Edge-Flux Certificates for Centered Discrete Maximal Variation

We study the open sharp variation conjecture Var(Mf) ≤ Var(f) for the centered discrete Hardy–Littlewood maximal operator on BV (Z). We do not prove the conjecture. Instead we prove a finite-support edge-flux reduction: peak excesses of Mf are compared with interval measures and converted by Abel summation into cumulative edge coefficients. This yields a signed residual certificate criterion with fractional touching budgets. We verify the criterion for two genuine two-floating families: the even three-spike chain and a parameter-stable D = 6 three-spike height-ratio family. We also record the compactified Abel identity and uniform ε- certificate principle needed for eventually constant data. Consequently, the full conjecture would follow from finite-support signed floating packing together with boundary-stable compactified certificates for eventually constant data.