Direct signed moments and explicit concentration bands for Collatz words

1. Contribution and relation to earlier work

For a critically tilted geometric Collatz word of length $s$, how much precision can one retain while obtaining nearly unit-dimensional upper concentration bounds for its actual source and endpoint? The accepted paper [1, Theorems C--E] proves that every dimension $d<1$ is available on some positive linear precision band. Its proof passes through a Rényi order close to one, and its resulting band is existential. The present paper supplies an explicit dimension--precision relation by a different argument: conditional signed-numerator atoms, direct even moments, and Fourier Hölder.

Write

$$ \alpha=\log_3 2,\qquad \rho=\log_2 3=1/\alpha,\qquad \theta=H_2(\alpha), \quad H_2(x)=-x\log_2x-(1-x)\log_2(1-x). \tag{1} $$

At joint precision $P=p+\rho h$, every $0<d<1$ is available on every fixed band $c_0s\le P\le vs$ with

$$ 0<c_0<v<c_d,\qquad c_d=\frac{\min(1-d,1/2)^2}{512}. \tag{2} $$

For $d=24/25$, a separate fourth-moment choice permits the larger upper slope $1/6250$. These bounds apply to joint modular atoms and to joint real intervals, under the original word law and separately at each total in any prescribed $o(s)$ window about $\rho s$. They retain the actual source and endpoint coefficients. Passing to canonical coordinates costs one common carry, hence at most a factor $s$.

The source interval estimate also gives an explicit gain $\gamma=5\cdot10^{-7}$ in uniform labelled interval compression and in the number of original sources that fail to halve in a specified finite time. The accepted paper [1] already proves existential power improvements and an exact final-count reduction. We use that reduction with the explicit concentration band and give a direct final-count recursion for compression. We do not claim a first power saving, a best exponent, or an effective onset.

The new finite input is Lemma 1 below. Its parity-triangular mechanism is related to the one-word coding in [2, Lemma 11.1], but it estimates a signed sum of independently conditioned words, with the conditioning cost displayed. This permits even moments without amplification through an unspecified Rényi order. The geometric-word coding is classical in this setting [3, Lemma 3.5]. The shared source/endpoint carry is due to Mahler [4, equation (6.14)], and the two arithmetic separation statements are imported from [1, equations (J11), (R6)]. Their ineffective threshold comes from the integral fixed-place Subspace Theorem [5, Theorem 2.3]. The original-source counting step uses the usual finite parity bijection [6] in exactly the form of [1, equations (R18)--(R20)].

Wren [7] has already used that unconditional real-source law and its final-count reweighting to improve fixed-barrier density rates. His rate gain depends on the unspecified concentration band. Theorem 1 makes this input quantitative, while Theorem 2 quantifies compression for translated intervals with a common clock and odd-step labels.

Thus the contribution is an explicit band theorem and its quantitative consumers, rather than new arithmetic separation or new parity coding. All modular and real specializations are proved together. The argument supplies upper concentration, not a lower occupancy bound at a prescribed endpoint or a descent guarantee for each positive integer. Collatz remains unresolved.

2. Word laws, statements, and imported arithmetic inputs

2.1. Raw and canonical coordinates

Let $e_1,\ldots,e_s$ be independent positive geometric variables with parameter $\alpha$: $\Pr(e_i=t)=\alpha(1-\alpha)^{t-1}$ for $t\ge1$. Set $E_j=\sum_{i\le j}e_i$, $E_0=0$, and $S=E_s$. For a coprime inverse, brackets denote the least nonnegative residue in the indicated modulus. Define

$$ \begin{aligned} B_s&=\sum_{i=1}^s3^{s-i}2^{E_{i-1}},\\ W_s&=\sum_{i=1}^s3^{s-i}[2^{-(S-E_{i-1})}]_{3^i},\\ Z_s&=\sum_{i=1}^s2^{E_{i-1}}[-3^{-i}]_{2^{S-E_{i-1}}},\\ u_s&=[W_s]_{3^s},\qquad n_s=[Z_s]_{2^S}. \end{aligned} \tag{3} $$

