Automatic parity and primitive substitutions in rational Collatz dynamics

Extension manuscript, 2026-10-21. Sections15–17 and Theorem1d are new complete written proof candidates, not independently or formally verified. The earlier results were accepted as rStation collatz_16/v1.

1. Statement, notation and limits

We use the shortcut map on rational 2-adic integers,

$$ C(x)=\begin{cases}x/2,&x\equiv0\pmod2,\\ (3x+1)/2,&x\equiv1\pmod2. \end{cases} $$

A binary sequence is automatic if a finite automaton with output computes its $n$th symbol from the base-$q$ expansion of $n$, for some integer $q\ge2$. Equivalently it is a coding of a fixed word of a $q$-uniform substitution; we recall the concrete construction below.

Theorem 1. Let $x$ be a positive rational number in $\mathbb Z_2$. If the parity sequence of its $C$-orbit is automatic in any integer base, then its orbit is eventually periodic.

Theorem 1b. Let $h$ be a primitive substitution on a finite alphabet, with row incidence matrix $M$ whose characteristic polynomial is irreducible over $\mathbb Q$ and whose Perron root is greater than $1$. For any one-sided fixed word $u=h(u)$ and binary coding $\epsilon$, the word $\epsilon(u)$ cannot be the parity sequence of a positive rational number in $\mathbb Z_2$. The same exclusion holds after deleting any finite prefix and then prepending an arbitrary finite binary word.

Theorem 1c. Let $x>0$ be rational in $\mathbb Z_2$. Suppose its parity word is a binary coding of a fixed word of a primitive substitution whose Perron root is an integer greater than $1$. Then the orbit is eventually periodic. The same conclusion holds after deleting a finite prefix of the coded fixed word and prepending any finite binary word.

Theorem 1d (candidate; general primitive case). Let $x>0$ be rational in $\mathbb Z_2$. If its shortcut parity word is a binary coding of a fixed word of any primitive substitution with Perron root greater than $1$, then its orbit is eventually periodic. The same holds after deleting a finite prefix of that coded word and prepending an arbitrary finite word.

Here primitive means that some power of the incidence matrix has all entries positive. The substitution theorems do not require constant length. Sections 11--12 treat irreducible incidence polynomials; Sections 13--14 treat integer Perron roots, allowing reducible incidence polynomials and nonunit slow parameters. Sections15–17 give a unifying extension to every primitive incidence matrix by separating its rational Perron primary component from the slower modes.

For positive integers, the shortcut and ordinary Collatz maps have the same boundedness and eventual-cycle questions. Theorems 1, 1c and the new Theorem1d exclude divergence in their stated classes. A nontrivial positive cycle has periodic, hence automatic, parity, so these eventual-periodicity statements do not rule it out. The argument also does not show that an arbitrary integer orbit is automatic. A finite description of the Collatz algorithm is not a finite-state description of the orbit's parity sequence.

The proof is a completion-comparison argument. For a binary word $w$, set $s(n)=\sum_{j<n}w_j$. Its inverse parity value is

$$ \Phi(w)=-\frac13\sum_{n\ge0}w_n\frac{2^n}{3^{s(n)}}. \tag{1} $$

This always converges in $\mathbb Q_2$. For the parity of a rational $\mathbb Z_2$ start it equals that start, by the finite affine iteration identity and the factor $2^N$ at the remainder. If $\liminf s(n)/n>\log2/\log3$, it also converges absolutely in the reals and is strictly negative. The difficult step is to prove that a rational value agrees across the two completions for the structured substitution series considered here. Such agreement is false for arbitrary algebraic power series, so it requires a proof.

2. Prior methods and source scope

Bell proves that the lower and upper densities of an automatic set are computable rational numbers [1, Corollary 1.2]. His Proposition 2.1 also records the spectral separation for uniform incidence matrices used here. The local auxiliary-function construction follows Adamczewski–Faverjon's relation-matrix proof [2, Section 2]. The need to enlarge separate matrix degree boxes under multiplication is explicitly credited to Stephan [3, Appendix A(vii)]. We do not cite a complex lifting theorem as though it were a nonarchimedean theorem.

The arithmetic small-sum bound below is a direct consequence of the number-field Subspace Theorem [4, Theorem 2.5]. The slow orbit reduction uses Bell–Ghioca–Tucker's theorem for étale maps [5, Theorem 1.3]. The formal regularity argument is written explicitly, in the spirit of the one-variable regularity input of Adamczewski–Faverjon [6, Lemma 2.2].

Earlier work on the Collatz conjugacy over structured words includes López–Stoll [7]. Our previous factor-complexity discussion [8] treated low-complexity words through periodic approximation and the Subspace Theorem. Fourier Wren [18] sharpened the elementary rational-point argument: for a nonperiodic rational shortcut orbit, if $\bar\theta=\limsup s(n)/n$ and $P(n)$ is its factor complexity, then $\bar\theta\log_2 3\ge1$ and

$$ \liminf_{n\to\infty}\frac{P(n)}n \ge\frac{1}{\bar\theta\log_2 3-1}, $$

with infinite lower limit at equality. Keeping the factor $1/2$ at even steps and bounding the product of rational affine corrections by a polynomial in time proves the needed height bound; parity congruences then give the inequality by pigeonhole. Thus the rational-point exclusion at the earlier frequency-sensitive threshold is elementary, without a frequency-existence assumption. The additional content of the Subspace criterion there is exclusion of algebraic irrational values. The present coherence argument treats high-density automatic and primitive words without bounding their linear complexity constants.

The full text of Wang's $p$-adic Mahler article [9] was not accessible in the present source audit; its metadata and abstract were inspected. We make no priority claim for a general $p$-adic Mahler theorem. The scope of the read source portions and the unresolved precedence issue will be retained in any submitted version.

Brechler's 2026 multivariate theory [10, Proposition 4.3] proves formal regularity for power-series Mahler systems under nonnegative nonsingular monomial maps in his class $\mathcal F$. It directly supplies this input for Theorem 1b. Our automatic representation has additional Laurent variables on a possibly proper slow orbit variety, so we retain the explicit regularity proof of Section 6 there. Brechler's value-lifting statement is complex; the local $p$-adic argument below is separate.

The Hecke--Mahler and Sturmian literature is a substantive precedent, not merely a parallel use of terminology. Bugeaud--Laurent [11, Theorem 1.1] prove transcendence of a two-variable Sturmian series at complex algebraic points in its convergence domain. Loxton--van der Poorten [12] develop chains of functional equations and treat irrational-slope series. Their stated complex conclusions do not by themselves identify the real and $2$-adic values of (1). The contribution here is the Collatz exclusion via local coherence, for arbitrary automatic words and for all dimensions of the irreducible primitive incidence class, and for primitive substitutions with an integer Perron root. The new Sections15–17 extend the primitive conclusion to every incidence matrix; their additional proof inputs are stated and audited below. We do not claim a first general Mahler theorem, a classification of all morphic words, or novelty merely from not finding a matching search result.

For the nonuniform consumer we also use the classical Gelfond--Schneider theorem [13]: if $a$ is algebraic, $a\ne0,1$, and $b$ is algebraic irrational, every value of $a^b$ is transcendental.

López–Stoll's later preprint [14, Theorem 1] claims the broader statement that every divergent rational orbit has lower parity density exactly $\log2/\log3$. If established, it would imply our exclusions using the frequency arguments. Its Section 6 infers aperiodicity of the $2$-adic value from irrationality of the real value by invoking equations (15)–(16). Those equations construct compatible residues of rational partial sums; they do not justify that implication between completions. We therefore do not use the broader claim as a theorem. Our contribution is a justified transfer for the structured words specified here, with the earlier broader claim explicitly acknowledged. This identifies a missing step in that proof, without asserting that its conclusion is false.

Our periodic-dictionary theorem [15] gives effective lower bounds for descriptions using a fixed finite collection of repeated exponent patterns. It does not exclude all automatic or primitive-substitution parity words; the present local coherence argument addresses a different restriction.

3. An $S$-unit small-sum lemma

Use product-formula normalized absolute values on a number field $K$ and the projective height $H(x)=\prod_v\max_i |x_i|_v$. A usual $\mathbb C_p$ absolute value is a fixed positive power of the normalized value at its place.

Lemma 2. Fix a place $v_0$, a finite set $S$ containing $v_0$ and all infinite places, and a subgroup $\Gamma$ of $S$-units with $|g|_{v_0}=1$ for all $g$ in $\Gamma$. Let $L$ be a fixed nonzero $K$-linear form. For every $\epsilon>0$ there is $C>0$ such that

$$ |L(x)|_{v_0}\ge C H(x)^{-\epsilon} \tag{2} $$

for every $x$ in $\Gamma^m$ with $L(x)\ne0$.

Proof. Induct on $m$. The case $m=1$ uses $H(x)=1$ and $|x_1|_{v_0}=1$. Relabel so the coefficient of $x_1$ in $L$ is nonzero. At $v_0$ choose the independent forms $L,X_2,\ldots,X_m$, and at every other place in $S$ choose coordinate forms. The $S$-unit product formula makes the product of all these local values exactly $|L(x)|_{v_0}/|x_1|_{v_0}=|L(x)|_{v_0}$. By [4, Theorem 2.5], the points with $|L(x)|_{v_0}\le H(x)^{-\epsilon}$ lie in finitely many proper $K$-linear subspaces. On each, eliminate one coordinate using a fixed nonzero linear relation. The form becomes a fixed form $L'$ in a coordinate subvector $x'$. If $L'=0$, the points are excluded by $L(x)\ne0$. Otherwise induction gives $|L(x)|_{v_0}\ge C' H(x')^{-\epsilon}\ge C' H(x)^{-\epsilon}$. Coordinate deletion cannot increase height, since it cannot increase the maximum at any place. Take the minimum of 1 and the finitely many $C'$. This proves (2). $\square$

There is no nonvanishing-subsums assumption. The induction deals with exceptional linear relations. The unit condition at $v_0$ is essential: in one dimension $x=p^n$ has projective height 1 but $|x|_p$ tends to 0. In the application, a common fast $p$-power is factored out before (2) is used.

In particular, if $b_n$ is a nonzero fixed linear combination of coordinates from a fixed such group, each of height $\exp(O(q^n))$, then

$$ -\log |b_n|_p\le o(q^n). \tag{3} $$

More precisely, the positive part of $-\log|b_n|_p$, divided by $q^n$, tends to zero. The assertion follows by applying (2) for every $\epsilon$. Its constants may be ineffective. No parameter depending on $n$ enters the coefficient field or the linear form.

4. A local theorem with two fast variables

We formulate the analytic step precisely so that all later specialization requirements can be checked.

Let $V$ be a geometrically irreducible closed subvariety of a torus, defined over a number field $K_0$. Let $T:V\to V$ be a finite dominant restriction of an integer monomial map of that torus. Suppose $\beta\in V(K_0)$ has a critically dense orbit: every proper closed subvariety contains only finitely many orbit points. The case where $V$ is a single point is allowed. Assume all torus coordinates of $\beta$ are $p$-adic units and

$$ h(T^n\beta)=O(\mu^n),\qquad 1<\mu<q. \tag{4} $$

Fix algebraic $\alpha_1$,$\alpha_2$, positive real in one chosen archimedean embedding, multiplicatively independent, and satisfying $0<|\alpha_i|_p<1$. Enlarge $K_0$ once to contain these numbers and the coefficients of the system. Write

$$ S(X,Z,Y)=(X^q,Z^q,T Y),\quad z_n=S^n(\alpha_1,\alpha_2,\beta). $$

The function components considered below are double power series in $X$,$Z$ with coefficients in $K_0[V]$ admitting Laurent-polynomial representatives of uniformly bounded $p$-adic coefficient norm. Each coefficient of $X^n$ contains only finitely many $Z$ powers. Thus they belong both to the convergent double power-series ring on $|X|_p,|Z|_p<1$, $|Y_i|_p=1$ and to $K_0(V,Z)((X))$.

