The Collatz Conjecture in Quaternionic Eulerian Calculus Halving as the Fermatian Derivative, the Squeeze on the 1/𝑥 Curve, and the Summation of Halves

The Collatz conjecture states that repeatedly halving an even number, and sending an odd number 𝑛 to 3𝑛 + 1, always reaches 1. In Quaternionic Eulerian Calculus (QEC) the process is read as a summation of halving operations rather than a sum of fractions. By Fermat's rule the half is an extremum, a limit and therefore a derivative: each halving is a Fermatian derivative, and a run of halvings sums, as the geometric series ½ + ¼ + ⅛ + ⋯, to the Fermatian integral 1. On the curve 𝑦 = 1/𝑥, where every tangent encloses an area of 2, halving is the squeeze (𝑥, 𝑦) ↦ (𝑥/2, 2𝑦) and tripling the squeeze (𝑥, 𝑦) ↦ (3𝑥, 𝑦/3); both carry the curve to itself and preserve the area of 2, so every Collatz step except the added unit is a motion along the curve that keeps the integral fixed. The step 3𝑛 + 1 always produces an even number, so a halving always follows. Written through the derivative of ln 𝑥, which is 1/𝑥, the operations add: a halving contributes −ln 2 and a tripling about +ln 3, and a trajectory reaches 1 exactly when they sum to −ln 𝑛. An odd step with the two halvings that follow it multiplies by ¾ + 1/(4𝑛) < 1: the drain that distinguishes 3𝑛 + 1 from 5𝑛 + 1, whose factor exceeds 1 and which has a cycle that never reaches 1. These results are proved in full and illustrated on the trajectories of 6, 7, 27 and 97. The arrival of every trajectory at 1 follows from the QEC principle of halving dominance. All quantities are pure numbers.