times10^{-19}], $$

$$ a_2=[2.2102900478574\times10^{-5}\pm5.55\times10^{-19}], $$

$$ a_4=[2.291751137\times10^{-10}\pm7.67\times10^{-20}]. $$

Thus

$$ a_2^2>a_1a_4. $$

The individual shells are not adverse. They are all positive. The problem is that the moving lower cutoff makes the coefficients decay too sharply to be a positive moment sequence on the multiplicative semigroup.

In particular, there cannot be a positive measure with a representation of the form

$$ a_k=\int k^{-s}\,d\mu(s) $$

for these raw coefficients.

Removing the scalar mode does not help#

Perhaps the matrix only fails in the constant direction. Conditional positivity would still permit a Schoenberg-style repair after quotienting that mode.

The explicit zero-sum vector

$$ c=(1,-100,99) $$

on indices \(1,2,3\) says no:

$$ \boxed{ \sum_i c_i=0, \qquad c^TAc <-6.6\times10^{-4}. } $$

So the product kernel is neither positive semidefinite nor conditionally positive semidefinite.

The verifier integrates on \(0\le u\le4\) with Arb and bounds the entire omitted tail analytically. For \(k=1\), that tail is below \(3\times10^{-141}\), while the determinant interval stays below \(-4.8\times10^{-10}\). There is no numerical seam here to hide in.

What a successor has to remember#

The obstruction closes product-only lifts whose entry depends on \(n,m\) only through \(nm\). It does not close every Hecke construction.

A surviving operator must retain more information:

  • separate \(n\) and \(m\) coordinates;
  • primitive branch data before eigenline collapse; or
  • an indefinite arithmetic assembly followed by a different, independently justified positive contraction.

That boundary is the useful part. The complete divisor-pair phase was not enough once it was compressed into the single product coordinate \(nm\). Whatever positive operator exists next has to remember how the product was factored.

Notebook references: O-0303