Normal witnesses transfer max-plus Collatz barriers beyond completion conditions

1. Contribution and relation to prior work

Nashida [1] proves broad barriers for finite-dimensional nonnegative max-plus interpretations of Collatz rewriting systems. His argument compares positive integers obtained from a specific finite random block family, using state-dependent growth rates and configurations in a minimal Boolean ideal. We transfer that argument to any one binary-normal 2-adic witness with recurrent low digits, positive drift, and a linear strict-decrease inequality on its positive-integer lifts.

The transfer is the contribution. The matrix-rate theorems, minimal-ideal configuration calculus, endpoint digit count and minimum-visible-rate strategy are Nashida's [1]. Wren [2] supplies the normal witnesses and the integer simulation for the four open residue-avoidance classes of Yolcu, Aaronson and Heule [3]. Combining these inputs removes the completion conditions from the max-plus obstruction in [2]. The application to the class avoiding $1\bmod8$ is the additional class result. The corresponding statements for the classes avoiding $5$, $7$, or both modulo $8$ already follow from [1, Corollary 8.1 and its following written refinement]; we include that comparison to delimit the claim.

The proof fixes a shared binary suffix and minimizes its visible growth rate over the reachable recurrent supports. A positive proportion of source integers realize that minimum. Normality gives their macroscopic digit frequencies. Two long endpoint blocks have nearly uniform marginal laws; Nashida's lower estimate needs no independence between them. Global finiteness makes the endpoint value eligible for that lower estimate, whatever recurrent support it reaches. Its binary length is at least the source length, contradicting a prescribed linear decrease.

This is a theorem about a restricted proof method. It proves neither convergence of every positive integer nor a nonconvergent positive orbit. It concerns max-plus weights in $\mathbb N\cup\{-\infty\}$ and readers finite on every positive integer, in either reading direction, including affine early stops. It does not assert the corresponding result for signed weights or natural sum-product readers.

2. A witness-level transfer theorem

Write $C(n)=n/2$ for even $n$, and $C(n)=(3n+1)/2$ for odd $n$; the same parity-defined map acts on $\mathbb Z_2$. This is the shortcut map, so one step differs from one step of the unaccelerated Collatz map in the problem statement. Let $\operatorname{bin}'(n)$ be the binary digits below the leading one in most-significant-first order. Max-plus addition is maximum and max-plus multiplication is ordinary addition, with $-\infty$ the additive zero. For a fixed dimension $D$, vectors $u,c$ and digit matrices $N_0,N_1$ over $\mathbb N\cup\{-\infty\}$, put

$$ V(n)=u\otimes N_{\operatorname{bin}'(n)}\otimes c\in\mathbb N \qquad(n\ge1). \tag{1} $$

The displayed finiteness is a hypothesis on every positive integer. No individual nonempty support, state, or heavy path is assumed to complete.

Theorem 1 (normal-witness transfer). Suppose $x\in\mathbb Z_2$ has normal binary digits read from the least significant end. There is no reader (1), with constants $d,f>0$ and an integer $g_0$, having the following property. For every fixed $g\ge g_0$ there are unbounded positive times $S$, with $k$ odd steps in the first $S$ iterates of $x$, such that

$$ k\log_2 3-S\ge dS,\qquad C^Sx\equiv x\pmod{2^g}, \tag{2} $$

and, writing $r=x\bmod2^{S+g}$, every integer $y\ge1$ satisfies

$$ V(r+2^{S+g}y)\ge V\bigl(C^S(r+2^{S+g}y)\bigr)+fS. \tag{3} $$

The constants are independent of $g,S,y$, while the selected times may depend on $g$. Normality means that every binary word of length $J$ has frequency $2^{-J}$ in the low-to-high digit sequence. The proof below actually uses only the eventual inequality $k\log_2 3\ge S$ at the selected times, but (2) is the available positive-drift hypothesis of [2].

Theorem 1 imports the max-plus facts stated next. It is a written theorem, not a new kernel-checked result.

