The Self-Trapping Selectivity Principle: Zeolite Shape-Selectivity and Pt–Sn Ensemble Effects.

webpage: The Self-Trapping Selectivity Principle: Zeolite Shape-Selectivity and Pt–Sn Ensemble Effects. Note on this version: V6 is a new version of V5 (10.5281/zenodo.22851704), same paper and scope. It decides the normalization of the self-trapping radius that V5 left open, derives the relation behind it from the continuum DNLS equation (machine-checked), and limits the width law to one-dimensional channels after simulation showed that two- and three-dimensional cages have a self-trapping threshold instead. CHANGES_v5_to_v6.md lists every change. Sousa et al. (2014) posed a specific mechanistic question that remains unanswered: HZSM-5 and HMCM-22 have nominally identical acid chemistry and both possess 10-ring apertures, yet ethanol conversion over them gives markedly different product distributions, coke locations and deactivation profiles. This paper proposes a mechanism and states it as a theorem: a reaction pathway's spatial extent is bounded by a DNLS self-trapping radius r*. At fixed conserved norm P (the quantity the DNLS conserves; P = 1 for one adsorbed intermediate), r*_P = 4aJ/(λP), with a a lattice length scale, J the inter-site coupling and λ the on-site binding energy; the earlier form r* = a√(J/λ) is kept as the fixed-peak-amplitude convention. The transition-state selectivity is σ = 1 − (r*/r_pore)², i.e. σ_P = 1 − 16(J/(λP))²(a/r_pore)² at fixed norm. The profile A·sech(x/w) solves the stationary continuum DNLS equation exactly when 2Ja² = λA²w²; this is proved in Lean in both directions, and with the norm relation it gives w = 4Ja/(λP). The width law holds in quasi-one-dimensional channels such as ZSM-5's; in simulated two- and three-dimensional cages there is no continuous width but a threshold in λP/J (about 6–7 in 2D and 8–12 in 3D on a bare lattice), which bears directly on the MCM-22 supercage. The contact manifold used for coordinates has a Reeb flow that is volume-preserving and does not itself confine anything; confinement is modelled on DNLS self-trapping and tagged as a model. Pt–Sn ensemble effects (Corollary 1) are treated as a second instance of the same bound. Every claim carries a status tag (VERIFIED / DERIVED / SIMULATION / MODEL / OPEN). The core is formalized in Lean 4 (main file 23 theorems, no sorry, up from 13 in V5; compiled by the author under Lean/Mathlib v4.32.0 and v4.14.0). Open items are stated as open: the quantitative fit to Sousa et al.'s selectivity data (not publicly accessible); the continuum limit and the norm relation used in the derivation; the mapping of λ, J and P to measurable quantities and temperature; and the definition of the operator pipeline C/K/F/U as operators on L²(X_cat). What was cut from earlier drafts stays cut: the "Coherence Bridge," the extrudate-pellet corollary, the "helical phase" prediction, and the unified-theory / clean-energy framing.