Summer of Math Exposition

The Collatz Conjecture is one of the simplest problems to state in all of mathematics: take any whole number, and if it’s even, divide by 2; if it’s odd, multiply by 3 and add 1. Repeat. The conjecture says that no matter what number you start with, eventually the sequence reaches 1. Despite its simplicity, the problem has remained unsolved for nearly a century. What makes Collatz so mysterious is the mix of two different rules — halving steps versus the “3x + 1” step — which creates irregular, chaotic-looking orbits. In this video, I present a way of collapsing Collatz into a single-rule function: f(x) = 3x + 2ⁿ, where 2ⁿ is the largest power of 2 dividing x. This approach folds the halving steps back into the triple step, treating them as part of a unified successor process along the 2-adic ring. In effect, each prime power of two acts as a successor modulator in a base-2 system. By doing so, the halving steps no longer branch into separate paths — instead, they align along the power-of-two axis, forming the structural backbone of the network. Through this lens, the usual “branching” Collatz tree undergoes an inversion: rather than chaotic bifurcation, the system exhibits convergence toward the 2-adic modular axis. This reframes the dynamics as a kind of gravitational collapse, where exponential decay replaces exponential branching. The key equivalence is that completing one cycle of “halving steps + 3x + 1” is algebraically the same as applying “3x + 2ⁿ” in a single move. This unification transforms Collatz into a deterministic sieve, where all orbits compress and converge toward the powers of two. ----------Abstract---------- In this video, I present a way of collapsing Collatz into a single rule function: f(x)=3x+2ⁿ where, 2ⁿ ∣ x and 2ⁿ⁺¹∤ x (2ⁿ is the largest 2ⁿ that divides x. This approach folds the halving steps back into the triple step as a unit successor modulator along the 2-adic ring by taking advantage of the smallest prime factor, 2ⁿ, and its position as the next successor to 1 and 3 so that, at each iterative step, x's prime factor, 2ⁿ, can be used as a successor modulator in a counting system of base log 2. This returns continuity to the network, eliminating bifurcation of branches and the fractalization of halving steps because the trivial cycle undergoes coordinate inversion becoming a hyperbolic "Penrose-boundary" represented by the power of 2 axis,—or 2-adic ring— the building blocks of which are all the halving steps. The inversion of the network combined with the elimination of halving steps also inverts the growth through bifurcation to a decay through convergence and changes the order from an exponential decay 'rate' to exponential decay resembling a gravitational collapse toward the modular boundary. So, instead of alternating between two different processes, we can see Collatz as one consistent successor function. This reveals an algebraic equivalence: completing a single rotation of halving and then applying “3x+1” is the same as applying “3x+2ⁿ” directly. This becomes a deterministic sieve as the branches/orbits of the Collatz tree converge to a single modular axis at 2ⁿ. https://doi.org/10.5281/zenodo.16733351 https://doi.org/10.5281/zenodo.16819899