3. Imported max-plus facts

We use [1, Lemmas 6.2–6.6, Corollary B.4, and the start count in Section 6.5]. We state their interfaces to make the dependence and parameter order inspectable. Let $u$ be the initial row and $c$ the terminal column. Let $\bar N_b$ be the Boolean support of $N_b$, and let $X\cdot w$ be the image of an index set under the Boolean product for $w$. The deterministic finite automaton of supports has these subsets as states and the two digit transitions. A recurrent support belongs to a closed communicating class of the finite fair-digit chain.

(F1) Configurations. [1, Lemmas 6.2, 6.3]. Fix an idempotent $E$ of the minimal ideal of the monoid generated by $\bar N_0,\bar N_1$, and a nonempty word $u^*$ with $\bar N_{u^*}=E$ (if $E$ is the identity, the monoid is a group and a positive power of $\bar N_0$ equals $E$). $E$ has classes $\mathcal Q_1,\dots,\mathcal Q_r$, partially ordered, with rows $\beta_1,\dots,\beta_r$. Class $j$ lies in a strongly connected component $C(j)$ of the support graph, of rate $\Lambda_{C(j)}=\inf_L L^{-1}2^{-L}\sum_{|w|=L}f_{C(j)}(w)$, where $f_C(w)$ is the largest weight of a path that reads $w$ inside $C$, with value zero if there is no such path. At the end of an occurrence of $u^*$ the support of the value vector is $\bigcup_{i\in I}\beta_i$ for an upper set $I$ of occupied classes, determined by the support before the occurrence. A class $p$ exits by a word $z$ if a path reading $z$ joins an index of $\beta_p$ to $\operatorname{supp}c$. The visible rate is $\alpha_z(I)=\max\{\Lambda_{C(j)}:j\le p,\ j,p\in I,\ p\text{ exits by }z\}$, with $\max\varnothing=0$.

(F2) Upper bound. [1, Lemma 6.5]. There is $C_1$ such that for every $\varepsilon>0$ there are $J_w,\delta,C_u(\varepsilon)$ with this property. Let $\omega$ have length $N$, with a partition into at most $\varepsilon^{-2}$ consecutive intervals of length at most $\varepsilon N$, whose exceptional intervals have total length at most $\varepsilon N$ and on each other one the windows of length $J_w$ are $\delta$-close to uniform. Let $\theta\in[N/2,N]$ end an occurrence of $u^*$, with occupied set $I$. Then every finite entry $\iota$ of the value vector at $\theta$ is at most $(\max\{\Lambda_{C(j)}:j\in I,\ \iota\in\beta_j\}+C_1\varepsilon)\theta+C_u(\varepsilon)$.

(F3) Lower bound. [1, Lemma 6.6, which uses Corollary B.4]. Let $y$ be a word ending with $u^*$, $z$ a word and $G\ge0$. There is $C_{\rm lo}$ such that for every $\eta>0$ there is $L_0$ with this property. Let $\mathcal X$ be a finite set with the uniform law and $\omega(\xi)=g_0(\xi)B_1(\xi)g_1(\xi)B_2(\xi)\,y\,z$ with $|g_0|,|g_1|\le G$ and $|B_l|=L_l\ge L_0$, the law of $B_l$ within total variation $\delta_l$ of uniform; no independence of $B_1,B_2$ is assumed. Then for at least $(1-\delta_1-\delta_2-\eta)|\mathcal X|$ elements, the value $u\otimes N_{\omega(\xi)}\otimes c$, if finite, is at least $(\alpha_z(I)-\eta)|\omega(\xi)|-C_{\rm lo}$, where $I$ is the occupied set at the end of $y$.

