The Row Sums Were the Method, Not the Theorem: a Machine-Checked Chain from a Positive Weight to Exponential Decay of Correlations, and a Misattributed Uniformity Wall

A formal Lean 4 development revisits a two-dimensional spatial transfer kernel whose earlier uniform spectral argument relied on constant row sums. We show that the loss of row-sum constancy obstructs that method, not the conclusion. With no sorry and no project axiom, we prove twelve machine-checked theorems.The chain establishes the sharp field-uniform one-bond influence envelope tanh J; assembles these bounds into a finite-volume Dobrushin matrix; proves a volume-independent resolvent estimate under row-sum bounds alpha < 1 without assuming constant rows; mechanises Dobrushin's comparison inequality with the attained 1/4 covariance constant; constructs Gibbs measures, heat-bath kernels and intrinsic influence matrices from arbitrary strictly positive finite weights; and specialises to anisotropic Ising interactions. For L x T rectangles with free boundary, the condition 2 tanh|beta| + 2 tanh|gamma| <= alpha < 1 yields exponential decay of correlations with beta, gamma, alpha and the prefactor fixed before the volume quantifiers.The transport into the operator formulation is closed: an exact finite band identity relating endpoint covariances to matrix elements; an abstract theorem showing that a common exponential decay rate for band covariances - a hypothesis on finite path measures, carrying no operator, norm or spectrum - implies a uniform positive gap for a family of projected transfer operators; a Perron-boundary tilt identity that preserves the decay rate while absorbing boundary costs into extent-dependent constants; an exact currying identification of the free strip measure with the rectangle Ising measure; and the resulting corollary: inside the window there is one m > 0 bounding the projected transfer operator of the coupled kernel's normalised Perron data by exp(-m) at every extent, with m = -log alpha.Numerical measurements at L <= 12 provide counterevidence to attributing spectral degeneracy solely to nonzero spatial coupling. No infinite-volume state, thermodynamic limit or boundary-condition independence is constructed; the window is sufficient, not sharp. The underlying Dobrushin mathematics is classical; the contribution is a non-vacuous, reproducible mechanisation and composition of the full chain, ending at a volume-uniform operator gap. No consequence for Yang-Mills theory is claimed.