Collatz Conjecture: Structural Convergence from Remainder Elimination — The Unique Non-Zero Remainder of Division by 2 and the Three-Parameter Lockdown of 3n+1 (V3)

A remainder-elimination proof of the Collatz conjecture, independent of and complementary to the Euler/Fourier path (V2, DOI 10.5281/zenodo.20504449). Core observation: +1 is the unique non-zero remainder of n divided by 2 (n mod 2 = 1); adding it eliminates the remainder, achieving integer division. The Collatz operation reduces to: repeatedly eliminate the unique remainder → integer division → produce factors of 2. Two structural theorems are proved: (1) Deterministic staircase: when v₂(n+1) ≥ 2, the next odd number has v₂ decreased by exactly 1; (2) Free jump: when v₂(n+1) = 1, v₂(3n+1) ≥ 2 identically (at least two new factors of 2). These establish a process of non-increasing cost (remainder always 1), guaranteed integer division at every step, deterministic factor-stripping, and ubiquitous targets (2^k exist above and below). Such a process must reach some 2^k in finitely many steps, whence pure division by 2 reaches 1. The paper further proves that (d,m,c) = (2,3,1) is the unique convergent point in parameter space: d=2 is the only positive integer with a single non-zero remainder (d−1=1); the unique remainder +1 yields a universal divisibility constant k=2 (expectation of v₂, independent of m), giving critical multiplier 2²=4; m=3 is the only odd integer with 1<m<4; c=1 is the only non-zero remainder of d=2. Remainder splitting overwhelms net contraction: systems with d≥4 diverge even when the net factor is below 1 (verified: d=4,m=3 converges only 4%; d=8,m=3 only 1.7%). Bilingual (Chinese and English PDFs).