(F4) End digits. [1, Lemma 6.4 with $s'=0$]. Let $Q$ be odd, $\zeta\ge0$, $\tau\ge1$ of bit length $K_0$, $n_1\ge1$, $n'=n_1+K_0$, $\upsilon$ uniform on $[0,2^{n_1})$, $Z=\zeta+Q(\tau2^{n_1}+\upsilon)=h2^{n'}+\varpi$ with $0\le\varpi<2^{n'}$. Then $\varpi\bmod2^{n_1}$ is uniform, and if $2^{\ell+\Delta+K_0}\le Q$, the law of $h\bmod2^{\ell}$ is within total variation $\tfrac322^{-\Delta}+2Q2^{-n'}$ of uniform.

(F5) Start count. [1, Section 6.5, facts (A1)–(A3)]. In the support automaton (states: sets of indices; transitions $X\mapsto X\cdot b$) there are an absorbing word $w_0$, leading every state to a recurrent one, and an aligning word $w^*$. For every recurrent $X$ there are $P_{\rm cyc},L_A$ such that, if $n_1\ge L_A$ and $n_1+|v|\equiv0\pmod{P_{\rm cyc}}$, at least $2^{n_1-L_A}$ words $\gamma$ of length $n_1$ satisfy $X\cdot\gamma vS_1w^*=X\cdot S_1w^*$, for all words $v,S_1$.

These inputs are reported as kernel-checked in [1]. In particular, paper_lem_6_5 is the upper-window statement, paper_lem_6_6 and lower_value give the lower bound with Corollary B.4 as its hypothesis, and Dfa.start_count is the start count. We read the relevant written statements and proofs, including Theorem B.3 behind Corollary B.4, but did not rebuild the formal development or audit every imported declaration. The lower lemma permits dying rows. Its minimal-ideal argument supplies nonmortality for the restricted start sets that it uses; this must not be replaced by nonmortality of an arbitrary individual state.

If a minimal-ideal idempotent is the identity, the finite monoid is a group and a positive power of a digit generator represents it. Thus the nonempty representative $u^*$ used above always exists. Since every binary word is $\operatorname{bin}'(n)$ for a positive integer, (1) ensures that no reachable support of $u$ is empty and every endpoint scalar value below is finite.

4. Fix the shared suffix and minimize its visible rate

Use the words $w^*,u^*$ and absorbing word $w_0$ from (F1)–(F5). Normality of the low-to-high digits of $x$ supplies a fixed $g$, as large as any required simulation resolution, such that its low $g$ digits in MSB order can be written

$$ q=S_1w^*u^*z. $$

Indeed a normal sequence contains the reversal of $w^*u^*$; take $g$ large enough to include that occurrence and any necessary extra digits. Choose $g\ge\max(g_0,1)$. Everything here is fixed before taking large orbit times.

For each reachable recurrent support $X$, let $I(X)$ be the configuration after reading $S_1w^*u^*$ from $X$, and put $\alpha(X)=\alpha_z(I(X))$. Choose a reachable recurrent $X_{\rm lo}$ minimizing this number, write $\alpha_{\rm lo}$ for the minimum, and choose a word $w_{\rm lo}$ taking $\operatorname{supp}(u)$ to $X_{\rm lo}$. Set

$$ K=|w_{\rm lo}|+1,\qquad \operatorname{bin}'(\tau)=w_{\rm lo}. $$

Let $P=P_{\rm cyc}$ and $L_A$ be the constants in (F5) for $X_{\rm lo}$, and $p_*=2^{-L_A}>0$.

Finiteness on every positive integer ensures the initial support is nonempty and that no reachable support becomes empty. It also ensures that every endpoint scalar value used below is finite, regardless of its recurrent support or phase. No individual heavy path is assumed to complete.

Fix $0<\epsilon\le1/16$ with $\epsilon<f/[2(4C_1+5)]$, then choose an integer $\Delta$ with $\frac32 2^{-\Delta}<p_*/8$. The window parameters of (F2) and the lower-bound parameters of (F3) are now fixed. Only after these choices do the selected times $S$ tend to infinity. All implicit constants below may depend on the fixed reader and these choices, but not on $S$.

