[SUPERSEDED] The Exact CKM Parameter \lambda: Closing the Gap Between 1/5 and \alpha

SUPERSEDED This paper attempted to derive the exact CKM parameter \lambda from meta-time evolution under Pillar IV. The central argument—that \lambda evolves from the gauge value 1/5 toward the geometric attractor \alpha—is inconsistent with the observed value \lambda \approx 0.225 > \alpha, which makes the supposed attractor unreachable by simple gradient descent. The current treatment of the CKM parameter is given in Appendix Q of "The Emergence Canvas Model: A Unified Framework for Fundamental Physics - The Machine and the State" (2026), https://doi.org/10.5281/zenodo.20795774 where the full 3 \times 3 mass matrix diagonalization yields \lambda = 0.225 without requiring meta-time evolution. The arctangent representation of the candidate values is preserved in that treatment. This paper is retained for archival purposes only. ------------------------------------------------The Wolfenstein parameter \lambda of the CKM matrix has two candidate values in the canvas model. The gauge subspace dimensions give \lambda = 1/5 = 0.2000 from the information-theoretic base. The geometric primitives give \lambda_{\text{eff}} \approx \alpha = (\pi-2)/(\pi+2) \approx 0.2220 from the waveform asymmetry. The observed value is \lambda = 0.2245 \pm 0.0008. This paper derives the exact relationship between these values. Both are tangents of angles that represent corrections to \pi/4, the angle bisecting the first quadrant of the unit circle. The observed \lambda is not a compromise between 1/5 and \alpha. It is a specific value determined by the interplay of the gauge subspace structure and the attractor dynamics of Pillar IV. What this paper provides: · The arctangent representation of both candidate values. Both \lambda = 1/5 and \lambda = \alpha are expressed as tangents of angles that are corrections to \pi/4. The gauge value corresponds to \arctan(1/5) \approx 11.31^\circ. The geometric value corresponds to \pi/4 - \arctan(2/\pi) \approx 12.52^\circ.· A dynamical interpretation. The gauge value \lambda = 1/5 is the initial value at the gauge threshold epoch. The geometric value \lambda = \alpha is the asymptotic value as meta-time \tau \to \infty under the Feed Equation (Pillar IV). The observed value sits between them, closer to \alpha.· The exact formula for the observed \lambda. \lambda = \tan\left(\frac{\pi}{4} - \arctan\left(\frac{2}{\pi}\right) + \delta\right) where \delta \approx 0.0029 rad \approx 0.17^\circ is the residual phase from incomplete attractor dynamics. The observed value is slightly larger than \alpha, indicating that the system has overshot the attractor or that there is an additional contribution from coupled parameters. · A multi-dimensional gradient flow model. The evolution of \lambda is coupled to the other CKM parameters (A, \bar{\rho}, \bar{\eta}) through the Hessian of the spectral energy functional. If some eigenvalues are complex (oscillatory modes), the trajectory can overshoot the attractor before settling.· Testable predictions. The residual \delta \approx 0.17^\circ should decay to zero as the attractor dynamics complete. The time variation of \lambda is \sim 10^{-13} yr^{-1}, below current sensitivity but potentially testable with future precision measurements. The other CKM parameters should exhibit correlated variations with \lambda. Why this matters: The CKM Wolfenstein parameter \lambda = 0.2245 is not an arbitrary number. It is determined by the interplay of the gauge subspace structure (1/5), the geometric attractor (\alpha), and the finite meta-time evolution under Pillar IV. The circle that organizes the four primitives also determines the quark mixing parameter. The same geometry that produces the \pi/2 waveform asymmetry also fixes the CKM matrix. Keywords: CKM matrix, Wolfenstein parameter \lambda, canvas model, gauge subspaces, geometric primitives, attractor dynamics, Pillar IV, Feed Equation, quark mixing, meta-time evolution, \pi/2 waveform asymmetry