Ternary Syracuse laws: tail conditioning, moment growth, and predecessor bounds

1. Problem and contributions

The random affine series used in the study of the Syracuse map can be evaluated both in the real numbers and in the $3$-adic integers. Its two values retain different information about the same exponent sequence. This paper studies how conditioning the real value to be large changes the distribution of the ternary value, including the depth at which a finite ternary quotient detects the change. We also determine higher-moment growth rates and derive target-uniform predecessor bounds using the known exact integrability threshold of the original ternary density.

Let $(e_j)_{j\ge1}$ be independent with

$$ P(e_j=e)=2^{-e},\qquad e\ge1, $$

and define

$$ a_j=\frac3{2^{e_j}},\quad D_k=\sum_{j=1}^k e_j,\quad A_k=\prod_{j=1}^k a_j=\frac{3^k}{2^{D_k}},\quad B_k=\frac13\sum_{j=1}^k A_j. \tag{1} $$

The limits $R=\lim B_k$ in $\mathbb R$ and $Z=\lim B_k$ in $\mathbb Z_3$ exist almost surely. The real convergence follows from $E\log a=\log(3/4)<0$, and ternary convergence from $|a_j|_3=1/3$. Write $q(r)=P(R>r)$.

The classical real-tail change of measure is the product law

$$ P_*(e_j=e)=\frac3{4^e}. \tag{2} $$

Let $\sigma$ be the law of $Z$ under $P_*$ and $\sigma_k$ its projection modulo $3^k$. Set

$$ \mu=\log3-\frac43\log2>0,\qquad v=\frac23\log2. \tag{3} $$

All logarithms without a subscript are natural. Total variation is normalized as $\|\lambda-\nu\|_{\rm TV}=\sup_E|\lambda(E)-\nu(E)|$.

Theorem 1 (ternary transition). If $r\to\infty$ and

$$ \frac{\mu k-\log r}{v\sqrt{k}}\longrightarrow s\in\mathbb R, \tag{4} $$

then

$$ \left\|\mathcal L(Z\bmod3^k\mid R>r)-\sigma_k\right\|_{\rm TV} \longrightarrow\Phi(s). \tag{5} $$

Equivalently, the center in quotient depth is $(\log r)/\mu$ and the window width is $v\mu^{-3/2}\sqrt{\log r}$. Their numerical coefficients are $5.733417$ and $6.343874$, respectively. Below any fixed smaller fraction of the center the distance tends uniformly to zero; above any fixed larger fraction it tends uniformly to one.

The main arithmetic contribution is the matching lower bound after projection to a ternary residue. Projection alone supplies only an upper bound. We construct a separating set from residues that admit a short exponent-sum representation and bound its conditional probability by counting the exact rational numerators of the same affine prefixes.

We also prove a quantitative coding statement. The exponent sum can be estimated from the depth-$k$ residue with an error having exponential tails, uniformly in $k$, under $P_*$. The conditional entropy of the entire length-$k$ word given that residue is bounded independently of $k$. Consequently $\sigma$ is exact dimensional with dimension

$$ d_* =\frac{(8/3)\log2-\log3}{\log3}=0.682479\ldots. \tag{6} $$

These are statements about random affine laws. No typical-path statement is transferred to a deterministic positive-integer Collatz orbit.

For the original law, write $\rho_k$ for its density on the depth-$k$ ternary quotient, lifted to $\mathbb Z_3$ with Haar probability measure. El Jaouhari already proves the quadratic energy bound and exact $L^p$ threshold, as detailed in Section 2. We give alternative proofs of $\|\rho_k\|_2^2\le12k^2$, membership in $L^p$ for $1\le p<2$, and failure of $L^2$. Our branch-overlap formula is another proof of positive quadratic-energy increments, also known from his work. These regularity statements are not claimed as new. They imply that the root mean square of the primitive depth-$k$ Fourier coefficients has exponential decay rate $(\log3)/2$. This is a mean-square statement; it does not determine the extremal Fourier coefficient.

We also determine every finite-level moment's exponential growth rate: zero for $1<p\le2$ and $\log(3^{p-1}/(2^p-1))$ for $p>2$. The maximum density has exponential growth rate $\log(3/2)$. Appendix A also determines the power exponent $-2$ of the limiting density's upper tail, with a quantitative lower bound. A finer-modulus form of the collision estimate also gives a direct Fourier-to-mixing reduction with fixed polynomial losses (Section 9.1). The resulting mixing conclusions are already known.

There is also an application to actual integer predecessors. Combining this known $L^p$ regularity with Mazur's weighted-path construction, and choosing a suitable nonperiodic predecessor of size $O(a)$, gives for each $\varepsilon>0$ a constant $c_\varepsilon>0$ such that the lower natural density of ordinary Collatz predecessors of $a$ is at least $c_\varepsilon a^{-2-\varepsilon}$ whenever $3\nmid a$. The coefficient is uniform in $a$. The finite count holds whenever its cutoff $Y$ satisfies $Y\ge C_{\rm on}a$, for one absolute constant $C_{\rm on}$, independent also of $\varepsilon$. Using Shaik's explicit primitive-decay recipe, Section 9.2 also gives the single subpower loss $c a^{-2}\exp[-C\log(3+a)/\log\log(9+a)]$. The external weighted-path input is stated precisely in Section 8.2, along with its verification status.

2. Relation to previous work

Tao introduced the finite Syracuse laws and proved fine-scale mixing for the original geometric exponent distribution [1]. In particular, his estimate implies that the ternary limit has a density with respect to Haar measure. Sum-conditioned and joint formulations of that mixing retain the division exponent; Allikvere's results [2] are a relevant precedent. Those results concern the original law and its central exponent sums. The rare real-tail conditioning considered here selects the different law (2), whose ternary limit is singular.

The scalar tail asymptotic and the change of measure (2) are classical Kesten–Grincevičius–Goldie theory [3,4]. Conditioned-path comparisons with the associated positive-drift walk go back at least to Asmussen [5]. Buraczewski, Collamore, Damek and Zienkiewicz study perpetuity exceedance times, including the critical logarithmic time and the half-tail asymptotic [6]. Blanchet, Lam and Zwart use the same change of measure in rare-event simulation [7]. The full-prefix estimate below is an elementary consequence of this framework. We do not claim novelty for the tail tilt, its drift, or the typical exceedance time.

Brofferio identifies the Poisson boundary of rational affine random walks through their contracting completions [8]. Under (2), only the ternary completion contracts: the real drift is positive and the $2$-adic drift is positive. Thus the full ternary value is the natural maximal boundary. The almost-sure coding recovery proved below is consistent with, and substantially covered by, that boundary theory. Our finite-quotient estimate is quantitative: it bounds the information lost at precision $3^{-k}$ by a constant. We include an elementary proof adapted to the rational numerator formula needed for Theorem 1.

Evaluating Collatz-related affine series in different completions also appears in Siegel's numen framework [9]. The contribution here is the Gaussian transition for the ternary quotient under a real-tail event, with a residue-measurable separating set and an explicit finite counting bound. We have not located this conclusion in the cited work. Our access to Asmussen [5] was limited to its abstract; the scalar renewal input is verified from the explicit statements in [4,6].

The offset injectivity used in Lemma 3 is Tao's Lemma 6.2 [1]; we give a numerator reconstruction for that known fact. Tao's Section 6 also uses collision estimates on stopped high-probability slices. El Jaouhari [10, Sections 5.4–5.5] already proves the global quadratic energy upper bound, positive energy increments, and the exact strong $L^p$ range $1\le p<2$. His density $f$ is relative to Haar probability on the units: $\rho=(3/2)f$ there and zero on nonunits. Consequently his energy bound $E_k^{\rm unit}\le8k^2$ gives exactly $\|\rho_k\|_2^2\le12k^2$ in our normalization. Our exponential-weight fiber estimate and branch-overlap proof provide alternative derivations of these known statements, not a priority claim. His stronger uniform atom floor also implies his positive energy increments. We read the relevant written proofs but did not rebuild the formal development.

