Exact Determination of the Autocorrelation Constant C₁c_ The Eigenvalue Equation tan x = x and the π-Factorization Loop

This paper provides an independent exact determination of the Barnard-Steinerberger autocorrelation constant C1c within the intrinsic-fracture framework. C1c is a cross-domain constant between the additive domain (equi-spaced grids, integer weights) and the multiplicative domain (Wiener-Khintchine spectrum, autocorrelation). The exact limit constant is determined as D0 = 1/(2(1+theta0)) = 0.410767488, which is locked by the sinc negative-lobe extremum of the eigenvalue equation tan x = x. The smallest positive root is x* = 4.4934094579, yielding theta0 = 0.2172336282. This demonstrates that the sinc negative lobe is the pi-system projection of the intrinsic fracture. For the lower bound, the pure pi-factor closed form starts at 1/pi = 0.318309886 (the arcsine construction). The field's external constructions (R2026: 0.4103287, Kos2026: 0.4103334, vidigoat: 0.4103433) exhibit an end-loading signature, which is the discrete manifestation of the arcsine boundary singularity. The pinching gap between the upper and lower bounds is approximately 4.24 x 10^-4, which is of the same order as the intrinsic fracture delta (approx. 2.41 x 10^-4). Using the interpenetration notation, the final determination is denoted as C1c = D0 = L-up-arrow-D-down-arrow-U, meaning the lower bound L approaches from below while the upper bound U presses from above, jointly locking the limit constant D0. The criterion type is "convergent-extremum," which is isomorphic but distinct from the "zero-locking" type of percolation criticality. The determination requires no function-space search; the convergence direction is fixed directly by the eigenvalue equation and the end structure. Both bounds are completely recomputable digit-by-digit via exact integer fractions and pi-factorization.