Complete Large-κ Channel-Cluster Classification for a Concentrating Two-State Ensemble

Abstract

A predecessor analysis of a concentrating two-state flea ensemble showed that, at fixed 0<λ₀<∞, the naive boundary point (κ,λ)=(∞,λ₀) has no unique limiting channel. Here we ask the separate channel-theoretic question of classifying all cluster points of the same exact two-state family as κ→∞, for one fixed probability density g∈L¹(R) and nonnegative scaled time λ. A uniform-in-angle axis reduction converts the problem to one complex coherence parameter. The exact variables θ=κλ and τ=λ/κ then separate a leading phase from the residual scale: finite τ is governed by Ψg(τ)=∫g(x)e^{-iτx²/2}dx, whereas τ→∞ gives complete dephasing by a fixed-L¹ van der Corput argument without moment, smoothness, bounded-variation, compact-support, or support-gap assumptions. We prove exhaustive pathwise classification and show that, when λ is allowed to vary with κ→∞, the complete family-level cluster set is exactly the closed unit disk of x-axis contraction/rotation channels. Every disk point is realized by an explicit admissible path. Consequently every finite-λ₀ boundary fibre is the unit circle, while the (∞,∞) fibre is the full disk. The result concerns this exact two-state channel family only; it does not assert a full Schrödinger double-well theorem, a general Born-rule derivation, or an individual-outcome mechanism.