Every Salem polynomial with exactly six nonzero coefficient units belongs to one of twelve infinite families or is one of \(126\) sporadic polynomials in McKee and Smyth's table.
Unconditionally.
Their published search was exhaustive above Salem number \(1.01\). The remaining gap was the possibility of an unseen length-six polynomial below that cutoff. The whole closure is therefore one inequality:
$$ \boxed{\tau>1.01.} $$
The route to it is delightfully indirect. First show that at least one sixth of the polynomial's degree survives after all cyclotomic factors are removed. Then make prime \(17\) charge that surviving factor through Frobenius and a field norm.
Six terms collapse to one normal form#
Let \(S\) be a monic Salem polynomial of coefficient length six and degree \(N\). Reciprocity or antireciprocity arranges its six coefficient units into three symmetric pairs.
Evaluating at \(1\), and differentiating there in the antireciprocal case, forces both free signs to be negative. After removing a common exponent gcd, every possibility has the primitive form
$$ \boxed{ S_{n,k,d,\varepsilon} =z^n(z^k-z^{k-d}-1)+\varepsilon(z^k+z^d-1), } \tag{1} $$
with
$$ n\ge k\ge d>0,\qquad \gcd(n,k,d)=1. $$
For \(\varepsilon=1\), the strict inequality \(n>k+2d\) is necessary. At equality the polynomial factors completely into
$$ (z^k-1)(z^d-1)(z^{k+d}-1), $$
which is a rather efficient way for a proposed Salem polynomial to admit that it is made entirely of roots of unity.
The cyclotomic part cannot eat more than five sixths#
A cyclotomic root of (1) gives a vanishing sum of six roots of unity. The weight-six classification reduces every such root to one of a small number of shapes:
- three opposite pairs;
- two \(R_3\) triangles;
- the minimal order-\(30\) configuration;
- or one of ten positive-dimensional torsion blocks.
Repeated cyclotomic factors are excluded separately by solving
$$ S(\zeta)=\zeta S'(\zeta)=0. $$
The decomposable cases reduce to \(360\) exact tangency systems; the order-\(30\) case reduces to \(112\) residue systems. None survives. So the cyclotomic part is squarefree.
The same root-of-unity classification bounds its total degree \(C\):
$$ \boxed{C\le\frac{5N}{6}.} \tag{2} $$
Every large torsion block has degree at most \(N/2\). A violation of (2) beyond degree \(1000\) would require two large blocks, hence two bounded integer relation planes. The independent planes are exhausted on \(212{,}484\) primitive rays; the dependent planes die by exact phase contradictions. The finite range through degree \(1000\) is checked exactly.
If \(D\) is the degree of the minimal Salem factor, squarefreeness and (2) give
$$ D=N-C\ge\frac N6. \tag{3} $$
That one sixth is all prime \(17\) needs.
Frobenius turns one sixth into 1.01#
Let \(\tau>1\) be the Salem number. For a monic polynomial of coefficient length \(L\), Frobenius modulo a prime \(p>L\) gives
$$ S(\tau^p)\equiv S(\tau)^p=0\pmod p. $$
Taking the field norm over the degree-\(D\) Salem factor, while bounding every archimedean conjugate by the coefficient length, yields
$$ \tau\ge \left(\frac pL\right)^{D/(pN)}. \tag{4} $$
Now use
$$ L=6,\qquad p=17,\qquad D/N\ge1/6. $$
Equation (4) becomes
$$ \tau\ge \left(\frac{17}{6}\right)^{1/102}. $$
And the old decimal cutoff is crossed by the exact integer comparison
$$ \boxed{ 17\cdot100^{102}>6\cdot101^{102}. } $$
Therefore
$$ \boxed{ \tau\ge\left(\frac{17}{6}\right)^{1/102}>1.01. } $$
I keep staring at the exponent \(102\). It is not pretty in the ornamental sense. It is pretty because every part of it has a job:
$$ 102=17\cdot6. $$
The prime supplies the norm charge; the cyclotomic argument supplies the surviving one sixth.
The finite table was already the whole table#
McKee and Smyth proved that all but finitely many length-six Salem polynomials lie in twelve explicit infinite families, and their exhaustive interval algorithm found exactly \(126\) sporadic examples above \(1.01\).
The inequality above proves that there are no examples below their cutoff. So the published list was not merely conditional on Lehmer's conjecture:
$$ \boxed{ \text{twelve infinite families plus 126 sporadics is complete.} } $$
The external ingredients are exactly the published above-\(1.01\) classification and the classification of minimal vanishing sums of roots of unity through weight six. The new squarefreeness tables, torsion-block degree bound, relation-ray exhaustion, Frobenius norm arithmetic, and final cutoff all replay exactly in the attached packet.
Notebook references: R-0355