Local Spectral Structure and Reset Geometry in a Chamberland Collatz Extension and a Forced Exponential Parameter Family

This work studies a smooth complex extension of Collatz-type dynamics through two complementary lenses: a local holomorphic analysis of the Chamberland interpolation and a global numerical study of a forced exponential parameter family. On the analytical side, the paper derives explicit local expansions near the attracting fixed point at the origin, introduces a domain-restricted tanh⁡\tanhtanh chart, and computes the normalized Koenigs linearization coefficient exactly, obtaining the spectral fingerprint b2=π2.b_2=\pi^2.b2=π^2. On the global side, it studies a companion cosine gate GGG adapted to the numerical “Collatzscape” experiments, including the exact reset identity G(2)=1G(2)=1G(2)=1 and the closure relation G(c)ln⁡c=ln⁡2,G(c)\ln c=\ln 2,G(c)lnc=ln2, which yields a distinguished real parameter c∗c_\astc∗ and an exact frozen two-point orbit {c∗,2}\{c_\ast,2\}{c∗,2}. The paper is intentionally cautious in scope. It does not claim a proof of the Collatz conjecture or a rigorous global classification of basin structure. Instead, it cleanly separates: exact local results, exact structural identities for the frozen model, and numerical observations from the forced non-autonomous system. In particular, it records binary and fractional survival maps in the complex parameter plane as reproducible numerical diagnostics, while avoiding stronger claims than the current mathematics supports.