Exact Synchronized Coalescence Clocks and Peak-Envelope Transfer for a Positive-Density Family of Consecutive Collatz Trajectories

A dyadic-coordinate translation and extension of LaDue's criterion for consecutive Collatz trajectories (v4.1, August 2026).

For the full map F(x) = 3x+1 (odd) / x/2 (even), write an odd integer as a = 2^t · u − 1 with u odd and set e = v_2(3^t · u − 1). The condition e = 1 is exactly LaDue's same-parity criterion under the dictionary p = t+1, 2q+1 = u, ℓ = a+1. The local criterion and its equal-stopping-time consequence are prior results.

The present contribution begins after that identification. The paper proves:

(1) Minimal synchronized coalescence clocks: 2t+3 under the full map and t+3 under the semi-accelerated map, with strict pre-coalescence inequality at every earlier synchronized time.

(2) A fixed-point-free involution on all positive odd integers outside 5 (mod 8), with every orbit being an adjacent-floor pair whose consecutive starts synchronize.

(3) Exact counts on every complete dyadic block: the involution domain contains 3·2^{K−3} odd integers and the oriented lower floors contain 2^{K−2}, giving natural density 1/8 for the induced consecutive even starts.

(4) Exact finite-horizon peak envelopes: with z = (3^t·u − 1)/2 and Y = 3z+1, the synchronized maxima are max{4z, H_r(Y)} and max{12z+4, H_r(Y)}, yielding an exact three-regime ratio formula in [1, 4] without assuming boundedness or global Collatz convergence.

(5) An exact release-residue identity, the joint Haar law of floor and release valuations, and a rigorous parity-cylinder realizability guard separating affine map identities from realized trajectory identities.

All mathematical statements are unconditional and make no use of global Collatz convergence.

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11B83 — Special sequences and polynomials (Collatz)

11A07 — Congruences; primitive roots; residue systems

37A45 — Relations with number theory and harmonic analysis