Sign-Quotiented Doubling Modulo n: Cycle Structure, 2-Adic Synchronization, Lifting, and Forward-Closed Subsets — Four Working Papers

Multiplication by 2 on Z/nZ after identifying x with −x, studied for every n. Write n = 2^s m with m odd and r(d) = min{r ≥ 1 : 2^r ≡ ±1 mod d}.

I. Cycle structure. For odd n the map is a permutation; its cycles are one fixed point and, for each divisor d > 1, φ(d)/(2r(d)) cycles of length r(d). II. 2-adic synchronization. r(d) = ord_d(2)/2 exactly when all v2(ord of 2 modulo the prime-power factors) are equal and positive; the criterion depends only on the prime divisors. III. 2-adic lifting. The full functional graph: G_{2^s m} ≅ Lift_s(G_m) — the odd part gives the cycle core, the factor 2^s universal rooted trees; the subsets of the complete binary layer closed toward the root are the plane binary trees, with B_0 = x, B_{h+1} = x(1 + B_h)^2. IV. Forward-closed subsets. The polynomial P_n(x) counting subsets S with D(S) ⊆ S factorises as (1 + Z_s(x)) ∏_{d | m, d > 1} (1 + T_s(x)^{r(d)})^{φ(d)/(2r(d))}; P_n(1), for n = 1..200, is given in the dataset doi:10.5281/zenodo.22991226.

Prior work (version 1.1.0). Sign-quotiented doubling is the Chebyshev map T₂ in disguise (x = y + 1/y). The structure of papers I and III is known for the cyclic groups of orders p ± 1 (Vasiga–Shallit 2004) and q ± 1 (Qureshi–Panario 2019); it is given here for every modulus, with self-contained proofs and no claim of priority. The criterion of paper II is very probably classical. The enumeration of forward-closed subsets (paper IV) appears to be new.

Working manuscripts. Papers I–II are at version 3, III–IV at version 4. The study of the coefficients of N_h = B_h − B_{h−1} (exact height) is a separate working preprint, not included. Two scripts check the structure and the factorisation for 1 ≤ n ≤ 200 (0 mismatch); the CSV of P_n(1) is included.

This doubling generates the vocabulary of axes of La Livrée d'Hermès; the case n = 12 is Theorem 8 of the companion note doi:10.5281/zenodo.22965031.

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