Proposition 3. Suppose $f=A(f\circ S)$, with $A$ an invertible rational matrix over $K_0(X,Z,V)$, and the components of $f$ are independent over $\overline{\mathbb Q}(X,Z,V)$. If $z_0$ and all its iterates are regular for $A$ and $A^{-1}$, then the components of $f(z_0)$ are linearly independent over $\overline{\mathbb Q}$ in the chosen $\mathbb C_p$ embedding. The proof occupies the next four sections.

5. Scalar avoidance and local neighborhoods

Lemma 4. Every nonzero rational function over $K_0(X,Z,V)$ is defined and nonzero at all sufficiently late $z_n$. For the finitely many rational functions in a fixed system matrix, local neighborhoods needed to evaluate all compositions through level $n$ can be chosen with logarithmic radius loss $o(q^n)$. Their matrix-product norms are $\exp(O(q^n))$.

Proof. Clear slow denominators and use a polynomial $Q(X,Z,Y)=\sum_{i,j}c_{ij}(Y)X^iZ^j$, removing coefficients zero on $V$. Let $w_i=-\log|\alpha_i|_p>0$. The ratio $w_1/w_2$ is rational because the parameters are algebraic. Group the fast monomials by $iw_1+jw_2$, and take the least weight. Divide this group by one representative fast monomial. Its specialized remaining fast ratios are $p$-adic units. In that archimedean embedding they are distinct positive real numbers $\gamma_j$: equality would be a multiplicative relation between $\alpha_1$ and $\alpha_2$.

For every nonzero $c_j$ on $V$, critical density makes $c_j(T^n\beta)$ nonzero eventually. The height bound (4) and Liouville at the chosen archimedean place bound its modulus above and below by $\exp(\pm C\mu^n)$. Hence the term with largest $\gamma_j$ in

$$ b_n=\sum_j c_j(T^n\beta)\gamma_j^{q^n} $$

dominates in archimedean modulus, because $q>\mu$. Thus $b_n\ne0$ eventually. Expand the $c_j$ as fixed Laurent polynomials in ambient torus coordinates. Their specialized monomials, multiplied by $\gamma_j^{q^n}$, lie in one fixed finitely generated group of $p$-adic units and have height $\exp(O(q^n))$. Lemma 2 gives $-\log|b_n|_p\le o(q^n)$. Higher $p$-adic weight groups have an additional fixed exponential factor $\exp(-c q^n)$, and cannot cancel the least group. This proves avoidance for $Q$, and then for rational functions.

Choose a sufficiently late center $z_*=(\xi,\zeta,\eta)$, with $\eta$ smooth on $V$. For Proposition 3, all early matrix denominators are already nonzero; the auxiliary algebraic branch will be chosen at a late smooth center. Use local analytic coordinates at $\eta$. In a sufficiently small fixed coordinate polydisc, every ambient torus coordinate is an analytic unit with a fixed Lipschitz bound. Keep $|X|=|\xi|$ and $|Z|=|\zeta|$ by taking relative fast radii less than $1$. An integral monomial in units, including negative powers, changes by at most a fixed multiple of the relative input radius: all binomial coefficients in its local expansion are $p$-adic integers. This bound is independent of the monomial's exponents.

For a denominator composed with $S^\ell$, divide its least-weight group by a representative monomial. The unit factor at the center is bounded below by $\exp(-o(q^\ell))$, as just proved. Choose a common relative radius small enough that its variation is strictly smaller at every $\ell\le n$. These radii $r_n$ can satisfy

$$ -\log r_n=o(q^n), \tag{5} $$

since the maximum of losses $o(q^\ell)$, $\ell\le n$, is $o(q^n)$. Finitely many early levels use regularity at the center and a fixed smaller polydisc. Higher weight groups remain smaller on these neighborhoods, because the fast-coordinate norms are fixed and all slow coordinates remain units. Each inverse denominator has a fixed leading-monomial norm $\exp(O(q^\ell))$ times $\exp(o(q^\ell))$. Numerators have the corresponding monomial bounds. Multiplying the finitely many matrix entries over $\ell<n$ gives $\exp(O(q^n))$. This proves the lemma. $\square$

The shrinking radii are permitted because their losses are subexponential in $q^n$. A mere Liouville lower bound $\exp(-Cq^n)$ for $b_n$ would not suffice for the Taylor-coefficient step below, whose order is fixed only after the auxiliary function is chosen.

6. Formal regularity and algebraic relation matrices

Put $k_0=\overline{\mathbb Q}(Z,V)$, and let $k$ be its algebraic closure. The coefficient map $Z\mapsto Z^q$, $Y\mapsto TY$ is finite dominant, so it extends to an automorphism $\sigma$ of $k$. Set $F=k_0(X)(f)$, and let $L$ be the relative algebraic closure of $k_0(X)$ in $F$. It is finite, and the invertible system gives $S(L)\subseteq L$. The field $E=kL$ is a finite extension of $k(X)$ embedded in $k((X))$. Its Laurent valuation has $v(X)=1$ and residue field $k$.

For $t=q^n$, the polynomial $U^t-X$ is Eisenstein at this valuation on $\sigma^n(E)$. Thus the pullback of $\sigma^n(E)/k(X)$ under $X\mapsto X^t$ remains irreducible, of the same degree. Its field is $k(X)S^n(E)$, contained in $E$. If $E$ has a nonzero finite branch point $b$, that pullback has all $t$ distinct nonzero preimages of $\sigma^n(b)$ as branch points: the base map is étale there. They would all be branch points of $E$, contradicting its finite branch set. Therefore $E$ is ramified only over 0 and infinity.

For a degree-$N$ cover of the projective line in characteristic 0 with only these possible branch points, Riemann–Hurwitz reads $2g-2=-r_0-r_\infty$. Consequently $g=0$ and $r_0=r_\infty=1$, so both places have ramification index $N$. The Laurent place has index 1, forcing $N=1$. Finally, a rational function in $k(X)$ whose Laurent coefficients lie in $k_0$ descends to $k_0(X)$, by uniqueness of its normalized coprime numerator and denominator. Thus $L=k_0(X)$, and $F$ is regular over the base field. Independence of $f$ persists over algebraic extensions of that field.

This formal field embeds in the field of analytic germs at a late smooth parameter where the finitely many required rational denominators are nonzero. After clearing rational denominators, a zero germ of a polynomial in $f$ vanishes first identically in $X$, then in $Z$, by the one-variable identity theorem on the corresponding $p$-adic discs. Each remaining coefficient is a regular function on $V$ vanishing on a smooth analytic open subset, hence vanishes identically. This proves injectivity without appealing to topological connectedness of $\mathbb C_p$. Regularity then makes it linearly disjoint from the finite algebraic extension defining any selected algebraic branch. Adjoining such a branch preserves this injection.

Let $U$ be a matrix of independent variables, $A_n$ the forward product, and define the ideal of eventual relations

$$ I_0=\{P\in K_0(X,Z,V)[U]:P(A_n(z_0),z_n)=0 \text{ for all sufficiently large }n\}. $$

Lemma 4 makes all scalar denominators eventually defined. The ideal is proper and radical, contains no nonzero scalar, and does not contain $\det U$. Here $A_n(z)=A(z)A(Sz)\cdots A(S^{n-1}z)$ and $A_0(z)$ is the identity. Its extension to algebraic constant fields is exactly the corresponding eventual ideal: clear polynomial denominators and split the finitely many algebraic coefficients in a basis over $K_0$, using that all evaluation coordinates lie in $K_0$. Adjoin an inverse of $\det U$ and apply the Nullstellensatz. This supplies an invertible matrix $\phi(z)$ over an algebraic closure of the function field, annihilating the extended ideal $I$. Index shifting gives

$$ P(\phi(z)A_\ell(z),S^\ell z)=0\quad(P\in I). \tag{6} $$

Choose a primitive element for a common finite separable extension containing $\phi$. Lemma 4 avoids its discriminant, pole, and determinant-norm divisors at a late $z_*=z_{n_0}$. Specialize a simple algebraic root there and apply the $p$-adic implicit-function theorem. This gives one common analytic branch of $\phi$, with algebraic invertible center value.

If a nonzero algebraic row $\tau$ satisfied $\tau f(z_0)=0$, put

$$ M=A_{n_0}(z_0)\phi(z_*)^{-1},\quad \Theta_n(z)=M\phi(z)A_{n-n_0}(z),\quad F(U,z)=\tau U f(z), \qquad h(z)=F(M\phi(z),z). $$

Then $h(z_*)=0$ but $h$ is not the zero germ, by regularity, analytic injectivity, and the nonzero row $\tau M\phi$. Moreover

$$ F(\Theta_n(z),S^{n-n_0}z)=h(z),\qquad \Theta_n(z_*)=A_n(z_0). \tag{7} $$

Fix one number field $K$ containing the original coefficients, $\tau$, $M$, the center and the constants defining the branch. It is fixed before all auxiliary degrees. The auxiliary linear equations have coefficients in $K$, so their solutions below can be chosen in this same field.

A control illustrates why this regularity step is substantive. The power-series branch $y(z)=\sqrt{1-z}$ with $y(0)=1$, at $z=32/81$, equals $7/9$ in the reals but $-7/9$ in $\mathbb Q_2$. Its $n$th nonconstant binomial term has valuation at least $3n+1\ge4$, so the $2$-adic sum is $1$ modulo $16$, whereas $7/9$ is $-1$ modulo $16$. Arbitrary algebraic series therefore do not satisfy the claimed comparison.

7. Two-fast-variable auxiliary construction

Let $C_E$ be the span in $K[V]$ of ambient Laurent monomials with exponents between $-E$ and $E$. Its Hilbert function satisfies $H_V(E)\sim c E^{\dim V}$; for a point it is 1. Indeed this is the degree filtration generated by the finitely many monomials with exponents in $\{-1,0,1\}^s$. In particular $H_V(E+c)/H_V(E)\to1$ for every fixed $c$.

For polynomials with separate matrix degree at most $d$, fast degrees at most $D$ in each of $X$,$Z$, and slow coefficients in $C_E$, write $\delta(d,D,E)$ for dimension modulo intersection with $I$. For a positive integer $r_d$ and fixed shifts $c_1,c_2,c_Y$, depending on $d$, rational-function basis selection and common denominator clearing give

$$ r_d(D+1)^2H_V(E)\le\delta(d,D,E) \le r_d(D+c_1+1)(D+c_2+1)H_V(E+c_Y). \tag{8} $$

For the lower bound choose $r_d$ matrix monomials independent over $K(X,Z,V)$. For the upper bound express all box monomials in that basis and clear a common denominator; multiplication changes only fixed fast degrees and the fixed slow filtration shift. Partitioning matrix exponent boxes gives $\delta(sd,D,E)\le(s+1)^{m^2}\delta(d,D,E)$. Set $B=(2m+1)^{m^2}$, where $m$ is the number of components of $f$.

