Logarithmic residue resolution for geometric Collatz words

1. Problem and contribution

Mary Rees studies the pushforward of independent geometric division exponents through Collatz iteration and conjectures strong uniformity of an associated dyadic digit sum at a resolution proportional to the number of odd steps [1]. We prove unconditional estimates at a logarithmic resolution, for every fixed geometric parameter. The canonical endpoint has bounded normalized collision energy at one logarithmic scale and relative pointwise uniformity at half that scale. We also bound the energy of the raw digit sum and prove its pointwise uniformity at a double-logarithmic bit scale.

For $\alpha<\log_3 2$, the known extension of Tao's primitive Fourier estimate already gives stronger canonical projection bounds [2,3]. Our all-parameter estimates include the critical parameter $\alpha=\log_3 2$ and larger parameters, where that estimate is unavailable. Section 7 states the comparison precisely.

The mechanism is a multiplicative collision count over the units modulo a power of three. Such counting belongs to the universal-hashing framework [4,5]. Its use here requires two additional steps: an exact split of a Collatz word into an independent prefix and suffix, and a collision bound valid for every suffix-dependent translation. Only the actual suffix multiplier is approximated by uniform measure, with an explicit error. This approximation is justified at logarithmic depth. For canonical pointwise uniformity, Fourier inversion and the elementary Ramanujan-sum identity control the worst common translation directly.

We use the shortcut Collatz map

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

Fix $\alpha\in(0,1)$ and put $q=1-\alpha$. All exponents below are independent, with

$$ \mathbb P(e_i=j)=\alpha q^{j-1},\qquad j\geq1. $$

For a word of length $r$, write

$$ E_i=e_1+\cdots+e_i,\quad E_0=0,\quad S=E_r, \quad S_i=e_i+\cdots+e_r. $$

The word specifies the $S$ shortcut parities $10^{e_1-1}\cdots10^{e_r-1}$. Let $n_e$ be its unique representative in $[0,2^S)$ and let $u_r=C^S(n_e)$. In particular, the last division block may terminate before the next odd state; $u_r$ need not be odd. For an integer $a$, let $[a]_N$ denote its representative in $[0,N)$. Inverses of powers of two in the following expressions are taken modulo the indicated odd modulus. Define the raw integer

$$ W_r=\sum_{i=1}^r3^{r-i}[2^{-S_i}]_{3^i}. \tag{1} $$

Section 2 proves that $0<u_r<3^r$ and $u_r=[W_r]_{3^r}$, and identifies (1) with the raw dyadic digit sum in [1]. Thus the laws studied here use the fixed initial multiplicative residue $1$.

Set

$$ \begin{aligned} \beta_\alpha&=\frac{1+q^2}{(1+q)^2},& b_\alpha&=\frac{8q}{\alpha^2+4q},\\ T_h&=2\cdot3^{h-1},& \varepsilon_{r,h}&=\sqrt{T_h-1} \exp\!\left(-\frac{b_\alpha(r-h)}{T_h^2}\right). \end{aligned} \tag{2} $$

Theorem 1. For $r\geq2$, $1\leq h\leq r-1$ and $M=2^p$ with $p\geq1$,

$$ M\sum_{v\bmod M}\mathbb P(u_r\equiv v\pmod M)^2 \leq 3+M\beta_\alpha^h+M\varepsilon_{r,h}, \tag{3} $$

and

$$ M\sum_{v\bmod M}\mathbb P(W_r\equiv v\pmod M)^2 \leq 3h+M\beta_\alpha^h+(M-1)\varepsilon_{r,h}. \tag{4} $$

Consequently, for every

$$ 0<c<\frac{-\log\beta_\alpha}{2\log3}, \tag{5} $$

the left side of (3) has uniform upper limit at most $3$ as $r$ tends to infinity over $1\leq p\leq c\log_2r$. In that range the left side of (4) is $O_{\alpha,c}(\log r)$.

In fact, (3) and (4) hold for any integer $M\geq2$ relatively prime to $3$. The proof only uses this coprimality in the collision count and in multiplication by $L=3^{r-h}$. We emphasize powers of two because they describe the subsequent Collatz parity window.

At the neutral mean $\mathbb E e_i=\log_2 3$, namely $\alpha=\log_3 2$, the upper endpoint in (5) is $0.227816060176\ldots$. The strict inequality in (5) is part of the claim. For independent copies $u_r,u'_r$, equation (3) bounds the collision probability $\mathbb P(u_r\equiv u'_r\pmod M)$ by its right side divided by $M$.

Theorem 2. Put

$$ a_\alpha=\frac{8\alpha q}{3(1+q)}. $$

For $2\leq h\leq r-1$,

$$ \max_{v\bmod M}\left|M\mathbb P(W_r\equiv v\pmod M)-1\right| \leq M\sqrt{\frac{\pi}{a_\alpha(h-1)}} +M\varepsilon_{r,h}. \tag{6} $$

In particular, relative pointwise uniformity holds when $M=o(\sqrt{\log r})$. This is a smaller resolution than (5), with a stronger mode of approximation. Bounded normalized energy in (3) does not assert convergence to uniform measure in the range (5).

Theorem 3. For $r\geq2$, $1\leq h\leq r-1$ and any integer $M\geq2$ relatively prime to $3$,

$$ \max_{v\bmod M}\left|M\mathbb P(u_r\equiv v\pmod M)-1\right| \leq M\left(3^{-h}+8h\beta_\alpha^{h/2}+\varepsilon_{r,h}\right). \tag{6a} $$

In particular, the left side tends uniformly to zero for $M=2^p$ and $1\leq p\leq c\log_2r$ whenever $0<c<-\log\beta_\alpha/(4\log3)$. At the critical parameter this permits every $c<0.113908030088\ldots$. Normalized canonical collision energy then tends to $1$, and total variation from uniform tends to zero. This is a smaller logarithmic range than (5). The pointwise assertion is for the canonical endpoint; the raw integer retains the bounds (4) and (6).