The pair $(n_s,u_s)$ is the canonical positive source and endpoint for the prescribed division word [3]. The last division block can be truncated, so its endpoint need not be odd. As in [1, (J7)--(J8)] and [4, equation (6.14)],

$$ \begin{gathered} 2^SW_s-3^sZ_s=B_s,\\ a_s=\lfloor W_s/3^s\rfloor=\lfloor Z_s/2^S\rfloor, \quad 0\le a_s<s,\\ (n_s,u_s)=(Z_s-a_s2^S,W_s-a_s3^s). \end{gathered} \tag{4} $$

For clarity, the raw duality follows term by term: if $A=[2^{-D}]_{3^i}$, then $(2^DA-1)/3^i$ lies in $[0,2^D)$ and is $[-3^{-i}]_{2^D}$. The equality of the two carries additionally uses the positive canonical source identity from [3], as detailed in [1, Section 7.10.2]. It is one carry, with no independence assumption.

For integers $p,h\ge0$, set

$$ \mathcal G_{h,p}=(\mathbb Z/3^h\mathbb Z)\times (\mathbb Z/2^p\mathbb Z),\qquad P=p+\rho h. \tag{5} $$

The modular vector is $M_s=(Z_s\bmod3^h,W_s\bmod2^p)$, with canonical version $\bar M_s=(n_s\bmod3^h,u_s\bmod2^p)$. The real vectors are

$$ \begin{aligned} V_s^{\rm end}&=(Z_s\bmod3^h,W_s/3^s),& \bar V_s^{\rm end}&=(n_s\bmod3^h,u_s/3^s),\\ V_s^{\rm src}&=(Z_s\bmod3^h,Z_s/2^S),& \bar V_s^{\rm src}&=(n_s\bmod3^h,n_s/2^S). \end{aligned} \tag{6} $$

Theorem 1 (explicit concentration bands). Fix $0<d<1$ and $0<c_0<v<c_d$, with $c_d$ as in (2). There are constants $C,s_0$ such that for all $s\ge s_0$ and $c_0s\le P\le vs$,

$$ \begin{aligned} \max_x\Pr(M_s=x)&\le C2^{-dP},\\ \sup_{a,t}\Pr\{V_s^\sigma\in\{a\}\times[t,t+2^{-p})\} &\le C2^{-dP}\qquad(\sigma\in\{\rm src,\rm end\}). \end{aligned} \tag{7} $$

Each canonical vector satisfies the corresponding bound $Cs2^{-dP}$. For every prescribed $G(s)=o(s)$, all these bounds also hold separately under the law conditional on $S=T$, uniformly over integer totals $|T-\rho s|\le G(s)$. For $d=24/25$ the value $c_d=1/6250$ is also admissible. The constants and onset may depend on the fixed parameters and on $G$, and are ineffective. The cases $h=0$ and $p=0$ are included; the lower band excludes $P=0$.

The intervals in (7) concern finite atomic real laws. There is no claim of a density, unit dimension on one fixed band, or a positive atom lower bound. The wider bands for fixed Rényi orders in [1, Theorems B--E] are different assertions and are not replaced here.

2.2. Actual chunk contributions

For a chunk in positions $k+1,\ldots,k+\ell$, put $U=E_k$, $m=S-U$, $n=k+\ell$, and $F_j=E_{k+j}-E_k$. Its total is $T=F_\ell$. Define

$$ \begin{aligned} R_Z&=\sum_{j=1}^{\ell}2^{F_{j-1}} [-3^{-(k+j)}]_{2^{m-F_{j-1}}},\\ R_W&=\sum_{j=1}^{\ell}3^{\ell-j} [2^{-(m-F_{j-1})}]_{3^{k+j}},\\ B&=\sum_{j=1}^{\ell}3^{\ell-j}2^{F_{j-1}}. \end{aligned} \tag{8} $$

The termwise identity used in (4) yields the exact relation

$$ 2^mR_W-3^nR_Z=B,\qquad 0<R_Z<\ell2^m,\quad 0<R_W<\ell3^n. \tag{9} $$

The actual modular contribution is $(2^UR_Z\bmod3^h,3^{s-n}R_W\bmod2^p)$. Its real counterpart is $(2^UR_Z\bmod3^h,R_W/3^n)$ or $(2^UR_Z\bmod3^h,R_Z/2^m)$, respectively. In particular, normalization removes the exterior multiplier exactly in the real coordinate.

