Resolution-Independent Bond-Breaking Dynamics in Triangulated 2-Spheres via an Absolute Physical Length Scale

We introduce a resolution-independent fracture criterion for discrete simplicial manifolds, replacingthe standard relative strain-ratio snap threshold with an absolute physical displacementcondition (rphys). Standard thresholds are pathological: the physical tear distance shrinks asN−1/2 with vertex count, causing cascade dynamics to become artifacts of the discretization.The rphys condition—analogous to the process-zone length of cohesive fracture models [10, 11] andthe horizon length of peridynamic formulations [22]—defines an absolute gap below which bondsdo not yield regardless of mesh density. The primary result is a constant-density ensemble onS2 (5% defect density, cascades run to topological equilibrium): the fracture fraction is flat atϕrphys = 0.0949 ± 0.0001 across N = 5,000–200,000 (a factor of 40, 7 seeds per resolution; perresolutionstd falls monotonically from 9.2×10−4 at N=5,000 to 1.13×10−4 at N=200,000). A20-seed sub-study at N ∈ {5,000; 10,000; 20,000; 60,000} gives RSMrphys = 0.0036 (95% bootstrapCI [0.0012, 0.0079]) versus standard RSM = 0.884 (95% CI [0.880, 0.887]) — non-overlapping confidenceintervals provide formal statistical confirmation of resolution-independence over a factor of 40in N. The standard condition requires N ≈ 20,000 to reach the same plateau — a 4× gain in resolutionefficiency (4-point RSM: 0.760 → 0.233, 3.3× improvement). The plateau is engine-stable:re-running the same code on a fresh build of the canonical engine reproduces every per-seed cellbit-exactly (0/122 cells deviating; this work). A fixed-defect-count study across N ∈ {2000–40 000}confirms the mechanism (RSM: 0.674 → 0.338, 2.0× improvement). The criterion is validated in4D on the pentachoral Regge engine: the severance length ℓsever follows the predicted A+B N−1/dscaling across N ∈ {2,000–20,000} nodes per spatial slice, yielding a resolution-independent rphyswith coefficient of variation CV = 1.7%. A geometric outcome test on dynamically formed topologicalvoids [17] confirms that the curvature balance |C−/C+| converges monotonically toward theGauss–Bonnet prediction (0.805 → 0.865 → 0.980, 7–10 seeds per resolution), demonstrating thatrphys-regulated cascades produce resolution-independent physics, not merely resolution-independentsnap counts. This study establishes a scale-controlled foundation for dynamic topology in discretespacetime models, with rphys acting as a nonlocal regularization length that remains finite as latticespacing vanishes.