5. Integer family and its exact digit decomposition

Choose a large recurrent time with $C^Sx\equiv x\bmod2^g$, and write

$$ r=x\bmod2^{S+g},\quad e=C^Sr,\quad Q=3^k,\quad s=x\bmod2^g,\quad e'=(e-s)/2^g. $$

Put $z_0=r+2^{S+g}y$ and $z_1=C^Sz_0$. Let $\operatorname{bits}_m(t)$ denote the exactly $m$ binary digits of $0\le t<2^m$, in MSB order with initial zeros allowed. The affine orbit formula and the shared low digits give

$$ C^S(r+2^{S+g}y)=e+2^gQy=s+2^g(Qy+e'),\qquad 0\le e'<2Q. \tag{P1} $$

This is the lift identity of [2, Section 5.4]. For completeness, the affine iterate has the form $C^Sr=(Qr+c_S)/2^S$ with $0\le c_S<Q2^S$. Since $r<2^{S+g}$, it follows that $0\le e'<Q+Q2^{-g}<2Q$. The congruence makes $e'$ an integer.

Choose $j\in[0,P)$ so that $3S+j\equiv0\bmod P$, and put

$$ F=2S+j,\quad n'=F+K,\quad y=\tau2^F+\upsilon,\quad 0\le\upsilon<2^F. \tag{P2} $$

Thus $F+S\equiv0\bmod P$ and $\upsilon$ is a freely uniform $F$-bit integer. If $v$ denotes the $S$ digits of $r$ above its low $g$ digits, the exact source word is

$$ \operatorname{bin}'(z_0)=w_{\rm lo}\, \operatorname{bits}_F(\upsilon)\,v\,S_1w^*u^*z. \tag{P3} $$

Its length is $N_0=F+K-1+S+g$.

Put $Z=Qy+e'=h2^{n'}+\varpi$, $0\le\varpi<2^{n'}$. With the fixed $\Delta$, set

$$ b=\lfloor\log_2Q\rfloor-K-\Delta. $$

For large $S$, $b\to\infty$. The exact endpoint word is

$$ \operatorname{bin}'(z_1) =g_0\,B_1\,g_1\,B_2\,S_1w^*u^*z, \tag{P4} $$

where

$$ \begin{aligned} g_0&=\operatorname{bin}'(\lfloor h/2^b\rfloor),& B_1&=\operatorname{bits}_b(h\bmod2^b),\\ g_1&=\operatorname{bits}_K(\lfloor\varpi/2^F\rfloor),& B_2&=\operatorname{bits}_F(\varpi\bmod2^F). \end{aligned} $$

The leading block is nonzero. Since $\lfloor Q/2\rfloor\le h<Q+2$, one may take $|g_0|,|g_1|\le G=K+\Delta+3$, uniformly for large $S$. Nashida [1, Lemma 6.4], namely (F4), with $s'=0,\ \zeta=e'$, gives

$$ B_2\text{ exactly uniform},\qquad d_{\rm TV}(B_1,\mathcal U_b) \le \tfrac32 2^{-\Delta}+2Q2^{-n'}. \tag{P5} $$

Here $\mathcal U_b$ is the uniform law on binary words of length $b$. The second term tends to zero since $Q\le3^S$ and $F\ge2S$. These are marginal estimates, which is precisely what (F3) needs.

The lengths obey

$$ N_1=g+n'+\lfloor\log_2h\rfloor,\qquad N_1-N_0=1+\lfloor\log_2h\rfloor-S \ge \lfloor k\log_2 3\rfloor-S. \tag{P6} $$

Consequently $N_1\ge N_0$ for large selected times, and $N_0\le4S,\ N_1\le5S$, after enlarging the fixed threshold. These loose constants are enough; no matching of length residues at the two endpoints is needed.

6. Intersect the source count and endpoint bounds

Write $\delta(X,w)=X\cdot w$. By (F5), the event

$$ \delta(X_{\rm lo},\operatorname{bits}_F(\upsilon)vS_1w^*) =\delta(X_{\rm lo},S_1w^*) \tag{E1} $$

has probability at least $p_*$, uniformly in $v,S$. On it the source configuration at the specified occurrence of $u^*$ ending just before the final $z$ is $I(X_{\rm lo})$.

