The 3-adic limit of the Krasikov-Lagarias inequalities, and an improved lower bound for the 3x+1 problem

Let T(n) = n/2 for n even and (3n+1)/2 for n odd, and for a not congruent to 0(mod 3) let pi_a(x) be the number of integers 1 <= n <= x whose forward orbitunder T reaches a. Krasikov and Lagarias bounded pi_a(x) from below byexhibiting a feasible solution of a linear program L^NT_k(lambda), the exponentobtained being gamma = log_2 lambda; their Theorem 2.2 is uniform in the levelk. Whether the method reaches the exponent 1 - equivalently, whether thecritical values lambda_k tend to 2 - has been open since Applegate and Lagariasposed it in 1995. Our main result reduces that question to a statement about a single operator.The substitution m = 3v + 2 collapses the whole k-indexed family into onepositive operator (Bc)(v) = (1/4) c(2 + 4v) + w(v) c(Nv) on C(Z_3), in which v -> 2 + 4v is a 3-adic isometry and N expands Haar measure by exactly3 - which is precisely where the loss in the method comes from. We show thatrho_k(2) is the Collatz-Wielandt value of B over functions locally constant atlevel k-1, so that rho_inf = lim_k rho_k(2) = lim_n q_n^(1/n) = sup_n q_n^(1/n), where q_n = min_v (B^n 1)(v), is the lower spectral radius of B. The level k disappears, and every positive cgives a rigorous lower bound on rho_inf that is its own certificate. Two suchbounds are proved here in exact integer arithmetic, with no floating point atany stage: the canonical, search-free choice c = sum_{j = 171/200 at n = 20, and an optimised level-K vector givesrho_inf >= 0.9652. We also prove 3/2 <= rho(B) <= (3 + sqrt(10))/4, and thateach 4 rho_k(2) is an algebraic integer, 4 rho_2(2) being the real root ofx^3 = 3x + 7. Numerically q_n decays like a power of n rather thangeometrically, which points to rho_inf = 1 and so to an affirmative answer; weset that evidence against a competing extrapolation that points the other way,and claim neither. On the computational side we compute the critical exponents gamma_k for k <= 23to nine significant figures and obtain pi_a(x) >= x^0.9294 from a feasiblesolution of L^NT_23(lambda) with 3^22 = 31,381,059,609 variables, verified inexact integer arithmetic, coefficients included. Allikvere has independentlyreached level k = 20 and obtained pi_a(x) >= x^0.9145 in work deposited beforeours; where the levels overlap, his certificate and ours are different vectors,found and verified by different protocols, and agree closely on the quantitiesthey share. Priority note. J. Allikvere (doi:10.5281/zenodo.21728793, 1 August 2026)reached level k = 20 and improved the 2003 record to x^0.9145 before this work;that priority is his, and Section 1 of the manuscript says so. The presentdeposit reaches k = 23.