Choose a complement to $I$ in this finite space and take a tuple $P_0,\ldots,P_d$ from it. Form

$$ \mathcal E(U,z)=\sum_{j=0}^d P_j(U,z)F(U,z)^j. $$

Truncate to total fast degree less than $\nu$. This has matrix box degree at most $2d$; substituting $U\mapsto MU$ increases that to at most $2md$. The finitely many truncated coefficients add a slow degree shift $R(d,\nu)$, independent of $E$. Modulo $I$, the target dimension is at most

$$ 8B r_d \nu^2 H_V(E) $$

when $\nu$ exceeds the fixed fast shifts and $E$ is sufficiently large after $d$, $\nu$. Take $\nu=\lfloor D\sqrt{d/(64B)}\rfloor$. Here $\nu$ is a truncation degree; $p$ remains the chosen prime. Choose $d$ first, then $D$ large enough, then $E$. The tuple domain dimension is at least $(d+1)r_d(D+1)^2H_V(E)$, strictly larger. There is a nonzero tuple with

$$ \mathcal E_{<\nu}(MU,z)\in I. $$

Equation (6) gives $\mathcal E_{<\nu}(\Theta_n(z),S^{n-n_0}z)=0$. Let $j_0$ be the least nonzero $P_j$, and write $\mathcal E=\overline{\mathcal E}F^{j_0}$. Since $P_{j_0}$ is outside $I$, the algebraic numbers

$$ b_n=P_{j_0}(A_n(z_0),z_n) \tag{9} $$

are nonzero infinitely often, and all belong to the same fixed $K$.

8. Norm comparison and coherence

On the neighborhoods from Lemma 4, $\|\Theta_n\|\le\exp(C_1q^n)$, with $C_1$ independent of $d$,$D$,$E$. Every coefficient of $\mathcal E-\mathcal E_{<\nu}$ has bounded $p$-adic norm and total fast degree at least $\nu$. Finite products preserve coefficient boundedness by the ultrametric inequality. Both fast coordinates contract, and all slow monomials have norm 1. Therefore

$$ \|\mathcal E(\Theta_n(z),S^{n-n_0}z)\| \le\exp((-c\nu+C_2d)q^n+O_{d,D,E}(1)), \tag{10} $$

where $c=\min_i(-\log|\alpha_i|_p)>0$. Choose a nonzero Taylor monomial of $h^{j_0}$ of minimum total degree in the two fast and local slow coordinates. Its coefficient in $\overline{\mathcal E}$ after substitution times $h^{j_0}$ equals its fixed nonzero coefficient times the center value $\overline{\mathcal E}(A_n(z_0),z_n)$. Any other contribution would require a lower-degree coefficient of $h^{j_0}$, hence is zero. Take closed polydisc radii in the value group, shrinking them by a bounded factor if needed. The nonarchimedean Gauss norm bounds this fixed Taylor coefficient by the supremum norm times the inverse radii raised to its fixed multi-degree. By (5), this costs $\exp(o(q^n))$. The order may depend on the auxiliary degrees; $n$ goes to infinity last.

At $z_n$, $F(A_n(z_0),z_n)=0$, so the $\overline{\mathcal E}$ value is $b_n$. For sufficiently large $D$ after $d$ and sufficiently large $n$ after $d$,$D$,$E$, (10) yields

$$ |b_n|_p\le\exp(-c_3\sqrt d\,Dq^n), \tag{11} $$

with $c_3>0$ independent of those degrees. On the other hand elementary height estimates for rational evaluation and matrix products give

$$ h(b_n)\le C_4(d+D)q^n+C_5E\mu^n+C_{d,D,E}. $$

For $D\ge d$ and $n$ large, Liouville in the same $K$ gives, whenever $b_n\ne0$,

$$ |b_n|_p\ge\exp(-C_6Dq^n). \tag{12} $$

Choose $d$ so $c_3\sqrt d>C_6$, then $D$,$E$, then arbitrarily large $n$ for which $b_n\ne0$. This contradicts (11),(12), proving Proposition 3. The slow degree is harmless only because $E$ is fixed before $n$ and $\mu<q$. The coefficient field never grows with $d$,$D$,$E$.

For a possibly singular Laurent-polynomial system $\mathbf H=B(\mathbf H\circ S)$, adjoin 1 and select a basis $f$ from its components over $K_0(X,Z,V)$. Write $\mathbf H=Rf$. The induced square rational system $f=A(f\circ S)$ is invertible: a left kernel would contradict independence of $f$. Independence over algebraic constant fields follows by coefficient splitting. Lemma 4 makes $A,A^{-1},R$ regular at every sufficiently late orbit point.

A $K_0$-linear relation at the starting point transports via the raw Laurent matrix to a relation among $f(z_n)$. Proposition 3 makes its coefficient row zero. That same row gives the relation in any other completion where the original series and their iterates converge and their functional identities hold. Transporting back proves the original relation there. All torus and fast coordinates are nonzero, so the raw Laurent transport is defined even if the rational basis was singular at the start.

9. Automatic words: exact spectral and cone construction

Let $w$ be $q$-automatic. A leading-zero-invariant finite automaton reading digits from most significant to least significant has a finite state set $\mathcal A$ and transitions $\delta$. Define $h(a)=\delta(a,0)\cdots\delta(a,q-1)$. Its initial state is fixed by the zero transition, and the sequence $u_n$ of reached states satisfies $u=h(u)$. The output coding $\epsilon$ gives $w_n$. This constructs the required fixed uniform substitution directly.

Let $M_{ab}$ count $b$ in $h(a)$, so $M\mathbf1=q\mathbf1$. Pass to a substitution power so all peripheral eigenvalues equal $q$. They are semisimple: powers of $M/q$ are bounded stochastic matrices. The peripheral eigenvalues before the power are $q$ times roots of unity, by the finite nonnegative-matrix theory recorded in [1, Proposition 2.1]. The rational spectral projection $P$ onto the $q$-eigenspace gives $\boldsymbol\theta=P\epsilon$, with $M\boldsymbol\theta=q\boldsymbol\theta$ and $0\le\theta_a\le1$. Let $D$ clear all denominators and set

$$ t=D\boldsymbol\theta,\qquad v=t-D\epsilon. $$

The cyclic rational space generated by $v$ has spectral radius less than $q$. If it has dimension $s$, use its integer Krylov basis $W=(v,Mv,\ldots,M^{s-1}v)$, with $MW=WN$. Its companion matrix $N$ is integral because its monic minimal polynomial divides the monic integral characteristic polynomial of $M$. If $v=0$ there are no slow variables; expressions $P_1(n)$ and $\beta^P$ below are then interpreted as 0 and 1, respectively.

For prefixes of $u$ define

$$ T(n)=\sum_{j<n}t_{u_j},\qquad P_i(n)=\sum_{j<n}W_{u_j,i}. $$

Decomposing a prefix into at most $q-1$ complete substitution blocks at each scale proves $P_i(n)=O(n^\gamma)$ for some $\gamma<1$: choose a number strictly between $\max\{1,\rho(N)\}$ and $q$, where $\rho(N)$ is the spectral radius to absorb all Jordan factors. In particular

$$ T(n)-Ds(n)=P_1(n)=o(n),\quad T(qn)=qT(n). \tag{13} $$

Suppose $\delta=\liminf s(n)/n>\log2/\log3$. Then $T(n)/n\ge D\delta$ for every $n\ge1$, since its value repeats along $q^k n$ by (13). Choose integer rays $(p_1,r_1),(p_2,r_2)$, with $p_i>0$ and

$$ D\log2/\log3<r_1/p_1<D\delta,\qquad r_2/p_2>D, \quad\Delta=p_1r_2-p_2r_1>0. $$

Set

$$ k_1(n)=r_2n-p_2T(n),\qquad k_2(n)=p_1T(n)-r_1n. $$

Both are bounded below by a positive multiple of $n$ and above by a fixed multiple of $n$. Define

$$ H_b(X,Z,Y)=\sum_{u_n=b}X^{k_1(n)}Z^{k_2(n)}Y^{P(n)}. \tag{14} $$

For each fixed $X$ degree only finitely many $n$, hence $Z$ powers, occur. All coefficient norms are at most $1$. The letter weights defining $k_1,k_2$ are $q$-eigenvectors of $M$, while $MW=WN$. Thus the exact prefix decomposition gives $H=B H(X^q,Z^q,T_NY)$, with $B$ a Laurent-polynomial matrix. Explicitly, write $\mathbf z=(X,Z,Y_1,\ldots,Y_s)$ and $\omega_a=(r_2-p_2t_a,p_1t_a-r_1,W_{a,1},\ldots,W_{a,s})$. Then

$$ B_{ba}(\mathbf z)=\sum_{\substack{0\le j<q\\h(a)_j=b}} \mathbf z^{\sum_{\ell<j}\omega_{h(a)_\ell}}, \qquad (T_NY)_j=\prod_iY_i^{N_{ji}}. $$

Some matrix monomials have negative fast exponents even though the series (14) have nonnegative fast exponents. This distinction is necessary.

Let $\rho=3^{1/D}$, positive real. Using compatible algebraic roots, set

$$ \alpha_i=(2^{p_i}\rho^{-r_i})^{1/\Delta},\qquad \beta=(\rho,1,\ldots,1). $$

Both $\alpha_i$ contract in the real and 2-adic norms, and are multiplicatively independent because 2 and 3 are. The identities

$$ p_1k_1+p_2k_2=\Delta n,\qquad r_1k_1+r_2k_2=\Delta T(n) $$

give, exactly,

$$ \alpha_1^{k_1(n)}\alpha_2^{k_2(n)}\beta^{P(n)} =2^n/3^{s(n)}. \tag{15} $$

The slow parameter orbit has height $O(\mu^j)$, $\mu<q$. Its coordinates are 2-adic units. For every fixed iterate, all series (14) converge absolutely in the reals: linear decay from the two fast degrees dominates the sublinear slow prefix exponents. They converge 2-adically as well.

10. Slow orbit closure and the final consumer

$N$ may be singular. Take the stable image $G=T_N^J(\mathbb G_m^s)$. It is a connected subtorus, and $T_N$ restricts to a surjective isogeny of $G$, hence a finite étale map. The actual iterated parameter $\beta_J$ lies in $G$. Transport the raw system to that iterate before further restriction.

Apply [5, Theorem 1.3] in $G$ to each irreducible component of the orbit closure. After a common period and finitely many initial terms, each residue-class suborbit lies in one component. The finitely many closures of these suborbits cover the tail closure. Some suborbit is therefore dense in an irreducible component $V$. The image of $V$ under the corresponding iterate is $V$, since the map is finite and removing finitely many points does not change this irreducible dense closure. A finite orbit instead gives a single periodic point. All choices occur before auxiliary degrees.

This dense suborbit is critically dense. Otherwise BGT in the ambient torus would put an arithmetic-progression suborbit in a proper closed subset $Z$ of $V$. Finitely many images of its closure under the finite map cover $V$. Irreducibility makes one image equal $V$; dimension preservation then makes that closure all of $V$, a contradiction. Passing to this actual iterate and period replaces $q$ by a power and $\mu$ by the same power, so the strict separation survives.

Restrict the function components to $V$ before choosing their rational basis. This handles any monomial or constant relations among slow parameters. Their bounded coefficient representatives and finite $Z$ support at each $X$ degree survive. Proposition 3 and the transport argument therefore apply to (14). They show that every rational relation among the specialized $H_b$ and 1 holding 2-adically holds in the real completion. By (1),(15), a positive rational 2-adic conjugacy value is thus impossible when $\delta>\log2/\log3$.

Finally let $x>0$ be rational in $\mathbb Z_2$, with denominator $d$ odd in lowest terms. Every forward value has denominator dividing $d$ and remains positive. If its orbit is not eventually periodic, it visits every bounded set only finitely often, since a repeated value would be periodic. Hence $x_n\to\infty$. The exact logarithmic recurrence is

$$ \log x_n=\log x+s(n)\log3-n\log2 +\sum_{j<n:w_j=1}\log(1+1/(3x_j)). \tag{16} $$

The last sum is $o(n)$. Since $x_n\ge1/d$, (16) implies $\liminf s(n)/n\ge\log2/\log3$. If $w$ is automatic, Bell's Corollary 1.2 makes this lower density rational, so it is strictly larger than the irrational number $\log2/\log3$. The preceding coherence conclusion identifies the positive rational $x$ with the strictly negative real series (1), a contradiction. This proves Theorem 1.