Fourier Wren [11] computes finite energy increments and studies extremal Fourier decay. Its Section 6 leaves all-level mean-square control open. Corollary 10 records a consequence of the now-known regularity estimates: the exponential rate of that mean square. It does not prove a bounded prefactor or the conjectured extremal rate. The additional results here are the projected Gaussian tail-conditioning transition, finite-precision coding estimate, higher-moment spectrum, quantitative power-$2$ density tail, and target-uniform counting consequences. We found no corresponding statements in [10]; its counting theorem has target-dependent onset and subpower loss in source size.

Mazur proves positive predecessor density for every fixed target coprime to $3$ [12], using a weighted construction from his preceding paper [13]. His general-target statement leaves uniform dependence on the target outside its scope. We use that weighted construction and replace its coarse density-height truncation with Hölder's inequality. A two-sibling argument bounds the selected nonperiodic seed by a constant times the target. This yields the uniform power $a^{-2-\varepsilon}$. Wirsching's preliminary manuscript [14] proposes the stronger $c/a$ bound and states three remaining conjectures; our result does not resolve that conjecture.

Two further predecessors clarify the attribution. Mghirbi [15, Lemma 6.1] already uses two direct siblings in a favorable residue to choose a nonperiodic seed. Shaik [16, Lemmas 1.4--1.5 and Theorem 10.10] gives that selection in a finite root-pool framework, and [16, Proposition 10.1] already converts residue moments to integer mass by Hölder. We do not claim those mechanisms as new. Shaik's finite moment certificates and block propagation give exponential upper ceilings; [16, Appendix D.3] explicitly leaves polynomial moments at orders $3/2$ and $2$ open. El Jaouhari's global quadratic bound and uniform $L^p$ norms supply that regularity. The exponent $2+\varepsilon$ in our uniform target bound is a consequence of those known estimates and the weighted-path input. We include the short seed and Hölder arguments to make that consequence inspectable with the simpler input (34).

3. Scalar tail and prefix likelihood

The recursion is

$$ R\overset d=a(R'+1/3), \qquad R'\text{ independent of }a. \tag{7} $$

Put

$$ m(u)=Ea^u=\frac{3^u}{2^{1+u}-1}. \tag{8} $$

Then $m(1)=1$, $m'(1)=\mu>0$, and $E\log a<0$. The multiplier and additive term $a/3$ are positive and bounded above. The affine maps have no common fixed point. The additive group generated by the support of $\log a$ contains both $\log2$ and $\log3$, so it is dense. Thus the law is nonarithmetic in the renewal sense. Although its support lies in a translate of $(\log2)\mathbb Z$, it does not lie in $h\mathbb Z$ for any $h>0$; [4, Section 2] explicitly distinguishes these two conventions.

The scalar theorem therefore yields

$$ q(r)\sim\frac{C}{r},\qquad C=\frac{E[(aR'+a/3)-aR']}{E(a\log a)} =\frac1{3\mu}. \tag{9} $$

The numerator is $E(a/3)=1/3$. This subtraction is pointwise before expectation and does not require a finite first moment of $R$.

On a length-$k$ prefix, $dP_*/dP=A_k$. Under $P_*$ the variables $\log a_j$ have mean $\mu$, variance $v^2$, and finite third absolute moment. Since $R=B_k+A_kR'$ with an independent suffix, the conditional prefix likelihood relative to $P_*$ is exactly

$$ L_{r,k}=\frac{q((r-B_k)/A_k)}{A_kq(r)}, \qquad \operatorname{TV}_{\rm prefix}=E_*(1-L_{r,k})_+. \tag{10} $$

Here $q(u)=1$ for $u\le0$. The negative-part formula follows from $E_*L_{r,k}=1$.

Lemma 2 (prefix transition). Under (4), the quantity in (10) converges to $\Phi(s)$.

Proof. Let $S_j=\log A_j$, $S_0=0$. For $0<\theta<1$, $E_*e^{-\theta\log a}=m(1-\theta)<1$. Reversal of independent increments and the nonnegative-supermartingale maximal inequality give

$$ P_*\left(\max_{0\le j\le k}S_j-S_k>h\right)\le e^{-\theta h}. \tag{11} $$

Take $h=k^{1/4}$. On $S_k\le\log r-h$ and drawdown at most $h/2$,

$$ \frac{A_k}{r}\le e^{-h},\qquad \frac{B_k}{r}\le\frac{k}{3}e^{-h/2}=o(1). $$

The argument $(r-B_k)/A_k$ tends uniformly to infinity there. From (9), $L_{r,k}=1+o(1)$ uniformly on this region. On $S_k\ge\log r+h$, (10) instead gives $0\le L_{r,k}\le2e^{-h}/C=o(1)$ for large $r$. The intermediate strip has probability $o(1)$ by the central limit theorem, since $h=o(\sqrt{k})$. Hence

$$ \operatorname{TV}_{\rm prefix} =P_*(S_k>\log r)+o(1)\longrightarrow\Phi(s). \tag{12} $$

Using the bounded negative part in (10) avoids a uniform-integrability claim for the likelihood itself. $\square$

For clarity about rates, define $\eta(x)=\sup_{u\ge x}|u q(u)/C-1|$. Taking $h=8\log k$ and $\theta=1/2$ in the same proof, and applying Berry–Esseen to the middle strip, gives for sufficiently large $k,r$

$$ \left|\operatorname{TV}_{\rm prefix} -\Phi\left(\frac{\mu k-\log r}{v\sqrt{k}}\right)\right| \ll\frac{\log k+1}{\sqrt{k}}+\eta(k^8/2)+\eta(r). \tag{13} $$

Indeed the bad drawdown probability is at most $k^{-2}$, while the lower region has $B_k/r\le k^{-3}/3$ and $(r-B_k)/A_k\ge k^8/2$. Equation (13) reduces the error to the classical scalar renewal error; it supplies no new rate for $\eta$.

4. Exact numerator counting and the projection lower bound

Every prefix has the form

$$ B_k=\frac{C_k}{2^{D_k}},\qquad C_k=3^{k-1}+3^{k-2}2^{e_k}+\cdots+2^{e_2+\cdots+e_k}. \tag{14} $$

Lemma 3 (coding and counts). For fixed $k$, the pair $(D_k,C_k)$ determines the exponent word, as follows also from Tao's offset injectivity [1, Lemma 6.2]. Consequently:

1. For fixed $z\bmod3^k$, fixed $D_k=d$, and $B_k\le Kr$, there are at most $Kr2^d/3^k+1$ possible words.
2. If $E_{k,m}$ is the set of residues admitting a word with $D_k\le m$, then $|E_{k,m}|\le4^m/3^k$.

Proof. When $k\ge2$, $v_2(C_k-3^{k-1})=e_k$. Subtracting $3^{k-1}$ and dividing by $2^{e_k}$ gives the numerator for length $k-1$. Repetition recovers $e_k,\ldots,e_2$, and $D_k$ recovers $e_1$. The case $k=1$ is immediate.

Since $C_k\equiv2^d z\pmod{3^k}$, the first assertion counts positive integers in one arithmetic progression up to $Kr2^d$. For the second, choose one word for each residue in $E_{k,m}$. Each chosen word has $P_*$-probability $3^k4^{-D_k}\ge3^k4^{-m}$; their probabilities sum to at most one. If $m<k$, the set is empty. $\square$

The numerator bound retains the congruence and the real prefix constraint on the same word. This is the ingredient that projection alone lacks.

