For the 2D massive Gaussian Free Field (GFF) on a periodic square lattice with covariance ?????? = [(−Δ?? + ??2 ) −1 ](??, ??), we study the Cohen–Steiner stability bound ‖?? − ?? ̃‖∞ under a real‑space block‑spin renormalization‑group (RG) step ?? ̃. We establish two results. Theorem 1 (Closed‑form sup‑norm). For uniform 2 × 1 block‑spin averaging, the sup norm of ?? − ?? ̃at the diagonal block‑corner entry is the closed‑form com‑ bination ‖?? − ?? ̃‖∞(??2 ;??) = 3 4 ????(0; ??2 ) − 1 2 ????(??1 ; ??2 ) − 1 4 ????(??1 + ??2 ; ??2 ), where????(??; ??2 )is the 2D massive latticeGreen’s function. In the continuum limit ?? → ∞, ?? → 0, this converges to the rigorously identiϐiable constant 1/8 + 1/(4??) ≈ 0.20461, using the lattice potential kernel values ??(??1 ) = 1/4 (Spitzer 1976) and ??(??1 + ??2 ) = 1/?? (Stöhr 1955). Theorem 2 (Universal IR log coefϐicient — 2D lattice with non‑degenerate quadratic IR propagator). For any symmetric local block‑spin kernel ?? on any 2D lattice ℒ whose inverse propagator ??L (??) admits a non‑degenerate quadratic small‑?? jet ??L ∈ ℝ 2×2 (positive deϐinite, the Bravais lattices form a natural class but the result uses only the analytic jet structure), deϐine the connected part of the sup‑norm bound as ΔRG (??; ??, ℒ) ∶= ‖?? − ?? ̃‖∞(??;∞; ??, ℒ) − ‖?? − 1?? ̃‖∞(0;∞; ??, ℒ). The leading small‑?? IR expansion is ΔRG (??; ??, ℒ) = −????,ℒ ⋅ ??2 log(1/??) − ????,ℒ ⋅ ??2 + ??(??4 log ??), with the universal coefϐicient ????,ℒ = tr(??−1 L ????) 4??√det ??L . Here ???? ∈ ℝ 2×2 is the quadratic jet of the kernel’s sup‑form factor ????(??) = ?? ⊤?????? + ??(?? 4 ). The lattice geometry enters only through the two scalars det ??L and tr(??−1 L ????). For square lattice (??L = ??) and uniform ?? × ?? block, ???? = (?? 2 − 1)/(24??) is recovered as a special case. The universal coefϐicient depends on?? only through the rotation‑invarianttrace of its quadratic jet — a ϐinite number of bits of kernel information. The ϐinite scheme constant ???? depends on full lattice combinatorics; it is non‑universal and is rele‑ gated to Appendix A. Empirical conϐirmation across fourteen test cases — ten on the square lattice (three uniform isotropic blocks, three rectangular anisotropic blocks, four weighted non‑uniform kernels) and four on the triangular lattice (one square‑ lattice regression cross‑check plus three triangular‑lattice kernel variants) — produces UV‑side log‑slope ratios within 10 −5 of unity to the predicted ????,ℒ, conϐirming the universal coefϐicient formula to essentially analytic precision. A pair of structurally distinct weighted kernels (center‑peaked Tent vs center‑ empty Edges‑only) with identical Σ?? produce indistinguishable IR log slopes, demonstrating that the universal coefϐicient is sensitive to the kernel weights only through the trace of their covariance matrix. Square vs triangular lattice tests at ϐixed kernel produce ?? values differing by factor ∼ 2.6, both predicted exactly by the lattice‑geometry factor ??−1 L /√det ??L . The interpretation in physical language is that the connected RG‑running contribution to the geometric drift equals the integrated β‑function over accumulated RG time, with prefactor ??(??)/(2??) controlled by the lattice geometry of the kernel and the universal 2D massless Green’s function IR singularity. Analytic result for ??UV at rectangular kernels. Via a Bessel‑integral representa‑ tion of the Spitzer–Stöhr lattice potential and the diagonal closed form ??(??, ??) = (1/??)∑ ?? ??=1 1/(2??−1) — in particular ??(2, 2) = 4/(3??), which unlocks the pre‑ viously inaccessible 3 × 3 baseline ?? (3,3) UV = 14/81 + 88/(243??) — the massless baseline ?? (??) UV admits elementary closed forms for all ϐive tested rectangular block kernels. Empirical identiϐication (PSLQ‑based candidate) for ????. PSLQ at 500‑bit on the next‑order Spitzer potential ??(??), satisfying a discrete Poisson equation Δ??(??) = ??(??) − ??0 with ??0 = 5 log 2/(4??) identiϐied at the same precision, yields candidate closed forms for ???? at the ϐive rectangular kernels in the basis {1, log 2/??} — no Catalan, no Euler–Mascheroni, no higher zeta values appear at 2the integer‑coefϐicient cutoff ∼ 10 9 probed here. These identiϐications are empiri‑ cal conjectures pending an analytic derivation; we record them as PSLQ‑identiϐied candidates rather than theorems (§6 Note + Open Problem O5).