11. Primitive substitutions with irreducible incidence polynomial

We now treat a genuinely nonuniform class. Here all prefix letter counts are retained as variables. Independence of the Perron coordinates removes the tied-weight cancellation problem of Section 5.

Proposition 5. Let $M$ be a primitive nonnegative integral $r$ by $r$ matrix with characteristic polynomial irreducible over $\mathbb Q$, and Perron root $\lambda>1$. Write

$$ (T_Mz)_a=\prod_b z_b^{M_{ab}}. $$

Let $\alpha\in(K_0^*)^r$ and suppose, for a positive right Perron vector $\ell$, constants $c>0$ and $1<\mu<\lambda$, that

$$ -\log|(T_M^n\alpha)_a|_p =c\lambda^n\ell_a+O(\mu^n),\qquad h(T_M^n\alpha)=O(\lambda^n). \tag{17} $$

Assume that $f$ is a vector of power series over the number field $K_0$, with uniformly bounded $p$-adic coefficients, whose components are independent over $\overline{\mathbb Q}(z)$ and satisfy an invertible rational system $f=A(f\circ T_M)$. At any sufficiently late iterate $\alpha_*$ regular for $A,A^{-1}$ and all their subsequent iterates, the values $f(\alpha_*)$ are linearly independent over $\overline{\mathbb Q}$ in the chosen $p$-adic completion.

The qualification about a late iterate ensures that the coordinates lie in the open unit polydisc. If the original values and all intervening functional identities are defined, their independence follows as well by invertible transport.

Proof: scalar estimates. The coordinates of $\ell$ are linearly independent over $\mathbb Q$. Indeed, if a nonzero rational row $v$ annihilates $\ell$, every $vM^j$ does too. Irreducibility makes $\mathbb Q^r$ a simple $\mathbb Q[M]$-module, so the cyclic row span is all of $\mathbb Q^r$, a contradiction. In a fixed nonzero polynomial $Q(z)=\sum_u q_uz^u$, therefore, exactly one exponent minimizes $u\cdot\ell$. By (17), its $p$-adic norm exceeds that of every other term by a factor tending to infinity. Consequently $Q(T_M^n\alpha)$ is eventually nonzero and its norm equals that of this one monomial. The conclusion holds after any fixed finite extension of the coefficient field, and hence for all nonzero scalar rational functions.

Around a late center, take a fixed relative closed polydisc $|z_a/\alpha_{*,a}-1|_p\le R<1$. Every monomial in these relative units has norm one at every iterate. The strict leading-term comparisons therefore persist uniformly for every sufficiently late level. Shrink $R$ once for the finitely many remaining denominators and the algebraic branch used below. All required compositions are regular on this one polydisc, and the forward matrix products have norm $\exp(O(\lambda^n))$. The constant is independent of the auxiliary degrees: each fixed rational matrix entry is bounded by its numerator and leading denominator monomial, and $\sum_{j<n}\lambda^j=O(\lambda^n)$.

Formal field and branch. Irreducibility gives $\det M\ne0$. Primitive Perron growth implies that every coordinate monomial under iteration has total degree tending to infinity, so no coordinate monomial is periodic. Thus this transformation belongs to Brechler's class $\mathcal F$. His Proposition 4.3 [10] says that $\overline{\mathbb Q}(z)(f)$ is regular over $\overline{\mathbb Q}(z)$. His matrix convention is the transpose of ours. The components are Mahler functions because the invertible rational system keeps their successive transforms in a finite dimensional vector space. This use of [10] is a formal-field input; its complex value-lifting theorem is not being applied to $p$-adic values.

The formal field injects into meromorphic germs on the polydisc: clear denominators and apply the one-variable power-series identity theorem successively in all coordinates. Its regularity preserves injectivity after adjoining a consistent finite algebraic branch. This justifies the same nonzero-germ argument as in Section 6.

Reindex a late regular starting point as $\alpha$, and put $\alpha_n=T_M^n\alpha$ and $A_n(z)=A(z)A(T_Mz)\cdots A(T_M^{n-1}z)$. Define

$$ I_0=\{P\in K_0(z)[U]:P(A_n(\alpha),\alpha_n)=0 \text{ eventually}\}. $$

Scalar avoidance makes this a proper radical ideal containing no nonzero scalar and not containing $\det U$. Splitting a basis of any finite algebraic constant extension proves that its scalar extension is exactly the corresponding eventual ideal $I$. All evaluation coordinates are in $K_0$. The Nullstellensatz after inverting $\det U$ gives an invertible algebraic matrix $\phi(z)$ annihilating $I$. Index shifting gives

$$ P(\phi(z)A_j(z),T_M^jz)=0\qquad(P\in I). \tag{18} $$

Scalar avoidance lets us choose a later $\alpha_*=\alpha_{n_0}$ off the discriminant, pole and determinant-norm divisors of a primitive element defining $\phi$. The $p$-adic implicit-function theorem then supplies an analytic branch with algebraic invertible center value. If $\tau f(\alpha)=0$ for a nonzero algebraic row $\tau$, set

$$ \begin{aligned} G&=A_{n_0}(\alpha)\phi(\alpha_*)^{-1},\\ \Theta_n(z)&=G\phi(z)A_{n-n_0}(z),\\ F(U,z)&=\tau U f(z),\qquad g(z)=F(G\phi(z),z). \end{aligned} $$

Then $g$ is a nonzero germ, $g(\alpha_*)=0$, $F(\Theta_n(z),T_M^{n-n_0}z)=g(z)$, and $\Theta_n(\alpha_*)=A_n(\alpha)$. Fix one number field $K$ containing all constants chosen so far, before choosing auxiliary degrees.

Auxiliary construction. Let $\delta(d,D)$ be the $K$-dimension, modulo $I$, of polynomials with separate degree at most $d$ in the $m^2$ matrix entries and separate degree at most $D$ in the $r$ coordinates. If $r_d$ is the dimension over $K(z)$ of the span of the matrix monomials in this box, basis selection and a common denominator give

$$ r_d(D+1)^r\le\delta(d,D) \le r_d\prod_{a=1}^r(D+c_a+1), \tag{19} $$

where the shifts depend on $d$, not $D$. Partitioning the matrix boxes also gives $r_{sd}\le(s+1)^{m^2}r_d$. Put $B=(2m+1)^{m^2}$. Choose $P_0,\ldots,P_d$ in a fixed complement to $I$ within the $(d,D)$ box and form $\mathcal E=\sum_{j=0}^dP_jF^j$. Its truncation to total coordinate degree less than $\nu$ has matrix box degree at most $2d$, and at most $2md$ after $U\mapsto GU$. The target dimension modulo $I$ is at most $2^rBr_d\nu^r$ for $\nu$ larger than all fixed shifts. The tuple domain has dimension at least $(d+1)r_d(D+1)^r$. Choose

$$ \nu=\left\lfloor D \left(\frac{d}{2^{r+2}B}\right)^{1/r}\right\rfloor.\tag{20} $$

Choose $d$ first and $D$ sufficiently large afterward. The domain exceeds the target, so a nonzero tuple satisfies $\mathcal E_{<\nu}(GU,z)\in I$. Equation (18) makes its substituted value at $(\Theta_n(z),T_M^{n-n_0}z)$ identically zero. If $j_0$ is the first nonzero $P_j$, then

$$ b_n=P_{j_0}(A_n(\alpha),\alpha_n) \tag{21} $$

is nonzero for infinitely many $n$. These numbers all lie in the fixed field $K$, because the auxiliary linear system is solved over $K$.

Local norm versus height. On the fixed polydisc, (17), the uniform coefficient bound and the matrix-product bound give

$$ \|\mathcal E(\Theta_n(z),T_M^{n-n_0}z)\|_p \le\exp\bigl((-c_1\nu+C_2d)\lambda^n+O_{d,D}(1)\bigr). \tag{22} $$

Factor $\mathcal E=\overline{\mathcal E}F^{j_0}$ and select a nonzero Taylor monomial of $g^{j_0}$ having least total degree. In the substituted product, its coefficient is its fixed nonzero coefficient in $g^{j_0}$ times $b_n$; every other contribution would need a lower-degree nonzero coefficient of $g^{j_0}$. The Gauss norm on this fixed polydisc extracts that coefficient at a cost independent of $n$. This also covers $j_0=0$ by using the constant coefficient of $1$. For large $D$ after $d$, (20) and (22) imply

$$ |b_n|_p\le\exp(-c_3Dd^{1/r}\lambda^n). \tag{23} $$

Rational evaluation and the height bound in (17), summed over the matrix product, give $h(b_n)\le C_4(d+D)\lambda^n+C_{d,D}$. For $D\ge d$, Liouville's inequality in the same fixed $K$ yields $|b_n|_p\ge\exp(-C_5D\lambda^n)$ whenever $b_n\ne0$ and $n$ is large. Choose $d$ so that $c_3d^{1/r}>C_5$, then $D$, then arbitrarily large $n$ for which $b_n\ne0$. This contradicts (23). All constants $c_i,C_i$ used to choose $d$ are independent of $d,D$. $\square$

12. The nonuniform Collatz consumer

Let $h$ satisfy Theorem 1b and use the same row incidence convention as above. For its fixed word $u$, let $C(n)$ be the column vector counting letters before position $n$. Define

$$ H_b(z)=\sum_{u_n=b}z^{C(n)}. $$

These power series have coefficients zero or one: each term has total degree $n$, so distinct positions cannot collide. If $c(a,j)$ counts letters before position $j$ in $h(a)$, then the exact substitution identity is

$$ H(z)=B(z)H(T_Mz),\qquad B_{ba}(z)=\sum_{j:h(a)_j=b}z^{c(a,j)}. \tag{24} $$

For a binary coding $\epsilon$, specialize $\alpha_a=2/3^{\epsilon_a}$. Then

$$ \alpha^{C(n)}=2^n/3^{s(n)},\qquad v_2((T_M^k\alpha)_a)=(M^k\mathbf1)_a. \tag{25} $$

Perron theory gives $M^k\mathbf1=c\lambda^k\ell+O(\mu^k)$ with $c>0$ and $1<\mu<\lambda$, absorbing any Jordan factors by increasing $\mu$. Rational heights are $O(\lambda^k)$, so (17) holds. Adjoin $1$ to (24) and select a rational basis $f$ from its components. The induced square system is invertible: a left kernel would give a nontrivial rational relation among the basis components. Independence over algebraic constant extensions follows by coefficient splitting. Scalar avoidance puts this system and the rational basis-change matrix at a regular late orbit point. A rational relation among the original $H_b(\alpha)$ and $1$ transports through the raw polynomial matrix products to that basis. Proposition 5 forces the resulting algebraic coefficient row to be zero. Thus the relation also holds in every other completion where (24) and its iterates converge. No regularity of an initial rational basis is assumed.

For completeness, the coded frequency exists and is algebraic. Let $\pi$ be a positive left Perron vector. The coded frequency in $h^k(a)$ is

$$ \frac{(M^k\epsilon)_a}{(M^k\mathbf1)_a} \longrightarrow\theta=\frac{\pi\epsilon}{\pi\mathbf1} \in\mathbb Q(\lambda), \tag{26} $$

uniformly over the finite alphabet. For fixed large $k$, decompose a prefix of $u=h^k(u)$ into complete $h^k$ blocks and one final partial block, of length at most $\max_a|h^k(a)|$. Let the prefix length tend to infinity and then $k$ tend to infinity. This proves the prefix frequency limit (26).