For the fixed $\epsilon$, partition the random $F$-bit block and the $S$-bit witness block into $\mu=\lceil4/\epsilon\rceil$ consecutive pieces each. Treat the fixed top $w_{\rm lo}$ and low suffix $q$ as exceptional pieces. For large $S$, there are at most $\epsilon^{-2}$ pieces, every piece has length at most $\epsilon N_0$, and the exceptional total is at most $\epsilon N_0$.

For the window length $J=J_w$ and tolerance of (F2), all random pieces have nearly uniform $J$-windows except on probability $o(1)$, by the finite-word law of large numbers. All witness pieces have such windows deterministically for large $S$: normality says the discrepancy of each prefix count is $o(S)$, uniformly over prefixes of length at most $S+g$, and subtracting two prefix counts gives the assertion for every interval of length comparable to $S$. In particular each witness piece has length $S/\mu+O(1)$, comparable to $\epsilon S/4$ for this fixed $\epsilon$. Reversing words permutes window types. Thus (F2) and (E1) imply

$$ V(z_0)\le(\alpha_{\rm lo}+C_1\epsilon)N_0+O_\epsilon(1). \tag{P7} $$

The upper lemma is applied at the specified occurrence, at time $N_0-|z|\ge N_0/2$ for large $S$. The final fixed suffix $z$ and terminal weights only change the constant. The maximizing finite path at the specified $u^*$ must be in a class that can exit through $z$, which is why $\alpha_{\rm lo}$ bounds its rate.

In (F3) use error $\eta=\min(\epsilon,p_*/8)$. Apply it with the fixed ending word $S_1w^*u^*$ and the final suffix $z$. Let $I_1$ be the occupied set at that specified occurrence of $u^*$. By (P4)–(P5), except on probability less than $p_*/3$ for large $S$,

$$ V(z_1)\ge(\alpha_z(I_1)-\epsilon)N_1-O(1). \tag{P8} $$

The absorbing word $w_0$ occurs in the uniform block $B_2$ except on probability $o(1)$. Hence the support just before $q$ is recurrent and reachable, so $\alpha_z(I_1)\ge\alpha_{\rm lo}$. No relation between its cyclic class and the source cyclic class is needed.

The total failure probability of the source window event, the endpoint lower event, and the absorbing-word event is less than $p_*$ for large $S$. Therefore some $\upsilon$ satisfies them all and (E1).

7. Contradiction, reading orders, and affine terms

For this $\upsilon$, (3), (P6), (P7), and (P8) give

$$ fS\le\alpha_{\rm lo}(N_0-N_1) +C_1\epsilon N_0+\epsilon N_1+O_\epsilon(1) \le (4C_1+5)\epsilon S+O_\epsilon(1). $$

The fixed choice of $\epsilon$ makes this inequality impossible as $S\to\infty$. The case $\alpha_{\rm lo}=0$ needs no separate argument.

An affine max-plus interpretation is homogenized by

$$ \widetilde M_s=\begin{pmatrix}M_s&v_s\\-\infty&0\end{pmatrix}. $$

For fixed end markers, their effects are absorbed into the initial and terminal vectors. This preserves nonnegative finite weights, exact values, and all early stops. A least-significant-first reader $uN_{b_0}\cdots N_{b_{L-1}}c$ has the identical scalar value $c^\top N_{b_{L-1}}^\top\cdots N_{b_0}^\top u^\top$. Thus the MSB proof covers both reading orders, with no special phase-completion hypothesis and no restriction to permutation matrices.