For disjoint selected chunks, condition on their totals and on every letter outside them. We call this information the environment. It fixes $S$, $U,m,n$ for every chunk, and all exterior contributions. The internal words are independent uniform positive compositions of their respective totals. The actual contributions therefore add independently, with all coefficients retained. No independence is asserted after mixing over environments.

2.3. Separation inputs, with their full hypotheses

We use the following two results from [1]. They are stated explicitly because their size hypotheses are needed in the moment argument.

Modular separation [1, (J11)]. Fix $c,C,K,\delta>0$. For all sufficiently large $s$, no integers $m,n,p,h,a,b,D$ satisfy

$$ \begin{gathered} 0\le m,n,p,h\le Cs,\quad P=p+\rho h\ge cs,\\ 2^{m+p}\ge2^P,\quad3^{n+h}\ge2^P,\\ D=2^{m+p}a-3^{n+h}b,\quad |a|\le Ks3^n/2^p,\quad |b|\le Ks2^m/3^h,\\ 0<|D|\le2^P e^{-\delta s}. \end{gathered} \tag{10} $$

Real separation [1, (R6)]. With the same fixed parameters and any additional fixed $L\ge1$, no integers satisfy, for all sufficiently large $s$,

$$ \begin{gathered} 0\le m,n,p,h\le Cs,\quad P=p+\rho h\ge cs,\\ 2^m\ge2^P,\quad3^{n+h}\ge2^P,\\ D=2^ma-3^{n+h}b,\quad |a|\le Ks3^n,\quad |b|\le Ks2^m/3^h,\\ \text{either }|a|\le L3^n2^{-p} \text{ or } |b|\le L2^m/(3^h2^p),\\ 0<|D|\le2^Pe^{-\delta s}. \end{gathered} \tag{11} $$

The published real statement has $L=1$. Its proof allows every fixed $L$: take $X_1=2^ma$, $X_2=3^{n+h}b$. At the real place use $X_1-X_2$ and the small coordinate, and at two and three use the coordinate forms. The product of the six norms is at most $L|D|/(2^p3^h)\le Le^{-\delta s}$. In (10) the corresponding product is at most $Ks|D|/(2^p3^h)$. Both are exponentially small compared with the norm $\max(|X_1|,|X_2|)=e^{O(s)}$. The integral fixed-place Subspace Theorem [5, Theorem 2.3] confines the vectors to finitely many rational lines. It does not require primitive vectors. Zero coefficients are excluded by the lower bounds on the powers. On the remaining vectors, $|X_1/X_2-1|\le e^{-\delta s}$, so eventually only the diagonal is possible; there $D=0$. This also explains why the thresholds are ineffective and why a fixed moment order can be absorbed in $K,L$.

For a modular collision of fixed signed-copy sums of one actual chunk, the unit multipliers can be removed. Equation (9) then gives (10), with $a$ the signed $R_W$ difference divided by $2^p$ and $b$ the signed $R_Z$ difference divided by $3^h$. For a real collision, the source residue again gives a factor $3^h$, while closeness of the real coordinate supplies one of the small-width alternatives in (11). These are statements about copies of one conditional chunk law.

