Sharp Worst-Case Contraction for the Two-Dimensional Mönkemeyer Algorithm

For a binary word w, let Delta_w be the projective triangle generated by the two-dimensional Farey-Monkemeyer algorithm, and let delta_n be the maximum Euclidean diameter of Delta_w over all words of length n. We prove the exact worst-case contraction law: delta_0 = delta_1 = sqrt(2), and delta_n = sqrt(6)/(ceil(n/2)+1) for every n >= 2. The extremal value is attained by the eventually parabolic family 01 0^(n-2) and by its first-symbol reflection. For the upper bound we derive exact denominator and weighted edge-energy recurrences and prove the sharp weighted diameter inequality b_w c_w diam(Delta_w)^2 <= 6. The geometric argument reduces this inequality to the nonnegativity of two context-resolved block defects, whose nonnegativity is in turn reduced to a finite reset-renewal theorem. The released transition archive contains 450 exact rational positivity certificates: 324 for the auxiliary energy-safety rows De and Df, and 126 for the spread row Dh. The renewal interface is completed by eighteen exact lifted-row transport witnesses, an exact two-class Dh transfer lemma, and parameterized initial-anchor identities. A unified release driver invokes independent exact rational checkers that reconstruct the source and target polynomials, verify every archived certificate identity, check all positive-semidefinite Gram contributions, replay the structural renewal identities and transfer lemmas, and confirm complete coverage of the reset-anchor domains. No floating-point estimates or finite-depth extrapolations enter the proof. Consequences. A two-level plateau delta_{2k+1} = delta_{2k+2} = sqrt(6)/(k+2), and the sharp asymptotic rate limit n*delta_n = 2*sqrt(6). Consequently no constants C > 0 and 0 < lambda < 1 satisfy delta_n <= C*lambda^n for all n: the polynomial rate is caused by a parabolic block and rules out uniform exponential contraction.