Simplex reciprocal sum

A useful identity adjacent to the mass spectrum

The identity below concerns the count $S(n,d)=\binom{n+d-1}{d}$ used by the V2 mass module. It supplies a closed-form check and an exact truncation error for sums of inverse counts. No new geometry or particle assignment is needed.

For positive integer $n$ and integer $d>1$,

$$\frac{1}{S(n,d)} =\frac{d}{d-1} \left[\frac{1}{S(n,d-1)}-\frac{1}{S(n+1,d-1)}\right].$$

To verify this, factor $(d-1)!/[n(n+1)\cdots(n+d-2)]$ from the difference in brackets. The remaining factor is $(d-1)/(n+d-1)$, giving $d!/[n(n+1)\cdots(n+d-1)]$ after multiplication by $d/(d-1)$.

Summing the difference gives

$$\sum_{n=1}^{N}\frac{1}{S(n,d)} =\frac{d}{d-1} \left[1-\frac{1}{S(N+1,d-1)}\right],$$

and hence

$$\sum_{n=1}^{\infty}\frac{1}{S(n,d)}=\frac{d}{d-1}, \qquad \sum_{n=N+1}^{\infty}\frac{1}{S(n,d)} =\frac{d}{(d-1)S(N+1,d-1)}.$$

The remainder tends to zero for every $d>1$. For $d=1$, $S(n,1)=n$, so the corresponding harmonic series diverges.

Adjacent-depth recursion

The factorial definition also gives the multiplicative recursion

$$\frac{S(n,d)}{S(n,d+1)}=\frac{d+1}{n+d},\qquad S(n,d+1)=\frac{n+d}{d+1}S(n,d).$$

At fixed $n>1$, $S(n,d)$ strictly increases with $d$; for $n=1$ all counts equal one. At fixed $d$, the ratio decreases with $n$. These are count identities for any positive indices, usable alongside the additive Pascal recursion without assigning physical occupation weights.

What the trace interpretation requires

If a specified positive operator has exactly one eigenvalue $\mu_d S(n,d)$ per positive integer $n$, with no additional degeneracy, then its inverse trace is

$$\operatorname{Tr}(A_d^{-1})=\frac{d}{(d-1)\mu_d}.$$

This is conditional on that operator and counting measure. The formula does not establish that the master Dirac operator has this spectrum or that all these bands are occupied particles. Restricting to the occupied particle indices gives a different finite sum; including a massless zero mode makes an ordinary inverse undefined.

If each band is instead assigned multiplicity $S(n,d)$, each band's contribution to the inverse trace is $1/\mu_d$, and the infinite trace diverges. This simple example is why multiplicities must be stated. Oscillator excitation energies and cumulative counts also remain distinct, as in the existing mass analysis.

Verification and related work

The spatial structure checks compare direct rational finite sums with the telescoping expression across several depths and truncations. The all-$N$ proof is the algebra above, rather than numerical extrapolation.

• Nested spatial tools — companion tools for spatial evolution and normalization.