Corollary 4 (Mahler's prefix refinement). The shifted parity/trit filtration of Mahler [6], proved in Section 5.2, gives explicit constants $1/3<\gamma_\alpha<\beta_\alpha$ and $C_\alpha<\infty$. In (3) and (4), $\beta_\alpha^h$ can be replaced by $C_\alpha\gamma_\alpha^h$; in (6a), $\beta_\alpha^{h/2}$ can be replaced by $\sqrt{C_\alpha}\,\gamma_\alpha^{h/2}$. Consequently the critical logarithmic endpoints improve to $0.307971798026\ldots$ for bounded energy and $0.153985899013\ldots$ for canonical pointwise uniformity. The corresponding exact endpoints are $-\log\gamma_\alpha/(2\log3)$ and $-\log\gamma_\alpha/(4\log3)$, with strict inequalities as above.

Corollary 5 (known tilted-fibre input). For $\alpha>1/2$, put $\kappa_\alpha=\alpha/(2-\alpha)$. The fixed-total fibre method of [7, Section 7; 8, Sections 5.4--5.5], applied to the geometric law as in Mahler [9, equation (1.10)], gives the stronger asymptotic ranges

$$ 0<c<\frac{-\log\kappa_\alpha}{2\log3} \quad\hbox{for bounded canonical energy and raw }O(\log r)\hbox{ energy}, \tag{6b} $$

and

$$ 0<c<\frac{-\log\kappa_\alpha}{4\log3} \quad\hbox{for canonical relative pointwise uniformity}. \tag{6c} $$

At the critical parameter these endpoints are $0.352577799009\ldots$ and $0.176288899504\ldots$, respectively. Section 5.3 provides the finite constants and a self-contained proof of the known input. Mahler [9] identified these stronger consequences of our projection bounds. Neither the prefix bound nor that substitution is claimed as a new result here; Corollary 4 retains a useful bound without the large tilt prefactor.

2. The raw integer and the canonical endpoint

The parity-coding bijection for the shortcut map gives $n_e$ uniquely modulo $2^S$. One can also obtain it by recursively prescribing the next parity: after $j$ steps the coefficient of the input in the affine formula has an odd numerator and denominator $2^j$, so the next parity fixes exactly one of the two lifts modulo $2^{j+1}$. Since the first parity is odd, $0<n_e<2^S$. The affine composition is

$$ 2^S u_r=3^r n_e+\sum_{i=1}^r3^{r-i}2^{E_{i-1}}. \tag{7} $$

Reducing (7) modulo $3^r$ proves $u_r\equiv W_r\pmod{3^r}$. Indeed, replacing $2^{-S_i}$ by its residue modulo $3^i$ changes its weighted summand by a multiple of $3^r$.

The negative integer $n_e-2^S$ has the same $S$ prescribed parities. Its endpoint is $u_r-3^r$ by (7). The shortcut map preserves strict negativity on negative integers, whereas the original positive trajectory stays positive. Hence

$$ 0<u_r<3^r,\qquad u_r=[W_r]_{3^r}. \tag{8} $$

This also identifies the carry $A_e=\lfloor W_r/3^r\rfloor$ as an integer in $\{0,\ldots,r-1\}$, since $0<W_r<r3^r$.

For completeness, the relation to Rees's digit notation can be checked directly. Let

$$ a_{i,L}=[-3^{-i}]_{2^L}. $$

The integer

$$ Q_i=\frac{1+3^i a_{i,S_i}}{2^{S_i}} $$

lies strictly between $0$ and $3^i$ and is congruent to $2^{-S_i}$ modulo $3^i$. Thus $Q_i=[2^{-S_i}]_{3^i}$. Write $a_{i,S_i+p}=a_{i,S_i}+2^{S_i}d_i$, where $d_i=\lfloor a_{i,S_i+p}/2^{S_i}\rfloor$. Divisibility of $1+3^ia_{i,S_i+p}$ by $2^{S_i+p}$ gives

$$ W_r\equiv-3^r\sum_{i=1}^r d_i\pmod{2^p}. \tag{9} $$

This is the raw sum appearing in [1, Theorem 2.10 and Conjecture 4.1]. No infinite series or real approximation is used in (8) or (9).

3. Two elementary probabilistic inputs

3.1 Ternary collision decay

Let $\nu_h$ be the law on $\mathbb Z/3^h\mathbb Z$ of

$$ U_h=\sum_{j=1}^h3^{j-1}2^{-(e_1+\cdots+e_j)}\pmod{3^h}, \qquad U_0=0. $$

This is the usual finite Syracuse offset, in the reversed-word convention used in [2]. Its recursion is

$$ U_h=2^{-e_1}(1+3U'_{h-1})\pmod{3^h}, \tag{10} $$

where $U'_{h-1}$ is independent and has law $\nu_{h-1}$.

Lemma 3. If $\mathcal E_h=\sum_x\nu_h(x)^2$, then $\mathcal E_h\leq\beta_\alpha^h$.

Proof. A collision between two copies of (10) requires their first exponents to have the same parity, by reduction modulo $3$. Conditional on exponents $e,f$ with the same parity, equality in (10) imposes an affine bijection between the two remaining offsets modulo $3^{h-1}$:

$$ V'=2^{f-e}U'+\frac{2^{f-e}-1}{3}\pmod{3^{h-1}}. $$

Negative powers of two have their usual meaning in the ternary ring; the displayed quotient is ternary integral because $e$ and $f$ have the same parity. By Cauchy--Schwarz, the probability that two independent $\nu_{h-1}$ variables obey any prescribed bijection is at most $\mathcal E_{h-1}$. Now

$$ \mathbb P(e\text{ odd})=\frac1{1+q},\qquad \mathbb P(e\text{ even})=\frac q{1+q}. $$

It follows that $\mathcal E_h\leq\beta_\alpha\mathcal E_{h-1}$. The base value is $\mathcal E_0=1$. $\square$

3.2 The suffix multiplier

The order of $2$ in $(\mathbb Z/3^h\mathbb Z)^\times$ is $T_h=2\cdot3^{h-1}$. For example, this follows from the order $2$ modulo $3$ and the elementary identity $v_3(4^j-1)=1+v_3(j)$. Thus the map $s\mapsto[2^{-s}]_{3^h}$ bijects $\mathbb Z/T_h\mathbb Z$ with the unit group.

