The ds : 3 → 4 Problem in Stochastic Rupture: Proved Results, Reformulation, and Exploratory Investigations

The Stochastic Rupture (SR) framework predicts ds = 2 (UV, single Planck sheet) and ds → 3 (cubic stacking). This paper addresses the step ds : 3 → 4. We prove four results within SR: the causal diamond volume scales as D(T) ∼ T⁴; the SR field equation implies local causality via the Cattaneo equation; the dissipative (Onsager– Machlup) sector has ds = 5/2; and the undamped vibrational sector has ds = 4. We then establish that the ds = 4 regime is ultraviolet (near-Planck), not macroscopic, leading to a precise reformulation: the central open problem is whether SR dynamics can generate an emergent (3 + 1)-dimensional structure. The second part reports exploratory numerical investigations of triadic-closure network dynamics, which are not derived from SR but are motivated by SR organization principles. These reveal a condensation–fragmentation phase structure, a simplex cascade, and two empirical laws: τ ∝ k −1 (exact, from random-walk theory) and Tmin ∝ d 0.70 G k −0.64 min (universal across 100 networks, possibly (dG/kmin) 2/3 ). No independent simplicial contribution beyond degree was found. Limitations of each result are stated explicitly.