3. Conditional signed-numerator atoms

Lemma 1. Let $w$ be a uniform positive composition of $T$ into $\ell$ parts, where $1<\ell<T$, and let $B(w)=\sum_{j=1}^{\ell}3^{\ell-j}2^{F_{j-1}}$. For $q$ independent copies $w_1,\ldots,w_q$, any signs $\epsilon_i\in\{-1,1\}$, and $0<b<1$, put

$$ \kappa_q(b)=\frac{1+|1-2b|^q}{2},\qquad \pi=\Pr\{\operatorname{Bin}(T-1,b)=\ell-1\}. \tag{12} $$

Then

$$ \max_a\Pr\left\{\sum_{i=1}^q\epsilon_iB(w_i)=a\bmod2^T\right\} \le\frac{\kappa_q(b)^{T-1}}{\pi^q}. \tag{13} $$

In particular, for $b=(\ell-1)/(T-1)$,

$$ \max_a\Pr\left\{\sum_i\epsilon_iB(w_i)=a\bmod2^T\right\} \le T^q\kappa_q(b)^{T-1}. \tag{14} $$

The same upper bound applies to a specified exact integer sum.

Proof. Force a parity bit at position zero to be one, and let the other $T-1$ bits be independent Bernoulli variables of parameter $b$. For this auxiliary law keep the scalar $3^\ell$ fixed before generating the bits. If $0=t_1<\cdots<t_K<T$ are the positions of the ones, define the auxiliary numerator modulo $2^T$ as

$$ \sum_{j=1}^{K}3^{\ell-j}2^{t_j}. \tag{15} $$

For $K>\ell$, negative powers of three mean odd units modulo $2^T$. The use of the fixed scalar is essential: changing it with the final count would destroy the following prefix argument.

At position $t\ge1$, the coefficient of the possible new term is odd and determined by the earlier input bits. Earlier terms determine the carry into this position; future terms have larger two-adic valuation. Thus, conditional on all $q$ earlier input prefixes, the next bit of the signed sum is the XOR of $q$ independent Bernoulli bits, up to a determined complement. Negative signs have the same effect modulo two. Its larger probability is $\kappa_q(b)$. Conditioning only on earlier sum bits mixes such probabilities and cannot increase their maximum. The first sum bit is fixed, so induction bounds every residue by $\kappa_q(b)^{T-1}$.

Now condition each one-count to be $\ell$. The $q$ events are independent, of joint probability $\pi^q$, and their conditional law is precisely that of $q$ uniform positive compositions of $T$ into $\ell$ parts. Dividing by this probability proves (13). At $b=(\ell-1)/(T-1)$ the integer $\ell-1$ is a binomial mode. Since there are $T$ possible counts, $\pi\ge1/T$, giving (14). An exact integer-sum event is contained in its residue event. $\square$

The lemma concerns signed sums. It asserts neither independent ordinary digits of a conditioned numerator nor independence of actual integer trajectories.

4. One moment argument for modular and real outputs

4.1. Parameters and conditional moments

Put $\varepsilon=\min(1-d,1/2)$. Let $q$ be the least positive even integer with $3^{-q}\le\varepsilon/16$, and choose

$$ N=\left\lceil\frac{512}{\varepsilon^2}\right\rceil+q-1,\qquad \ell=\lfloor\alpha(1-\varepsilon/4)P\rfloor,\qquad H=\varepsilon P/16. \tag{16} $$

A chunk is good if its complete prefix path satisfies $|F_j-\rho j|\le H$ for $1\le j\le\ell$. Every good total obeys

$$ (1-5\varepsilon/16)P-\rho\le T\le(1-3\varepsilon/16)P. \tag{17} $$

For all sufficiently large $P$, uniformly over these totals, $1/2\le(\ell-1)/(T-1)\le2/3$. Thus $\kappa_q\le(1+\varepsilon/16)/2$, and (14) implies

$$ T^q\kappa_q^{T-1} \le8P^q2^{-(1-7\varepsilon/16)P} =:\mathcal E_\varepsilon(P). \tag{18} $$

Also, on good words,

$$ 0<B\le\ell3^{\ell-1}2^H \le\operatorname{poly}(P)2^{(1-3\varepsilon/16)P} \le2^{(1-\varepsilon/8)P} \tag{19} $$

eventually. Appendix A gives the uniform ratio and constant checks.