Lemma 4 (small conditional mass of short representations). Let $P_r=P(\cdot\mid R>r)$, $K\ge1$, $H,h>0$, and

$$ t=\log_2(3^k/r),\qquad m=\lfloor t-H\rfloor, \qquad\delta=2(3/4)^8. $$

Then

$$ P_r(Z\bmod3^k\in E_{k,m}) \le\frac{q(Kr)}{q(r)}+\frac{\delta^k}{q(r)} +\frac{2^{-h}+8k(K+2^h)4^{-H}}{r q(r)}. \tag{15} $$

Proof. The conditional probability of $A_k\ge r2^h$ is at most $2^{-h}/(r q(r))$, by $dP=A_k^{-1}dP_*$ on prefixes and by bounding the conditional suffix probability by one. Also $P_r(R>Kr)=q(Kr)/q(r)$. Finally, $E_P(4/3)^{e_1}=2$, so Chernoff gives $P_r(D_k>8k)\le\delta^k/q(r)$.

On the remaining event, $B_k\le R\le Kr$ because all real summands are positive, and $d=D_k>t-h$. A word of total sum $d$ has original probability $2^{-d}$. By Lemma 3, for every fixed residue and sum its conditional contribution is at most

$$ \frac{Kr/3^k+2^{-d}}{q(r)} \le\frac{(K+2^h)r}{3^kq(r)}. $$

There are at most $8k$ possible sums. Multiplying by $|E_{k,m}|\le4^m/3^k$ and using $4^m\le(3^k/r)^2 4^{-H}$ proves (15). $\square$

Proof of Theorem 1. Set

$$ H=\lceil4\log_2k\rceil,\qquad h=H/2,\qquad K=k^2. $$

Under (4), $\log r=\mu k+O(\sqrt{k})$. The first term in (15) is $O(k^{-2})$ by (9); the last is $O(k^{-2})$. The middle tends to zero because

$$ \mu+\log\delta=-1.433893\ldots<0. $$

Thus $P_r(E_{k,m})=o(1)$. But

$$ \sigma_k(E_{k,m})\ge P_*(D_k\le m) =P_*(S_k\ge\log r+H\log2+O(1))\longrightarrow\Phi(s). $$

This proves the lower bound in (5). The upper bound follows by projecting the full-prefix laws in Lemma 2. Monotonicity of quotient TV with respect to $k$ extends the transition to the two stated outer regimes. $\square$

5. Bounded information loss at finite precision

Let $d_k(z)$ be the smallest exponent sum of a length-$k$ word coding the residue $z$, and define under $P_*$

$$ \Delta_k=D_k-d_k(Z\bmod3^k)\ge0,\qquad V_k=\frac{C_k}{3^k}=\frac{B_k}{A_k}. \tag{16} $$

Only residues in the image of the coding map need a value of $d_k$.

Theorem 5 (uniform decoding error and entropy). There are positive constants $c,C_0$ independent of $k$ such that

$$ P_*(\Delta_k\ge u)\le C_0e^{-cu}\quad(u\ge0), \qquad H(e_1,\ldots,e_k\mid Z\bmod3^k)\le C_0. \tag{17} $$

Proof. For every $0<\theta<1$, reversal of independent increments and subadditivity give

$$ E_*V_k^\theta \le\frac{3^{-\theta}}{1-m(1-\theta)}=:M_\theta<\infty. \tag{18} $$

In fact $V_k$ has the law of $(1/3)\sum_{j=0}^{k-1}\exp(-S_j)$ under $P_*$.

For fixed $z,d$, Lemma 3 bounds the number of words with $V_k\le K$ by $K+1$. Since all words with sum $d$ have tilted probability $3^k4^{-d}$, for integer $u\ge0$

$$ \begin{aligned} P_*(\Delta_k\ge u,V_k\le K) &\le\frac43(K+1)4^{-u} \sum_z3^k4^{-d_k(z)}\\ &\le\frac43(K+1)4^{-u}. \end{aligned} \tag{19} $$

For the last inequality, select one minimum-sum word for each residue. Taking $K=4^{u/(1+\theta)}$ in (19) and using (18) proves the exponential tail in (17).

The residue, $\Delta_k$ and $\lfloor V_k\rfloor$ determine $D_k$ and $C_k$, hence the word by Lemma 3. The exponential tail bounds the entropy of $\Delta_k$ uniformly. The positive fractional moment in (18) bounds the entropy of $\lfloor V_k\rfloor$ uniformly: compare its law with the normalized reference probabilities proportional to $(j+1)^{-1-\theta/2}$ and use nonnegativity of relative entropy. Subadditivity of entropy completes the proof. $\square$

Corollary 6 (exact dimension). The measure $\sigma$ is exact dimensional, with dimension $d_*$ in (6), and

$$ H(\sigma_k)=kH(P_*)+O(1). \tag{20} $$

Proof. The entropy identity gives (20). For the almost-sure statement, let $W_k=(e_1,\ldots,e_k)$ and

$$ I_k=-\log P_*(W_k\mid Z\bmod3^k). $$

Its expectation is bounded by Theorem 5. Markov's inequality and Borel–Cantelli on $k=j^2$ show that $I_k/k\to0$ along this subsequence, almost surely. The strong law gives $-k^{-1}\log P_*(W_k)\to H(P_*)$. Therefore

$$ -\log\sigma_k(Z\bmod3^k)=-\log P_*(W_k)-I_k $$

divided by $k$ tends to $H(P_*)$ along $j^2$. Monotonicity of nested ball probabilities and $(j+1)^2/j^2\to1$ extend the limit to all $k$. Finally $H(P_*)=(8/3)\log2-\log3$. $\square$

The full ternary value determines the infinite tilted exponent word almost surely. To see this directly, put $h_0=H(e_1\mid Z)$. The invertible affine recursion and independence of the suffix give $H(W_k\mid Z)=kh_0$. Conditioning on $Z$ gives no more entropy than conditioning on its residue, so Theorem 5 forces $h_0=0$. Iterating recovers all exponents. We regard this as an elementary manifestation of the boundary theory discussed in Section 2, rather than a new general Poisson-boundary theorem.

6. Joint tail law and the full ternary space

Corollary 7 (angular tail law). Let $Y$ have the Pareto distribution $P(Y>t)=t^{-1}$ for $t\ge1$, independently of $Z_*\sim\sigma$. Then

$$ \mathcal L((R/r,Z)\mid R>r)\Longrightarrow\mathcal L(Y,Z_*). \tag{21} $$

For every finite $r>0$, however,

$$ \left\|\mathcal L(Z\mid R>r)-\sigma\right\|_{\rm TV}=1. \tag{22} $$

Proof. For each fixed quotient, prefix likelihood convergence follows from Lemma 2 and projection, or directly from (10). Thus for a ternary cylinder $E$ and fixed $t\ge1$,

$$ P(R/r>t,Z\in E\mid R>r) =\frac{q(tr)}{q(r)}P(Z\in E\mid R>tr) \longrightarrow t^{-1}\sigma(E). $$

Regular variation gives radial tightness, and $\mathbb Z_3$ is compact. These tail cylinders determine (21).

For (22), let $\rho_k$ be the original ternary law's level-$k$ density with respect to Haar probability measure. Tao's fine-scale estimate gives $\|\rho_n-\rho_m\|_1\ll_A m^{-A}$ for $n\ge m$, hence an $L^1$ limit density. The conditional law at finite $r$ is dominated by this original law divided by $q(r)>0$, so is absolutely continuous. Corollary 6 makes $\sigma$ Haar singular. Therefore the two full-space measures are mutually singular. $\square$

Equivalently, the joint tail measure away from radial zero is $C x^{-2}\,dx\,\sigma(dz)$. The asymptotic independence in (21) applies after real-tail conditioning; the original pair $(R,Z)$ is not asserted independent. Equations (5) and (22) concern different orders of scale.

7. Known critical regularity and further moment consequences

Let $m_3$ be Haar probability measure on $\mathbb Z_3$, and set $\rho_k(z)=3^kP(Z\equiv z\pmod{3^k})$. These are the conditional expectations of the original density $\rho$ from Section 6. Put $E_k=\|\rho_k\|_2^2$, with $\rho_0=1$ and $E_0=1$.

