Compressing the coefficient frontier of the Collatz map to one anchor, one path per source, and a closure that claims exactly what it has

The Collatz conjecture is open, and nothing here bears on it directly. Terras’s coefficient stopping time \(\tau_c(n)\) is the first \(k\) at which the multiplier \(3^u/2^k\) of the first \(k\) steps of the map \(n \mapsto n/2, (3n+1)/2\) falls below one; that it is finite for every \(n \ge 2\) is the coefficient-stopping conjecture. A Collatz programme compresses this conjecture in six documents. The set of integers not yet settled at depth \(k\) is a union of residue cylinders over a ballot tree; its zeta-weight is an exact finite Hurwitz sum; every progression tail is negligible; so the weight tends to zero exactly when the least unsettled integer \(m_k\) tends to infinity. In the accelerated code the least anchor becomes a canonical 2-adic source; at every node exactly one exponent preserves the source, so the tree of codes collapses to one path per source; every unit of extra valuation is paid out of a Sturmian budget of density \(\log_2 3 - 1\); and the programme declares this “A line” closed — as a reduction, not a proof, which is what it claims.

We check every document’s mathematics exactly and measure what each leaves open. The least anchor escapes 23 times below \(2^{32}\), to 2,788,008,987, and each measured \(m_k\) is converted into a rigorous bound on the true infinite weight. The same anchor sequence appears in both coordinate systems; a proposed parity route is priced as equivalent to the conjecture, not cheaper; the continued-fraction gate opens on only two depths in seven orbits; the finite-local no-go’s witness is ten orders of magnitude more expensive than necessary; and every finite orbit measured ends its subcritical life having spent all but one or two units of its Sturmian credit.

Not claimed anywhere in this paper: the Collatz conjecture; Terras’s coefficient-stopping conjecture; that \(m_k \to \infty\) — 23 escapes are what a bounded sequence could also show; that any infinite subcritical spine exists or does not; and novelty of the documents’ mathematics.

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 documents this paper rests on: the coefficient frontier and its reduction to the minimum anchor, the accelerated code, the 2–3 bridge, the zero-lift edge and the spine, the deficit ledger, the closure with its no-go and saturation reduction, and the handoff to the next line.

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 every document with independently written code in exact arithmetic, measured the anchor sequence and converted it into rigorous bounds, extended the anchor table and found the two coordinate systems agree, priced the parity route and the no-go’s witness, measured the continued-fraction gate and the end-of-life saturation, checked the closure’s scope and its citation, formalized the no-go and the dichotomy in Lean, built and drilled the instruments, and wrote this paper. Nothing in the documents’ 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 measurement of what each reduction leaves, and the exact checks.