Contour-count indicator fields for visible pole clusters in meromorphic continuation

We develop a contour-count indicator method for visible pole clusters in outward meromorphic continuation from circular boundary data.

The method starts from determinant characteristics built from positive Fourier coefficients.

In the pure finite-pole model, the correct determinant characteristic factors into a polynomial whose zeros are the reciprocals of the exterior poles.

In the presence of a holomorphic background, finite sampling, and noise, roots of individual determinants are unstable and are used only as local evidence.

We aggregate this evidence into a scalar indicator field on the reciprocal pole plane: at each sampling point, the field records the fraction of determinant orders and shifts for which a small contour centered at that point encloses exactly one empirical determinant zero.

The resulting field plays the role of a sampling-type imaging functional for pole visibility.

Fixed superlevel sets give visible-pole clusters, while zero-dimensional persistent homology is used only as a threshold-robust post-processing step.

We prove deterministic results linking pure-pole contour counts, Rouché stability, indicator-field contrast, fixed-threshold component recovery, and persistence-gap stability.

These results explain why isolated poles with sufficient residue and separation generate stable high-value components, whereas weak, close, boundary-near, or noise-dominated poles may give low, short-lived, or merged components.

The framework is a cluster-certification and imaging method, not an unconditional all-pole recovery procedure.