Place the $N$ chunks consecutively and centrally in the $s$ letters. Since $N<515/\varepsilon^2$, $\alpha<2/3$, and $P<c_ds=\varepsilon^2s/512$,

$$ N\ell<\frac{515}{\varepsilon^2}\frac{2P}{3} <\frac{1030}{1536}s<\frac34s. \tag{20} $$

For all large $s$, the unselected prefix and suffix each contain at least $s/9$ letters. For each selected chunk, therefore, $m\ge s/9$ and $n\ge s/9$. Since $P\le s/2048$, we have $m\ge P$ and $\rho n\ge P$. On the clock cap $S\le4s$, all the arithmetic parameters in (10)--(11) are $O(s)$. Their lower power hypotheses hold, including the cases $h=0$ or $p=0$.

Fix an environment with $S\le4s$. For modular outputs let $f_i$ be the law of the $i$th actual contribution in $\mathcal G_{h,p}$, restricted to good words. For real outputs use the quantized law

$$ Q_p(A,x)=(A,\lfloor2^px\rfloor) \tag{21} $$

on $(\mathbb Z/3^h\mathbb Z)\times\mathbb Z$, again restricted to good words. These are subprobabilities, not normalized good-word laws. Their supports are finite at the fixed environment.

For either group, with normalized Haar measure on its compact dual,

$$ \int|\widehat f_i|^q \tag{22} $$

is the probability that the sums of $q/2$ independent outputs on each side agree, with all $q$ words good. This is a balanced moment of one conditional law.

In the modular case, a collision gives (10) as described after (11). In the real case, a collision of quantized sums gives equality of source residues and an unwrapped real signed difference of absolute value less than $q2^{-p}$. Thus (11) applies with $L=q$. In either case the corresponding numerator difference $D$ satisfies $|D|\le q2^{(1-\varepsilon/8)P}$. If $D\ne0$, then, because $P\ge c_0s$, this contradicts the appropriate separation statement for a fixed positive $\delta$. Consequently $D=0$.

After this implication we may drop the good-word restriction in the numerator event. The conditional words are uniform compositions of one fixed total, so Lemma 1 and (18) yield

$$ \int|\widehat f_i|^q\le\mathcal E_\varepsilon(P),\qquad \|f_1*\cdots*f_q\|_\infty \le\prod_{i=1}^q \left(\int|\widehat f_i|^q\right)^{1/q} \le\mathcal E_\varepsilon(P). \tag{23} $$

The middle inequality is Fourier inversion followed by Hölder. It combines $q$ different actual chunk laws; these laws need not be identically distributed. The preceding moment, in contrast, uses copies of one law. No exterior coefficient has been replaced by a surrogate coefficient.

For real outputs, if the sum of $q$ real contributions lies in a specified interval of length $2^{-p}$, the sum of their individual cell indices has at most $q+1$ possible values. Its joint interval mass is therefore at most $(q+1)\mathcal E_\varepsilon(P)$. Further convolution with subprobabilities preserves this bound for arbitrarily translated intervals. It also preserves the modular atom bound. The real variable is the unwrapped value throughout this step; canonicalization is deferred to Section 4.3.

4.2. Recovering all words

We establish the unconditional estimates uniformly for geometric parameters $\alpha'$ in a sufficiently small fixed neighborhood of $\alpha$, chosen so that

$$ \alpha'>.63,\qquad |1/\alpha'-\rho|\le\varepsilon/128. \tag{24} $$

The conditional composition laws and the preceding moment proof do not change. A maximal exponential inequality for the entire prefix path gives

$$ \Pr_{\alpha'}(\text{bad chunk}) \le2\exp(-\varepsilon^2P/512),\qquad \Pr_{\alpha'}(S>4s)\le e^{-s/5}. \tag{25} $$

Appendix A.2 proves these numerical bounds including the drift from $1/\alpha'$ to $\rho$.

At each fixed environment satisfying the clock cap, expand the $N$ chunk laws into good and bad subprobabilities. Each term with at least $q$ good factors has the bound (23), with an additional factor $q+1$ in the real case. There are at most $2^N$ terms, a constant depending only on the fixed dimension. A term with fewer than $q$ good factors requires at least $N-q+1=\lceil512/\varepsilon^2\rceil$ actual chunks to be bad.

