Packing deficits by congruence, and an exact certificate for a transport exponent

The Collatz conjecture is open, and nothing here bears on it directly. This paper concerns one quantity a Collatz programme uses to control a hypothetical divergent orbit: along a segment of the accelerated orbit that begins at a value \(y\) and ends where the orbit’s slack first drops, the endpoint is \(z = 2^{-D}\,\Pi\) times \(y\), with \(D > 0\) and a multiplicative correction \(\Pi = \prod (1 + 1/(3Y_j))\) over the segment’s states. The endpoint can lie above \(y\) only if \(\Pi\) is large, and four consecutive rounds of the programme bound \(\Pi\) by envelopes growing like a power of the segment’s length \(L\).

The exponent falls from \(1/6\) to \(1/9\), to \(4/45\), to \(1373/25856 \approx 0.0531\), and every step is decided in integers or rationals. The second step is one congruence: \(3n + 1 \equiv 1 \pmod 3\), so no accelerated image is divisible by three, and the states pack at spacing three rather than two. The third is an exact identity that telescopes. The fourth is not an estimate at all but the value of a finite dual certificate, which we print in full; a reader can verify it with no computation of ours. We separate the premises carefully: the first two envelopes hold on every such segment, and were tested on tens of thousands of them; the last two are proved under the premise \(z > y\), which describes the hypothetical object and which no real segment we built satisfies, so on real orbits they are consistent but untested. We give every proof the paper relies on, record what the programme’s own records got wrong, and report how the instruments were made to fail.

Not claimed anywhere in this paper: the Collatz conjecture; that a divergent orbit exists or does not; that any of the four exponents is optimal; that the transport exponents \(\alpha_h\) tend to zero as the modulus grows; that the conditional envelopes of §§6–7 were tested on a real orbit — their premise is met by none; and any exponent that passes through the programme’s Diophantine input \(\rho_\star = 4.1164\). The rounds also derive root-source and dense-overlap exponents from that input; the programme later withdrew it, and those exponents are left out of this paper entirely.

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 four rounds this paper rests on: the multiplicative identity and the consecutive-odd envelope; the sieve, the admissible positions and the span theorem; the reciprocal-flow balance, the mod-9 cost, the span bound and the static countermodel; and the transport identity, the dual certificates and the exponent they give.

Neo.K set the programme’s target and its method, and wrote its documents with Aletheia.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — rechecked each round with independently written code in exact arithmetic, measured every premise separately and so found which statements real orbits can test, found that the Gamma forms are exact, eliminated the countermodel’s closed forms into a checkable relation, verified the certificates and computed their tails, quoted them into an evidence file so that they could be printed, corrected its own record of the third round’s conditional statements, built and drilled the instruments, and wrote this paper, including the proofs as written out here. Nothing in the rounds’ mathematics is this line’s invention.

Priority, stated plainly. No result here is claimed as new mathematics. The envelopes, the identities, the certificates and the exponents are the programme’s. What is offered as this line’s own is the separation of premises, the exact checks, and the record of what held where.