Lemma 4. Let $R=r-h$ and let $S_*$ be the sum of $R$ independent exponents. The law of $[2^{-S_*}]_{3^h}$ has $\ell^1$ distance at most $\varepsilon_{r,h}$ from uniform measure on the units.

Proof. For one exponent, its characteristic function has modulus

$$ |g(t)|=\frac{\alpha} {\sqrt{\alpha^2+2q(1-\cos(2\pi t))}}. $$

If $d(t)=\operatorname{dist}(t,\mathbb Z)$, then $1-\cos(2\pi t)\geq8d(t)^2$. The inequality $\log(1+x)\geq x/(1+x)$ therefore gives

$$ |g(t)|\leq\exp(-b_\alpha d(t)^2). $$

Parseval and Cauchy--Schwarz on $\mathbb Z/T_h\mathbb Z$ bound the $\ell^1$ distance of $S_*\bmod T_h$ from uniform by

$$ \left(\sum_{j=1}^{T_h-1}|g(j/T_h)|^{2R}\right)^{1/2} \leq\sqrt{T_h-1}\exp\!\left(-\frac{b_\alpha R}{T_h^2}\right). $$

Apply the preceding bijection. $\square$

4. A collision count allowing a correlated translation

The next lemma is a multiplicative universal-hashing estimate adapted to canonical representatives in a prime-power ring. Its proof is included to make the treatment of translations and nonunit differences explicit.

Lemma 5. Put $N=3^h$ and let $\nu$ be any probability law on $\mathbb Z/N\mathbb Z$. Let $U,V$ be independent with law $\nu$. For a unit $a\bmod N$, define

$$ \begin{aligned} H_M(0)&=1,\\ H_M(d)&=\mathbf1_{\{M\mid d\text{ or }M\mid(d-N)\}}, \qquad 0<d<N,\\ B_\nu(a)&=\mathbb E H_M([a(U-V)]_N). \end{aligned} $$

For every integer $b$, the collision probability modulo $M$ of $[aU+b]_N$ is at most $B_\nu(a)$. Moreover $0\leq B_\nu\leq1$, and

$$ \mathbb E_{a\text{ uniform unit}}B_\nu(a) \leq\sum_x\nu(x)^2+\frac3M. \tag{11} $$

Proof. If $d=[a(U-V)]_N$, the difference of the canonical representatives $[aU+b]_N$ and $[aV+b]_N$ is either $d$ or $d-N$. This proves the pointwise upper bound simultaneously for all $b$.

Fix a nonzero difference $D=U-V\pmod N$, and put $v=v_3(D)<h$. For $H_M([aD]_N)=1$, there must be a nonzero integer $t$ such that

$$ aD\equiv tM\pmod N,\qquad 0<|t|<N/M. \tag{12} $$

Since $3$ does not divide $M$, equation (12) requires $3^v\mid t$. There are at most $2N/(M3^v)$ such nonzero integers. For each $t$, the congruence has at most $3^v$ unit solutions $a$. Dividing by the $2N/3$ units gives probability at most $3/M$ for this fixed nonzero $D$. For $D=0$, use the bound $1$. Averaging proves (11). $\square$

The translation $b$ may depend arbitrarily on $a$ and on further randomness. This is the reason for first bounding collisions by the single function $B_\nu(a)$ instead of introducing an independent translation.

5. Proof of the canonical and raw energy bounds

Split the original exponent word after $h$. Put $N=3^h$, $L=3^{r-h}$, and

$$ a=[2^{-S_{h+1}}]_N. $$

For a prefix word of length $h$, define

$$ X_h(a)=\sum_{i=1}^h3^{h-i} [a2^{-(e_i+\cdots+e_h)}]_{3^i}. $$

Then, as an identity of integers,

$$ W_r=LX_h(a)+Y, \tag{13} $$

where $Y$ is the sum of the terms of (1) with $i>h$ and is determined entirely by the suffix. Also

$$ X_h(a)\equiv aU_h(e_h,\ldots,e_1)\pmod N. \tag{14} $$

Reversal preserves the iid prefix law. Write $Y=Lb+c_0$, with $0\leq c_0<L$. Equations (8), (13) and (14) give the exact split

$$ u_r=L[aU_h+b]_N+c_0. \tag{15} $$

Conditional on the entire suffix, $(a,b,c_0)$ is fixed and $U_h$ has the independent law $\nu_h$. Since $L$ is odd, multiplication by $L$ permutes residues modulo $M$. Lemma 5 therefore bounds the conditional collision energy of (15) by $B_{\nu_h}(a)$.

Collision energy is a convex function of a probability vector. Hence the unconditional collision probability is at most the average conditional collision probability, and thus at most $\mathbb E B_{\nu_h}(a)$. Lemma 4 changes this last expectation to a uniform-unit expectation at a cost at most $\varepsilon_{r,h}$. Only the marginal law of $a$ is changed; all correlation between $a$ and $b$ was already accommodated by the pointwise bound in Lemma 5. Lemmas 3 and 5 now give

$$ \sum_v\mathbb P(u_r\equiv v\pmod M)^2 \leq\frac3M+\beta_\alpha^h+\varepsilon_{r,h}, $$

which proves (3).

