Local integer histories in sharp Collatz completion hulls

1. The completion question

Finite parity words have exact integer cylinders, and prescribed Collatz words have rational periodic realizations [1]. We study a conjunction of these facts: can one actual integer history pass the sharp joint convex test supplied by all rational completions of its observed past and future, while failing every discrete completion in that same family?

We construct such histories on infinitely many near-critical rows. Their observed incoming and outgoing depths grow logarithmically. The observed odd states are distinct positive integers prime to three, with a strict minimum at the common root. The entire observed vector lies inside the exact joint rational-completion hull, and the root approaches the full canonical rational cap. A specified later step nevertheless descends below the root and violates every completion's next-one deadline. The number of such roots in a shrinking relative neighborhood of the cap tends to infinity.

The affine formulas, canonical coordinates and rational root bounds are established tools [1,2]. Our contribution is their simultaneous realization in this explicit sharp completion family, together with the certified later failure and quantitative multiplicity. Section 5 gives a separate finite joining calculation explaining why independently selectable integer pieces do not automatically supply their intervening edges. We claim no barrier for additional observations, nonconvex tests or arbitrary integer methods.

Write

$$ T(x)=\begin{cases}x/2,&x\text{ even},\\(3x+1)/2,&x\text{ odd},\end{cases} \qquad \lambda=\log_2 3. $$

On rational states with odd denominator, parity means parity of the numerator. A binary word has shortcut length $S$, odd count $k$, and odd-source positions $0=p_0<p_1<\cdots<p_{k-1}<S$. Put

$$ D=2^S-3^k,\quad C(w)=\sum_{t=0}^{k-1}3^{k-1-t}2^{p_t},\quad X(w)=C(w)/D. \tag{1} $$

When $D>0$, $X(w)$ generates a positive rational periodic profile with the prescribed parity word. Indeed the branch composition is $(3^kx+C(w))/2^S$, and cyclic versions of (1) give all its positive states and their required parities. This is the classical rational-cycle correspondence [1, Theorem 2.1]. A canonical word satisfies

$$ p_t\le c_t:=\lfloor St/k\rfloor. \tag{2} $$

Its root is bounded by the mechanical root

$$ X_{\rm mech}=C_{\rm mech}/D,\qquad C_{\rm mech}=\sum_{t=0}^{k-1}3^{k-1-t}2^{c_t}. \tag{3} $$

These are the canonical coordinates of [2]; the convention agrees with papers 24 and 26 [3,4]. Here $S$ counts shortcut steps, not ordinary steps.

2. A common completion family and its exact hull

Assume a determinant-one bracket

$$ \frac pq<\lambda<\frac Sk,\qquad qS-pk=1,\qquad k\ge2,\quad p,q\ge1,\quad 0<\varepsilon:=S-k\lambda\le1. \tag{4} $$

Then $\gcd(S,k)=1$ and $3/2<S/k<2$. The upper inequality follows because $S\ge2k$ and $p<2q$ would imply $qS-pk\ge k>1$. Set $z=S-k$. Fix outgoing depth $b\ge1$ and incoming depth $h\ge1$ with

$$ b+h+2\le k,\qquad h+1\le z. \tag{5} $$

Let $\mathcal F$ consist of the canonical position lists with

$$ p_t=t\ (0\le t\le b),\qquad p_{k-j}=S-2j\ (1\le j\le h+1). \tag{6} $$

Thus the observed outgoing accelerated exponents are $b$ ones and the observed incoming exponents are $h$ twos. The extra fixed tail step will ensure that the oldest observed odd state is prime to three.

The minimum-root assertion below overlaps the upper-spine state-ordering result of paper 27/v1 [8, Theorem 3(2)]. We give the determinant-one argument needed for this completion family.

Lemma 1. Every canonical rational profile under (4) is rooted at its unique numerical minimum. The family $\mathcal F$ has coordinatewise least and greatest words $w_-,w_+$, where

$$ \begin{aligned} p_t^-&=t &&(0\le t<k-h-1),\\ p_t^+&=\min\{c_t,z-h-1+t\} &&(b+1\le t\le k-h-2), \end{aligned} \tag{7} $$

and the fixed coordinates are as in (6). Consequently $\underline X:=X(w_-)$ and $\overline X:=X(w_+)$ are attained root extrema.