Theorem 8 (known sharp integrability threshold [10]). For all $k\ge1$,

$$ E_k\le12k^2. \tag{23} $$

The limiting density satisfies $\rho\in L^p(m_3)$ for every $1\le p<2$ and $\rho\notin L^2(m_3)$. Moreover, there are $c>0$ and $k_0$ such that $E_k-E_{k-1}\ge c$ for $k\ge k_0$.

Alternative proofs. The assertions above are established in [10] in unit-Haar normalization. The following proofs are included for the mixed identities needed in Sections 7–9.

Proof of the upper bound. Let $N_d(z)$ count length-$k$ words with $D_k=d$ and residue $z$. By Lemma 3, their positive numerators are distinct integers in one class modulo $3^k$. In increasing order, the $j$th numerator is greater than $(j-1)3^k$. Therefore

$$ N_d(z)^2\le N_d(z)+2\sum_{w:D_k=d,\,Z=z}V_k(w). $$

For $1<u<2$, weighted Cauchy--Schwarz gives

$$ \begin{aligned} E_k &=3^k\sum_z\left(\sum_{d\ge k}N_d(z)2^{-d}\right)^2\\ &\le\frac{u^{1-k}}{u-1}\,3^k \sum_{z,d\ge k}u^dN_d(z)^2 4^{-d}\\ &\le\frac{u^{1-k}}{u-1}\,E_*[u^{D_k}(1+2V_k)]. \end{aligned} \tag{24} $$

Finite truncation and Tonelli justify the nonnegative sums. Define

$$ G(u)=E_*u^e=\frac{3u}{4-u},\qquad Q(u)=E_*\frac{u^e}{a}=\frac{u}{2-u}. $$

The representation of $V_k$ used in (18), retaining its joint dependence on $D_k$, gives the exact mixed moments

$$ E_*u^{D_k}=G(u)^k,\qquad E_*[u^{D_k}V_k] =\frac13\sum_{j=0}^{k-1}G(u)^{k-j}Q(u)^j. \tag{25} $$

Take $u=1+1/(4k)$. Then $G(u)\le Q(u)$ and $u^{1-k}/(u-1)\le4k$. Bernoulli's inequality gives

$$ Q(u)^k =\frac{(1+1/(4k))^k}{(1-1/(4k))^k} \le\frac{e^{1/4}}{3/4}<\frac{16}{9}. $$

For the strict inequality, $\log(4/3)>1/4$. Combining (24)--(25),

$$ E_k\le\frac{64}{9}k\left(1+\frac{2k}{3}\right) \le\frac{320}{27}k^2<12k^2. \tag{26} $$

Proof of the $L^p$ inclusion. Tao's estimate supplies $\|\rho_n-\rho_m\|_1\ll_A m^{-A}$ for $n\ge m\ge1$ and every $A>0$. Equation (23) gives $\|\rho_{2m}-\rho_m\|_2=O(m)$. If $1<p<2$, let $\vartheta=2/p-1\in(0,1)$. Interpolation yields

$$ \|\rho_{2m}-\rho_m\|_p \ll_{A,p}m^{-A\vartheta+1-\vartheta}. \tag{27} $$

Choose $A>(1-\vartheta)/\vartheta$ and sum over $m=2^j$. The dyadic densities converge in $L^p$, and their $L^1$ limit is $\rho$. The endpoint $p=1$ was established in Section 6.

Proof of the $L^2$ exclusion. For $F_e(x)=(3x+1)/2^e$, define

$$ (L_e f)(z)=3\,2^{-e}\,1_{F_e\mathbb Z_3}(z)\, f((2^ez-1)/3). $$

The Haar Jacobian of $F_e$ is $1/3$. The transfer operator $L=\sum_{e\ge1}L_e$ satisfies $\rho=L\rho$ and $\rho_{k+1}=L\rho_k$. For any nonnegative $f\in L^2$,

$$ \|L_e f\|_2^2=3\,4^{-e}\|f\|_2^2,\qquad \sum_{e\ge1}3\,4^{-e}=1. \tag{28} $$

If $\rho$ were in $L^2$, stationarity and expansion by Tonelli would force all cross terms between distinct $L_e\rho$ to vanish. Writing $E=\{\rho>0\}$, the sets $F_eE$ would be pairwise disjoint modulo null sets. Each has the same positive measure $m_3(E)/3$. Infinitely many such subsets of a probability space cannot be pairwise disjoint. This contradiction proves $\rho\notin L^2$.

Proof of positive energy increments. First $\rho>0$ almost everywhere on the units. Indeed stationarity gives $F_eE\subseteq E$ modulo null sets for every $e$. Let $C=3E+1$, a positive-measure subset of the unit group. The union of $2^{-e}C$ over $e\ge1$ has full measure in that group: multiplication by $2$ is ergodic, since its order modulo $3^j$ is $2\cdot3^{j-1}$. This union lies in $E$. The law has no mass on $3\mathbb Z_3$, so this also identifies the essential support.

Put $S(x)=4x+1$. Opposite-parity branches have disjoint ranges. For $e\ge1$ and $d\ge1$, the change of variable $z=F_e(x)$ gives

$$ \langle L_e f,L_{e+2d}f\rangle =3\,4^{-e-d}\int f(x)f(S^d x)\,dm_3(x). $$

Summing the diagonal terms using (28), and both orders of the cross terms, yields the exact identity

$$ E_{k+1}-E_k =2\sum_{d\ge1}4^{-d} \int\rho_k(x)\rho_k(S^d x)\,dm_3(x). \tag{29} $$

Since $x$ and $Sx$ are both units when $x\equiv1\pmod3$, the integral of $\min(\rho,M)(x)\min(\rho,M)(Sx)$ is positive for any $M>0$. The bounded truncations of $\rho_k$ converge to this truncation in $L^1$, and $S$ preserves Haar measure. Their product integrals therefore converge. Retaining only $d=1$ in (29) and using these truncations gives the claimed eventual lower bound. $\square$

Theorem 9 (moment spectrum). For $p>1$, put $M_p(k)=\int\rho_k^p\,dm_3$. Then

$$ \lim_{k\to\infty}\frac1k\log M_p(k) =\begin{cases} 0,&1<p\le2,\\ \log\bigl(3^{p-1}/(2^p-1)\bigr),&p>2. \end{cases} $$

For each fixed $p>2$, writing $\lambda_p=3^{p-1}/(2^p-1)>1$, there is $C_p<\infty$ such that $\lambda_p^k\le M_p(k)\le C_p k^{p-1}\lambda_p^k$. In addition,

$$ \lim_{k\to\infty}\frac1k\log\|\rho_k\|_\infty=\log(3/2). $$

Proof. The range $1<p\le2$ follows already by interpolation between $\|\rho_k\|_1=1$ and (23); no Fourier mixing input is needed for this exponential-rate statement. For $p>2$, the branch formula gives $\|L_e f\|_p^p=3^{p-1}2^{-ep}\|f\|_p^p$. Nonnegativity implies $M_p(k+1)\ge\lambda_p M_p(k)$, proving the lower bound. The function $\log\lambda_p$ has positive second derivative for $p>0$ and positive first derivative at $2$, and $\lambda_2=1$; hence $\lambda_p>1$ for $p>2$.

