Four trace identities on the diagonal prime operator: rational and character-modulated trace formulas

We exhibit four classical constants of analytic number theory as trace identities on the diagonal prime operator D acting on l^2(P), where P is the set of rational primes and D e_p = p e_p. The four constants are the twin prime constant C_2, the Meissel-Mertens constant M (in the combination M - gamma), the Artin constant C_Artin, and the Landau-Ramanujan constant K_LR. Each appears as tau(O) = log C for an explicit operator O = log f(D) with f holomorphic on a neighbourhood of sigma(D), where tau is the regularised trace defined in our Foundations of the Prime Machine. We observe a structural dichotomy:- Class 1a (rational symbols): C_2, M - gamma, C_Artin all arise from f in Q(z), with poles only at integers below or equal to 1, hence holomorphic on the spectrum.- Class 1b (character-modulated symbols): K_LR requires a character projector P_3 = (I - sin(pi D / 2))/2 that selects primes p congruent to 3 mod 4. The same projector mechanism, applied to the Feller-Tornier symbol, produces a second pair of Class 1b constants Q_1, Q_3 with Q_1 Q_3 = 2 Q_FT. All identities are verified numerically with convergence rate exactly 1/(X log X), matching the tail integral. The dichotomy is placed in a duality framework: Class 1a captures multiplicative invariants visible in any basis, Class 1b uses Dirichlet characters as the operational hook between the orbit-geometric (position) and L-function (momentum) descriptions of primes. The Chebyshev bias and the Hardy-Littlewood heuristic B_2 are shown to lie outside the trace machinery, on the L-function side of the duality. Numerical values:- log C_2 = -0.4150096547- M - gamma = -0.3157184520- log C_Artin = -0.9836176340- log(K_LR sqrt 2) = +0.0776787979- log Q_FT = -1.1312364184- log Q_1 = -0.1111, log Q_3 = -0.3270 License: Creative Commons Attribution 4.0 International (CC BY 4.0) Related works:- T. Krause, Foundations of the Prime Machine (Zenodo, 2026)- Prime Machine Goldbach JNT v11 (Zenodo, 2026)- Prime Machine Collatz JNT (Zenodo, 2026)- Prime Machine Polignac JNT v10 (Zenodo, 2026)- Prime Machine C2 JNT v3 (Zenodo, 2026)