III · Crystallization

The Collatz conjecture is open, and nothing here bears on it. This paper concerns one reformulation of it and the first proof obligation that reformulation names. Let \(\sigma(n)\) be the stopping time of the shortcut map \(T\) and \(Z_k(s) = \sum_{\sigma(n) > k} n^{-s}\) the weight of the integers not yet settled at depth \(k\). For a monotone certificate system on a countable domain, holding everywhere is equivalent to the exceptional weight tending to zero; so the conjecture is equivalent to \(Z_k(s) \to 0\) for any fixed \(s > 1\), and the hypothesis of monotonicity cannot be dropped.

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 Collatz conjecture is open, and nothing here bears on it. This paper concerns the finite structure of the map \(T(n) = n/2\) or \((3n+1)/2\): what \(k\) steps of it do to the integers that follow a given parity word \(w\) of length \(k\) with \(u\) odd steps. The answer is exact and complete. The word acts as one affine map \(x \mapsto (3^u x + b_w)/2^k\), whose correction \(b_w\) has a recursion, a closed form, a concatenation law, an upper-triangular matrix representation (multiplying in the order opposite to reading), and sharp bounds attained at the two extreme orderings. The integers that follow \(w\) are exactly one residue class modulo \(2^k\), and on it \(k\) steps carry \(r_w + 2^k a\) to \(m_w + 3^u a\): in the right coordinates the dynamics is the identity. The transport runs both ways, with an exact inverse, a legality congruence on the target, and an inverse fiber whose terminal family is \((4^j - 1)/3\).

The Collatz conjecture is open, and nothing here bears on it directly. A Collatz programme studies the stretch of an accelerated orbit from a value \(y\) to the first point where the orbit’s slack drops — its first crossing — and the additive correction \(B\) by which the endpoint exceeds the pure multiplicative prediction, \(2^Q z = 3^L y + B\). Five consecutive rounds of the programme prove statements about this correction and about the nested chains of such crossings. This paper separates the statements that hold on every real orbit from those that describe only the hypothetical divergent one, and decides the first kind in integers.

The Collatz conjecture is open, and nothing here bears on it directly. A Collatz programme studies how an accelerated orbit’s slack \(\delta_m = m\log_2 3 - K_m\) falls and recovers, and what a divergent orbit would have to look like if it existed. Five consecutive rounds prove statements of two kinds: identities and bounds that hold on every orbit, and theorems about a surviving first crossing — one where the slack has dropped and the value has not — which would itself refute Terras’s coefficient-stopping conjecture. On the first crossings of 99999 odd starting values, there is none. This paper checks the first kind exactly and the second as algebra, and reports three structural facts about the rounds.

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 Collatz conjecture is open, and nothing here bears on it directly. A natural strategy against a hypothetical divergent orbit is to average: take the empirical measures of its valuation sequence, pass to a limit, and look for a contradiction in the limit. Three consecutive rounds of a Collatz programme show, each from a different side, why this cannot work by itself. The limits exist and are shift-invariant with the critical mean — and two explicit countermodels, a Bernoulli measure and the mechanical word of slope \(\log_2 3\), satisfy every such condition, so no contradiction can come from them alone. The information that would separate an integer from a counterexample is the source lift: how the canonical 2-adic source of a growing code prefix moves. Compact normalized coordinates send every integer to the same point, erasing it again. And once a positive integer’s source freezes, its lift tail is zero forever, so no statistic of the source alone distinguishes a convergent orbit from a candidate.

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.

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 \(j\).

The Collatz conjecture is open, and nothing here bears on it. Its local theory — every finite parity word acts as one exact affine map on one residue class, in coordinates where the dynamics is the identity — is shown here not to depend on the numbers 3 and 1. For the map \(n/2\) or \((mn+r)/2\) with \(m\) and \(r\) positive and odd, every word of length \(k\) with \(u\) odd steps acts as \(x \mapsto (m^u x + b_w)/2^k\) on exactly one class modulo \(2^k\), carries it onto a progression of difference \(m^u\), and splits into a skeleton fixed by the counts and a correction fixed by the order; \(r\) only scales the correction. The binomial count of contracting classes then turns over at \(m = 4\): their proportion tends to one for \(m = 3\), is one for \(m = 1\), and tends to zero for every odd \(m \ge 5\).

