Lemma 53 ([Uniformity for one controlled family]). The mixed constructions and their finite dispersion operations have the controlled-family uniformity of Definition 18 ([Controlled families for one finite request]). For one fixed finite …

Fix the finite labels, derivative orders, and Taylor and transition budgets in the request. Choose common compact ordinary, layer, and state boxes with positive distance to the larger analytic domains. The field suprema and their finitely needed Cauchy bounds are then common. The initial amplitude choice gives a fixed contraction margin, say at most 1/2 , in the actual integral equations and in the layer-ring fixed-point equations. For the latter the graph inverse, the coefficientwise l/m≤2 estimate, and the bounded off-balance inverse give the same margin at every point satisfying the prescribed numerical conditions on |c| and the charge cutoff. These estimates involve values in these boxes, not the subsequent history of the ordinary test in the flag path parameter. The angular comparisons use the common finite real jets and Taylor remainders only through finite triangular sensitivity bounds. Their thresholds can therefore be chosen as common inequalities in the finitely many rates, gaps, and angular buffers. In the comparable case first fix κ and the side constant as in Equation; its vanishing remainder has the common modulus provided by the family controls. This gives common normalized action and pole-cone margins. The polygon length constants and stable comparison bounds are then common by Lemma 46 ([Continuation in the leading band]). A dependence on the fixed limiting slope is permitted here; common amplitude smallness over slopes is the stronger, separately proved assertion above. For the trees, geometric weights, counting constants, and the sorted-cost constant are determined by the field boxes and finite rate list. For each of the finitely many requested main charges, the finite low-cost separations and derivative orders give one common constant; the tail has the common factorial bound of Lemma 47 ([Ordered cost and contour bounds]). Its contour remainders require only specified inequalities such as L>K(1/ρ) and the finitely many cap-separation thresholds. The small-slope formal replacement uses the same geometric moment estimate at every point. Similarly, the finite formal norms in Lemma 52 ([Gevrey estimates for full charts]) use two fixed layer radii, a fixed state margin, and a common small formal weight. Taking the minimum of the finitely many permissible optimal proportions makes all factorial and optimal-error constants common. An already fixed convention for a shorter label is preserved by its exponentially small cutoff-comparison estimate; this does not redefine that label on a new test. On a meromorphic band, the finite lattice lists and maximal pole orders are determined by the labels and their derivative orders. The direct finite jet formula above gives common powers of the explicit pole losses, even when the numerical denominators vary. Lemma 48 ([Meromorphic losses and their derivatives]) retains the allowed D_ j-1 loss on the entire forward domain. Its safe-column proof is also uniform: the common real-jet and flag controls make Equation uniform, so a fixed forbidden lattice width and a fixed union bound leave columns in every required fixed-length interval after common numerical thresholds. The exceptional-collar width and sweep-speed estimates use the same constants; no later return to the germ is used to obtain their o(1) variation. Independent Cauchy radii are a common fixed exponential (-C f_j) times the explicit pole-preserving factor. Their finite powers have the common logarithmic bounds already proved. Taylor depths can therefore be chosen once for the request, before applying those losses. The moving-cutoff and optimal-order comparisons have the common costs in Equation and e^ f_k , respectively. Their absorption uses the common numerical gap and side inequalities in Equation and the simple-cost estimate. Fixed-width bins in the retained log modulus have uniformly bounded overlap. The same partitions in those bins have common derivative bounds after the stated triangular losses; hence both the number and precision constants of the source terms are common for this finite request. Finally, high real-jet agreement preserves these domains, angles, and evaluations of independent derivatives. The differentiated source bounds come from the frozen holomorphic formulas and their proved cutoff overlaps. Holomorphic ordinary inputs contribute no input defect; smooth sheets already constructed by Theorem 4.1 ([Packet calculus]) supply their own differentiated defects. For pullbacks, the finite chain-rule factors have exactly the raw, Cauchy, and controlled-sheet losses just bounded. Their logarithms are o(U) at intrinsic active cutoff sources by the preceding cost comparisons. Inherited sheet sources retain their own permitted costs; the comparisons in Definition 9 ([Packet data]) and Equation apply to them as well. At a level- p good join, keep a fixed current loss as C Z_p ; the remaining later-rate and local Cauchy losses are o(V_p) . Lemma 23 ([Propagation of differentiated source errors]) therefore preserves the source estimates, choosing a transition budget larger than its current losses or a sufficiently vertical collar for an independently established e^ -cV_p error. The common controls above make these choices uniform for this finite request. All the preceding estimates hold throughout the part of each full forward domain on which the stipulated common ordinary boxes and numerical inequalities hold. Entering a smaller germ neighborhood may be delayed by a different amount on each path. It affects a later request using that smaller box, not any constant, starting inequality, or source count for this fixed request. This proves the claimed uniformity.