Annular Shell Isolation Near Zeta Zeros

We describe a finite-range diagnostic for local isolation near nontrivial zeros of the Riemann zeta function. The construction was motivated by a real-axis ``simplex bracket'' for $\zeta(s)$ when $s>1$, in which the reciprocal triangular and tetrahedral totals,

\[

    \sum_{n\ge 1}\frac{1}{T_n}=2,

    \qquad

    \sum_{n\ge 1}\frac{1}{\binom{n+2}{3}}=\frac32,

\]

place $\zeta(2)=\pi^2/6$ inside the interval $(3/2,2)$. This real-axis bracket does not locate zeros and does not extend directly to the critical line, where $\zeta(s)$ is complex. It does, however, introduce the chamber arithmetic that later reappears in the zero-shell diagnostic:

\[

    m_{\mathbb R}(s)

    =

    \left\lfloor \frac{1}{\zeta(s)-1}\right\rfloor+1,

    \qquad

    \misolate(\rho)

    =

    \left\lfloor \frac{1}{|\zeta'(\rho)|h_\rho}\right\rfloor+1.

\]

For a simple nontrivial zero $\rho_j=1/2+i\gamma_j$, Taylor expansion gives

\[

    \zeta(s)=\zeta'(\rho_j)(s-\rho_j)+O\!\left((s-\rho_j)^2\right).

\]

Thus the zero-centered level shell $\abs{\zeta(s)}=1/m$ has local first-order radius

\[

    r_m(\rho_j)\approx \frac{1}{m\abs{\zeta'(\rho_j)}}.

\]

Comparing this radius to the nearest-neighbor half-gap

\[

    h_j=\frac12\min(\gamma_j-\gamma_{j-1},\,\gamma_{j+1}-\gamma_j)

\]

reduces local shell isolation to the product

\[

    P_j=\abs{\zeta'(\rho_j)}h_j,

    \qquad

    \misolate(\rho_j)=\left\lfloor \frac1{P_j}\right\rfloor+1.

\]

This is not a new zeta invariant; it is a compact finite-data coordinate obtained from the linearization and the local zero spacing.

Across a baseline block of $4519$ zeros, a disjoint block of $1000$ zeros, and three high-altitude sentinel bands of $250$ zeros each near $T=10^5,10^6,$ and $9\cdot 10^6$, the derivative magnitude $\abs{\zeta'(\rho_j)}$ and the local half-gap $h_j$ are positively rank-correlated. The strongest isolation outliers are dominated by adjacent close-zero pairs, where small gap and depressed derivative scale jointly suppress $P_j$. The reported maxima are finite-range observations only; no global bound is claimed or expected.