Average over environments and drop the clock cap in this latter nonnegative upper bound. The bad events concern disjoint original chunks and are independent under the unconditioned geometric law. Using (25) and summing over the finitely many possible bad subsets bounds their combined probability by $C_\varepsilon e^{-P}$. This independence is used before exact-total conditioning; it is not asserted for chunks conditioned only on the total $S$.

Let $\mathcal C_s(P)$ denote either the maximum raw modular atom or the largest raw joint interval mass. Uniformly over (24),

$$ \mathcal C_s(P) \le C_\varepsilon P^q2^{-(1-7\varepsilon/16)P} +C_\varepsilon2^{-P}+e^{-s/5}. \tag{26} $$

Since $d\le1-\varepsilon$, the first exponent exceeds $d$ by at least $9\varepsilon/16$. This strict margin absorbs its polynomial on $P\ge c_0s$. The clock exception is smaller still, because $P\le s/2048$. This proves the unconditional raw bound in (7).

4.3. Exact totals and the common carry

For a prescribed $|T-\rho s|\le G(s)=o(s)$, tilt to $\alpha'=s/T$. Uniformly over this window, the parameter eventually satisfies (24), and the conditional composition law given $S=T$ is unchanged. The negative-binomial identity and the binomial-mode bound give

$$ \Pr_{\alpha'}(S=T) =\alpha'\Pr\{\operatorname{Bin}(T,\alpha')=s\} \ge\frac{\alpha'}{T+1}\ge\frac1{4(s+1)} \tag{27} $$

eventually. Divide (26), including all its exceptional mass, by this probability. The additional polynomial factor is absorbed by the strict exponent margins. This proves every asserted critical exact-total raw bound; no total in the prescribed window is discarded.

For the canonical bounds, apply (4) at the fixed environment, before mixing. There $S$ is fixed, and the modular map subtracts one of at most $s$ known translations

$$ (a2^S\bmod3^h,a3^s\bmod2^p),\qquad 0\le a<s. \tag{28} $$

For either real vector it subtracts $(a2^S\bmod3^h,a)$; its quantized version subtracts $(a2^S\bmod3^h,a2^p)$. Partition by the actual carry and translate each piece. A union bound costs at most $s$ in the conditional good atom or interval estimate. The bad-word and clock exceptions can still be bounded by their full probabilities. Averaging therefore gives at most $s$ times the first term of (26), plus the same exceptions. The tilted comparison (27) applies to this complete bound as well. All polynomial factors are absorbed while retaining the stated $Cs2^{-dP}$ form. In particular, no $S$-dependent source translation is fixed after the environments have already been mixed.

4.4. The stronger fourth-moment instance

For $d=24/25$, replace (16) by

$$ q=4,\quad N=10000,\quad \ell=\lfloor49P/(50\rho)\rfloor,\quad H=P/100, \quad 0<c_0<v<1/6250. \tag{29} $$

Good totals lie between $.97P-\rho$ and $.99P$. For all sufficiently large $P$, the mode parameter lies in $[1/2,16/25]$, so $\kappa_4\le196513/390625=:\kappa$. The exact comparison

$$ 196513^{970}2^{961}<390625^{970} \tag{30} $$

gives the conditional moment bound $\kappa^{-3}P^4 2^{-.961P}$. Good numerators satisfy $B\le2^{.995P}$ eventually. The selected chunks occupy less than $.989408s$ letters, leaving at least $s/250$ letters on each side for all large $s$. This still gives $m\ge P$, $\rho n\ge P$. Thus both separation arguments apply with a strict exponential margin.

In a fixed neighborhood with $\alpha'>.63$ and $|1/\alpha'-\rho|\le1/10000$, the bad-chunk probability is at most $2\exp(-68P/10^6)$. At least $9997$ failures are required to leave fewer than four good chunks, and

$$ \frac{68\cdot9997}{10^6}=.679796>.97\ln2. \tag{31} $$

The complete analogue of (26) is consequently bounded by $CP^4 2^{-.961P}+C2^{-.97P}+e^{-s/5}$. Its strict gap above $.960$ absorbs the exact-total and canonical polynomial costs. Appendix A.3 records all the numerical margins. This completes Theorem 1.

