Structural Notes on the Collatz Dynamics: A Forced Gate, a Climbing-Run Characterisation, an Integer-Energy Reformulation, and a Depletion Law for Climbing Runs

ABSTRACT. This is a short expository note on the elementary structure of the Collatz map T. We give self-contained proofs of four facts and state precisely what they do and do not establish. First, a forced-gateobservation: because 16 is the only preimage of 8, every orbit that reaches the terminal cycle from outside it mustpass through 16 → 8 → 4 → 2 → 1, so any convergent orbit starting above 8 contains both 16 and 8. This is thefirst case of a parity alternation: 2k has the single preimage 2k+1 when k is odd, and the two preimages 2k+1 and(2k−1)/3 when k is even; hence an orbit can step onto the set of powers of two from outside only at an evenexponent, through one of the sparse doors (4m−1)/3 = 1, 5, 21, 85, … Second, an integer-energy reformulation:writing E(n) = log2 n, the conjecture is equivalent to the statement that every orbit eventually attains an integerenergy level, i.e. hits a power of two; we note explicitly that this equivalence is close to tautological and isoffered as a reformulation, not as progress. Third, a climbing-run characterisation: a maximal run of jconsecutive odd steps each having 2-adic valuation 1 occurs if and only if n ≡ −1 (mod 2j+1); hence below 2k thelongest such run has length k−1 and is attained only by n = 2k − 1, and the depth of the ensuing collapse is givenexactly by 1 + v2(3k − 1) via the p = 2 lifting-the-exponent formula. Fourth, a depletion law: with t(n) the numberof trailing 1-bits of n, one has t(S(n)) = t(n) − 1 exactly, so resistance is a stock consumed at the rate of one bitper odd step and never renewed within a run; the only infinite climb in the 2-adic integers is the fixed point −1,whose finite truncations are exactly the resistant family 2k − 1. An exact orbit identity yields the escape criterion:a bounded orbit must satisfy lim inf Jm/m ≥ log2 3, where Jm is the total number of halvings after m odd steps; anontrivial cycle forces J/m to approximate log2 3 at continued-fraction quality, which is the mechanism behindthe classical lower bounds on cycle length. None of these results proves the conjecture, and no novelty isclaimed: the facts are elementary and belong to the folklore of the 3x+1 literature. We also record the elementarydrift computation: the expected number of halvings triggered by one odd step is exactly 2, so the expected energychange per odd step is log2 3 − 2 ≈ −0.415 bits. Being a statement about natural density rather than aboutindividual orbits, this is simultaneously the precise form of the reason the conjecture is expected to hold and theprecise form of what remains open. The note closes with a methodological litmus test — the 5n+1 map has thesame 2-adic structure but admits a nontrivial cycle, so no argument resting on 2-adic locality alone can settleCollatz; applying the same drift computation to n qn+1 gives an average multiplier of exactly q/4 per odd step,so q = 3 is the only odd multiplier above 1 with downward drift