Set $q=p-1>1$. For fixed sum and residue, order the $N=N_d(z)$ numerators as $V_i=C_i/3^k>i-1$. Then

$$ N^p\le p\sum_{i=1}^N(1+V_i)^q \le p\,2^{q-1}\sum_{i=1}^N(1+V_i^q). $$

The first inequality uses $\sum_{i=1}^Ni^q\ge\int_0^N x^q\,dx=N^p/p$. Weighted Hölder with $u>1$, $t=u^q<2$, and $c_k(u)=u^{1-k}/(u-1)$ consequently gives

$$ \begin{aligned} M_p(k) &\le c_k(u)^q3^{kq}\sum_{z,d}t^dN_d(z)^p2^{-pd}\\ &\le p\,2^{q-1}c_k(u)^q\lambda_p^k E_p[t^{D_k}(1+V_k^q)], \end{aligned} $$

where $P_p(e)=(2^p-1)2^{-pe}$. Let $G_p(t)=E_p t^e=(2^p-1)t/(2^p-t)$ and tilt this product law to $P_{p,t}(e)=t^eP_p(e)/G_p(t)$. The remaining expectation is $G_p(t)^k E_{p,t}(1+V_k^q)$. Reversal expresses $V_k$ as $(1/3)\sum_{j=0}^{k-1}\prod_{i=1}^j a_i^{-1}$, and

$$ H_p(t):=E_{p,t}a^{-q}=\frac{2^p-t}{3^q(2-t)},\qquad H_p(1)=\lambda_p^{-1}<1. $$

For $u$ sufficiently close to one, $H_p(t)<1$. Minkowski gives, uniformly in $k$,

$$ (E_{p,t}V_k^q)^{1/q} \le\frac{1}{3(1-H_p(t)^{1/q})}. $$

Since $c_k(u)^q=[u/(u-1)]^qu^{-qk}$, the upper exponential rate is at most $\log\lambda_p+\log(G_p(t)/t)$. Letting $u\downarrow1$ proves the stated limit. More quantitatively, take $u=1+c_p/k$ with a sufficiently small fixed $c_p>0$. Then $H_p(u^q)$ stays strictly below one and $[G_p(u^q)/u^q]^k$ stays bounded, proving the polynomial prefactor. All word sums are nonnegative and can be justified by finite truncation.

Finally the all-one word gives $\rho_k(-1)\ge(3/2)^k$. On the finite quotient, $\|\rho_k\|_\infty^p\le3^kM_p(k)$, so for every fixed $p>2$ its upper exponential rate is at most $\log3-p^{-1}\log(2^p-1)$. Let $p\to\infty$ to obtain $\log(3/2)$. $\square$

Shaik [16, Appendix D.3] establishes existence and monotonicity of Rényi divergence rates using moment inheritance, and identifies the order-one rate as zero. Theorem 9 determines the rates at every higher order: divide the displayed moment rate by $p-1$. The underlying numerator argument is independent of Tao's fine-scale theorem, which is needed above to obtain bounded moments, rather than merely zero exponential rates, for $1<p<2$.

Corollary 10 (primitive Fourier mean square). Set $\chi_k(\xi)=E\exp(2\pi i\xi Z/3^k)$ and

$$ M_{2,k}=\left(\frac{1}{2\cdot3^{k-1}} \sum_{3\nmid\xi\bmod3^k}|\chi_k(\xi)|^2\right)^{1/2}. $$

For some positive constants $c_1,C_1$ and all sufficiently large $k$,

$$ c_1\,3^{-k/2}\le M_{2,k}\le C_1 k\,3^{-k/2},\qquad -\frac1k\log M_{2,k}\longrightarrow\frac{\log3}{2}. \tag{30} $$

Proof. Parseval and consistency of the quotient laws give $\sum_{3\nmid\xi}|\chi_k(\xi)|^2=E_k-E_{k-1}$. Apply Theorem 8. $\square$

No linear upper bound for $E_k$, bounded upper bound for its increments, or extremal Fourier decay rate is established. The $L^p$ estimates control the mass of small measurable sets by Hölder's inequality; they do not provide relative estimates at a prescribed moving integer.

8. Integer traces and polynomial predecessor density

8.1. A fixed-integer trace

The tilted law also has an exact positive-integer interpretation. For positive odd integers, let

$$ U(x)=\frac{3x+1}{2^{v_2(3x+1)}},\qquad \omega_k(x)=\frac{3^k}{2^{\sum_{j<k}v_2(3U^j(x)+1)}}. $$

Proposition 11. For every positive odd $a$ and every $k\ge0$,

$$ 3^k\sigma(a+3^k\mathbb Z_3) =\sum_{U^k(x)=a}\omega_k(x)^2. \tag{31} $$

If $a$ is nonperiodic, then

$$ \sum_{k\ge0}3^k\sigma(a+3^k\mathbb Z_3) \le\frac{\pi^2}{8}a^2. \tag{32} $$

For periodic $a$, the right side may be multiplied by at most $3a+1$.

Proof. Expose the prefix in inverse chronological order. Its first congruence is $2^{e_1}a\equiv1\pmod3$, giving the positive odd integer $x_1=(2^{e_1}a-1)/3$. Dividing out this congruence gives the next one. Induction recovers a positive odd predecessor path, with exactly the specified valuations. Conversely every such path supplies the prefix congruence. A length-$k$ word has tilted probability $3^k4^{-D_k}$; multiplying by $3^k$ gives $\omega_k^2$, proving (31).

The real affine identity gives $x\omega_k(x)\le a$. When $a$ is nonperiodic, each source can reach it at only one time. Summing over all odd positive sources therefore gives $a^2\sum_{x\ \mathrm{odd}}x^{-2}=\pi^2a^2/8$.

On a cycle, each repeated visit contributes a further factor $\lambda^2$, where

$$ \lambda=\prod_{u\text{ on the cycle}}\frac{3u}{3u+1} \le\frac{3a}{3a+1}<1. $$

The return sum is at most $1/(1-\lambda^2)\le3a+1$. $\square$

Thus every positive odd integer is a point where the normalized ball masses tend to zero and are summable over depths, despite the typical dimension being $d_*<1$. This does not provide a subquadratic bound uniform in a moving target $a$. In particular, (32) cannot be applied as though its right side were fixed along an unbounded forward orbit. Neither cycles nor divergent integer orbits are excluded by this paper.

8.2. Uniform polynomial dependence on the target

For a positive integer $a$, let $\mathcal P(a)$ be the set of positive integer starts whose ordinary Collatz orbit visits $a$. Write $\underline d$ for lower natural density.

Theorem 12 (polynomial predecessor density). Using the weighted-path input of Mazur [12, Proposition 3.1], for every $\varepsilon>0$ there is $c_\varepsilon>0$ independent of $a$ such that

$$ \underline d(\mathcal P(a))\ge c_\varepsilon a^{-2-\varepsilon} \qquad(a\ge1,\ 3\nmid a). \tag{33} $$

More precisely, there is an absolute $C_{\rm on}>0$, independent also of $\varepsilon$, such that

$$ \#\bigl(\mathcal P(a)\cap[1,Y)\bigr) \ge c_\varepsilon Y a^{-2-\varepsilon} \qquad(Y\ge C_{\rm on}a). $$

The constant $C_{\rm on}$ is finite but inherits very large absolute constants from the weighted-path construction.

Here is the exact imported input. There are universal integers $B_0,q$ with $q\ge1$ and a unit $y\pmod{3^q}$ with the following property. For every odd $R\ge B_0$ with $R\equiv y\pmod{3^q}$ and all sufficiently large real $X$, there exist $k\ge1$, $J\ge0$, a finite family $\mathcal W$ of exact positive odd valuation words ending at $R$, and selected weights $v_w$ such that

$$ \begin{gathered} 3^k\le X,\qquad X\le x_w<32X,\qquad0\le v_w\le\omega_w,\\ \frac23\le\sum_{w\in\mathcal W}v_w \sum_{j=0}^J4^{-j}r_k(\Psi_j(x_w)),\qquad \Psi_j(x)=4^jx+\frac{4^j-1}{3}. \end{gathered} \tag{34} $$

The source $x_w$ follows the word to $R$ with its exact valuations, and $\omega_w=3^{|w|}2^{-D(w)}$. The reference functions in [12] are $r_k=(2/3)\rho_k$, by [13, equation (2.2)] and [12, Appendix A.1]. The recursion used there has the same quotient law by reversal of the iid exponents. We take its auxiliary parameter $m=1$. The proof of the input permits $H(R,1)=A_*R$ with an absolute finite $A_*$, as follows.