5. Explicit original-source consequences

5.1. Statements and the exact final-count reduction

Let $C$ be the shortcut Collatz map on the nonnegative integers:

$$ C(x)=\begin{cases}x/2,&x\text{ even},\\(3x+1)/2,&x\text{ odd},\end{cases} \qquad C(0)=0. \tag{32} $$

Let $J_h(x)$ count odd steps before time $h$, and define

$$ G_h(m)=\sup_{a\in\mathbb Z_{\ge0}} \#\{(C^h(x),J_h(x)):a\le x<a+m\}. \tag{33} $$

Theorem 2 (explicit compression and halving exponent). Set $\gamma=5\cdot10^{-7}$ and $k=\lceil\log_2m\rceil$. There is an ineffective constant $K<\infty$ such that, for every integer $m\ge1$,

$$ G_{k^2}(m)\le K m^{\theta-\gamma}. \tag{34} $$

Among any $m$ consecutive positive integers, at most $K m^{\theta-\gamma}$ fail to reach a value strictly below half their own starting value by

$$ T(k)=k+\lfloor k/10000\rfloor \tag{35} $$

shortcut steps. For the ordinary Collatz map, at most $2T(k)$ steps suffice for the corresponding halving assertion. No statement is made about a particular member of the exceptional set.

We first state the exact counting input from [1, Section 8.3] with the assumptions needed here. Suppose the canonical real-source bound of Theorem 1 holds at some $\theta<d<1$ and band slope $c_d$. Choose $0<\delta<1$ with

$$ \frac{\delta}{\alpha(1+\delta)}<c_d, \qquad p=\lfloor\delta k\rfloor,quad T=k+p. \tag{36} $$

Then for every fixed $H\ge0$ and every interval $I$ of $2^k$ consecutive nonnegative integers,

$$ \#\{x\in I:J_T(x)\ge\alpha T-H\} \le C_H k2^{\theta k-(d-\theta)p} \tag{37} $$

for all sufficiently large $k$, uniformly in the interval location. There is no hypothesis about earlier counts or survival. We reproduce the reduction to make explicit where the original sources enter.

Write $x=q_0 2^T+b$, $0\le b<2^T$. The source interval meets at most two quotient blocks. For a fixed block, let $j$ be the initial number of zero parity bits, and let $r=J_T(x)>0$. Finite parity coding [6], or the division-word form [3], gives $r$ positive letters with total $S=T-j$. Their canonical source is $n_r=b/2^j$, and exactly

$$ \frac{n_r}{2^S}=\frac b{2^T}. \tag{38} $$

At fixed $q_0$ the original source determines the parity word injectively. The admissible $b$ form an interval of length at most $2^k$, so (38) lies in a real interval of length at most $2^{-p}$.

Each word with specified $j,r$ has geometric probability

$$ w_{T,j,r}=\alpha^r(1-\alpha)^{T-j-r},\qquad w_{T,j,r}^{-1}=(1-\alpha)^j2^{\theta T} z^{-(r-\alpha T)},\quad z=\frac\alpha{1-\alpha}>1. \tag{39} $$

For $\alpha T-H\le r\le T$, the ratio $p/r$ eventually stays in a fixed band strictly inside $(0,c_d)$, by (36). Apply the unconditional canonical real-source bound to the length-$r$ word law, with $h=0$. Drop only the exact-total restriction in the interval event. Its word count is at most $Cr2^{-dp}w_{T,j,r}^{-1}$. This step deliberately uses the unconditional theorem: the totals $T-j$ need not lie in a critical window for every summand.

