The Collatz Conjecture in Quaternionic Eulerian Calculus Halving as the Fermatian Derivative, the Squeeze on the 1/π₯ Curve, and the Summation of Halves
Year: 2026
Type: article
Abstract: The Collatz conjecture states that repeatedly halving an even number, and sending an odd number π to 3π + 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 Β½ + ΒΌ + β
+ β―, to the Fermatian integral 1. On the curve π¦ = 1/π₯, where every tangent encloses an area of 2, halving is the squeeze (π₯, π¦) β¦ (π₯/2, 2π¦) and tripling the squeeze (π₯, π¦) β¦ (3π₯, π¦/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 3π + 1 always produces an even number, so a halving always follows. Written through the derivative of ln π₯, which is 1/π₯, the operations add: a halving contributes βln 2 and a tripling about +ln 3, and a trajectory reaches 1 exactly when they sum to βln π. An odd step with the two halvings that follow it multiplies by ΒΎ + 1/(4π) < 1: the drain that distinguishes 3π + 1 from 5π + 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.
Author: Willie Johnson Jr.
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225807741
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