# 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

```bash
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