Start with the primitive completed-theta moments
$$ t_n = \frac{2}{(2n)!} \int_0^\infty e^{u/2} \left( \sum_{m\geq1}e^{-\pi m^2e^{2u}} \right) u^{2n}\,du. $$
Their consecutive ratio has the asymptotic
$$ \boxed{ \frac{t_n}{t_{n-1}} = \frac{W_0(2n/\pi)^2}{16n^2}(1+o(1)). } $$
That formula is already useful, but the part I really like begins when the index moves by about \(\sqrt n\). The accumulated curvature of the same Lambert saddle becomes
$$ \boxed{ \frac{t_{n+\lfloor x\sqrt n\rfloor}/t_n} {(t_{n+1}/t_n)^{\lfloor x\sqrt n\rfloor}} \longrightarrow e^{-x^2}. } $$
Then reverse one of two shifts:
$$ \boxed{ \frac{t_{n-r+s}t_n}{t_{n-r}t_{n+s}} \longrightarrow e^{2xy}, \qquad r\sim x\sqrt n,\quad s\sim y\sqrt n. } $$
Concavity has changed orientation and become a strictly totally positive kernel. On the integer mesh it is literally a geometric Vandermonde matrix. The determinant algebra is exact.
This is such a satisfying sequence of transformations:
$$ \text{double-exponential theta shell} \longrightarrow \text{Lambert saddle} \longrightarrow \text{Gaussian curvature} \longrightarrow e^{2xy} \longrightarrow \text{\(q\)-Vandermonde determinants}. $$
The last arrow is where an asymptotic shape suddenly turns into something I can factor with no asymptotics at all.
The first shell chooses the saddle#
Write
$$ T(u)=\sum_{m\geq1}e^{-\pi m^2e^{2u}}. $$
For large \(u\),
$$ T(u) = e^{-\pi e^{2u}} \left[ 1+O\left(e^{-3\pi e^{2u}}\right) \right]. $$
The first shell therefore has phase
$$ \phi_n(u) = 2n\log u+\frac u2-\pi e^{2u}. $$
It is strictly concave:
$$ \phi_n''(u) = -\frac{2n}{u^2}-4\pi e^{2u}<0. $$
Its unique saddle \(u_n\) satisfies
$$ \frac{2n}{u_n}+\frac12=2\pi e^{2u_n}. $$
If \(w_n=2u_n\), then
$$ \pi w_ne^{w_n}=2n+\frac{w_n}{4}, $$
so
$$ w_n=W_0(2n/\pi)+o(1). $$
All higher theta shells are exponentially smaller on the moving saddle window, and strict concavity kills the tails. The full arithmetic theta integral and the one-shell integral therefore have the same leading ratio at two consecutive orders.
There is also an exact identity hiding in the integral:
$$ \frac{I_n}{I_{n-1}} = \mathbb E_{n-1}[U^2]. $$
Concentration replaces \(U^2\) by \(u_n^2\), and factorial normalization contributes \(1/((2n)(2n-1))\). That is the whole consecutive-ratio constant. The Lambert argument is \(2n/\pi\), not a decorative approximation; using the wrong argument later creates order-one normalization errors that converge painfully slowly.
Curvature accumulates on the square-root window#
Let \(\mathcal F(x)\) be the real-order logarithm of the normalized primitive moment. Differentiating the saddle calculation gives
$$ \mathcal F''(x) = -\frac{2}{x}\frac{W_0(2x/\pi)}{1+W_0(2x/\pi)} +O\left( \frac{1}{xW_0(2x/\pi)^2} +\frac1{x^2} \right). $$
Put
$$ \gamma_n = \frac{W_0(2n/\pi)}{1+W_0(2n/\pi)}. $$
Exact discrete second-difference summation then gives, uniformly for \(k=o(n^{2/3})\),
$$ \log\left[ \frac{t_{n+k}/t_n}{(t_{n+1}/t_n)^k} \right] = -\frac{\gamma_n}{n}k(k-1) +O\left( \frac{k^2}{nW_0(2n/\pi)^2} +\frac{k^3}{n^2} \right). $$
Fixed \(k\) barely sees this curvature. At \(k\asymp\sqrt n\), the tiny second difference is summed about \(n\) times and becomes order one:
$$ -\frac{\gamma_n}{n}k(k-1)\longrightarrow-x^2. $$
That is why the heat law lives on the square-root scale. It is not a guessed central-limit window; it is the scale on which the Lambert saddle's curvature finally accumulates enough to be visible.