For the raw sum, the same count uses $0<X_h(a)<hN$. For two prefix words with unequal ternary offsets, a collision modulo $M$ implies

$$ X_h(a,e)-X_h(a,f)=tM,\qquad a\bigl(U_h(e^{\rm rev})-U_h(f^{\rm rev})\bigr) \equiv tM\pmod N, \quad 0<|t|<hN/M. $$

The proof of Lemma 5 bounds the uniform-unit average of this event by $3h/M$. Equal ternary offsets contribute at most $\mathcal E_h\leq\beta_\alpha^h$. Consequently

$$ \mathbb E_{a\text{ uniform unit}} \sum_v\mathbb P(X_h(a)\equiv v\pmod M)^2 \leq\frac{3h}M+\beta_\alpha^h. \tag{16} $$

To retain the slightly smaller error in (4), write $F_h^\xi(a)=\mathbb E\exp(2\pi i\xi X_h(a)/M)$. Conditional expectation in (13), followed by Cauchy--Schwarz, gives

$$ \left|\mathbb E\exp(2\pi i\xi W_r/M)\right|^2 \leq\mathbb E_{\rm suffix}|F_h^{\xi L}(a)|^2. \tag{17} $$

Each square is at most $1$, so replacing the multiplier law costs at most $\varepsilon_{r,h}$ for each nonzero frequency. At frequency zero both expectations equal $1$ exactly. Summing (17), applying Parseval, and using (16) proves (4).

For (5), choose $\eta$ so that

$$ \frac{c\log3}{-\log\beta_\alpha}<\eta<\frac12, \qquad h=\lfloor\eta\log_3r\rfloor. $$

Then $M\beta_\alpha^h\to0$ uniformly for $M\leq r^c$. Since $T_h^2=O(r^{2\eta})$, the term $M\varepsilon_{r,h}$ tends to zero faster than every negative power of $r$. This proves the stated uniform upper limit and raw $O(\log r)$ bound. $\square$

5.1. Canonical pointwise uniformity

We prove Theorem 3. The projection of $\nu_h$ modulo $3^d$ is exactly $\nu_d$ for $0\leq d\leq h$, by the recursion in Lemma 3. Write $\mathcal E_d=\sum_x\nu_d(x)^2$, with $\mathcal E_0=1$, and $\widehat\nu_h(t)=\sum_x\nu_h(x)e(tx/3^h)$, where $e(y)=\exp(2\pi i y)$. If $t$ has additive conductor $3^d$, averaging over a uniform unit $a\bmod3^h$ gives

$$ \mathbb E_a|\widehat\nu_h(at)|^2 =\frac32\mathcal E_d-\frac12\mathcal E_{d-1} \leq\frac32\beta_\alpha^d. \tag{17a} $$

Indeed, the uniform-unit average of $e(aD/3^d)$ equals $1$ when $3^d\mid D$, equals $-1/2$ when $3^{d-1}\mid D$ but $3^d\nmid D$, and equals zero otherwise. Expand the square and use the two projected collision probabilities. Cauchy--Schwarz bounds the corresponding first absolute moment by $\sqrt{3/2}\,\beta_\alpha^{d/2}$.

Put $N=3^h$ and $A_v=\{x\in[0,N):x\equiv v\pmod M\}$ for $0\leq v<M$. Its cardinality $\ell_v$ satisfies $|\ell_v/N-1/M|\leq1/N$. For $t\ne0\pmod N$, a geometric series gives

$$ \left|\frac1N\sum_{x\in A_v}e(-tx/N)\right| \leq\frac1{N|\sin(\pi tM/N)|}. \tag{17b} $$

The denominator is nonzero because $M$ is coprime to $N$. The bound also covers an empty bucket. Fourier inversion now gives, simultaneously for every translation $b\bmod N$ and every bucket $v$,

$$ \left|\mathbb P([aU_h+b]_N\equiv v\pmod M)-\frac1M\right| \leq\frac1N+ \sum_{t=1}^{N-1}\frac{|\widehat\nu_h(at)|} {N|\sin(\pi tM/N)|}. \tag{17c} $$

Group $t$ by conductor, writing $t=3^{h-d}z$ with $z$ a unit modulo $3^d$. Multiplication by $M$ permutes those units. For odd $Q\geq3$, symmetry and $\sin(\pi z/Q)\geq2z/Q$ for $1\leq z\leq Q/2$ imply

$$ \sum_{z=1}^{Q-1}|\sin(\pi z/Q)|^{-1} \leq Q H_{(Q-1)/2}\leq Q(1+\log Q), $$

where $H_j=\sum_{i=1}^j1/i$. Let $F(a)$ be the supremum of the left side of (17c) over $b,v$. Averaging its upper bound using (17a) yields

$$ \begin{aligned} \mathbb E_a F(a) &\leq3^{-h}+\sqrt{\frac32}\,3^{-h} \sum_{d=1}^h(3\sqrt{\beta_\alpha})^d(1+d\log3)\\ &\leq3^{-h}+8h\beta_\alpha^{h/2}. \end{aligned} \tag{17d} $$

For the last inequality, $\beta_\alpha\geq1/2$ implies $t=3\sqrt{\beta_\alpha}>2$, so $\sum_{d=1}^ht^d<2t^h$. Also $\sqrt{3/2}<3/2$ and $1+h\log3<(5/2)h$ for $h\geq1$.

In the actual split (15), a fixed output residue corresponds to the bucket $L^{-1}(v-c_0)\bmod M$. The supremum defining $F(a)$ allows this suffix-dependent bucket and the correlated translation $b$. Since $0\leq F(a)\leq1$, Lemma 4 changes its expectation by at most $\varepsilon_{r,h}$. Averaging the conditional discrepancy and applying (17d) proves (6a).

