Open Problems Behind the Wall a Coda to the One-Medium Series

Coda to the seven-paper One-Medium series. The series developed vacuum, particles, forces, redshift, thermodynamics, measurement, and nuclei as behaviors of a single topological medium, with explicit boundaries drawn wherever a computation was attempted and found to require mathematics the authors could not supply. This document collects what lies behind those boundaries and hands it over, written for mathematicians and numerical analysts with experience in topological solitons and constrained nonlinear field equations. Five problems are stated with the physics stripped to the minimum, each with its equations, pre-registered falsification criteria, and what a solution would decide: (P1) a constrained deformation solver for frustrated multi-defect knots that must not relax to the trivial sector; (P2) whether the calibration integer N = 273 can be selected by closed-orbit counting from the medium's time-dependent dynamics (four routes archived and blocked, a 1D scaling law and a measured 6.6% 3D self-consistency gap on record — a proof of non-existence is welcomed as an answer); (P3) the frequency-glue binding channel with a pre-registered ~3x three-body collective enhancement requirement; (P4) the amplitude-weighted charge profile of composite nuclei, confronting precision electron-scattering data; (P5) the fused-region shape problem gating the extension to 40Ca. All instruments, scripts, and per-run logs are public in the research record. The document also records, honestly, the collaboration that produced the series — one independent researcher and one AI system — and the specific limits of that working mode, which are exactly where these problems stand. Engagement carries no obligation to the framework's physics: refutations are as welcome as confirmations, and will be cited as such.