Proof. Root any positive profile at its actual minimum $m$. A proper prefix of length $j$ and odd count $a$ ends at a state at least $m$; its nonempty returning suffix has slope strictly less than one. Hence

$$ q(S-j)-p(k-a) =q\bigl((S-j)-(k-a)\lambda\bigr)+(k-a)(q\lambda-p)>0. $$

This integer and $qj-pa$ sum to one, so $qj-pa\le0$. For $j>0$ this also forces $a>0$, and $q(Sa-kj)=a-k(qj-pa)>0$. Thus that minimum cut is canonical. Coprimality makes the centered walk's values distinct at distinct physical cuts, giving uniqueness. The same argument identifies the minimum of a profile already displayed at a canonical cut.

For (7), every increasing list has $p_t\ge t$. The fixed tail leaves $p_t\le z-h-1+t$ before it, while (2) gives the other upper bound. Both bounds in the minimum increase by at least one. Their splice with the fixed head and tail is legal by (5); the tail obeys (2) since $S/k<2$. Thus both extrema belong to $\mathcal F$. Formula (1) is strictly increasing in every position. $\square$

The following quantitative bounds will locate integer roots in the hull:

$$ \begin{aligned} C(w_-)&=3^k\left[1-(2/3)^{k-h-1}\right] +2^S\left[1-(3/4)^{h+1}\right],\\ 1<\underline X&<\frac3{2^\varepsilon-1},\\ \overline X-\underline X &>\frac{k-b-2h-20}{6(2^\varepsilon-1)} \quad(k>b+2h+20). \end{aligned} \tag{8} $$

The first identity sums two geometric series. For $\underline X>1$, subtract $D$ and use $2-(2/3)^{k-h-1}-2^\varepsilon(3/4)^{h+1}\ge5/24$. For $b+1\le t\le k-2h-2$, (7) equals $c_t$, because $\lceil S(k-t)/k\rceil\ge(k-t)+(h+1)$. Each such numerator term is at least $3^{k-1}/2$. Therefore $\overline X\ge(k-b-2h-2)_+/[6(2^\varepsilon-1)]$; subtracting the upper bound for $\underline X$ proves (8).

3. Actual integer histories with impossible later completion

For a real variable $v$ define the observed odd states

$$ x_t(v)=(3/2)^t(v+1)-1\quad(0\le t\le b),\qquad x_{-j}(v)=1+(4/3)^j(v-1)\quad(1\le j\le h). \tag{9} $$

Include their intervening even states and let $\mathbf V(v)$ be the resulting indexed observation vector. Each coordinate is affine increasing in the same $v$. Every rational completion in $\mathcal F$ has precisely $\mathbf V(X(w))$ as its observed vector. Thus

$$ \operatorname{conv}\{\mathbf V(X(w)):w\in\mathcal F\} =[\mathbf V(\underline X),\mathbf V(\overline X)]. \tag{10} $$

The observed vectors lie on one affine line. Thus joint convex containment is immediate once a root lies between the attained endpoints; retaining all correlations introduces no further obstruction on this line. The content of the next theorems is the construction of integer roots in that interval with the additional itinerary, cap and later-failure properties.

Theorem 2. Put $M_{\rm drop}=2^{2b+3}3^{h+1}$. If $\overline X-\underline X>M_{\rm drop}$, there is an integer $\underline X<m<\overline X$, within $M_{\rm drop}$ of $\overline X$, with all of the following properties:

1. $\mathbf V(m)$ is one genuine positive integer itinerary across the displayed root. Its observed odd states are distinct and prime to three; $m$ is the strict minimum of its observed physical interval.
2. It occupies the relative interior of (10) and satisfies every rational completion's common root congruences modulo $2^{b+1}$ and $3^{h+1}$.
3. Its next accelerated exponent is $b+2$, and its next odd state is less than $m$. No word in $\mathcal F$ is its actual full forward itinerary.