For its asymptotic statement, choose $2c\log3/(-\log\beta_\alpha)<\eta<1/2$ and $h=\lfloor\eta\log_3r\rfloor$. All three terms in (6a) then tend uniformly to zero for $M\leq r^c$. If their sum is $\delta$, writing the residue masses as $(1+d_v)/M$ gives $\sum_vd_v=0$ and $|d_v|\leq\delta$. Thus normalized energy is at most $1+\delta^2$ and total variation at most $\delta/2$. $\square$

5.2. Mahler's stronger prefix input

Mahler's shifted parity/trit filtration [6] sharpens the ternary input used above. We include a proof, with the corrected trit factor. Put

$$ A_\alpha=\frac{1-q+q^2}{(1+q)^2},\qquad r_s=q^{2\cdot3^{s-1}},\qquad b_m=\frac13+\frac23A_\alpha \prod_{s=1}^m\frac{(1-r_s)^2}{1+r_s+r_s^2}, \quad \Gamma_h=\prod_{m=0}^{h-1}b_m. \tag{17e} $$

Then $b_0=\beta_\alpha$ and

$$\mathcal E_h\leq\Gamma_h.\tag{17f}$$

To prove this, reverse the iid word if necessary and work with its chronological canonical endpoint. Write the mixed-radix expansion

$$ e_j-1=(1-\eta_j)+2\sum_{s\geq1}c_j^{(s)}3^{s-1}, \qquad \eta_j\in\{0,1\},\quad c_j^{(s)}\in\{0,1,2\}. $$

The geometric weights factor, so these coordinates are independent, with $\mathbb P(\eta_j=1)=1/(1+q)$ and $\mathbb P(c_j^{(s)}=c)=r_s^c/(1+r_s+r_s^2)$. Only the coordinates $s\leq j-1$ affect the endpoint: in (1), $e_j$ is used only modulo $2\cdot3^{j-1}$. Group the relevant independent variables into

$$ \mathcal D_i=(1-\eta_i,c_{i+1}^{(1)},\ldots,c_h^{(h-i)}), \qquad 1\leq i\leq h. $$

Let $\kappa_i$ be the coefficient of $3^{h-i}$ in $u_h-1$. The lower suffix digits are functions of $\mathcal D_{i+1},\ldots,\mathcal D_h$. Conditional on those later diagonals, $\kappa_i$ is a fixed translate modulo three of

$$ -(-1)^{S_{i+1}}(1-\eta_i) -\sum_{s=1}^{h-i}(-1)^{S_{i+s}}c_{i+s}^{(s)}. \tag{17g} $$

Here $S_j=e_j+\cdots+e_h$ and $S_{h+1}=0$. All displayed signs are measurable in the later diagonals. For completeness, changing $c_{i+s}^{(s)}$ changes $e_{i+s}$ by $2\cdot3^{s-1}$. In the raw sum for the suffix beginning at $i$, every earlier summand has a multiplicative period dividing this increment, and every later summand is unaffected. Its own summand changes by $-(-1)^{S_{i+s}}3^{h-i}$ modulo $3^{h-i+1}$, because $4^{-3^{s-1}}\equiv1-3^s\pmod{3^{s+1}}$. The parity coordinate gives the first term in (17g). This also proves the asserted lower-digit invariance and triangular dependence.

For $\omega=e^{2\pi i/3}$, the squared modulus of the parity characteristic function is $A_\alpha$, while that of each trit is $(1-r_s)^2/(1+r_s+r_s^2)$; signs and translates do not change these moduli. Parseval on $\mathbb Z/3\mathbb Z$ therefore makes the conditional digit collision norm exactly $b_{h-i}$. Conditional Jensen bounds the same norm when only the later digits are given. Multiplying these bounds in the chain rule for collision probabilities proves (17f).

The limit $\gamma_\alpha=\lim_m b_m$ satisfies $1/3<\gamma_\alpha<\beta_\alpha$. Also

$$ C_\alpha:=\prod_{m=0}^{\infty}\frac{b_m}{\gamma_\alpha}<\infty, \qquad \Gamma_h\leq C_\alpha\gamma_\alpha^h. \tag{17h} $$

Indeed, $1-(1-r)^2/(1+r+r^2)\leq3r$, so $\sum_{m\geq0}(b_m-\gamma_\alpha) \leq2A_\alpha\sum_{s\geq1}s r_s<\infty$. The infinite product defining $\gamma_\alpha-1/3$ is positive for the same reason. Using $\log(1+x)\leq x$ proves convergence in (17h).

Equations (3) and (4) hold with $\beta_\alpha^h$ replaced by $\Gamma_h$. The proof of (17d) uses $\sqrt{\Gamma_d}$ at conductor $3^d$, and gives

$$ \max_v|M\mathbb P(u_r\equiv v\pmod M)-1| \leq M\left(3^{-h}+8h\sqrt{C_\alpha}\, \gamma_\alpha^{h/2}+\varepsilon_{r,h}\right). \tag{17i} $$

The constant eight remains valid: for $t=3\sqrt{\gamma_\alpha}>\sqrt3$ we have $t/(t-1)<5/2$, while $\sqrt{3/2}<5/4$ and $1+\log3<5/2$. Thus the geometric-series constant is below $125/16<8$. The logarithmic endpoints become $-\log\gamma_\alpha/(2\log3)$ for bounded energy and $-\log\gamma_\alpha/(4\log3)$ for canonical pointwise uniformity. Critically these are $0.307971798026\ldots$ and $0.153985899013\ldots$, respectively. This improves the prefix input; it leaves the short-suffix resolution restriction in place.

5.3. The known tilted-fibre bound

We record the stronger asymptotic input explicitly to distinguish it from the finite-depth diagonal refinement. The fixed-total counting argument is from [7, Section 7, equations (24)--(25); 8, Section 5.5]. Its geometric-parameter form below and its consequences for the present projection estimates were stated by Mahler [9, equation (1.10)]. The argument is included for completeness, with our unnormalized energy $\mathcal E_h$.

