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1 \clearpage
2 \section{A nonzero Hankel contact determinant}\label{sec:hankel}
3 \subsection{Independent moment directions replace one filter}
4 For an integer $r\ge1$, define
5 \[
6  \mathcal H_r(t)=\det\bigl(\Li_{i+j+1}(t)\bigr)_{0\le i,j<r}.
7 \]
8 It uses weights one through $2r-1$. The expression is a formal series over $\Q$ and an analytic function for $|t|<1$. No algebraicity at a rational argument is assumed.
9 
10 For the first two ranks,
11 \[
12  \mathcal H_1=\Li_1,\qquad
13  \mathcal H_2=\Li_1\Li_3-\Li_2^2.
14 \]
15 The rank-two determinant compares genuinely different moment directions. Its order-three contact is not inferred from the presence of weight three.
16 
17 \begin{theorem}[Cauchy--Binet expansion]\label{thm:hankel-expansion}
18 \[
19  \mathcal H_r(t)=\sum_{1\le k_1<\cdots<k_r}
20  t^{k_1+\cdots+k_r}
21  \frac{\prod_{i<j}(k_j-k_i)^2}{\prod_{i=1}^r k_i^{2r-1}}.
22 \]
23 \end{theorem}
24 \begin{proof}
25 Write $v_k=(1,k^{-1},\ldots,k^{-(r-1)})^{\mathsf T}$. The moment matrix is
26 \[
27  \sum_{k\ge1}\frac{t^k}{k}v_kv_k^{\mathsf T}.
28 \]
29 For a finite sum, Cauchy--Binet expands its determinant over $r$-element index subsets. The determinant of the chosen columns is a Vandermonde in $k_i^{-1}$. Squaring it and including the factors $t^{k_i}/k_i$ gives the stated coefficient. In detail,
30 \[
31  \prod_{i<j}(k_j^{-1}-k_i^{-1})^2
32  =\frac{\prod_{i<j}(k_j-k_i)^2}{\prod_i k_i^{2(r-1)}}.
33 \]
34 Passing to the formal infinite sum is coefficientwise legitimate. At a fixed power of $t$, only finitely many index subsets occur.
35 \end{proof}
36 Every coefficient in this expansion is positive. The formal cancellation implicit in the determinant is now resolved into a manifestly nonzero expansion.
37 
38 \begin{theorem}[Exact contact and first coefficient]\label{thm:hankel-contact}
39 Put $s_r=r(r+1)/2$. Then
40 \[
41  \ord_0\mathcal H_r=s_r,\qquad
42  [t^{s_r}]\mathcal H_r=\prod_{k=1}^r k^{-(2k-1)}.
43 \]
44 \end{theorem}
45 \begin{proof}
46 The unique increasing $r$-tuple of positive indices with least sum is $(1,\ldots,r)$, of sum $s_r$. Its coefficient is nonzero. Its Vandermonde numerator is $\prod_{j=1}^{r-1}(j!)^2$. Dividing this by $\prod_{k=1}^r k^{2r-1}$ leaves exponent $2k-1$ in the denominator at each $k$, giving the formula.
47 \end{proof}
48 For $r=2$ the first coefficient is $1/8$. For $r=3$ it is $1/(2^3 3^5)$. The contact orders are $1,3,6,10,\ldots$, the triangular numbers.
49 
50 \figplaceholder{16_contact_order}{The Hankel rank produces an exact triangular contact order. The first nonzero coefficient is proved and remains separate from the highest polylogarithmic weight.}
51 
52 \subsection{Real positivity and a bound uniform in rank}
53 \begin{theorem}[Positive real determinant]\label{thm:hankel-real}
54 For every $r\ge1$ and every real $0<t<1$,
55 \[
56  0<\mathcal H_r(t)\le e^{1/4}[-\log(1-t)].
57 \]
58 \end{theorem}
59 \begin{proof}
60 The Cauchy--Binet expansion consists of positive terms and contains the nonzero term $(1,\ldots,r)$, so positivity follows. Equivalently, the real moment matrix is positive definite: a nonzero polynomial of degree below $r$ cannot vanish at all the distinct nodes $k^{-1}$.
61 
62 Hadamard's inequality for the positive matrix gives
63 \[
64  \mathcal H_r(t)\le\prod_{i=0}^{r-1}\Li_{2i+1}(t)
65  \le[-\log(1-t)]\prod_{i=1}^{r-1}\zeta(2i+1).
66 \]
67 For the higher weights, $t^k\le1$ bounds the series by its zeta value. Since all summands are positive,
68 \[
69  \sum_{i\ge1}(\zeta(2i+1)-1)
70  =\sum_{k\ge2}\frac1{k(k^2-1)}=\frac14.
71 \]
72 The last sum telescopes using
73 \[
74  \frac1{k(k^2-1)}=\frac12\left(\frac1{k(k-1)}-\frac1{k(k+1)}\right).
75 \]
76 Finally $\log(1+x)\le x$ for $x\ge0$ bounds the zeta product by $e^{1/4}$, uniformly in its length.
77 \end{proof}
78 At $t=a/c$ and $t=b/c$, the bounds become $e^{1/4}\log(c/b)$ and $e^{1/4}\log(c/a)$. Thus these real period values are nonzero and have a simple logarithmic envelope. This is valuable analytic control. It is not an all-conjugate height theorem for an algebraic scalar.
79 
80 \subsection{The third complex chart has a branch boundary}
81 For fixed rank and a consistent upper or lower boundary branch at $x=c/a>1$, the classical integer-weight inversion formula expresses each $\Li_j(x)$ as a fixed degree-$j$ polynomial in a chosen $\log(-x)$ plus a sign times $\Li_j(1/x)$~\cite{DLMFPolylog}. The latter is bounded by $\zeta(j)$ for $j\ge2$. For $j=1$, use $\Li_1(x)=-\log(1-x)$ and $x-1=b/a$.
82 
83 Since $|\log(b/a)|\le\log c$ and $\log(c/a)\le\log c$, every entry has a bound by a constant times $(1+\log c)^j$. A determinant term has total weight
84 \[
85  \sum_{i=0}^{r-1}(i+\pi(i)+1)=r^2.
86 \]
87 Therefore
88 \[
89  |\mathcal H_r(c/a)|\le C_r(1+\log c)^{r^2}
90 \]
91 for each of those consistent branches. The constant may depend on $r$. No nonvanishing assertion at every argument in this third chart follows from the positive expansion on $0<t<1$.

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