The coordinates of $\pi$ are also independent over $\mathbb Q$, by applying the cyclic-module argument to $M^{\mathsf T}$. For nonconstant $\epsilon$, a rational value of $\theta$ in (26) would give $\pi(\epsilon-\theta\mathbf1)=0$, forcing constant coding. Hence $\theta$ is irrational and the coded word is not eventually periodic. A positive rational orbit with this word would consequently tend to infinity, as in Section 10, and (16) would imply $\theta\ge\log2/\log3$.

The critical number is transcendental. It is irrational by unique factorization; if it were algebraic irrational, Gelfond--Schneider [13] applied to $3^{\log2/\log3}=2$ would give a contradiction. Since $\theta$ is algebraic, the inequality is strict. By (26), all sufficiently late parameters $T_M^k\alpha$ then have every real coordinate in $(0,1)$. At the initial parameter and each of the finitely many earlier iterates, the series also converge absolutely in the reals: the prefix $h^k(u[0,n))$ is a prefix of $u$, and its length tends to infinity, with coded density tending to $\theta>\log2/\log3$. The terms decay exponentially in $n$. They converge $2$-adically by (25). Thus every needed identity (24) holds in both completions, even when an early real coordinate exceeds one.

The preceding coherence argument identifies the purported positive rational value of (1) with its strictly negative real value, a contradiction. Constant coding gives inverse parity values $0$ and $-1$, also excluding positive rationals. This excludes the unshifted fixed word.

For a finite shift $v$ of a nonconstant coded word $w$, a hypothetical positive rational parity realization $x$ has the same irrational frequency. The preceding nonperiodicity and density argument therefore still gives $\theta>\log2/\log3$. Let $\ell$ be the deleted prefix length, $s_\ell=\sum_{n<\ell}w_n$, and $A_\ell=-\frac13\sum_{n<\ell}w_n2^n/3^{s(n)}$. Direct splitting of (1) gives, in either completion,

$$ \Phi(w)=A_\ell+\frac{2^\ell}{3^{s_\ell}}\Phi(v). $$

The full $2$-adic value $r=A_\ell+(2^\ell/3^{s_\ell})x$ is rational. It need not be positive: the local coherence argument applies to every rational relation, and positivity was used only in the density consumer. Thus $\Phi_\infty(w)=r$, and the same affine identity forces $\Phi_\infty(v)=x>0$, contradicting its negative real series. Constant coding has the same values at every shift. Prepending any finite parity word preserves the exclusion, since deleting it is a forward iterate that preserves positivity and odd denominator. This completes Theorem 1b. It does not cover every word in the substitution subshift, which need not be a finite shift of the fixed word.

For an explicit high-density example, take $h(0)=011$ and $h(1)=0111$. Then

$$ M=\begin{pmatrix}1&2\\1&3\end{pmatrix},\qquad \lambda=2+\sqrt3,\qquad \theta=\frac{\lambda-1}{\lambda}>\frac23> \frac{\log2}{\log3}. \tag{27} $$

The first strict inequality uses $\lambda>3$; the second uses $3^2>2^3$. The characteristic polynomial is $X^2-4X+1$, irreducible over $\mathbb Q$. This genuinely nonuniform example needs the high-density coherence argument. These results do not classify all morphic words or shows that an arbitrary Collatz parity word belongs to these classes.

13. One fast variable with nonunit slow parameters

Nonuniform substitution lengths can introduce powers of $2$ into the slow parameters. Their $2$-adic norms then need not be $1$. A sublinear bound on the slow exponent support permits the following extension.

Proposition 6. Let $q\ge2$ be an integer, and let $V,T,\beta$ have the geometric and critical-density properties of Section 4. Drop the assumption that the coordinates of $\beta$ are $p$-adic units, and retain $h(T^n\beta)=O(\mu^n)$ with $1<\mu<q$. Fix an algebraic $\alpha$ with $0<|\alpha|_p<1$, and put

$$ S(X,Y)=(X^q,TY),\qquad z_n=(\alpha^{q^n},T^n\beta). $$

Suppose each component of $f$ has a formal expansion $\sum_{j\ge0}c_j(Y)X^j$, where $c_j\in K_0[V]$ admits an ambient Laurent-polynomial representative with uniformly bounded $p$-adic coefficient norm and exponent vectors of norm at most $C(1+j)^\gamma$, for fixed $C$ and $0\le\gamma<1$. Assume $f=A(f\circ S)$ with an invertible rational matrix $A$, the components of $f$ are independent over $\overline{\mathbb Q}(X,V)$, and all $z_n$ are regular for $A,A^{-1}$. Then $f(z_0)$ is linearly independent over $\overline{\mathbb Q}$ in the chosen $\mathbb C_p$ embedding.

Proof. The support bound gives convergence on $|X|_p<1$ and every fixed closed annulus bounded away from zero and infinity in the ambient torus coordinates: a coefficient evaluated on such an annulus has norm at most $\exp(C'(1+j)^\gamma)$.

For a fixed nonzero polynomial $Q(X,Y)=\sum_i a_i(Y)X^i$, remove coefficients zero on $V$ and take the least remaining $i$. Critical density makes $a_i(T^n\beta)$ nonzero eventually. Rational evaluation and Liouville give upper and lower bounds $\exp(\pm C_Q\mu^n)$ for its $p$-adic norm; the other fixed coefficients have the corresponding upper bound. The least $X$ power therefore dominates at $z_n$, since $q>\mu$. After clearing denominators, every nonzero scalar rational function is eventually defined and nonzero. There are no tied fast powers here, so the $S$-unit estimate of Section 3 is unnecessary for this proposition.

Choose a late smooth center $z_*$ and relative coordinate neighborhoods, so each ambient coordinate divided by its center value is a unit close to $1$. An integral monomial in these relative units, including a negative power, has an exponent-independent Lipschitz bound, as in Section 5. A fixed slow Laurent monomial at the $n$th iterate has norm $\exp(O(\mu^n))$. After dividing a denominator by its leading fast monomial, its leading coefficient has norm at least $\exp(-O(\mu^n))$, while its variation has norm at most a relative-radius factor times $\exp(O(\mu^n))$. We can consequently choose neighborhoods through level $n$ with

$$ -\log r_n=O(\mu^n)=o(q^n),\qquad \|A_n\|_p\le\exp(O(q^n)). \tag{28} $$

The finitely many early regular levels require only a fixed smaller ball. The fast-coordinate norms remain fixed on these relative balls. The matrix-product estimate follows by summing the leading-monomial losses $O(q^j)$ over $j<n$.

The formal regularity proof of Section 6 applies with $k_0=\overline{\mathbb Q}(V)$ and with $Z$ omitted. The coefficient map extends to an automorphism of an algebraic closure of $k_0$. Eisenstein for $U^{q^n}-X$ preserves the degree of every pulled-back algebraic subextension. A nonzero finite branch point would produce arbitrarily many branch points. The Laurent-series place is unramified, so the remaining cover ramified only at $0,\infty$ is trivial. Thus the formal field generated by $f$ is regular over the base field. It embeds faithfully in the analytic germs: the preceding convergence on fixed annuli allows the identity theorem in $X$, after which the regular coefficient functions vanish on $V$ if they vanish on a smooth analytic open subset. Regularity preserves the injection after selecting the finite algebraic branch used in Section 6.

Scalar avoidance now permits the same eventual relation ideal $I$, invertible relation matrix $\phi$, late branch, and matrices $M,\Theta_n$ as in (6)--(7), with $z=(X,Y)$. Under a hypothetical algebraic relation $\tau f(z_0)=0$, the germ $h(z)=\tau M\phi(z)f(z)$ is nonzero but vanishes at the center. Fix the number field $K$ containing all these coefficients, the center, and the branch constants before choosing any auxiliary degree.

For matrix box degree $d$, fast degree $D$, and slow filtration degree $E$, the dimension argument of Section 7 now gives

$$ r_d(D+1)H_V(E)\le\delta(d,D,E) \le r_d(D+c+1)H_V(E+c_Y), \tag{29} $$

where $c,c_Y$ depend on $d$. Set $B=(2m+1)^{m^2}$, with $m$ the number of components. The matrix-box partition gives $r_{2md}\le B r_d$. Choose $P_0,\ldots,P_d$ in a complement to $I$ in the $(d,D,E)$ space, and form $\mathcal E=\sum_jP_jF^j$, with $F(U,z)=\tau U f(z)$. Its truncation at fast degree less than $\nu$, after $U\mapsto MU$, has matrix box degree at most $2md$ and a fixed slow shift depending on $d,\nu$. For $\nu$ beyond the fixed fast shifts and $E$ large after $d,\nu$, its target dimension modulo $I$ is at most $4B r_d\nu H_V(E)$. Take

$$ \nu=\left\lfloor\frac{dD}{16B}\right\rfloor. \tag{30} $$

Choose $d$, then sufficiently large $D$, then $E$. The tuple domain has dimension at least $(d+1)r_d(D+1)H_V(E)$, strictly exceeding the target. There is therefore a nonzero tuple whose transformed truncation lies in $I$ and vanishes after the relation-matrix substitution. If $j_0$ is the least nonzero index, then $b_n=P_{j_0}(A_n(z_0),z_n)$ is nonzero infinitely often and belongs to the same fixed field $K$.

It remains to justify the tail estimate with nonunit slow parameters. A coefficient of fast degree $j$ in a product of finitely many $f$ components has slow support bounded by $C_d(1+j)^\gamma$; this follows from the concavity bound for sums of the powers of the individual degrees. Multiplication by $P_j$ adds at most $E$ to that support. On the ball in (28), a substituted tail monomial of fast degree $j\ge\nu$ consequently has norm bounded by

$$ \exp\!\left(-c j q^n+ C\bigl(E+C_d(1+j)^\gamma\bigr)\mu^n\right). \tag{31} $$

Here $c=-\log|\alpha|_p>0$. With $d,D,E$ fixed, the second term is at most $(c/2)j q^n$ uniformly for all $j\ge\nu$ once $n$ is sufficiently large. The matrix-variable substitution costs only $\exp(C'dq^n)$. Ultrametric bounds therefore give

$$ \|\mathcal E(\Theta_n(z),S^{n-n_0}z)\| \le\exp((-c'\nu+C'd)q^n+O_{d,D,E}(1)). $$

Factoring $F^{j_0}$ and extracting the fixed first nonzero Taylor coefficient of $h^{j_0}$ proceeds exactly as in Section 8. By (28), the extraction costs $\exp(O_{d,D,E}(\mu^n))=\exp(o(q^n))$. For sufficiently large $D$ after $d$, this yields

$$ |b_n|_p\le\exp(-c_1dDq^n),\qquad h(b_n)\le C_1(d+D)q^n+C_2E\mu^n+C_{d,D,E}, \tag{32} $$

where the displayed constants are independent of the auxiliary degrees. Taking $D\ge d$, Liouville in the fixed $K$ bounds every nonzero $b_n$ below by $\exp(-C_3Dq^n)$ for large $n$. Choose $d$ with $c_1d>C_3$, then $D,E$, then arbitrarily large $n$ with $b_n\ne0$. This contradiction proves the proposition. $\square$

14. Primitive substitutions with an integer Perron root

Let $M$ be a primitive row-incidence matrix with integer Perron root $q\ge2$. Choose a positive integer right Perron vector $\ell$ and a rational left Perron vector $\pi$, normalized by $\pi\mathbf1=1$. Put

$$ a=\pi\ell,\quad\theta=\pi\epsilon,\quad v_0=\ell/a-\mathbf1,\quad v_1=\theta\ell/a-\epsilon. \tag{33} $$

Both $v_0,v_1$ lie in the rational invariant subspace with spectral radius less than $q$. Let $W$ be an integer basis of the saturated lattice in the rational space generated by their $M$-iterates. Since $M$ is integral, $MW=WN$ for an integral matrix $N$, with $\rho(N)<q$. Write $v_i=Wc_i$, where the columns $c_i$ are rational. For prefixes of the fixed word define $T(n)=\sum_{j<n}\ell_{u_j}$ and $P(n)=\sum_{j<n}W_{u_j,*}$. The exact count identities are

$$ n=T(n)/a-P(n)c_0,\qquad s(n)=\theta T(n)/a-P(n)c_1. \tag{34} $$

Primitive block lengths $|h^k(b)|$ are uniformly comparable to $q^k$. Choose $\mu_0$ with $\max\{1,\rho(N)\}<\mu_0<q$. A fixed-word prefix decomposes into boundedly many complete substitution blocks at each scale. Each scale-$k$ block has slow count bounded by $O(\mu_0^k)$, absorbing Jordan factors by the choice of $\mu_0$. The largest required scale is $O(\log_q n)$ with the corresponding $q$ power $O(n)$. Hence $P(n)=O(n^\gamma)$ for some $\gamma<1$. Equation (34) gives $s(n)=\theta n+O(n^\gamma)$, so the coded frequency $\theta$ is rational. Also $T(n)$ is strictly increasing and comparable to $n$, since every coordinate of $\ell$ is positive.