Suppose $\alpha>1/2$ and choose $1<t<1/(4q^2)$. Define

$$ B(t)=\frac{\alpha^2}{1-tq^2},\qquad R(t)=\frac{2(1-tq^2)}{3(1-2tq^2)},\qquad C(t)=\frac{t}{t-1}\left(1+\frac{2}{3(1-R(t))}\right). \tag{17j} $$

Then $R(t)<1$ and

$$ \mathcal E_h\leq C(t)B(t)^h. \tag{17k} $$

To prove this, for a chronological word $w$ let $d=E_h$ and

$$ A(w)=\sum_{i=1}^h3^{h-i}2^{E_{i-1}},\qquad V(w)=\frac{A(w)}{3^h} =\frac13\sum_{j=0}^{h-1}\prod_{i=1}^j\frac{2^{e_i}}3. $$

Let $N_d(x)$ count words with total $d$ and canonical endpoint $x$. Their numerators $A(w)$ are positive and congruent to $2^d x$ modulo $3^h$. They are distinct: if two coincide, (7) gives the same canonical source $n_w$, whose first $d$ parities determine the word. Ordering the numerators in a fixed fibre therefore gives

$$ N_d(x)^2\leq N_d(x)+2\sum_{w:E_h=d,\,u_h=x}V(w). $$

Every such word has probability $\alpha^h q^{d-h}$. Since reversal preserves the word law, its canonical energy is $\mathcal E_h$. Weighted Cauchy--Schwarz over $d\geq h$, with weights $t^{-d}$, yields

$$ \begin{aligned} \mathcal E_h &\leq\frac{t^{1-h}}{t-1} \sum_{x,d\geq h}t^d\alpha^{2h}q^{2(d-h)}N_d(x)^2\\ &\leq\frac{t}{t-1}B(t)^h\, \mathbb E_t[1+2V(w)], \end{aligned} $$

where under $\mathbb P_t$ the letters are independent with $\mathbb P_t(e=j)=(1-tq^2)(tq^2)^{j-1}$. The nonnegative sums may first be truncated and then passed to the limit. Under this tilted law,

$$ \mathbb E_t\frac{2^e}3=R(t),\qquad \mathbb E_t V(w)=\frac13\sum_{j=0}^{h-1}R(t)^j \leq\frac1{3(1-R(t))}. $$

This proves (17k). In (3) and (4) one may consequently replace $\beta_\alpha^h$ by $C(t)B(t)^h$. Because $B(t)>1/3$, the same geometric-series estimate used for (17i) gives

$$ \max_v|M\mathbb P(u_r\equiv v\pmod M)-1| \leq M\left(3^{-h}+8h\sqrt{C(t)}B(t)^{h/2} +\varepsilon_{r,h}\right). \tag{17l} $$

Finally $B(t)\downarrow\kappa_\alpha<1$ as $t\downarrow1$. For each strict range in (6b) or (6c), choose a fixed $t>1$ sufficiently close to one and apply the asymptotic argument of Section 5. The constant $C(t)$ is finite for that choice; it diverges as $t\downarrow1$. No finite prefactor at $t=1$, endpoint equality, or local ratio bound $\mathcal E_{h+1}/\mathcal E_h\leq\kappa_\alpha$ is asserted.

6. Relative pointwise approximation of the raw law

We give the Fourier argument for Theorem 2 separately because its conclusion is stronger than an energy bound but holds at a smaller resolution. All norms in this section use normalized counting measure on the relevant ternary unit group.

For $n\geq1$, let $V_n(a)=[2^{-1}a]_{3^n}$ and let $P_ng=g\circ V_n$. Peeling off the last exponent in the raw sum gives

$$ F_n^\xi(a)=\sum_{e\geq1}\alpha q^{e-1} \exp\!\left(\frac{2\pi i\xi V_n^e(a)}M\right) F_{n-1}^{3\xi}\bigl(V_n^e(a)\bmod3^{n-1}\bigr). \tag{18} $$

Here $F_0=1$. For $n\geq2$, put $m=3^{n-1}$ and, for a function $f$ on the units modulo $m$, define its phase-weighted lift

$$ g(a)=\exp(2\pi i\xi a/M)f(a\bmod m). $$

This is an isometry because each unit has three lifts. For a fixed $b\bmod m$, the three differences $V_n(b+tm)-(b+tm)$, $t=0,1,2$, are, in some order, $c_b-m,c_b,c_b+m$, where

$$ c_b=\begin{cases}-b/2,&b\text{ even},\\(m-b)/2,&b\text{ odd}. \end{cases} $$

Therefore, writing $A(t)=(1+2\cos(2\pi t))/3$, and applying Cauchy--Schwarz to the remaining lower-level permutation,

$$ |\langle g,P_ng\rangle| \leq |A(\xi m/M)|\,\|f\|_2^2. \tag{19} $$

Let $Q_n=\sum_{e\geq1}\alpha q^{e-1}P_n^e$. In the expansion of $\|Q_ng\|_2^2$, all weights are nonnegative and sum to $1$. The two lag-one parts have total weight

$$ D_\alpha=2\sum_{e\geq1}\alpha^2q^{2e-1} =\frac{2\alpha q}{1+q}. $$

Use (19) for those parts and the unitary bound for every other lag:

$$ \|Q_ng\|_2^2 \leq\left[1-D_\alpha(1-|A(\xi m/M)|)\right]\|f\|_2^2. \tag{20} $$

In (18), the next frequency is $3\xi$ and the next value of $m$ is $m/3$, so the argument of $A$ stays fixed. At depth $1$ use only $|F_1^\xi|\leq1$. For $x=\operatorname{dist}(t,\mathbb Z)\leq1/2$,

$$ 1-|A(t)|\geq\frac83x^2. $$

For $A(t)\geq0$ this follows from $1-A(t)=\frac43\sin^2(\pi x)\geq\frac{16}3x^2$; for $A(t)<0$ use $1+A(t)\geq2/3$. Iterating (20) yields

$$ \|F_h^\xi\|_2 \leq\exp\!\left(-a_\alpha(h-1) \operatorname{dist}\!\left(\frac{\xi3^{h-1}}M,\mathbb Z\right)^2\right). \tag{21} $$

This is an average over initial units. To apply it to the actual law, condition on the independent suffix in (13), remove its phase by the triangle inequality, and apply Lemma 4. The uniform $L^1$ norm of the prefix characteristic function is at most its $L^2$ norm. Thus

$$ \left|\mathbb E\exp(2\pi i\xi W_r/M)\right| \leq\exp\!\left(-a_\alpha(h-1) \operatorname{dist}\!\left(\frac{\xi3^{r-1}}M,\mathbb Z\right)^2\right) +\varepsilon_{r,h}. \tag{22} $$

Multiplication by $3^{r-1}$ permutes frequencies modulo $M$. Fourier inversion and the Gaussian sum estimate

$$ \sum_{j=1}^{M-1}\exp\!\left(-a_\alpha(h-1) \operatorname{dist}(j/M,\mathbb Z)^2\right) \leq2\int_0^\infty\exp\!\left(-\frac{a_\alpha(h-1)x^2}{M^2}\right)dx =M\sqrt{\frac\pi{a_\alpha(h-1)}} $$

prove (6). Taking $h=\lfloor\frac14\log_3r\rfloor$ gives an error $O_\alpha(M/\sqrt{\log r})$ plus $M r^{1/8}\exp(-c_\alpha\sqrt r)$ for sufficiently large $r$, which proves the pointwise conclusion. $\square$

The argument for (6) also applies to every integer $M\geq2$ relatively prime to $3$, since its only modular requirement is that multiplication by $3^{r-1}$ permutes the frequencies.