Reverse a shift and the kernel becomes positive#
For backward and forward shifts \(r,s\geq0\), an exact rectangular curvature identity gives
$$ \log\frac{t_{n-r+s}t_n}{t_{n-r}t_{n+s}} = \frac{2\gamma_nrs}{n} +O\left( \frac{(r+s)^2}{nW_0(2n/\pi)^2} +\frac{(r+s)^3}{n^2} \right). $$
Thus \(r\sim x\sqrt n\) and \(s\sim y\sqrt n\) produce \(e^{2xy}\). For increasing nonnegative \(x_i,y_j\),
$$ \det[e^{2x_iy_j}]_{i,j=1}^m>0. $$
The proof is Cauchy--Binet applied to
$$ e^{2xy} = \sum_{k\geq0}\frac{2^k}{k!}x^ky^k. $$
Every generalized Vandermonde product in the expansion is nonnegative, and the \(k=0,\ldots,m-1\) term is strictly positive.
I find the sign reversal lovely. The original moment sequence is concave in the relevant logarithmic coordinate. Mixing one backward shift with one forward shift inserts a minus sign in the rectangular second difference, and the same concavity now yields a positive kernel.
On the lattice, every determinant factors#
Freeze the local heat model at
$$ q=e^{2\gamma/n}>1, \qquad H_{r,s}=q^{rs}, \qquad 0\leq r,s<m. $$
Then
$$ D_m(q) := \det[H_{r,s}]_{r,s=0}^{m-1} = \prod_{0\leq i<j<m}(q^j-q^i). $$
Equivalently,
$$ \boxed{ D_m(q) = q^{\binom m3} \prod_{d=1}^{m-1}(q^d-1)^{m-d}>0. } $$
After the Gaussian gauge
$$ \widetilde H_{r,s} = e^{-\gamma(r-s)^2/n}, $$
the adjacent principal cross-ratio is exactly
$$ \boxed{ \frac{\widetilde D_m\widetilde D_{m-2}} {\widetilde D_{m-1}^2} = 1-e^{-2\gamma(m-1)/n}. } $$
For \(m\sim x\sqrt n\), this reserve is only
$$ \frac{2\gamma x}{\sqrt n}(1+o(1)). $$
Positive, explicit, and tiny. This number is the boundary of the result.
The annoying factor of two#
The Lambert arguments also catch a normalization mistake that looks innocent at small rank.
If compact Jacobi coefficients satisfy
$$ 4ja_j-W_0(j/(2\pi))\longrightarrow0, $$
and
$$ \frac{h_n}{h_{n-1}}=(a_{2n-1}a_{2n})^2, $$
then the primitive ratio above forces
$$ \boxed{ \frac{h_n}{h_{n-1}} = \frac1{16} \left(\frac{t_n}{t_{n-1}}\right)^2 (1+o(1)). } $$
The previously proposed \(1/32\) is incompatible with those premises. In the exact Lambert model,
$$ 32\frac{h_n}{h_{n-1}} \left(\frac{t_{n-1}}{t_n}\right)^2 \longrightarrow2, $$
not \(1\). The convergence is logarithmically slow enough that ranks in the tens can look like they prefer the wrong constant. Rude little asymptotic.
Where the heat picture stops#
The theta moment theorems are unconditional, and the determinant formulas are exact for the limiting heat kernel.
What is not proved is determinant transfer at growing order. The entrywise error in the theta heat approximation is larger than the \(n^{-1/2}\) adjacent-minor reserve above. Ordinary determinant continuity can therefore destroy the sign while every entry still converges correctly.
The next theorem has to preserve cancellation: a Neville-factor expansion, a Casoratian identity, or an exact theta factorization that keeps the heat orientation automatically. Merely improving the prose around entrywise convergence will not do it.
So the current object is a clean local model with a quantified warning label. Primitive theta moments really do become a Lambert heat kernel on the square-root window, and that kernel has beautiful exact positive determinants. The reserve also tells us, with unusual precision, why the obvious transfer argument is not yet enough.
Notebook references: K-0133, K-0134, K-0135, O-0269