Holonomy–Integerization: Quantitative Stability, Cocycle Algebra, and a Readout–Transport Bridge

We solve the following quantitative margin-preserving alignment problem for phase signals on S¹. Given BV phase data γ, a uniform sup-error budget ε, and a required readout integrity (no false carries), we choose a smoothing degree N and a phase margin δ₀ that (i) preserve integerization and (ii) control the Skorokhod J₁ drift with explicit constants. Our solution produces a deterministic, copy-ready policy (N ≍ TV(γ)/ε, δ₀ = 2ε) and proves tight stability bounds. Mathematically, the paper bridges two areas: (A) time-resolved integerization (functorial, J₁-stable with Cₛₖ(δ₀) ≍ δ₀⁻¹), and (B) an optimal-transport-style jump-graph cost T_{δ₀} such that d_{J₁} ≃ T_{δ₀} on the margin class (two-sided control). The explicit defect of additivity defines a canonical carry 2-cocycle with scale-locality and a quantitative rigidity gap E₀(ρ, δ₀) ≳ δ₀(1 − ρ). A parabolic invariance window shows that heat flow preserves the readout for t ≲ (δ₀ / TV(u₀))² with explicit J₁ drift. All constants are consolidated in a one-screen table; sharpness examples show that the factor 2 and the δ₀⁻¹ scaling are optimal up to universal factors.