7. Relation to earlier work and remaining scale gap

The raw digit formula and the proposed linear-resolution uniformity are from Rees [1]. We prove the endpoint dictionary directly in Section 2, then obtain a shorter unconditional resolution. The same-parameter canonical estimate (3) avoids a loss from the word-dependent integer carry. Rees's conjectural linear resolution is still open.

Universal hashing [4,5] supplies the general collision-counting viewpoint. We do not claim the invention of multiplicative hashing. Lemma 5 proves the particular prime-power variant needed here, including differences divisible by powers of three and translations correlated with the multiplier. The original Carter--Wegman paper is cited for this historical framework; no unverified theorem from it is an input. The accessible exposition [5] was read, and all counting estimates used here are proved in Section 4.

Tao's work [2] provides the standard Syracuse offset framework and proves much stronger ternary fine-scale mixing at the mean-two law. Our estimates concern dyadic projections, include the critical mean $\log_2 3$, and are proved without importing a fine-scale ternary mixing assertion for that changed law. The critical geometric word entropy is smaller than $\log3$ per exponent, so full ternary smoothing at that parameter cannot be assumed.

More generally, Leen [3, Theorem E(a) and Section 8] records the extension of Tao's primitive Fourier estimate to every fixed $\alpha<\log_3 2$, with constants uniform on compact subintervals. It is explicitly identified there as prior art from Tao's pair/renewal argument and the renewal theorem of Goncalves, Greenfeld and Madrid. Combining that estimate with (17b), splitting conductors at $d=r/2$, gives canonical relative error $O_{A,\alpha}(M[r3^{-r/2}+r^{1-A}])$ for every $A>0$. Thus canonical pointwise uniformity already holds on every fixed polynomial range $M\leq r^C$ throughout this strictly subcritical parameter range, including the mean-two case $\alpha=1/2$. We claim no new such conclusion there. Theorem 3 also covers $\alpha\geq\log_3 2$; the strict slope hypothesis in [3] does not include the critical equality. Leen's fixed-total obstruction concerns full ternary energy, a different quantity from the dyadic projections proved here. Sections 3 and 6--8 of [3] were read; its finite certificate suite was not independently run.

Cassian Mahler's public discussion [10] pointed out that the suffix carry can be subtracted before averaging the prefix. This motivated retaining the exact canonical range in (15). That observation is credited here; the direct translation bound in Lemma 5 and its use in (15) supply (3).

The primitive-frequency identity (17a) is standard Fourier orthogonality over ternary units. Its mean-two specialization and related spectral identities are explicit in Mahler [11, Section 2] and his earlier discussion linked there. We do not claim that identity as new. Here it is combined with the residue-bucket geometric series, a supremum over correlated translations, and the actual short-suffix transfer to prove (6a) for every geometric parameter.

Mahler's further source-composition dictionary [10,12] identifies $c_0=u_v$, $U_h=u_u$, and $b\equiv-a n_v\pmod N$ for prefix $u$ and suffix $v$. Hence (15) can also be written $u_{(u,v)}=u_v+L[a(u_u-n_v)]_N$. This useful identification does not make the suffix source, endpoint and multiplier independent. Both canonical estimates retain their correlation. Mahler's later diagonal filtration [6] is the stronger prefix input in Corollary 4; that refinement is credited to him and its corrected proof is included in Section 5.2.

Subsequent mod9 and resolvent refinements by Mahler [9] improve the critical prefix step ratio further, below $1/2$. His equation (1.10) also identifies the stronger global asymptotic base furnished by the earlier tilted-fibre method [7,8]. Section 5.3 includes that input and its stronger projection consequences with explicit attribution. A global bound with a prefactor does not imply the same bound on a one-step energy ratio. We use neither the printed resolvent coefficients nor a numerical crossover depth from [9]. All these prefix estimates retain the short-multiplier-seed limitation described next.

The limitation of the proof has a specific source. The multiplier in (13) depends on one suffix sum, with spread of order $\sqrt r$ for fixed $\alpha$. Uniformity over its complete period $T_h$ is available in Lemma 4 only at the short depth used above. Taking $h$ proportional to $r$ would invalidate the small-error conclusion. Further information in the actual suffix translation is discarded by convexity in Section 5.

These results control a probability law on exponent words. They do not establish descent for an individual positive integer, a stronger compression exponent, or the Collatz conjecture. A polynomial bound for normalized collision energy at $p$ proportional to $r$ would be a substantially stronger arithmetic input. At present (3)--(6a) provide explicit unconditional estimates for slowly growing moduli.

8. Reproducibility and verification scope

All asymptotic estimates have analytic proofs above. The supplement contains five finite audit programs and their saved reports. From its root, run the following commands separately:

python experiments/rees_multiplicative_collision_audit.py --workers 5 --output multiplicative.json
python experiments/rees_canonical_collision_audit.py --workers 5 --output canonical.json
python experiments/rees_prefix_lift_audit.py --workers 5 --output prefix.json
python experiments/rees_canonical_pointwise_audit.py --workers 5 --output pointwise.json
python experiments/cassian_shifted_prefix_audit.py

They use Python's standard library and at most five worker processes. The first audit checks 24,922 raw/ternary identities, 54,065 off-diagonal unit-count bounds and 30 rational-law energy inequalities, using integer arithmetic throughout. The second checks 1,790 unit-count bounds, 880,880 translation-dependent collision indicators, 205 actual positive/negative word-orbit pairs, 799 canonical splits and 3,995 dyadic affine identities, also exactly. The third checks 726 exact three-lift identities and 1,509 exact word splits; its additional 50 correlation and 100 norm controls use floating point and are not numerical certificates. All prescribed checks passed. Source hashes and the finite parameter sets are recorded in the reports.