Fix the floor $b=2^{80}$, Fourier constant, caps $K_j=16+\lfloor j/100\rfloor$, and startup parameters $N,N_t,q$ of [13, Sections 8.2--8.3]. These precede the numerical root. The central pullback at stage $N$ has average at least $255/256$ over unit residues. Choose a residue with at least this mark. Every odd representative $R\ge16^b$ has the same mark: exact integer inverse paths are supplied by the physical floor. The subsequent central, terminal-transport and terminal-exclusion errors have the root-independent bounds displayed in [13, Section 8.3], respectively $F(j+1)^3r^j$ and twice $21F(j+1)^5r^j$, with $r=9999/10000$. Thus the marked mass is at least $175/256>2/3$ whenever $n\ge N+N_t$ and $X$ belongs to every endpoint's admissible external-scale interval. This is also the uniform calculation in [12, Appendix A.2].

Put $J_* =\max(N+N_t,20\cdot10^9)$ and write $\beta(j)$ for the floor sequence of [13]. Its Section 8.5 bounds every endpoint at generation $J_*$ by $2^{E(J_*)}R$, and every upper external-scale endpoint there by $A_*R$, where

$$ \begin{gathered} E(0)=0,\qquad E(j+1)=E(j)+17+\lfloor j/100\rfloor +2\beta(j)-\lfloor19\beta(j)/12\rfloor,\\ A_*=2^{E(J_*)+\lfloor2\beta(J_*)/3\rfloor}. \end{gathered} $$

These bounds include all histories. The scale intervals overlap after $20\cdot10^9$ independently of the root, since their sufficient spread condition involves only the floors and caps. The proof of [13, Lemma 6.2] therefore supplies a common admissible generation for every $X>A_*R$. The terminal reduction and modulus estimate of [12, Appendix A.3--A.4] give (34) throughout this range. All its depths satisfy $k\ge1$. The target-dependent coarse depth used later in [13, Section 8.4] is unnecessary here because the known $L^p$ bound holds at every depth.

This is a substantive external input about exact integer paths. It is not replaced here by congruence mixing. Appendix A of [12] derives it from [13] while retaining the central and terminal path restrictions. We have read the written construction and its counting interface. The source reports Lean verification for related main statements, but [12] expressly does not advertise Proposition 3.1 as a separately checked formal declaration. We have not independently rebuilt those dependencies.

Proof of Theorem 12. Fix $1<p<2$, put $p'=p/(p-1)$, and let $K_p=\|\rho\|_p<\infty$, supplied by Theorem 8. Since $\rho_k$ are conditional expectations of $\rho$, their $L^p$ norms are at most $K_p$. Every $\Psi_j$ permutes the full group $\mathbb Z/3^k\mathbb Z$, so Minkowski's inequality gives

$$ \|g_{k,J}\|_p\le\frac89K_p,\qquad g_{k,J}=\sum_{j=0}^J4^{-j}r_k\circ\Psi_j. \tag{35} $$

First suppose $R$ has no positive Syracuse return. The sources in $\mathcal W$ are distinct: unequal hitting times of $R$ would give a return, and equal hitting times determine the word. Set $\mathcal A=\{x_w:w\in\mathcal W\}$. The exact affine formula gives $x_w\omega_w\le R$. Each residue modulo $3^k$ occurs at most $33X/3^k$ times below $32X$, since $3^k\le X$. Retaining the selected weights in (34), then using Hölder on distinct integer sources, gives

$$ \begin{aligned} \frac23 &\le\frac RX\sum_{x\in\mathcal A}g_{k,J}(x)\\ &\le\frac RX|\mathcal A|^{1-1/p} \left(\sum_{X\le x<32X}g_{k,J}(x)^p\right)^{1/p}\\ &\le\frac RX|\mathcal A|^{1-1/p}(33X)^{1/p}\frac89K_p. \end{aligned} \tag{36} $$

It follows that $|\mathcal A|/X\ge (3/(4\,33^{1/p}K_pR))^{p'}$. Applying the construction at $X=Y/32$ for all sufficiently large integer $Y$ yields

$$ \underline d(\mathcal P(R))\ge \frac1{32}\left(\frac{3}{4\,33^{1/p}K_pR}\right)^{p'}. \tag{37} $$

It remains to choose $R\le C a$ with one universal $C$. We use the two-sibling construction of [15, Lemma 6.1; 16, Lemmas 1.4--1.5]. For any $a\ge1$ coprime to $3$, choose $e\in\{1,2\}$ so that $2^ea\equiv1\pmod3$ and put $R_j=(2^ea4^j-1)/3$ for $j\ge0$. These are distinct positive odd ordinary predecessors of $a$, all with next odd value equal to the odd part of $a$. The elementary identity $v_3(4^h-1)=1+v_3(h)$ gives $v_3(R_{j+h}-R_j)=v_3(h)$. Hence $j\mapsto R_j\pmod Q$, with $Q=3^q$, permutes the residues on every $Q$ consecutive indices.

Choose a fixed $J_0\ge1$ such that $(2\cdot4^{J_0}-1)/3\ge B_0$. There is $j\in[J_0,J_0+Q-1]$ with $R_j\equiv y\pmod Q$. Both $R_j$ and $R_{j+Q}$ are at least $B_0$ in the desired residue, and both are at most $Ca$, with $C=(4/3)4^{J_0+2Q}$. At most one can be periodic under $U$: if both were periodic, both would lie on the cycle containing their common next odd value, contradicting the unique predecessor on a cycle. Thus one is an eligible nonperiodic $R\le Ca$. This selection needs no bound on a hypothetical cycle's maximum and no algorithm deciding which sibling is nonperiodic.

Set $C_{\rm on}=64A_*C$. For every $Y\ge C_{\rm on}a$, we have $X=Y/32\ge2A_*Ca>A_*R$, so (36) applies at that cutoff. Since $\mathcal P(R)\subseteq\mathcal P(a)$, it gives a uniform coefficient times $Ya^{-p'}$. Take $p=(2+\varepsilon)/(1+\varepsilon)$ to obtain the finite assertion and hence (33). No lower restriction on $k$ depending on $p$ or $R$ was used. $\square$

The same exponent holds for predecessor sets under the shortcut map, since ordinary predecessors of $2a$ are shortcut predecessors of $a$. If $3\mid a$, the ordinary predecessor set is $\{2^j a:j\ge0\}$ and has density zero. Theorem 12 supplies a uniform linear cutoff, but it remains much larger than the target. On a hypothetical least-counterexample spine $a_t\ge a_0$, the choice $Y=a_0$ is outside its range. It gives neither a predecessor below that original minimum nor nonsummable relative mass along a divergent orbit.

9. Direct mixing reduction and a quantitative predecessor coefficient

9.1. A collision estimate at a finer ternary modulus

The same numerator bound gives a short alternative to the stopped-prefix collision argument in Tao's Section 6. The fine-scale mixing conclusions in this subsection are established results [1,2]; we record the explicit polynomial loss because it permits a quantitative use of [16, Appendix E].