The Collatz conjecture is open, and nothing here bears on it. For \(T(n) = n/2\) or \((3n+1)/2\), let \(\sigma(n)\) be the first \(j \ge 1\) with \(T^j(n) < n\). The conjecture is equivalent to \(\sigma(n) < \infty\) for every \(n > 1\), and a verification that \(\sigma\) is finite on \([2, N]\) proves the conjecture on \([1, N]\) and nowhere else. This paper makes that precise in the terms of the local affine atlas. The integers that follow a parity word \(w\) and have not descended within its length are one residue class cut by one exact height, \(\Omega_w \cap [1, h(w)]\), with \(h(w)\) read off the word’s contracting prefixes; a verification on \([2, N]\) is exactly the extinction of a finite frontier of such cylinders, at depth \(K(N) = \max_{n \le N} \sigma(n)\); and \(\sigma(n) = \infty\) exactly when \(n\) anchors an infinite hard branch of residue classes, so that the conjecture is the absence of such a branch at any integer above 1.

The Collatz conjecture is open, and nothing here bears on it. A Collatz programme studies how the accelerated odd map \(S(n) = (3n+1)/2^{v_2(3n+1)}\) can carry an orbit between two of its records along a near-critical bridge, one whose every suffix has valuation sum just above \(\beta\) times its length, \(\beta = \log_2 3\). Its last ten rounds turn that object into exact arithmetic, and this paper checks them in exact arithmetic: the real slack becomes an integer lift with a mechanical increment; the real cocycle left over becomes, after a change of variable, a strictly decreasing carry with no multiplier; every return of the orbit modulo \(3^k\) carries a certificate \((2^{Q} - 3^{L})\,r \equiv B \pmod{3^k}\); the quotient of a return is an integer defect with a semigroup law and a quotient-affine matrix that composes; a zero-defect return spends binary divisibility to buy ternary divisibility at the fixed rate \((Q, L)\); and the programme’s Diophantine input \(\rho^\star = 4.1164\), whose attribution it withdrew, is replaced by a finite continued-fraction statement that is certified here to 41 partial quotients — while the published review of the paper it cited shows the withdrawal’s reason does not hold.

The Collatz conjecture is open, and nothing here bears on it. A research programme studies a hypothetical divergent orbit through its slack, \(\delta_m = m\log_2 3 - K_m\), where \(K_m\) is the sum of the first \(m\) valuations of the Syracuse map, and through the renewal process of its suffix minima. Seven of its rounds build a theory of how sparse those renewals must be, and every quantitative step in them passes through one external input, a Diophantine exponent \(\rho^\star = 4.1164\) for \(\log_2 3\). The programme later withdrew that input, judging the cited theorem to concern \(\log 3\) rather than \(\log_2 3\). Its published review shows the cited result is a lower bound for linear forms in \(1, \log 2, \log 3\), which bounds \(\log_2 3\) directly; but the review prints the exponent as \(4.117\), rounded up, which does not reach \(4.1164\). This paper restores the input at \(4117/1000\), re-evaluates as exact rationals every exponent the rounds derived from it — each first reproduced exactly at \(\rho^\star\) — and finds each move the weaker way with every side condition intact. Around that, the structural core is checked where it is testable: the first-crossing intervals of any orbit are laminar and an annulus identity holds as an exact integer pair; a hierarchy of block exponents goes to zero while its constants explode, so that the new bound overtakes the old only at ratios no finite orbit approaches; the dense branch of the survivor space is empty; every suffix minimum has a first step of valuation 1; a record bound needing no premise is attained; and the programme’s correction bank, written in its own coordinates, is an identity in integers.