Quantitative Lemniscate Mergers near Clustered Zeros: Cancellation, Critical Values, and Numerical Resolution

A cluster of polynomial zeros generates small-scale merger events in the sublevel sets of the polynomial modulus. We give explicit estimates that retain cancellation among the inverse moments of the external zeros, rather than using their distances alone. These estimates bound pairwise minimax merger levels and preserve labelled root partitions under an explicit boundary barrier. A separate local theorem encloses a critical value using the quadratic stationary value and a third-derivative remainder. For external fields whose first nonzero inverse moment has order p, the resulting displacement and two value-prediction errors have orders εᵖ, ε²ᵖ, and ε³ᵖ, respectively. On a deterministic equal-degree experiment, moment-aware brackets certify 26 of 32 hierarchies, while distance-only brackets certify 12; ball arithmetic validates local critical-value enclosures in 63 of 64 cases. For the exact quadratic model, we prove a sufficient spacing condition for correct sampled root connectivity for every grid translation and orientation. A 5376-trial raster study illustrates the distinction between a geometric guarantee and a visually plausible image. This working draft provides complete proofs and reproducible computations; comprehensive novelty verification and independent mathematical review remain pending.