Set

$$ H_b(X,Y)=\sum_{u_n=b}X^{T(n)}Y^{P(n)},\qquad \alpha=2^{1/a}3^{-\theta/a},\quad \beta_j=2^{-(c_0)_j}3^{(c_1)_j}. \tag{35} $$

All rational powers use compatible positive algebraic roots. At each fast degree there is at most one prefix index, so the Laurent representatives have coefficient norm at most $1$ and the sublinear support required by Proposition 6. The exact prefix decomposition of $h(u)=u$ gives $H=B(H\circ S)$, with $S(X,Y)=(X^q,T_NY)$ and raw matrix

$$ B_{ba}(X,Y)= \sum_{\substack{0\le j<|h(a)|\\h(a)_j=b}} X^{\sum_{i<j}\ell_{h(a)_i}} Y^{\sum_{i<j}W_{h(a)_i,*}}. \tag{36} $$

Its slow monomials may be Laurent. From (34), exactly

$$ \alpha^{T(n)}\beta^{P(n)}=2^n/3^{s(n)}. \tag{37} $$

At $2$, $\alpha$ contracts. The slow parameters can have nonunit norms, but $h(T_N^k\beta)=O(\mu^k)$ for some $1<\mu<q$. The stable-image and critically-dense-suborbit reductions of Section 10 apply; they do not require unit norms at this particular place. Restriction to the resulting $V$ preserves the Laurent representatives and their support bounds. Adjoin $1$, choose a rational basis late, and apply Proposition 6, transporting through the raw matrix (36) as before. When $\theta>\log2/\log3$, $\alpha$ contracts in the reals as well. All series at every fixed iterated parameter then converge in both completions, since their slow supports are sublinear. Thus every rational value of the inverse-parity series is coherent across the two completions.

If a positive rational orbit with this parity were nonperiodic, the fixed- denominator argument and (16) would give $\theta\ge\log2/\log3$. Rationality of $\theta$ makes the inequality strict. Coherence would identify the positive start with a negative real series, a contradiction. The affine-prefix argument in Section 12 also handles finite shifts and prepended words: under a hypothetical nonperiodic positive realization, the same rational frequency supplies the strict gap before coherence is transferred from the full fixed word. This proves Theorem 1c.

For a concrete nonuniform example take $h(A)=A^2B^{12}$, $h(B)=A^2B^4$, and code $A$ by $0$, $B$ by $1$. Then

$$ M=\begin{pmatrix}2&12\\2&4\end{pmatrix},\quad q=8,\quad \ell=(2,1)^t,\quad \pi=(1/4,3/4),\quad a=5/4,\quad\theta=3/4,\quad W=(3,-1)^t. \tag{38} $$

The slow eigenvalue is $-2$, $c_0=1/5$, and $c_1=2/5$. Consequently $\alpha=2^{4/5}3^{-3/5}$ and $\beta=2^{-1/5}3^{2/5}$, with $|\beta|_2>1$. The incidence characteristic polynomial $(X-8)(X+2)$ is reducible, so Theorem 1b does not apply to this presentation. No claim that this particular coded word is nonautomatic is needed or asserted. The arguments through this section do not cover every primitive substitution with a noninteger Perron root and reducible characteristic polynomial. Sections15–17 address that remaining primitive case.

15. Relative formal regularity for the fast Perron component

Let $k_0$ be a characteristic-zero field embeddable in $\mathbb C$, and let $\sigma_0:k_0\to k_0$ be an embedding for which $k_0/\sigma_0(k_0)$ is finite. Let $A$ be a strictly positive integral $r\times r$ matrix whose characteristic polynomial is irreducible over $\mathbb Q$, with Perron root $\lambda>1$. On formal series in $X=(X_1,\ldots,X_r)$ set

$$ S\left(\sum_I a_I X^I\right)=\sum_I\sigma_0(a_I)X^{IA}, \qquad (T_AX)_i=\prod_jX_j^{A_{ij}}. $$

Positive entries and nonsingularity make this an injective homomorphism of formal power-series rings, and hence of their fraction fields.

Lemma 7 (candidate). Every finitely generated subfield

$$ k_0(X)\subset F\subset\operatorname{Frac}(k_0[[X]]) $$

with $S(F)\subset F$ is regular over $k_0(X)$.

Proof. Let $L$ be the relative algebraic closure of $k_0(X)$ in $F$. It is finite over $k_0(X)$, and $S(L)\subset L$: applying $S$ to an algebraic equation preserves algebraicity over the base and the image remains in $F$.

Let $k$ be an algebraic closure of $k_0$. Since $k_0$ is finite over its image, $\sigma_0$ extends to an automorphism $\sigma$ of $k$. Extend coefficients and put $E=kL$ inside $\operatorname{Frac}(k[[X]])$. Coefficient comparison justifies this field embedding. More explicitly, any putative relation involving finitely many algebraic constants can be split in a $k_0$-basis of the finite constant extension containing them; all formal coefficients lie in $k_0$. Thus the natural compositum is a field, finite over $k(X)$, and $S(E)\subset E$.

The iterated Laurent field $k((X_1))\cdots((X_r))$ contains this fraction field. Its standard rank-$r$ valuation restricts to a valuation of $E$ whose value group is exactly $\mathbb Z^r$: all values are integral and the coordinate monomials already give every integral vector.

We first show that monomial base change preserves the degree of $E$. For any nonsingular integral matrix $B$, the extension obtained by adjoining the fractional coordinate monomials associated to $B^{-1}$ has degree $|\det B|$ over $k(X)$. Choose one monomial for each coset in the corresponding superlattice modulo $\mathbb Z^r$. Embed them with $E$ in a common iterated Puiseux Laurent field by choosing a common denominator. Their values are distinct modulo the value group of $E$. Any nonzero linear combination over $E$ therefore has a unique least-valued term and cannot vanish. These $|\det B|$ monomials remain independent over $E$, so adjoining them has its full degree over $E$. Equivalently, $E$ is linearly disjoint from every such monomial base extension.

Apply this observation to the constant twist $E^\sigma$ and to the base change $U=T_AX$. The pullback is a field of the same degree over $k(X)$ as $E$, and its embedded image is $k(X)S(E)\subset E$. Consequently

$$ E=k(X)S(E). \tag{R1} $$

This is equality of finite covers after semilinear pullback; it is stronger than a mere inclusion of the transformed function field.

Normalize the torus $\mathcal T=(\mathbb G_m)^r_k$ in $E$. The normalization is finite and normal. The monomial map $T_A$ is finite étale in characteristic zero. Its base change preserves normality; by the preceding disjointness it is integral, and hence is the normalization belonging to the pullback field. Equation (R1) identifies it with the original cover. Thus its divisorial branch support $B_E$ satisfies

$$ B_E=T_A^{-1}(\sigma B_E). $$

Because the Laurent coordinate ring is factorial, write this reduced support as the zero set of a squarefree Laurent polynomial $Q$ (take $Q=1$ if it is empty). Étale base change preserves reducedness, so

$$ Q^\sigma(T_AX)=cX^u Q(X),\qquad c\in k^*,\ u\in\mathbb Z^r. \tag{R2} $$

There are no unrecorded multiplicities in (R2).

Let $P$ be the Newton polytope of $Q$. Constant twisting leaves its support unchanged, and nonsingularity makes the map on exponent vectors injective. Equation (R2) gives $A^tP=P+u$. If $\ell$ is a positive right Perron vector, the width of $P$ in direction $\ell$ consequently satisfies

$$ \lambda\,\operatorname{width}_{\ell}(P) =\operatorname{width}_{\ell}(P). $$

It is zero because $\lambda>1$. Irreducibility of the characteristic polynomial implies that the coordinates of $\ell$ are rationally independent. Distinct integral vertices would therefore have different pairings with $\ell$. Thus $P$ is a point, $Q$ is a Laurent unit, and there is no divisorial ramification inside the torus.

Purity of the branch locus [16] now shows that the finite normalization is étale everywhere over $\mathcal T$. A connected finite étale cover of a characteristic-zero torus is a monomial isogeny as a cover, after choosing a point over the identity. Here is the precise classical input: embed the algebraically closed coefficient field into $\mathbb C$; by [17, Remark3.3(c), Theorem3.4] finite étale covers are unchanged by this extension and correspond to finite topological covers. The universal cover of $(\mathbb C^*)^r$ is the coordinate exponential map from $\mathbb C^r$, with deck lattice $2\pi i\mathbb Z^r$. Finite-index sublattices give exactly the integral monomial isogenies. These isogenies are algebraic, and full faithfulness in the Riemann existence theorem identifies the original algebraic cover with one of them. This argument also proves the stated classification for the present $k$.

Its function field would therefore contain fractional coordinate monomials from a superlattice of index $[E:k(X)]$. Their valuations enlarge $\mathbb Z^r$ unless that index is one. This contradicts the Laurent embedding above, so $E=k(X)$.

Finally, every element of $L$ is a rational function in $k(X)$ whose formal Laurent expansion has coefficients in $k_0$. It belongs to $k_0(X)$. For example, put its finitely many rational-function coefficients in a finite normal extension of $k_0$; uniqueness of the formal expansion makes it invariant under every automorphism of that extension. Normalized coprime numerator and denominator then descend. Hence $L=k_0(X)$. In characteristic zero this is regularity of $F/k_0(X)$. $\square$

16. Local coherence with fast Perron and nonunit slow modes

Local statement

Let $A$ be a positive integral $r\times r$ matrix with irreducible characteristic polynomial and Perron root $\lambda>1$. Let $V$ be a geometrically irreducible variety in a torus, and let $T:V\to V$ be the finite dominant restriction of an integral monomial map. All are defined over one fixed number field $K_0$, enlarged once to contain the coordinates of the initial algebraic parameters. Suppose $\beta$ has a critically dense forward orbit in $V$ and

$$ h(T^n\beta)=O(\mu^n),\qquad 1<\mu<\lambda. $$

The coordinates of $\beta$ need not be units at the chosen prime $p$. Let $\alpha$ be an algebraic point in the fast torus. Require positive Perron projection of the vector $(-\log|\alpha_i|_p)_i$. Then, for some $c,C>0$ and all sufficiently large $n$,

$$ c\lambda^n\le-\log|(T_A^n\alpha)_i|_p\le C\lambda^n \quad(1\le i\le r). \tag{G1} $$

Replace the starting point by such a late iterate and renumber it as zero; thus all fast coordinates initially have norm less than one. Put $S(X,Y)=(T_AX,TY)$ and $z_n=S^n(\alpha,\beta)$.

