Suppose \(E(0)=1\), and look at the reciprocal series
$$ H(t)=\frac1{E(-t)}. $$
Imagine its first pole shell at modulus \(\rho\). The shell contains a positive real pole, perhaps repeated, together with one or more nonreal conjugate pairs, also perhaps repeated. This is exactly the collision that makes a naive dominant-pole argument annoying: the positive mode contributes a constant phase and can hide the oscillatory modes on the same modulus.
One positive-factor deflation is enough to remove that hiding place.
Deflation is a literal frequency notch#
If
$$ E(z)=\left(1+\frac z\alpha\right)F(z), $$
then
$$ \frac1{F(-t)} =\left(1-\frac t\alpha\right)\frac1{E(-t)}. $$
At the coefficient level this is the first difference
$$ h_n^{(\alpha)} =h_n-\alpha^{-1}h_{n-1}. $$
At the pole level, the factor \(1-t/\alpha\) cancels the pole at \(\alpha\) and multiplies every other residue at \(\beta\) by \(1-\beta/\alpha\). It is not vaguely “removing a zero.” It is a Christoffel transform that cuts one exact shell frequency.
The corresponding rectangular Toeplitz determinant becomes one formal orthogonal-polynomial evaluation:
$$ D_{r,k}(F) =(-1)^k p_{k,r-k}(\alpha^{-1})D_{r-1,k}(E). $$
That identity is useful by itself, but the shell theorem comes from what repeated poles do after the notch.
Repeated poles become a finite marginal-gain problem#
A pole of multiplicity \(m\) contributes an exponential-polynomial block \(P(n)x^n\) with \(\deg P=m-1\). In a size-\(k\) Hankel determinant, selecting \(r\) generalized modes from that block contributes polynomial degree
$$ r(m-r). $$
The marginal gains are therefore
$$ m-1,\ m-3,\ \ldots,\ 1-m. $$
For several pole blocks, the dominant determinant packets are found by taking the \(k\) largest entries from the union of these odd-step gain lists. A messy confluent asymptotic has become a tiny discrete sorting problem.
Now choose a gain level whose parity is absent from the positive block. If the positive multiplicity already has opposite parity from some nonreal multiplicity, use the original shell. If all parities agree, deflate one positive factor; its multiplicity drops by one and the parity flips.
At the selected marginal cut, every fully paired conjugate choice has phase one. The only remaining freedom is one extra generalized mode from one member of a conjugate pair. The leading term is a real trigonometric polynomial
$$ \sum_j\left(C_j\omega_j^n+\overline{C_j}\,\overline{\omega_j}^{\,n}\right) $$
with no constant frequency. It has mean zero and positive mean square, so it takes both signs infinitely often.
The explicit detector shift#
If \(q\) counts the earlier positive poles, \(M_*\) is the positive multiplicity after choosing the original or deflated branch, and the nonreal pair multiplicities are \(N_j\), define
$$ r(m)=\max\left(0,\left\lceil\frac{m-N_*}{2}\right\rceil\right). $$
Then the rectangular shift
$$ k_*=q+1+r(M_*)+2\sum_jr(N_j) $$
changes sign infinitely often.
The theorem absorbs arbitrary finite shell cardinality, repeated poles, positive/nonreal modulus collisions, and root-of-unity resonances. Its edge is equally exact: the shell must be finite and followed by a strict modulus gap. And this is a detector theorem, not the missing positivity theorem. To use it against off-axis Xi zeros, one still has to prove eventual nonnegativity for a sufficiently broad family of original and one-zero-deflated Xi minors.
Notebook references: C-0198, K-0057, K-0060, K-0061, C-0200