For $1\le r\le n$, set $d_0=\lceil n\log_2 3\rceil$ and

$$ f_d(z)=P(D_r=d,\ B_r\equiv z\pmod{3^n}),\qquad d\ge d_0. $$

Proposition 13 (finer-modulus collision and mixing reduction). Put $u=1+1/(4n)$, $G(u)=3u/(4-u)$ and $Q(u)=u/(2-u)$. Then

$$ 3^n\sum_{d\ge d_0,z}u^d f_d(z)^2 \le Q(u)^r+\frac23\sum_{j=0}^{r-1}G(u)^{r-j}Q(u)^j \le\frac{16}{9}(1+2r/3). \tag{38} $$

For $r<m\le n$, let $\delta$ bound every primitive Fourier coefficient of the Syracuse laws at depths $m-r+1,\ldots,n-r$. Then

$$ \operatorname{Osc}_{m,n} \le\sqrt{12}\,n\delta+2P(D_r<d_0). \tag{39} $$

Here oscillation is the $\ell^1$ distance from the probability vector to its average over each coset modulo $3^m$, as in [1]. If the same $\delta$ bounds the primitive transform uniformly under a twist $\exp(i\theta D_\ell)$, the joint oscillation retaining $D_n$ satisfies

$$ \operatorname{JOsc}_{m,n} \le\sqrt{48}\,n^{3/2}\delta +2P(D_r<d_0)+2P(D_n>4n). \tag{40} $$

Proof. For fixed $d,z$, let $N_d(z)$ count the prefix words. Their positive numerators are distinct and congruent modulo $3^n$, even though the words have length $r$. Hence $N_d(z)^2\le N_d(z)+2\sum C_r/3^n$. The left side of (38) is at most

$$ 3^n\sum_{w:D_r\ge d_0}u^{D_r}4^{-D_r} +2\sum_w u^{D_r}4^{-D_r}C_r \le E_Pu^{D_r}+2E_*[u^{D_r}V_r]. $$

For the first term use $3^n4^{-D_r}\le2^{-D_r}$ on the retained event. For the second use $P_*(w)=3^r4^{-D_r}$ and $V_r=C_r/3^r$. The mixed moment (25), $E_Pu^{D_r}=Q(u)^r$, and $r\le n$ give (38). In particular, at $u=1$ the unweighted collision sum is at most $1+2r/3$.

Split the full word after $r$ letters:

$$ X_n=B_r+3^r2^{-D_r}X'_{n-r}\pmod{3^n}, $$

where the suffix is independent. At a frequency with $j=v_3(\xi)<n-m$, the effective suffix depth is $n-r-j\ge m-r+1$. The retained law therefore has Fourier transform $\sum_{d\ge d_0}\widehat f_d(\xi)\chi_{n-r}(2^{-d}\xi)$, whose squared modulus is at most

$$ \delta^2\left(\sum_{d\ge d_0}u^{-d}\right) \sum_{d\ge d_0}u^d|\widehat f_d(\xi)|^2. $$

Sum over the high frequencies, apply Plancherel, and use $\sum_{d\ge d_0}u^{-d}\le4n$. Equation (38) bounds the result by $12n^2\delta^2$. The normalized $L^1$ norm is at most the $L^2$ norm. Removing the discarded prefix mass costs at most $2P(D_r<d_0)$, which proves (39).

For (40), let $g_t$ be the suffix subprobability with total exponent $t$. The Fourier transform of the joint slice of total $s$ is $\sum_d\widehat f_d(\xi)\widehat g_{s-d}(2^{-d}\xi)$. Parseval in $\theta$ gives $\sum_t|\widehat g_t(2^{-d}\xi)|^2\le\delta^2$. For a nonprimitive frequency, the discarded highest ternary digits contribute an independent sum twist of modulus at most one, so the same bound holds at the effective depth. Weighted Cauchy and (38) thus bound the sum of the squared Fourier transforms over all $s$ and high frequencies by $12n^2\delta^2$. Retain $s\le4n$, use Cauchy over at most $4n$ slices, and discard the upper tail to obtain (40). This does not bound a rare conditional slice uniformly. $\square$

For $r=\lfloor17n/20\rfloor$ and $m\ge0.9n$, the discarded lower sum tail is exponentially small and the effective suffix depths are at least $n/20$. Thus (39) and (40) recover the established high-scale mixing estimates from primitive decay with fixed polynomial losses. Projection and a geometric scale telescope give all scales; for the joint law the independent tail-sum kernel contracts $\ell^1$ and commutes with residue averaging. The twisted primitive input is supplied by the known pair-sum argument [1,2]. The point of Proposition 13 is the global collision estimate (38), not a new mixing conclusion.

9.2. One explicit subpower loss in the target

Corollary 14 (quantitative target dependence). Using [16, Theorem E.1] and the same weighted-path input (34), there are absolute $c,C>0$ such that

$$ \underline d\,\mathcal P(a) \ge c\,a^{-2}\exp\left[-C\frac{\log(3+a)}{\log\log(9+a)}\right] \qquad(3\nmid a). \tag{41} $$

The corresponding finite bound holds for every $Y\ge C_{\rm on}a$, with the same absolute onset constant as Theorem 12. This still does not give a $c/a^2$ coefficient or a source below the target.

Proof. Shaik's explicit primitive estimate [16, Theorem E.1 and (E.recipe)] gives $|\chi_n(\xi)|\le C_Bn^{-B}$ for every positive integer $B$ and primitive $\xi$. Its recipe implies $\log C_B\le\exp(C_0B)$ for an absolute $C_0$. To check this dependence, its $K,R$ are $O(B)$, while $\log A,\log H=O(B)$; the fixed $\epsilon=2^{-78}$ is an absolute constant. With $w_i=v_i+1$, its recurrence gives $w_{i+1}\le(H+3)w_i^4$, so $\log w_R\le4^R(\log w_0+\log(H+3)/3)=\exp(O(B))$. The subsequent $P,Z,M$ are polynomial in $w_R$ with coefficients of logarithm $O(B)$, and $C_B=(3M)^B$. This proves the asserted growth. We use the stated primitive theorem, whose Appendix E proof and recipe we have read; no independent formal rebuild is claimed.

Choose $B=\lfloor\log\log n/(2C_0)\rfloor$ for sufficiently large $n$. It follows that $|\chi_n(\xi)|\le\exp(-c_0\log n\log\log n)$ for some fixed $c_0>0$. This optimization is a consequence of the existing recipe. Applying (39) and telescoping scales gives constants $c_1,C_1>0$ with

$$ \sup_{n\ge m}\|\rho_n-\rho_m\|_1 \le C_1\exp(-c_1\log m\log\log m) \tag{42} $$

for sufficiently large $m$. Indeed, the polynomial loss is absorbed by the displayed rate, and the geometric scale sum is bounded by a constant times its first term. The same bound holds with $\rho_n$ replaced by $\rho$. This step only needs marginal mixing.

Let $R$ be the eligible nonperiodic root in (34). Choose an integer $L=L(R)$ making the right side of (42) at most $1/(88R)$. For large $R$ one can take

$$ L=\left\lceil\exp\left(K\frac{\log R}{\log\log R}\right)\right\rceil $$

with a sufficiently large absolute $K$: eventually $\log\log L\ge\tfrac12\log\log R$, so $K\ge8/c_1$ makes the exponential in (42) at most $R^{-4}$. Increase the fixed threshold to absorb $C_1$.

In (34), put $h=\min(k,L)$ and abbreviate $g_j=g_{j,J}$ from (35). If $k>L$, Haar invariance of every $\Psi_j$ gives $\|g_k-g_h\|_1\le(8/9)/(88R)$; if $k\le L$, the difference is zero. By (23), $\|g_h\|_2\le(8\sqrt{12}/9)L$ in either case. For the selected distinct sources $A\subset[X,32X)$, counting representatives modulo $3^k\le X$ yields

$$ \sum_{x\in A}|g_k-g_h|(x)\le33X\|g_k-g_h\|_1, \qquad \sum_{x\in A}g_h(x)\le\sqrt{33X|A|}\,\|g_h\|_2. $$

The marked mass and $\omega_w\le R/X$ now imply

$$ \frac23\le\frac RX\sum_{x\in A}g_k(x) \le\frac{8\sqrt{396}}9 RL\sqrt{|A|/X}+\frac13. $$

Thus $|A|/X\ge1/(2816R^2L^2)$, and setting $X=Y/32$ gives $\underline d\,\mathcal P(R)\ge1/(90112R^2L^2)$. The same credited sibling selection as in Theorem 12 gives $R\le C_{\rm seed}a$. Substitution proves (41) for large $a$; the fixed-$p$ bound of Theorem 12 covers bounded $a$ after reducing $c$. For the finite assertion, the same argument applies at every $Y\ge C_{\rm on}a$: the use of $h=\min(k,L)$ imposes no condition $k\ge L(R)$. Thus smoothing does not introduce an additional cutoff depending on the target. $\square$

This is an $L^1$ approximation followed by exact integer residue counting. It assumes no pointwise approximation at a selected integer. The primitive-decay constants inherited from [16] are extremely large, and (41) has the uniform finite-cutoff version just proved. Its subpower loss tends to zero in the exponent of $a$, but it supplies no universal descent mechanism.