Proof. Choose the integer CRT class

$$ 3^{b+1}(m+1)\equiv2^{b+1}+2^{2b+2}\pmod{2^{2b+3}}, \qquad m\equiv1\pmod{3^{h+1}}. \tag{11} $$

The first coefficient is invertible modulo its dyadic modulus, so this is one class modulo $M_{\rm drop}$. Take its largest member strictly below $\overline X$. The width hypothesis places it strictly above $\underline X$. It refines $m\equiv-1\pmod{2^{b+1}}$ and $m\equiv1\pmod{3^{h+1}}$. The fixed head and tail in (6), applied to (1), give exactly these two congruences for every rational $X(w)$; their denominators are coprime to six.

Equations (9) now give odd positive integers and precisely the stated accelerated exponents. Incoming odd states are $1\bmod3$, outgoing states other than $m$ are $2\bmod3$, and each side increases strictly away from $m$. Incoming even states are twice their odd receivers. Hence distinctness, three-unit type and the strict observed minimum hold. Equation (10) proves joint-hull membership on this same integer vector.

From (11), $v_2(m+1)=b+1$ and $v_2(3^{b+1}(m+1)-2^{b+1})=2b+2$. The next exponent is therefore $b+2$, and the next odd state equals

$$ y=(3/4)^{b+1}(m+1)-2^{-(b+1)}<m. \tag{12} $$

For the strict inequality put $r=b+1$. The elementary convexity bound $2(3/4)^r\le1+2^{-r}$ for integers $r\ge1$ gives $y-m=-(1-(3/4)^r)(m-1)+2(3/4)^r-2^{-r}-1<0$. The next odd input is at physical position $2b+2$, but (2) requires $p_{b+1}\le c_{b+1}<2b+2$. This disproves every discrete completion in $\mathcal F$, irrespective of its unobserved middle. $\square$

Only the displayed observation vector is asserted to be integral. Its eventual trajectory after (12) is unrestricted. The completion failure does not assume the Collatz conjecture.

4. Many false candidates near the full canonical cap

The full mechanical benchmark is prior: Fernández and Ibáñez [7, Theorem 7.3] prove Christoffel extremality for the cyclic minimum. Our construction approaches that benchmark while fixing the two-sided history.

Theorem 3. Along sufficiently late upper continued-fraction convergents $S/k$ to $\lambda$, set $b=h=\lfloor(\log_2 k)/4\rfloor$ and $r=b+2h+2$. There are at least

$$ \max\left\{0, \left\lfloor\frac{k^{(2-\lambda)/4}}{288\log 2}\right\rfloor-1 \right\} \tag{13} $$

distinct integer roots satisfying conclusions 1–3 of Theorem 2 and

$$ 0<\frac{X_{\rm mech}-m}{X_{\rm mech}}<\frac{4r+1}{k}. \tag{14} $$

In particular their number tends to infinity, their incoming and outgoing depths tend to infinity, and $m/X_{\rm mech}=1-O((\log k)/k)$.

Proof. The preceding lower convergent supplies (4), and $0<\varepsilon<1/k$. The bound $2^\varepsilon-1\le2(\log 2)\varepsilon$ and (8) imply hull width $>k(k-b-2h-20)/(12\log2)$. Also

$$ M_{\rm drop}\le24k^{(2+\lambda)/4}=o(k). \tag{15} $$

At most $r$ positions of $w_+$ differ from the mechanical positions. Every mechanical numerator term is less than $2^S/3$, while $C_{\rm mech}\ge k3^k/6$. Therefore

$$ 0\le\frac{X_{\rm mech}-\overline X}{X_{\rm mech}} <\frac{4r}{k},\qquad \frac{\underline X}{X_{\rm mech}}<\frac{18}{k},\qquad \frac{X_{\rm mech}}k>\frac{k}{12\log2}. \tag{16} $$

For $k>4r+19$, the interval $I=(\overline X-X_{\rm mech}/k,\overline X)$ lies strictly inside $(\underline X,\overline X)$. Every member of (11) in $I$ has the properties of Theorem 2, since those depend on its CRT class and interior position, not on being the nearest member. Their number is at least $\max\{0,\lfloor X_{\rm mech}/(kM_{\rm drop})\rfloor-1\}$. Equations (15)–(16) give (13), and the definition of $I$ gives (14). $\square$

Every convex set containing the projected rational completion family must contain (10), hence all these integer vectors. Consequently no additional convex inequality valid on this unchanged rational family can reject them. A further observation, an integer-specific restriction or a nonconvex test changes the relaxation and is outside this conclusion.