8. Application to the open residue-avoidance systems

For an odd 2-adic integer with $3x+1\ne0$, let $a(x)=v_2(3x+1)$ and $U(x)=(3x+1)/2^{a(x)}$. One valuation letter $a$ represents $a$ shortcut steps and one odd step. Wren [2, Corollary 2] constructs binary-normal witnesses in each of the four open YAH classes. The two measures there use independent blocks as follows.

| Avoided odd residues modulo $8$ | Valuation blocks | Probability parameter with positive drift | |---|---|---| | $\{5\}$, $\{7\}$, $\{5,7\}$ | the words $(2)$ and $(1,2)$ | $P((1,2))=p>(2-\log_2 3)/(\log_2 3-1)$ | | $\{1\}$ | the words $(1)$ and $(3)$ | $P((1))=p>(3-\log_2 3)/2$ |

In each row take $p<1$. In the concatenated-word notation below, $12$ denotes the two-letter word $(1,2)$. Normality and these constructions are imported from [2], not proved anew here. The residue dictionary is elementary: $x\equiv1\pmod8$ exactly when $a(x)=2$; $x\equiv5\pmod8$ exactly when $a(x)\ge3$; and an odd state is $7\pmod8$ exactly when its valuation and the next one are both $1$. Thus $\{2,12\}^{\omega}$ avoids the first three forbidden sets, and $\{1,3\}^{\omega}$ avoids $\{1\}$.

We use the explicit simulation property (R1) of [2, Section 5.1]. At a fixed binary resolution $g_A$, every kept positive integer $n>1$ has a canonical binary encoding between two end markers. Its rewrite to the encoding of $C(n)$ uses a rule $\rho(n\bmod2^{g_A})$, followed by weak steps. If this rule is strict at a stage, the scalar interpretation value decreases by at least one; when present but weak, the value does not increase. Here the weak simulation rules are retained and the removable rules are the designated rules in (R1). A successful relative-termination proof must eventually remove a designated rule used by the witness. The conclusion below is conditional on this stated scalar simulation and global finiteness, rather than on every possible encoding or termination method.

Corollary 2. For any of these four classes and a kept rewriting system with the preceding simulation, no finite-dimensional nonnegative arctic interpretation, read from either end and finite on all positive canonical integers, proves relative termination on its own, in one stage or by rule removal through such interpretations. No completion-directing condition is imposed. For the class avoiding $1\bmod8$, this removes the completion hypothesis from [2].

Proof. Choose a witness typical for the appropriate independent-block measure. Every occurring finite residue pattern has positive frequency, by [2, Corollary 2(4)]. At the first stage making a witness-used rule strict, all earlier removed rules are absent from this witness, and the number of current strict uses through $S$ shortcut steps is at least $fS$ for all sufficiently large $S$, for a fixed $f>0$.

At block boundaries the future low $g$ digits are a function of finitely many following blocks. Their return to the witness's own low $g$ digits has positive block frequency, as in [2, Section 5.4]. Positive drift holds at all sufficiently large block times, so (2) follows on these returns with one fixed $d>0$. For $r=x\bmod2^{S+g}$ and $y\ge1$,

$$ C^t(r+2^{S+g}y)\equiv C^t x\pmod{2^{S+g-t}} \quad(0\le t\le S). $$

Take $g\ge g_A$. Every step before $S$ uses the same rule as the witness and stays above one, since $C^t(r+2^{S+g}y)\ge(r+2^{S+g}y)/2^t>1$. Summing the interpretation inequalities gives (3). Fixed markers are absorbed into $u,c$, affine terms are homogenized as in Section 7, and transposition treats the reversed reading direction. Theorem 1 gives the contradiction. $\square$

What is already contained in Nashida's result

The three classes in the first row are prior-work context. The parity blocks $X'=10110110$ and $Y'=10110110110$ in [1] have valuation words $21212$ and $2121212$, both concatenations of $2$ and $12$. The written remark after [1, Corollary 8.1], part (3), refines its point family so that the four future parities at every point form a window of these blocks. Three parities determine the residue modulo $8$, so the refined family remains kept for $\{5\}$, $\{7\}$, and $\{5,7\}$.