10. Finite controls and reproducibility

The proofs above are analytic and do not use numerical fitting. A five-worker CPU diagnostic counts all exponent words by residue and total exponent sum, using integer arithmetic, through depth $12$ and sum $48$. It then evaluates information summaries in long-double arithmetic. The retained mass at depth $12$ omits $5.585\times10^{-14}$ of the tilted law. No interval certification of the floating summaries is claimed.

| Depth | Word entropy minus residue entropy | Posterior variance of $D_k$ | Unconditional variance | |---|---:|---:|---:| | 3 | 0.4922 | 0.4019 | 1.3333 | | 6 | 0.7164 | 0.5271 | 2.6667 | | 9 | 0.8397 | 0.6152 | 4.0000 | | 12 | 0.9100 | 0.6751 | 5.3333 |

These finite values suggested the coding estimate; Theorem 5 establishes the uniform conclusion independently. An independent Fraction-based enumerator through depth $5$ and sum $14$ checks numerator recovery, residue congruences, both counting bounds, the weighted quadratic-energy inequality chain (24)--(26), and six diagnostic summaries. The summaries agree with the integer-count implementation within $1.34\times10^{-15}$.

A separate five-worker exact-integer checker verifies the residue permutation, parity, height and shared-successor identities in the seed selection on 124,146 cases. The at-most-one-periodic-sibling argument is proved above; the checker does not decide hypothetical cycles.

A third exact-rational check tests the general moment inequalities and tilted mixed moments at orders $3,4,5$ through depth $5$ and sum $14$. These finite controls do not establish the asymptotic rates. A fourth checker verifies 20 finer-modulus collision cases and 7,504 exact prefix/suffix identities for Proposition 13. Independent sampled joint Fourier factorizations agree within $5.82\times10^{-18}$.

From the supplement root, run:

g++ -O3 -std=c++17 -fopenmp experiments/tilted_prefix_information.cpp -o tilted_prefix_information
OMP_NUM_THREADS=5 ./tilted_prefix_information 5 14 > results/tilted_prefix_information_small_20261020.json
python experiments/check_tilted_prefix_information.py
python experiments/check_polynomial_density_seed.py
python experiments/check_ternary_moment_spectrum.py
python experiments/check_finer_modulus_collision.py

The depth-$12$ diagnostic uses the same executable with arguments 12 48. It keeps a few hundred megabytes of working arrays in memory and writes only the small JSON summary. All supplied code is original and is released under the MIT license; result files are released under CC0. No external manuscript or credential is included in the supplement.

Appendix A. The power exponent of the limiting density tail

Proposition 15. There are $c>0$ and $t_0$ such that

$$ m_3\{\rho>t\}\ge\frac{c}{t^2\sqrt{\log t}}\qquad(t\ge t_0). \tag{43} $$

In particular,

$$ \lim_{t\to\infty}\frac{\log m_3\{\rho>t\}}{\log t}=-2. \tag{44} $$

This gives the power exponent, not a regularly varying asymptotic or a weak-$L^2$ upper bound.

Proof. Work first under $P_*$ and fix $0<\theta<1$. Put

$$ q_\theta=E_*[(2^e/3)^\theta] =\frac{3^{1-\theta}2^\theta}{4-2^\theta}<1. $$

Strict log convexity and equality at $\theta=0,1$ give the last inequality. Reversal preserves $D_k$ and gives the joint law $V_k=(1/3)\sum_{j<k}\prod_{i\le j}(2^{e_i}/3)$. Subadditivity therefore yields

$$ E_*[V_k^\theta 1_{D_k=d}] \le3^{-\theta}\sum_{j<k}q_\theta^j P_j(D_k=d), $$

where $P_j$ tilts the first $j$ letters by $(2^e/3)^\theta$. Under $P_j$ the letters are independent positive geometric variables with ratios either $1/4$ or $\beta=2^\theta/4\in(1/4,1/2)$. Every such sum has maximal atom $O(k^{-1/2})$, uniformly in $j$. Indeed, the squared modulus of a geometric characteristic function is

$$ \left(1+\frac{4\beta}{(1-\beta)^2}\sin^2(t/2)\right)^{-1} \le\left(1+\frac{16}{9}\sin^2(t/2)\right)^{-1}. $$

Fourier inversion of the product, using $\sin(|t|/2)\ge|t|/\pi$ on $[-\pi,\pi]$, bounds its largest atom by a Gaussian integral of order $k^{-1/2}$. For $|d-4k/3|\le1$, Stirling's formula applied to $P_*(D_k=d)=\binom{d-1}{k-1}3^k4^{-d}$ gives a lower bound $c_0k^{-1/2}$. Dividing the preceding moment estimate by this probability shows

$$ E_*[V_k^\theta\mid D_k=d]\le C_\theta \qquad(|d-4k/3|\le1), $$

uniformly for large $k$. Choose a constant $K$ such that the conditional probability of $V_k\le K$ is at least $1/2$. Then $P_*(D_k=d,V_k\le K)\ge c_0/(2\sqrt{k})$ on these slices.

Choose $\eta>0$ such that $E_0=\{\rho\ge\eta\}$ has positive Haar measure. Iterated stationarity and nonnegativity imply $\rho(F_w x)\ge A_w\rho(x)$ almost everywhere, where a length-$k$, sum-$d$ branch has $A_w=3^k2^{-d}$ and Haar Jacobian $3^{-k}$. For fixed $d,z$, at most $K+1$ words with $V_k\le K$ can have residue $z$, by numerator spacing. Select one for each distinct residue. The sets $F_wE_0$ so obtained lie in disjoint residue cells, each has measure $3^{-k}m_3(E_0)$, and each has density at least $\eta A_w$. Every selected-slice word has tilted probability $3^k4^{-d}$, so

$$ m_3\{\rho\ge\eta A_w\} \ge\frac{m_3(E_0)}{K+1}A_w^{-2} P_*(D_k=d,V_k\le K) \ge c_2 A_w^{-2}k^{-1/2}. $$

For large $t$, set $s=\log(2t/\eta)$, $k=\lceil s/\mu\rceil$, and $d=\lfloor(k\log3-s)/\log2\rfloor$. Then $|d-4k/3|\le1$, $2t/\eta\le A_w<4t/\eta$, and $k\asymp\log t$. This proves (43). The almost-everywhere branch inequalities can be imposed simultaneously on the countable family of finite words; no trace at an individual integer is used. Finally Markov's inequality and Theorem 8 give $m_3\{\rho>t\}\le\|\rho\|_p^p t^{-p}$ for every fixed $p<2$. Letting $p\uparrow2$ and combining with (43) proves (44). $\square$

Equation (42) gives the complementary quantitative upper estimate $m_3\{\rho>t\}\le Ct^{-2}\exp(C\log t/\log\log t)$ for large $t$: choose $L=\lceil\exp(K\log t/\log\log t)\rceil$ with $\|\rho-\rho_L\|_1\le t^{-1}$, and bound the tail by $4\|\rho_L\|_2^2/t^2+2\|\rho-\rho_L\|_1/t$. The gap between these bounds still allows an unbounded slowly varying factor. In particular, neither proposition proves the weak-$L^2$ estimate needed to remove the subpower loss in (41).

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