Structural Necessity of Threshold Conditioned Recurrence in Finite Capacity Substrates: A Bridge to Lithium Transport Constraints

This work provides the structural foundation for a multi‑paper program addressing the lithium transport constraint within a finite‑capacity continuous substrate framework. This manuscript develops a constraint‑closure bridge between the finite‑capacity substrate framework and the structural form of the lithium transport requirement. Its purpose is not to introduce a new dynamical model, propose a specific lithium‑depletion mechanism, or claim observational realization. Instead, it asks whether the structural ingredients required by lithium transport—threshold access, recurrence burden, confinement, bounded accumulation, and amplification—follow from the inherited admissibility constraints of the substrate program. The paper derives a lithium‑bridge sequence in which constraint closure leads to relaxation‑conditioned redistribution, recurrence persistence, state‑conditioned pathway weighting, migrating threshold proximity, layered confinement, bounded structured accumulation, amplification, and observer‑relative recurrence burden. Within this framework, the lithium transport requirement—characterized by a discrete activation threshold near 441.4 keV and a large effective interaction count of order N_eff ~ 10^10—is shown to possess the same structural form as the admissible continuation pathway derived from finite‑capacity, continuous, nonsingular substrate assumptions. The result is structural admissibility, not mechanism demonstration. The paper does not claim that lithium depletion is achieved, does not identify a realized astrophysical site, and does not modify standard nuclear physics, Big Bang nucleosynthesis, or CMB inference. It establishes the bridge architecture that later lithium‑specific work must test quantitatively. This manuscript is part of a coordinated research sequence. It establishes the structural bridge between the finite‑capacity substrate framework and the forthcoming lithium viability architecture. The subsequent Lithium Tool Paper and four supporting analyses are scheduled for release in 2026. Several cited works are forthcoming and are identified as such within the manuscript; DOI references will be added as those papers are released. A future journal‑oriented version may consolidate or revise citations once the full sequence is publicly available.