The odd-step Collatz path algebra: from diagonal coefficient object to arithmetic completion

The shortened Collatz map T has two affine branches with coefficients (1,0) and (3,1). Splitting the multiplier bit and the translation/integrality bit yields an ambient Klein four coefficient object V = F2^2, with F_(a,b)(x) = ((1 + 2a)x + b)/2. The classical map uses only the diagonal subgroup Delta = {(0,0), (1,1)} subset V, which is isotropic for the standard F2-bilinear bicharacter governing the Z2 x Z2 color sign rule.

Passing to the odd-step map C(x) = (3x + 1)/2^v2(3x + 1), we construct the path category Ec and its weighted algebra R[Ec] over Z[u,v]. The central identity is the affine path formula g_p(x) = chi(p)x + c(p), with chi(p) = 3^ell(p) 2^(-A(p)), together with Tao's algebraic offset rho(p) = sum from k = 1 to ell(p) of 3^(k - 1) 2^(-A_k). The two offsets satisfy distinct semidirect laws: c(pq) = chi(q)c(p) + c(q), and rho(pq) = rho(p) + chi(p)rho(q). The rho-law yields a canonical sigma-derivation and hence a local-unital Ore extension O_C = R[Ec][Theta; sigma, D_rho].

We compute the finite transformation monoids T_N on Z/3^N Z and prove: (i) a constant-collapse theorem, namely that a k-fold composition has image size exactly 3^(N - k) for k <= N, hence is constant for k >= N; (ii) a unique-fixed-point theorem, namely that every nonempty affine composition has exactly one fixed point modulo 3^N; and (iii) an exact cardinality formula |T_N| = 2 * 3^(N - 1) * (3^(N - 2) + 1) for N >= 2. The unique-fixed-point theorem implies exact, N-independent traces Tr(L_(beta,N)^n) = (3^beta / (2^beta - 1))^n and a one-pole finite dynamical zeta function. The normalized beta = 1 operator is exactly the Markov operator whose iterates generate Tao's Fourier quantity.

At the symbolic level every finite exponent word is admissible; the odd 2-adic itinerary map epsilon_2: Z_2^odd -> N_(>=1)^N is a bijection; and the beta = 1 Gibbs measure is Tao's geometric(1/2) law. The positive-integer image is characterized by the stabilization criterion: the canonical least positive odd representatives of the prefix residue classes must eventually become constant.

On the analytic side we construct the compact 3-adic diagonal D = C(Z_3), the branch corner isomorphisms rho_a: D -> e_a D, and the arithmetic crossed-product algebra A_C = D rtimes_rho N_(>=1)^*, together with its time evolution, gauge torus, arithmetic diagonal, associated germ-groupoid model, and semilocal {2,3,infinity} embedding. The path grading yields commuting derivations and several Hopf lifts: primitive derivations, Hasse-Schmidt/NSymm, bamboo Dyson-Schwinger series, and finite Myhill-Nerode recognition bialgebras.

Finally, the logarithmic cocycle delta(p) = A(p) log 2 - ell(p) log 3 factors through a fixed toric Kummer 1-motive, while the angular sum Theta(p) = sum over k of arctan(3^(k - 1) 2^(-A_k)) is governed by a torsion criterion in Q(i)^x.