5. A separate joining calculation for selectable integer pieces

The following elementary companion comes from the project's partial-integer realization work. It concerns a different assignment from Theorems 2–3. Suppose actual integer paths in $m\ge3$ consecutive blocks can be chosen independently by $h_i\in\{0,\ldots,Q-1\}$ for an integer $Q\ge1$, with block indices read cyclically modulo $m$ and entrance $y_i(h_i)=y_i(0)-2^{b_i}h_i$. Block $i$ has $k_i$ odd sources and its last odd input $y_{e_i}$ is followed by $g_i$ even inputs before the next entrance. Here $b_i$ is its cylinder length including its last assigned odd parity bit. Internal paths are assumed genuine for every permitted lift. The unassigned cut joins exactly when

$$ \begin{aligned} J_i(h)&=\frac{2^{1+g_i}y_{i+1}(h_{i+1})-3y_{e_i}(h_i)-1}{2}\\ &=J_i(0)+3^{k_i}h_i-2^{b_{i+1}+g_i}h_{i+1}=0. \end{aligned} \tag{17} $$

This follows by affine propagation; the two coefficients are coprime. Fix $h_0=0$ and choose the other lifts uniformly and independently. For a cut not adjacent to block zero, its coprime solution line has at most $1+\lfloor(Q-1)/A_i\rfloor$ points in the $Q$-square, where $A_i=\max(3^{k_i},2^{b_{i+1}+g_i})$. Each of the two adjacent cuts has probability at most $1/Q$. Thus the number of assignments in which every selected cut fails is at least

$$ Q^{m-1}\left[ 1-\frac2Q- \sum_{i\notin\{0,m-1\}} \frac{1+\lfloor(Q-1)/A_i\rfloor}{Q^2} \right]. \tag{18} $$

The union bound needs no independence between cut events. If its bracket tends to one, almost every assignment in this specified product fails every cut despite genuine block interiors. Feasibility of that product, all numerical caps and any cross-block distinctness must be supplied separately; (18) does not create them. Neither construction in this paper asserts a full ordinary integer cyclic itinerary.

6. Relationship to other limitations

Paper 26 [4] constructs phase-injective principal downsets and geometrically large minimal representatives. Its quantitative rational-state witnesses do not assert the two-sided integer histories or the sharp joint completion hull proved here. Conversely our finite observed hull does not establish phase-minimality or the full-phase accommodation of other project constructions.

Paper 33/v1 [9, Theorem 1] gives two-sided prefix/suffix certificates for phase-injective downsets, extending paper 26. Those certificates concern residue injectivity and full legal descent; they do not assert the joint completion hull or the integer histories with later failure constructed here.

Sylve Recursions [5] constructs arbitrarily long expanding integer prefixes at a fixed finite arithmetic fingerprint, using negative rational anchors. Prescribed-word shadowing and CRT are established mechanisms. Here the reference profiles have positive determinant and positive rational minima; the additional content is the exact common completion hull, its cap geometry and the later failure of that very completion family.

Wren's drift-certificate characterization [6] concerns residue-avoidance graphs and negative periodic pins. Our conclusion is instead a limitation of one explicitly projected positive rational-completion family. Neither statement rules out arbitrary finite arithmetic arguments.

Authorship and provenance

This paper reports human-directed, AI-generated research in the Collatz Project. The main construction was developed in the CR17 BISON17 lineage; the separate joining calculation was developed in RP03. Other project agents supplied source and scope reviews. Aaron Kyle Solomon directs the work and accepts responsibility for the submitted claims, sources and presentation. Murmuration represents the publishing workflow. Internal AI semantic review is distinct from conventional external peer review and formal proof verification. All essential arguments are given above.

References

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