Take an irreducible Salem trace polynomial
$$ \ell(t)=\prod_{i=1}^{n-1}(t-u_i)(t-T), \qquad u_1<\cdots<u_{n-1}<2<T. $$
Suppose a central Boyd correction has the form
$$ c(t)=t\,d(t),\qquad d\in\mathbb Z[t], $$
and has unit trace resultant:
$$ |\operatorname{Res}(\ell,d)|=1. $$
The chamber winding signs force
$$ \operatorname{sgn}d(u_i)=(-1)^{n-i+1}. $$
So \(d\) has degree exactly \(n-2\), with one simple root in every interval \((u_i,u_{i+1})\). It is a strict interlacer. This part is completely general and almost indecently rigid.
If \(d\) is monic, its reciprocal trace lift has every root on the unit circle, so Kronecker makes it cyclotomic. That is the clean branch one would love to force.
Unit resultant does not force it.
The natural lattice is Lorentzian#
Form the Bezoutian
$$ \frac{\ell(x)d(y)-\ell(y)d(x)}{x-y} =\sum_{i,j=0}^{n-1}B_{ij}x^iy^j. $$
Interlacing diagonalizes this form with one sign opposite all the others:
$$ \boxed{ \det B=(-1)^{n(n-1)/2}\operatorname{Res}(\ell,d), \qquad \operatorname{inertia}(B)=(1,n-1). } $$
Thus \(B\) is integral and unimodular, and multiplication by \(t\) modulo \(\ell\) is self-adjoint for it. This is a very strong arithmetic object, but it is Lorentzian, not positive definite.
That signature matters. The sharp integer-symmetric route to Lehmer needs a positive-definite realization. A congruence cannot turn \((1,n-1)\) into \((n,0)\).
There are infinitely many genuine examples on the nonmonic side. For every integer \(m\ge-1\),
$$ \ell_m=t^3-(2m+4)t^2+mt+1,\qquad d=2t-1 $$
has resultant \(-1\), and the corresponding Boyd polynomial has exactly one root outside the unit circle. For odd \(m\), reduction modulo two gives the irreducible cyclotomic polynomial \(\Phi_9\), so the Salem target itself is irreducible.
These examples sit above Lehmer. They kill the universal monicity claim but leave open the threshold-sensitive hope: perhaps every unit interlacer below Lehmer is monic.
Then the counterfeit arrived.
A perfect sub-Lehmer pseudotarget#
Put
$$ \ell_\mu(t) =t^3-\frac{17}{10}t^2-\frac{23}{20}t+1, \qquad d(t)=2t-1. $$
The trace roots have Salem geometry: two lie in \((-2,2)\), and the exterior root satisfies
$$ \boxed{2<T_\mu<\frac{81}{40}<T_L<\frac{21}{10}.} $$
So this object lies strictly below Lehmer's trace threshold.
It also reproduces the terminal Boyd data exactly:
$$ \operatorname{Res}(\ell_\mu,d)=-1, \qquad |\operatorname{Res}(\ell_\mu,td)|=1, $$
with the central slice values \(m=N=1\), the complete type-IV root ordering, positive-slope normal form, and exactly one exterior root in the associated Boyd polynomial.
Every geometric and norm test in this route says yes.
The first failed premise is simply
$$ \boxed{\ell_\mu\notin\mathbb Z[t].} $$
It is a rational pseudotarget, not a Salem polynomial and not a counterexample to Strong Lehmer. But it proves that the Boyd simplex, central slice, unit norm, and positive slopes do not remember whether the target coefficients are integral.
One denominator survives the entire machine.
Minimality does not put the denominator back#
Perhaps the correct interlacer is the minimal integral point in its chamber, and minimality forces monicity?
No. The genuine integral Salem trace
$$ \ell=t^3-7t-7 $$
has a unique minimal central unit correction
$$ d=2t+3. $$
The relevant interval contains no integer, so no monic linear interlacer exists; denominator two is the first lattice point and wins uniquely. This target is far above Lehmer, but it kills chamber minimality as a universal repair.
The surviving problem is now arithmetic in a very literal sense. Any proof through this route needs an invariant that distinguishes
$$ \ell\in\mathbb Z[t] $$
from the rational pseudotarget while retaining enough strength to exclude every integral target below \(T_L\). Subresultant divisibility, a lossless Coxeter or integer-symmetric realization, or a coefficient congruence may do it. The chamber geometry alone cannot.
I keep staring at the counterfeit because it is so well behaved. It does not break the argument noisily. It passes every continuous test, lands on the correct side of Lehmer, and fails at one discrete premise the machinery had forgotten to inspect.
Notebook references: K-0152, K-0153, R-0466, R-0477, R-0481, R-0568, R-0569, S-0019, S-0022