AN EXACT 2-ADIC DEPTH INVARIANT FOR FINITE CHRISTOFFEL PRODUCTS IN THE 3x + 1 CARRY EQUATION

Let T be the accelerated 3x+1 map on odd integers and D a finite valuation word over {1,2}. The periodization D^∞ is the valuation word of a unique rational Q(D) with odd denominator, and the characteristic Sturmian word c_α of slope α = log₂3 determines a 2-adic integer Ξ_α, the Sturmian carry constant. This note studies the bit depth ℓ(D) = v₂(Q(D) + Ξ_α), which by the Bernstein–Lagarias isometry equals the length of agreement of D^∞ with c_α, and gives an algorithm that computes it exactly for every finite product of lower Christoffel blocks of the convergents of α. The algorithm compares both words against the floor pattern of a lead fraction on the Stern–Brocot path of α. A single Departure Lemma, proved from the Farey-interval property, locates where c_α leaves that pattern for every lower ancestor of α (not only convergents): at the denominator of the next lower ancestor, upward by one, with the prefix equal to the Christoffel block of that ancestor. A Periodic Tracking Lemma gives the discrepancy of the periodized word in the affine form M(j) = m_r − t·w with integer drift w = qS_V − pL_V, so the word’s first departure has a closed form (Wrap Horizon Theorem), with w = 0 giving perpetual agreement. The trichotomy tail / ceiling / simultaneous departure then determines ℓ(D), and simultaneous departure forces a restart on the collapsed prefix. The Complete Algorithm Theorem states that the procedure terminates and returns ℓ(D), with the number of restarts bounded by the number of lower ancestors P/Q of α with ⌊Qα⌋ < ℓ(D). Termination is proved from the Fundamental Lemma alone and does not use the irrationality of Ξ_α. The single-block case is the depth law ℓ(X_n) = p_n − 1 for upper and p_n + p_{n+1} − 1 for lower convergents (q_n, p_n), n ≥ 3. Its lower-convergent half is the exponent sequence of López and Stoll’s 2-adic series for the conjugacy value of the characteristic word (Integers 9, 2009, Theorem 1); the upper-convergent half and the statement as an exact agreement depth of words are the author’s. Three worked examples are given: a saturating family (ℓ = 83), a phase-dependent tail (X₇³X₅²X₉⁴, ℓ = 335), and a restart (X₅⁴X₄ = X₇, ℓ = 568); all values are confirmed by an exact-arithmetic script included with the deposit. This note consolidates, with proofs only, the finite-Christoffel sector of the author’s series record “Structured Sectors of the Collatz Carry Equation” (concept DOI 10.5281/zenodo.20556483), whose validation data, tier ledgers, window-recycling certificates and censuses remain there. No statement about the Collatz conjecture is made or implied: the words treated form a zero-entropy class, and the invariant measures 2-adic proximity to Ξ_α, not integrality. The irrationality and transcendence of Ξ_α, first proved by Allikvere (July 2026) and independently by the author, are not used.