Simplex heat trace

Specify the tower being summed

Let a positive diagonal operator $A_d$ have one eigenvalue

$$\lambda_n=\mu S(n,d),\qquad n\ge1,\qquad \mu>0,$$

for each index, with no extra degeneracy. Define

$$K_d(t)=\sum_{n=1}^{\infty}e^{-t\mu S(n,d)},\qquad t>0.$$

Here $t$ is a heat parameter in inverse-eigenvalue units, not the shared physical time coordinate. This is an analytic diagnostic for the proposed count sequence; it does not identify $A_d$ with the spatial Dirac operator.

A bound that proves the asymptotic

For integer $d\ge1$,

$$\frac{n^d}{d!}\le S(n,d)\le\frac{(n+d-1)^d}{d!}.$$

Put $a=\mu t/d!$ and $f(x)=e^{-ax^d}$. Its integral is

$$I_d(t)=\int_0^\infty f(x)\,dx =\Gamma\!\left(1+\frac1d\right) \left(\frac{d!}{\mu t}\right)^{1/d}.$$

Since $f$ decreases, its positive-integer sum lies between $I_d-1$ and $I_d$. The shifted sum beginning at integer $d$ differs by at most $d-1$. Sandwiching $K_d$ between these two sums gives

$$\max(0,I_d(t)-d)\le K_d(t)\le I_d(t).$$

In particular,

$$K_d(t)= \Gamma\!\left(1+\frac1d\right) \left(\frac{d!}{\mu t}\right)^{1/d}+O(1) \quad(t\downarrow0).$$

For a numerical cutoff $L$, the omitted positive tail obeys

$$0\le K_d(t)-\sum_{n=1}^{L}e^{-t\mu S(n,d)} \le\int_L^\infty e^{-ax^d}\,dx =\frac{\Gamma(1/d,aL^d)}{d\,a^{1/d}},$$

where the final expression uses the upper incomplete gamma function. Thus both the leading approximation and finite numerical sum have explicit error bounds.

Different generating functions count different things

The ordinary index generating function is also exact:

$$\sum_{n\ge1}S(n,d)z^n=\frac{z}{(1-z)^{d+1}}, \qquad |z|<1.$$

It weights the index by the count; the heat trace instead places the count in an exponential. Their singular exponents need not agree. Likewise, comparison with $n^{-ds}$ shows that $\sum_n S(n,d)^{-s}$ converges absolutely exactly when $\operatorname{Re}s>1/d$.

The heat exponent $1/d$ is therefore a property of this one-eigenvalue-per-index tower. It is not evidence that the corresponding spatial dimension equals $d$. For example, the actual round-sphere Dirac spectrum has eigenvalues linear in its level and nontrivial multiplicities, as detailed in round-sphere Dirac counts. V2's physical depth is the number of ordinary macroscopic spatial directions in its principal-symbol construction. This diagnostic does not replace that construction or change any depth.

What this does not regularize

The bounds establish the displayed trace and convergence region. They do not establish analytic continuation at $s=0$ or finiteness of all interaction sums. Even if an operator zeta function is regular there, its regularized determinant changes under rescaling:

$$\log\det\nolimits_\zeta(\alpha A) =\log\det\nolimits_\zeta A+\zeta_A(0)\log\alpha.$$

This follows by differentiating $\zeta_{\alpha A}(s)=\alpha^{-s}\zeta_A(s)$. A finite value at zero alone neither determines its derivative nor removes normalization-scale dependence.

The mode counting checks check direct heat sums against both bounds and verify the index generating-function coefficients.

Related

• Simplex reciprocal sum — gives the exact $s=1$ value for $d>1$.
• Counting mode tools — companion counting and spectral tools.