Near-Critical Registration Capture and a Born-Frequency Limit in Canonical ACRF-4: A Conditional Theorem, Native Critical-Surface Tests, and Finite-Size Convergence

This paper establishes a conditional Born-frequency limit within canonical ACRF-4 and tests each bridge of the proposed measurement mechanism without inserting a Born probability rule. The result begins from a previously timestamped finding that complementary linear cross-registration channels produce a canonical receiver-successor consequence partition closely following cos^2(Delta):sin^2(Delta), while a deterministic whole-channel first-threshold mechanism fails. The present work replaces that failed mechanism with a native ACRF critical surface derived from the exact local stability of the centered receiver odd mode. The corresponding multiplier is Lambda=(alpha-beta p_S) sech^2[(alpha+beta p_S-mu/2)m_S], with readiness margin q=1-Lambda. A Near-Critical Registration Capture Theorem is proved conditionally: if a macroscopic ready apparatus has a smooth nonzero density of microreceivers near q=0, if weak canonical consequence shifts native stability linearly as DeltaLambda=kappa Gamma+o(Gamma), and if complementary channels are exchangeable, then the newly committed microreceiver fraction is linear in Gamma. The macroscopic capture ratio therefore approaches the normalized canonical consequence ratio, and in the weak-coupling two-channel construction approaches cos^2(Delta). The theorem premises are then tested numerically using the frozen canonical ACRF-4 successor and retained-edge update. The source-induced retained receiver-edge response scales as |Delta p_S| proportional to Gamma^0.999977. Native near-critical crossing count is linear in aggregate canonical consequence with R^2=0.996247 in a power-enhanced test. A direct seven-angle closure test gives an underlying consequence-ratio error below 2.3 x 10^-5 and a finite native crossing-ratio mean absolute error of 0.0150. An independent finite-size study finds the crossing-ratio error scales as N^-0.4816 and, more directly, as N_cross^-0.4916, consistent with inverse-square-root finite-count convergence rather than a persistent systematic departure. Two negative controls are retained as part of the result: a whole-channel threshold race produces winner-take-all behavior, and a fixed positive microscopic consequence threshold creates a weak-coupling dead zone. The paper therefore isolates the remaining RT proof obligations to deriving the physical near-critical apparatus ensemble, proving the coarse-grained relation between canonical retained structure and detector memory, and extending the construction beyond two-channel measurements.