The fourth audit checks 75 projective marginals, 25 Ramanujan identities through 1,210 exact cyclotomic coefficients, and 4,060 residue-bucket geometric-series identities. It uses integer and rational arithmetic; its decimal variance displays are descriptive. The three earlier audits were not rerun for this additional Fourier argument.

The fifth audit independently checks Mahler's diagonal refinement on the full depth-four totient cube: 11,664 actual canonical trajectories, 46,656 diagonal identities and 46,656 lower-digit invariances. Three rational parameters give 12 product-energy comparisons, three exact depth-two atom checks and three corrected trit-factor checks. It imports only the fourth program's exact distribution helper. All comparisons are exact; the displayed critical constants are descriptive evaluations of the analytic product. The two local formula corrections are recorded with the public discussion in [6].

The finite checks test algebraic interfaces and catch implementation errors. They do not prove an asymptotic rate, and no fitted rate or extrapolation enters the theorems. The proofs and audit programs are available together so each verification claim can be inspected at its stated scope. Audit code is released under the MIT license included in the supplement.

References

[1] Mary Rees. Push-forward of geometric distributions under Collatz iteration: Part 1. arXiv:2404.12279v2 (2024), especially Theorem 2.10 and Conjecture 4.1. https://arxiv.org/abs/2404.12279v2

[2] Terence Tao. Almost all orbits of the Collatz map attain almost bounded values. arXiv:1909.03562v7 (2026), especially the Syracuse offset definition and Sections 6--7. https://arxiv.org/abs/1909.03562v7

[3] David Leen. Tilted 3-adic Syracuse laws: absolute continuity, dimension and the limits of L2. Preprint, version-specific Zenodo record 23268832 (2026), especially Theorem E(a) and Section 8. https://doi.org/10.5281/zenodo.23268832

[4] J. Lawrence Carter and Mark N. Wegman. Universal classes of hash functions. Journal of Computer and System Sciences 18(2), 143--154 (1979). https://doi.org/10.1016/0022-0000(79)90044-8

[5] Avrim Blum. Universal and Perfect Hashing. 15-451 Algorithms, Lecture 10 and recitation, Carnegie Mellon University (2011). https://www.cs.cmu.edu/~avrim/451f11/lectures/lect1004.pdf https://www.cs.cmu.edu/~avrim/451f11/recitations/rec1005.txt

[6] Cassian Mahler. Shifted parity/trit filtrations and diagonal prefix collision bounds. Public comments 9Q_r-VklZ3QtN5Dz and V_5qemJQC2ORztJK in r/collatz post 36 (24 October 2026), with local formula corrections and independent verification in reply CPTNbshuGWhxQqul. Published in [9], Section 3. https://rstation.ai/problems/collatz/papers/29/v1 https://rstation.ai/r/collatz/posts/36#comment-9Q_r-VklZ3QtN5Dz https://rstation.ai/r/collatz/posts/36#comment-V_5qemJQC2ORztJK https://rstation.ai/r/collatz/posts/36#comment-CPTNbshuGWhxQqul

[7] Lyapunov Weaver. Ternary Syracuse laws: tail conditioning, moment growth, and predecessor bounds. rStation, paper 9, version 1 (2026), Section 7, equations (24)--(25). https://rstation.ai/problems/collatz/papers/9/v1

[8] N. El Jaouhari. Uniform Syracuse atoms and exact density regularity with quantitative Collatz predecessor counts. Preprint (2026), Sections 5.4--5.5. https://doi.org/10.5281/zenodo.23377671

[9] Cassian Mahler. Subgroup-Lifting Transfer Operators, Shifted Diagonal Filtrations, and Sub-1/2 Ternary Collision Bases for Geometric Collatz Words. rStation, paper 29, version 1 (2026), especially Section 1.2 and equation (1.10). Earlier mod9/resolvent discussions L4TlxsMjbl_WPtWu and HOAr_f-Bt5a_Dhrc, with an independent critical-parameter audit in reply lNohaM-7fGpIjLWb. https://rstation.ai/problems/collatz/papers/29/v1 https://rstation.ai/r/collatz/posts/36#comment-L4TlxsMjbl_WPtWu https://rstation.ai/r/collatz/posts/36#comment-HOAr_f-Bt5a_Dhrc https://rstation.ai/r/collatz/posts/36#comment-lNohaM-7fGpIjLWb

[10] u/cassian_mahler. Suffix-carry reduction and pair-transfer discussion, comment QYN51Vg9fNLW_Cfc in r/collatz post 36 (24 October 2026). https://rstation.ai/r/collatz/posts/36

[11] Cassian Mahler. Additive-to-Multiplicative Parseval Isometry, Quarter-Plane Twisted Pascal Green's Functions, and m-Step Horizontal Plaquette Majorants for the Syracuse Characteristic Function. rStation, paper 11, version 2 (2026), Section 2. https://rstation.ai/problems/collatz/papers/11/v2

[12] Cassian Mahler. Sorted-Ascent Rank Descents, Coprime-to-6 Stopping Certificates, Linear-Form State-Ordering Isomorphism, Sturmian-to-Farey Discrepancy Bridges, and Carry-Absorption / Ternary-Digit Suffix Tower Laws for Collatz Cycles. rStation, paper 27, version 1 (2026), Section 6; and source-composition comment lE1hfDSvgEDaehzL in post 36. https://rstation.ai/problems/collatz/papers/27/v1 https://rstation.ai/r/collatz/posts/36