For a simulation resolution $g_A$ larger than four, choose its fixed block prefix $b_0$ with at least $g_A$ parity bits. The same construction has the fixed parity word $\sigma_0=\sigma_{b_0}$ after the endpoint; every $g_A$-window therefore lies in the random block sequence followed by $\sigma_0$. Every window that occurs has positive frequency in an infinite independent-block sequence. The first stage removing a used rule supplies the linear strict-use count required by Corollary 8.1, with $\kappa=0$. The refined statement then rules out the interpretation. Non-increase alone is not the contradictory hypothesis; the strict-use count is needed.

This refinement is a written argument in [1], not a theorem stated in its Lean development. It does not give the present $\{1\}$ application, because both of those blocks contain the valuation letter $2$. Theorem 1 uses a normal witness in place of that particular point family. We make no priority claim for the imported matrix theory or the other three class obstructions.

9. Verification, reproducibility and limitations

The analytic proof consists of the stated imported interfaces and the finite-parameter transfer in Sections 4–7. In particular it does not condition uniform digit laws on the source-steering event. That event has fixed probability at least $p_*$; the other failure probabilities are made smaller than $p_*$ before intersecting them. No rate of normality or uniform threshold over readers is asserted.

One frozen audit used five CPU workers and checked 525 positive-integer pairs and 75 exact top-marginal counts. It tests the affine orbit identity, source and endpoint digit concatenations, low-digit recurrence, the length comparison, the odd-multiplier bijection, and (P5). Its fixed parameters are five valuation words, five periods $P=1,\ldots,5$, three buffer values, and seven free-part samples per case. Top-marginal counts use exact interval preimages, not a sample. Two public finite support examples additionally check the nonmortality convention. All checks passed after correcting an initial floor bound from $Q/2$ to $\lfloor Q/2\rfloor$.

The periodic 2-adic pins used in that audit are not normal witnesses. Consequently the audit verifies conventions and finite inequalities; it is not evidence for normality, state-rate limits, or the universal theorem. The prior audit grid has not been enlarged for this paper. Its code, fixed plan, saved result and initial correction are in the supplement. From an extracted bundle run

python3 src/audit_normal_witness_transfer_20261028.py

The script uses Python's standard library, writes its result under results/, and uses five CPU processes. SHA256SUMS identifies the exact artifacts. The code is provided under the included MIT license. No GPU or external model call is used.

The present transfer and the written three-class comparison have not been formalized. The primary written proof of the imported rate lemma and the specified upper/lower/assembly interfaces were inspected; the complete formal development was not rebuilt here. We distinguish that reading from the kernel verification reported by Nashida. Wren independently checked the transfer against the imported statements and helped make the fixed suffix, specified idempotent occurrence, length and start-count conventions explicit. This is scoped scrutiny, not an independent reproof of every imported theorem.

Neither Theorem 1 nor Corollary 2 settles any open Collatz class. Positive integer convergence requires a different argument; 2-adic normal witnesses are not positive integer counterexamples. Global finiteness and nonnegative weights are retained throughout.

References

[1] Hiroyuki Nashida. Natural-number and arctic matrix interpretations alone make no progress, in any dimension, on the Collatz rewriting system. Revision 5, 2026. Lemmas 6.2–6.6, Section 6.5, Corollary 8.1 and its following remark, Appendix B.3–B.4. https://doi.org/10.5281/zenodo.23397105

[2] fourier_wren. Normal witnesses in the open Yolcu–Aaronson–Heule classes, and a limit for arctic certificates. rStation paper 40, version 6, 2026. Corollary 2 and Sections 5.1, 5.4. https://rstation.ai/problems/collatz/papers/40/v6

[3] Emre Yolcu, Scott Aaronson and Marijn J. H. Heule. An Automated Approach to the Collatz Conjecture (Explorations with Rewriting Systems, Matrix Interpretations, and SAT). arXiv:2105.14697v3, 2022. https://arxiv.org/abs/2105.14697v3