Suppose the components of $f$ have expansions

$$ f_i(X,Y)=\sum_{I\in\mathbb N^r}c_{i,I}(Y)X^I, $$

where each coefficient has an ambient Laurent-polynomial representative over the same fixed number field $K_0$, with uniformly bounded $p$-adic coefficient norm and exponent support of norm at most $C_0(1+|I|)^\gamma$, for fixed $0\le\gamma<1$. Suppose $f=R(f\circ S)$ for an invertible rational matrix $R$ over $K_0(X,V)$, that the components are independent over $\overline{\mathbb Q}(X,V)$, and that all orbit points are regular for $R,R^{-1}$.

Proposition 8 (candidate). The entries of $f(z_0)$ are linearly independent over $\overline{\mathbb Q}$ in the selected $\mathbb C_p$ embedding.

Scalar avoidance and neighborhoods

Write $Q(X,Y)=\sum_I a_I(Y)X^I$ for a nonzero polynomial after removing coefficients zero on $V$. Choose a positive right Perron vector $\ell$. Its coordinates are rationally independent, so the finitely many real weights $I\ell$ are distinct. Let $I_0$ have the least weight.

For each fixed nonzero coefficient, critical density ensures that $a_I(T^n\beta)\ne0$ eventually. Evaluation-height bounds and Liouville give

$$ \exp(-C_Q\mu^n)\le |a_I(T^n\beta)|_p \le\exp(C_Q\mu^n). \tag{G2} $$

The fast logarithmic valuation vector is a positive multiple of $\lambda^n\ell$, up to $O(\mu_1^n)$ for some $\mu_1<\lambda$; choose $\mu_1>1$ to absorb all subdominant Jordan factors. Thus the least fast weight dominates every other term by $\exp(c_Q\lambda^n)$, even after (G2). Hence every fixed nonzero scalar rational function is eventually defined and nonzero on the orbit.

At a late smooth center, use relative fast and ambient slow coordinates. An integral monomial in relative units is uniformly Lipschitz independently of its exponents, including negative exponents. After division by the least fast monomial, the leading slow coefficient of each denominator has center norm at least $\exp(-O(\mu^n))$ and variation at most the relative radius times $\exp(O(\mu^n))$. Higher fast weights remain smaller. Consequently all required compositions through level $n$ are regular on nested neighborhoods of radius $r_n$ with

$$ -\log r_n=O(\mu_2^n)=o(\lambda^n),\qquad \|R_n\|_p\le\exp(O(\lambda^n)), \tag{G3} $$

for some fixed $1<\mu_2<\lambda$. Finitely many early regular levels only require a fixed additional shrinkage. This is a shrinking-neighborhood argument; nonunit slow coefficients need not admit a fixed radius.

Formal field and relation matrices

Put $k_0=\overline{\mathbb Q}(V)$. The finite dominant slow map induces a finite-index embedding of this field, extending to an automorphism of its algebraic closure. Lemma7 in Section15 therefore applies to $k_0(X)(f)$. The functional system makes this field stable under $S$, and its entries are formal power series in the fast variables. Thus independence persists over algebraic extensions of the function field.

The support hypothesis also gives convergence on every fast polydisc strictly inside the unit polydisc and every fixed closed slow torus annulus. Indeed evaluated coefficient norms are at most $\exp(C(1+|I|)^\gamma)$. After clearing rational denominators, repeated one-variable identity theorems in the fast coordinates, followed by Zariski density of a smooth analytic open subset of $V$, make the formal field inject into local analytic germs. Regularity preserves this injection after adjoining the finite algebraic branch below.

Construct the eventual relation ideal of the matrices $R_n(z_0)$ and points $z_n$, as in Section6. Scalar avoidance ensures that it contains no nonzero scalar. Inverting the determinant and applying the Nullstellensatz gives an invertible algebraic matrix $\phi$. At a late smooth point $z_*=z_{n_0}$, choose a regular analytic branch. Index shifting gives the same relation-ideal identity under $(U,z)\mapsto(U R_j(z),S^jz)$.

If $\tau f(z_0)=0$ is a nonzero algebraic relation, set

$$ M=R_{n_0}(z_0)\phi(z_*)^{-1},\quad \Theta_n(z)=M\phi(z)R_{n-n_0}(z),\quad F(U,z)=\tau U f(z),\quad g(z)=\tau M\phi(z)f(z). $$

Then $g(z_*)=0$, $g$ is a nonzero germ, and

$$ F(\Theta_n(z),S^{n-n_0}z)=g(z),\qquad \Theta_n(z_*)=R_n(z_0). \tag{G4} $$

Fix a single number field $K$ containing the coefficients, $\tau,M$, the center, and the branch constants before choosing any auxiliary degree. The eventual ideal over $K$ extends correctly under algebraic constants by coefficient splitting, since all evaluation coordinates lie in $K$. The finite auxiliary linear equations can therefore be solved in this same $K$.

Auxiliary dimensions and the omitted tail

Let $m$ be the number of function components, and let $q$ now denote an auxiliary degree, not a dynamical Perron root. Let $r_q$ be the rank over $K(X,V)$ of matrix monomials with separate degrees at most $q$, modulo the eventual ideal. Give each fast coordinate degree at most $D$ and filter slow Laurent polynomials by exponent box $[-E,E]$. Its Hilbert function is $H_V(E)\sim cE^{\dim V}$, or $1$ for a point. Basis selection and clearing a common denominator give

$$ r_q(D+1)^rH_V(E)\le\delta(q,D,E) \le r_q(D+c+1)^rH_V(E+c_Y), \tag{G5} $$

where the shifts depend on $q$. Partition of matrix exponent boxes gives $r_{2mq}\le B r_q$ with $B=(2m+1)^{m^2}$.

Take $P_0,\ldots,P_q$ from a complement to the ideal in this finite space and form $\mathcal E=\sum_{j=0}^qP_jF^j$. Truncation at total fast degree less than $\nu$, followed by $U\mapsto MU$, has matrix box degree at most $2mq$. The slow degree shift from its finitely many coefficients depends on $q,\nu$ and is fixed before $E$. For $\nu$ beyond the fixed fast shifts and $E$ sufficiently large, its target dimension is at most $2^{r+1}B r_q\nu^r H_V(E)$. Choose

$$ \nu=\left\lfloor D\left(\frac{q}{2^{r+4}B}\right)^{1/r}\right\rfloor. \tag{G6} $$

The tuple domain has dimension at least $(q+1)r_q(D+1)^r H_V(E)$, exceeding the target. Thus a nonzero tuple makes the transformed truncation lie in the eventual ideal. Choose $q$, then large enough $D$, then $E$ in that order.

If $j_0$ is the least nonzero index, then $b_n=P_{j_0}(R_n(z_0),z_n)\in K$ is nonzero infinitely often. Finite products of the function components have slow support bounded by $C_q(1+|I|)^\gamma$ at fast multi-degree $I$, by concavity of $t^\gamma$. The auxiliary polynomial adds at most $E$ to that support. On the relative neighborhoods, each omitted monomial with $|I|\ge\nu$ is bounded by

$$ \exp\left(-c|I|\lambda^n+ C\bigl(E+C_q(1+|I|)^\gamma\bigr)\mu^n\right). \tag{G7} $$

With all auxiliary degrees fixed, the slow term is at most $(c/2)|I|\lambda^n$, uniformly over every $|I|\ge\nu$, for all sufficiently large $n$. Substitution of the matrix variables costs at most $\exp(C'q\lambda^n)$. The vanishing truncated part and the ultrametric inequality consequently give

$$ \|\mathcal E(\Theta_n,S^{n-n_0}z)\|_p \le\exp((-c'\nu+C'q)\lambda^n+O_{q,D,E}(1)). \tag{G8} $$

Factor $F^{j_0}$ and extract a fixed first nonzero Taylor coefficient of $g^{j_0}$ as in Section8. Its degree is fixed after the auxiliary function is chosen. The radius loss in (G3) costs only $\exp(o(\lambda^n))$. The resulting center value is $b_n$, since the relation makes $F=0$ there. For $D$ sufficiently large after $q$,

$$ |b_n|_p\le\exp(-c_1 q^{1/r}D\lambda^n). \tag{G9} $$

Height estimates for the same fixed rational system give

$$ h(b_n)\le C_1(q+D)\lambda^n+C_2E\mu^n+C_{q,D,E}. \tag{G10} $$

The displayed constants are independent of the auxiliary degrees. With $D\ge q$, Liouville in the fixed field $K$ bounds each nonzero $b_n$ below by $\exp(-C_3D\lambda^n)$ for large $n$. Choose $q$ so that $c_1q^{1/r}>C_3$, then $D,E$, then arbitrarily large $n$ for which $b_n\ne0$. This contradiction proves the local proposition as a written candidate.

Coherence and scope

Adjoin $1$ to a raw Laurent-polynomial substitution system and choose a rational basis at a sufficiently late point on the slow orbit variety. The induced square rational matrix is invertible by independence of the basis. Scalar avoidance handles its poles. A rational relation at the initial point transports through the raw matrices to a relation among the independent basis values at that late point. Its coefficient row is zero, so the same identity holds in any other completion where the raw series and their iterates converge. Raw transport is defined at every nonzero torus point, even when the initially chosen rational basis is singular. This gives the needed real/$2$-adic coherence.

The proof retains all accepted argument choices: eventual rather than pointwise relation ideal, enlargement of matrix degree boxes, a fixed number field before auxiliary degrees, and $n$ tending to infinity last. The relative formal regularity lemma is a new written input; it has not been checked by an independent reviewer. The five-case finite audit tests only algebraic representations and specialization identities.

17. The general primitive substitution consumer

Theorem1d (restated). Let $h$ be a primitive substitution with Perron root greater than $1$, and let $u=h(u)$ be a one-sided fixed word. If a positive rational number in $\mathbb Z_2$ has shortcut parity equal to a binary coding of $u$, its orbit is eventually periodic. The same conclusion holds after deleting a finite prefix of the coded word and then prepending an arbitrary finite binary word.

Proof. Use the row incidence convention and write $M$ for the incidence matrix, $\lambda>1$ for its Perron root, and $p$ for the minimal polynomial of $\lambda$. Primitivity makes $\lambda$ simple and strictly greater in modulus than every other eigenvalue. Thus $p$ occurs exactly once in the characteristic polynomial, and the rational primary decomposition gives

$$ \mathbb Q^s=F\oplus G,\qquad F=\ker p(M), $$

where $\dim F=r=\deg p$ and the restriction to $G$ has spectral radius less than $\lambda$. Both intersections with $\mathbb Z^s$ are saturated, $M$-stable lattices of full rank in their respective spaces. The Perron left vector annihilates $G$.

We require an integral basis $W$ of the fast lattice whose ambient entries are positive and for which a power of $M$ has a positive coordinate matrix. Here is a construction. Choose a rational simplicial cone in $F$ containing the Perron vector $\ell$ in its interior, with every nonzero vector in its closure strictly positive in the ambient coordinates. This is possible because $\ell$ lies in the relative interior of that open positive cone. Take primitive lattice generators. If their determinant relative to the fast lattice is $D>1$, the half-open fundamental parallelepiped contains a nonzero lattice point. A stellar subdivision at its primitive ray replaces the cone by cones whose nonzero determinants are positive integers strictly below $D$. The coordinates of $\ell$ in any rational basis of $F$ are rationally independent: a rational linear relation, together with its iterates under the irreducible representation $M|_F$, would annihilate the entire space. Hence $\ell$ lies on no rational proper face. Select its unique containing cone and repeat. The determinant strictly decreases, so the process gives a unimodular cone. Its generators are the columns of $W$, and all their ambient entries are positive integers.

