Sparse lifts cannot keep up: the packing barrier for subcritical Collatz orbits, and where it stops

The Collatz conjecture is open, and nothing here bears on it directly. A Collatz orbit that never descended would have to stay subcritical forever: after \(m\) accelerated steps its total valuation \(K_m\) stays below \(m\log_2 3\). Measure how far it lags the mechanical bound by the deficit \(d_m = \lfloor m\log_2 3 \rfloor - K_m\). Four consecutive rounds of a Collatz programme prove that such an orbit cannot keep its deficit small: two equal blocks of valuations force the states before them into one residue class modulo a power of two, so repeated blocks force states far apart, while blocks of a thin-deficit word are too few to avoid repeating — their number grows like \(\Lambda_\gamma^r\) with \(\Lambda_\gamma = (1+\gamma)^{1+\gamma}/\gamma^\gamma = 2.8395\ldots < 3\), \(\gamma = \log_2 3 - 1\).

The consequence is a barrier on the record deficit \(D_N\): \(\limsup D_N/\log_2 N \ge 0.01\), improved by counting occurrences rather than repeats to the sharp \(c_{\mathrm{pack}} = 0.0358567600\ldots\), and by Stirling’s prefactor to a second-order term \(d_{\mathrm{pack}} = 0.3689789787\ldots\); a further lever, the prefix constraint, provably buys nothing at first order, and the branch is closed. We recompute every constant from scratch by a different method to 80 digits or more, check the finite arithmetic the arguments rest on, and eliminate the excluded families on explicit instances. The barrier itself is about infinite orbits and is the programme’s. We also record a counting error — paths counted where words were defined — that this line reproduced faithfully with two agreeing implementations and did not notice.

Not claimed anywhere in this paper: the Collatz conjecture; that a subcritical orbit exists or does not; that \(c_{\mathrm{pack}}\) or \(d_{\mathrm{pack}}\) is the true sharp constant — each is optimal only for its envelope; that the four levers the rounds list and do not pull are useless; and novelty of the elementary steps.

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 sparse-lift squeeze and its entropy constant, the occurrence packing and \(c_{\mathrm{pack}}\), the prefix-constraint no-gain and \(d_{\mathrm{pack}}\), and the pointed/unpointed correction with which the branch was closed — the correction found by the programme itself.

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 — recomputed every constant by an independent method, checked the finite arithmetic and the eliminated families on instances, measured how far the first scheme could be pushed before the next round pushed it, implemented the corrected counts from the definitions after missing the error, built and drilled the instruments, and wrote this paper. Nothing in the rounds’ mathematics 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 independent recomputation, the measured reach of each scheme, and an honest record of an error two implementations shared.