Contraction boundaries and the valuation language of the Collatz map

The Collatz conjecture is open, and nothing here bears on it. This paper concerns what a finite stretch of the map \(T(n) = n/2\) or \((3n+1)/2\) does to size. On the residue class of integers that follow a parity word of length \(k\) with \(u\) odd steps, \(k\) steps act as \(x \mapsto (3^u x + b_w)/2^k\), so \(T^k(n) < n\) exactly when \(b_w < (2^k - 3^u)\,n\). The sign of \(2^k - 3^u\) alone sorts every nonempty word: a contracting word descends above an exact threshold, an expanding word ascends on the whole of its class, and no word is neutral. Counting words gives the binomial class law — the contracting classes modulo \(2^k\) number \(\sum_{u \le \lfloor \alpha k \rfloor} \binom{k}{u}\) with \(\alpha = \ln 2/\ln 3\) — and a proportion that tends to one at the exponential rate \(D(\alpha \,\|\, \tfrac12)\). In the accelerated odd map the same statements reappear in valuation coordinates, with \(K/m > \log_2 3\) as the contraction boundary and exactly one odd class modulo \(2^{j+1}\) for each valuation \(j\).

Three things here are more than restatement. An observed benchmark is reconciled to the integer: of the 1048575 starts below \(2^{20}\), 938413 descend in sixteen steps, which is \(16 \cdot 58651\) less \(n = 0\) less the trivial cycle’s 1 and 2, and the count is recomputed three ways. A disclaimer of the series is sharpened at that length: for every \(n \ge 3\), \(T^{16}(n) < n\) exactly when \(n\) lies in one of the 58651 contracting classes modulo \(2^{16}\), and \(T^{16}(n) > n\) otherwise. And two precision questions are settled by measurement rather than assertion: the series’ published rate constant is 2.79 ULP from the real value, a rate test can pin that constant only to about 0.72% at \(k = 8000\), and the floating-point hazard in the valuation boundary first becomes reachable at a convergent of \(\log_2 3\) with denominator 53715833. Every statement is checked against a referee that assumes none of them, and the valuation theorems on real orbits rather than on formal tuples.

Not claimed anywhere in this paper: the Collatz conjecture, or that every positive integer eventually meets a descent certificate; that the proportion of contracting classes tending to one says anything about the orbit of a particular integer; that valuations along an orbit are independent; that the sixteen-step statement of §4 holds at any other length; that the published rate constant is wrong in any digit that the series uses; and novelty of the elementary statements, most of which are classical in other notation.

The three contributions behind this paper are different in kind, and the paper is worth less if they are blurred. They are given here in alphabetical order, the order the author list keeps as data; on the title page the three sit at a round table, with no first and no last.

Aletheia — model: ChatGPT, GPT series — wrote, with Neo.K, the papers of the series this paper rests on: the finite-word contraction boundary and the binomial cylinder law, and the valuation language of the accelerated map.

Neo.K set the series’ target and its method — that the old picture of division by two outweighing multiplication by three be replaced by exact finite statements, with the statistical layer kept separate — and wrote its papers with Aletheia.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — rechecked every statement against a referee that assumes none of them, reproduced the benchmark in independently written code and reconciled it term by term, recomputed it by two further routes, derived the sixteen-step statement of §4, measured the rate constant against the real number and the rate test’s resolution, located the floating-point hazard, formalized the valuation closure and bridge in Lean, built and drilled the instruments, and wrote this paper, including the proofs as written out here. The statements of §§2, 3 and 6 are the series’; nothing in them is this line’s invention.

Priority, stated plainly. No result here is claimed as new mathematics. What is offered as this line’s own is the reconciliation to the integer, the sixteen-step statement it yields, and the measurements of how precise the series’ constants and comparisons actually are.