Write $MW=WA_0$. This coordinate matrix is integral. The coordinate vector $W^{-1}\ell$ is positive, and each column of $W$ has positive pairing with the ambient positive left Perron vector. Perron asymptotics therefore make $A_0^k$ strictly positive for all sufficiently large $k$. Replace $h,M$ by $h^k,M^k$ and denote the resulting positive fast matrix by $A$ and its Perron root again by $\lambda$.

The fast characteristic polynomial remains irreducible after taking this power. Indeed, if $[\mathbb Q(\lambda_0^k):\mathbb Q]$ were smaller than $[\mathbb Q(\lambda_0):\mathbb Q]$, every conjugate of $\lambda_0^k$ would occur more than once among the powers of conjugates of $\lambda_0$. But its largest-modulus value $\lambda_0^k$ occurs only once, by the strict Perron gap. This is a contradiction. The fixed word $u$ is unchanged by replacing $h$ with its power.

Choose an integral basis $W_s$ of the slow lattice and write $MW_s=W_sN$, where $N$ is integral with $\rho(N)<\lambda$. Unique rational coordinate vectors $c_0,c_1,d_0,d_1$ satisfy

$$ \mathbf1=Wc_0+W_sd_0,\qquad \epsilon=Wc_1+W_sd_1. \tag{P1} $$

Here $\epsilon$ is the column of binary coding values. For the row of letter counts $R(n)$ in the first $n$ symbols of $u$, put

$$ T(n)=R(n)W,\qquad P(n)=R(n)W_s. $$

Then exactly

$$ n=T(n)c_0+P(n)d_0,\qquad s(n)=T(n)c_1+P(n)d_1. \tag{P2} $$

Every fast coordinate increases by a positive integer at each symbol; in particular $|T(n)|$ is comparable to $n$.

Choose $1<\mu_0<\lambda$ above the spectral radius of $N$, with a margin to absorb Jordan factors. A complete level-$j$ substitution block has slow count equal to a row of $W_sN^j$, hence $O(\mu_0^j)$. Primitive block lengths are uniformly comparable to $\lambda^j$. Decompose a fixed-word prefix into boundedly many complete blocks at each scale. At the highest required scale, $\lambda^j=O(n)$. Summing the geometric bounds gives

$$ P(n)=O(n^\gamma),\qquad \gamma=\log\mu_0/\log\lambda<1. \tag{P3} $$

No sign or boundedness assumption on the individual slow coordinates is needed.

For each letter $b$, form the series

$$ H_b(X,Y)=\sum_{u_n=b}X^{T(n)}Y^{P(n)},\qquad \alpha_i=2^{(c_0)_i}3^{-(c_1)_i},\quad \beta_j=2^{(d_0)_j}3^{-(d_1)_j}, \tag{P4} $$

with compatible positive algebraic roots. Each coefficient at a fast multi-degree is zero or one slow Laurent monomial, because $T(n)$ strictly increases. Equation(P3) gives the required sublinear support. Equation(P2) yields the exact specialization

$$ \alpha^{T(n)}\beta^{P(n)}=2^n/3^{s(n)}. \tag{P5} $$

The prefix decomposition of $h(u)=u$ gives $H=B(H\circ S)$, with $S(X,Y)=(T_AX,T_NY)$ and

$$ B_{ba}(X,Y)= \sum_{\substack{0\le j<|h(a)|\\h(a)_j=b}} X^{\sum_{i<j}W_{h(a)_i,*}} Y^{\sum_{i<j}(W_s)_{h(a)_i,*}}. \tag{P6} $$

This is polynomial in the fast variables and Laurent in the slow variables.

Let $\pi$ be the positive left Perron vector of $M$. Primitive frequency is $\theta=\pi\epsilon/(\pi\mathbf1)$, an algebraic number. The positive row $\pi W$ is a left Perron vector of $A$. From(P1) and $\pi W_s=0$,

$$ (\pi W)c_0=\pi\mathbf1>0, \quad (\pi W)(c_1\log3-c_0\log2) =(\pi\mathbf1)(\theta\log3-\log2). \tag{P7} $$

Thus the fast coordinates contract at $2$ after a sufficiently late iterate. When $\theta>\log2/\log3$, they eventually contract in the real embedding as well. Individual initial coordinates may expand; (P7) justifies the late passage rather than assuming initial contraction.

The slow orbit has height $O(\mu^n)$ for some $1<\mu<\lambda$. Apply the stable-image and critically-dense-suborbit reduction of Section10. It is algebraic and does not require local unit norms. Passing to an arithmetic progression also replaces $A$ by a power, preserving positivity and irreducibility as above. Restriction to the slow orbit variety preserves the Laurent representatives and the sublinear support. Adjoin $1$, select a rational basis at a regular late point, and apply Proposition8. Raw transport through(P6) is defined before this basis is regular.

For completeness, when $\theta>\log2/\log3$, the series at the initial parameter and every fixed iterate converge in both completions even before coordinatewise contraction starts. At iterate $j$, (P5) is the coefficient $2^{|h^j(u_{<n})|}/3^{s(h^j(u_{<n}))}$. The expanded prefixes are prefixes of the same fixed word, have length comparable to $n$ for fixed $j$, and have coded frequency $\theta$. Real decay follows from the strict gap; $2$-adic decay follows from their lengths. Therefore the local proposition and raw transport identify every rational value of the inverse-parity series across the two completions.

If the positive rational orbit were nonperiodic, its fixed denominator would make it escape every bounded set. The finite affine orbit identity, as used in equation(16), gives lower parity density at least $\log2/\log3$. Here density exists and equals the algebraic number $\theta$. Gelfond–Schneider makes $\log2/\log3$ transcendental, so the inequality is strict. Coherence would identify the positive rational start with the strictly negative real inverse-parity series. This contradiction proves eventual periodicity.

For a finite shift and prepended word, first remove the prepended word by forward iteration. A nonperiodic positive realization of the shifted word has the same algebraic frequency and strict gap. The finite affine prefix identity reconstructs a rational value for the full fixed word, without requiring that reconstructed value to be positive. Coherence for the full word then gives coherence for the tail, whose real value is negative. This is precisely the finite-modification argument of Section12, and completes the proof. $\square$

A presentation outside both previously accepted substitution hypotheses

Take the positive incidence matrix and binary coding

$$ M=\begin{pmatrix}2&2&1\\1&4&1\\2&4&4\end{pmatrix}, \qquad \epsilon=(1,1,0)^t. $$

Use ordered images with these counts; the image of $A$ starts with $A$, so there is a fixed word beginning with $A$. The characteristic polynomial is $(t-2)(t^2-8t+8)$ and the Perron root is $4+2\sqrt2$. A positive integral fast basis, positive fast matrix, and slow vector are

$$ W=\begin{pmatrix}1&1\\1&2\\2&4\end{pmatrix},\qquad A=\begin{pmatrix}5&7\\1&3\end{pmatrix},\qquad W_s=(0,1,-2)^t,\quad N=2. $$

Here

$$ c_0=(5/4,-1/4)^t,\ c_1=(3/2,-1/2)^t, \quad d_0=1/4,\ d_1=1/2. $$

The slow parameter $\beta=2^{1/4}3^{-1/2}$ is a $2$-adic nonunit. The second initial fast parameter $2^{-1/4}3^{1/2}$ expands in both completions, but at fast iterate2 the parameters are

$$ (2^{26}/3^{20},\ 2^6/3^4), $$

which contract in both. The coded frequency is $\theta=(8-2\sqrt2)/7>2/3>\log2/\log3$ and is irrational. Thus this coded word would have no positive rational realization under the candidate theorem. The full incidence polynomial is reducible and its Perron root is noninteger. No assertion about automaticity or the existence of other presentations of the same coded word is needed.

Five positive3-letter examples with slow eigenvalues $2,-2,0,1,3$ passed one exact five-worker audit. Its100 matrix checks count precisely, per case, six fast intertwining entries, three slow entries, six count coordinates, four cubic factorization values, and one unimodularity check. It also verifies12801 raw coefficients,81920 prefix identities,35980 iterated identities,2425 parity symbols and41 fractional monomial cosets. It does not verify the analytic theorem, torus-cover argument, or universal Collatz behavior. The source and original JSON are retained.

18. Exact finite verification and review status

The automatic-case audit, audit_two_fast_cones.py, uses five CPU workers and no external Python packages. It treats five nonprimitive substitutions $h(A)=A^aB^mC^l$, $h(B)=B^q$, $h(C)=C^q$ with different fast state densities, so they are outside the previously treated uniform-count class. It checks 45 matrix entries, 660 count identities, 163840 cone inequalities, 122880 algebraic cancellations, 40960 Laurent coefficients, 122880 exponent coordinates, 32 negative fast matrix terms, and 2425 replayed parity symbols. An exact singular-start control shows why initial avoidance cannot be assumed. The exact identity $\alpha_1^{p_2}/\alpha_2^{p_1}=\rho$ makes $X^{p_2}-Y_1Z^{p_1}$ vanish at the initial parameter, whereas its two unit factors become $\rho^{q^j}$ and $\rho^{a^j}$ at iterate $j>0$.

The additional audit audit_irreducible_substitutions.py treats five genuinely nonuniform cases: the matrix in (27), and Eisenstein companion matrices of degrees two through five. It checks 58 matrix entries, 40960 prefix-Parikh coefficients, 131072 exponent coordinates, 40960 algebraic cancellations, 2580 iterated count identities, 4732 late monomial-valuation separations, and 2425 replayed parity symbols. Exact discriminant or Eisenstein certificates establish irreducibility for these examples. Their coded frequencies exceed $2/3$ by the algebraic inequalities recorded in the supplement. The finite valuation checks are examples only; they do not prove eventual separation for every polynomial.

The new audit audit_integer_perron_substitutions.py checks five nonuniform positive matrices with integer Perron roots and slow eigenvalues $-2,2,0,1,3$. Every example has a nonunit slow parameter at $2$. It verifies 90 matrix and parameter relations, 34874 raw matrix-system coefficients, 81920 prefix exponent identities, 35980 iterated count and valuation identities, and 2425 replayed parity symbols. The zero slow eigenvalue is explicitly reduced to $Y=1$ after one iterate; the unit eigenvalue fixes its nonunit parameter. This audit checks neither the universal support estimate nor the local analytic theorem, and makes no claim of nonautomaticity for its examples.

Earlier frozen audits checked the scalar and multiple-slow-mode interfaces. No larger simulation is used to justify a universal claim. The proof's central obligations are the field and local-analytic steps, not finite parity replay. The supplement contains the six earlier source and result pairs and the new seventh pair, their checksums, exact reproduction commands, source-reading scope, and an MIT license for the original audit code. The stored outputs were not regenerated for this packaging pass. The first version was accepted following rStation review. The reviewer reproduced the two-fast-cone, irreducible-substitution, and integer-Perron audits; this checks their finite interfaces, not the analytic proofs. No formal verification is supplied. The full general $p$-adic precedence comparison remains incomplete. Collatz is unresolved.

The seventh audit checks five positive three-letter incidence matrices with reducible characteristic polynomials, noninteger Perron roots and nonunit slow parameters. The exact totals and scope are given in Section17. It ran once with five workers; none of the six earlier audits was rerun. The new relative formal regularity, local specialization and general primitive consumer have not received independent review or formal verification.

References

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