Sum over valid $j$ and extend to $j\ge0$, costing $1/\alpha$. The sum of $r z^{-(r-\alpha T)}$ over $r\ge\alpha T-H$ is at most $C_HT$. The all-zero word has $r=0$ and is outside the event for large $k$. The two quotient blocks cost only a factor two. Hence

$$ \#\{x\in I:J_T(x)\ge\alpha T-H\} \le C_HT2^{\theta T-dp}, \tag{40} $$

which is (37). This is an unnormalized count of actual original sources; no independent law is assigned to a survivor-selected set.

5.2. A direct recursion for labelled compression

In addition to (36), suppose $c(1+\delta)<1$, where $c=\log_2(3/2)$, and choose

$$ 0<\gamma<\min(\theta,(d-\theta)\delta),\qquad \eta=\theta-\gamma>0. \tag{41} $$

Consider an arbitrary original interval $[a,a+2^k)$. For each fixed low count $R<\alpha T-H$, the full shortcut affine formula places all its time-$T$ images in a real interval of length at most

$$ 2^{\rho R-p}+(3/2)^T. \tag{42} $$

Indeed the slope is $3^R2^{-T}$, and every length-$T$ parity word has nonnegative additive intercept at most $(3/2)^T-1$. The latter bound follows by applying the maximal update $u\mapsto(3/2)u+1/2$ from initial intercept zero. The common left endpoint in (42) can be taken as $3^R2^{-T}a$. Thus the location of the source interval is retained without an extra location factor.

An integer interval of width $2^{\ell_R}$ covers these images for

$$ \ell_R=\left\lfloor\max(\rho R-p,cT,1)\right\rfloor+3. \tag{43} $$

For sufficiently large fixed $H$ and then sufficiently large $k$, $\ell_R\le k-3$ for every low count: the first argument of the maximum is less than $k-\rho H$, and the second is below $k$ by a fixed linear margin. It follows also that $T+\ell_R^2\le k^2$.

We prove $G_{k^2}(2^k)\le K2^{\eta k}$ by strong induction in $k$. For a fixed low count $R$, apply the induction hypothesis to the covering interval at scale $\ell_R$. Its previous label $R$ is added to the subsequent odd-count label. Padding the resulting pairs to the common time $k^2$ is deterministic and cannot increase their cardinality. Thus this count contributes at most $K2^{\eta\ell_R}$.

Using $2^{\eta\max(A,B,1)}\le 2^{\eta A}+2^{\eta B}+2^\eta$ and summing over integer low counts gives the normalized total cost

$$ \sum_{R<\alpha T-H}2^{\eta(\ell_R-k)} \le\frac{2^{\eta(3-\rho H)}}{1-3^{-\eta}} +O(k)2^{-\eta[1-c(1+\delta)]k}+O(k)2^{-\eta k}. \tag{44} $$

The first term is the geometric sum in $R$; all count labels have been included. Choose $H$ to make this term less than $1/4$. Then choose $k$ large enough that the complete low-count cost is less than $1/2$.

The high-count sources are bounded by (37), and each source contributes at most one labelled pair. After normalization by $2^{\eta k}$, their cost is, up to a bounded rounding factor,

$$ O_H(k)2^{[\gamma-(d-\theta)\delta]k}, \tag{45} $$

which tends to zero by (41). Enlarge $K$ to handle this term and the finite initial range of the induction. This proves the desired bound at dyadic widths. For general $m$, round up to $2^k<2m$; the common clock remains $k^2$, and the rounding costs a constant.

This recursion uses (37) directly. It differs from the two-window compression proof in [1, Section 8.1] and accounts for both the full affine intercept and the labelled time budget.

5.3. Original-source halving

If $x\ge2^k$ has not reached a value below $x/2$ by time $T$, every state before that time is at least $x/2$. The exact identity used in [1, equation (R21)] is

$$ \frac{C^T(x)}x=\frac{3^r}{2^T} \prod_{\substack{0\le j<T\\C^j(x)\text{ odd}}} \left(1+\frac1{3C^j(x)}\right),\qquad r=J_T(x). \tag{46} $$

The product is between one and $\exp(2T/(3x))$. Since the left side is at least $1/2$, one has $r\ge\alpha T-2$ for all sufficiently large $k$, uniformly in $x\ge2^k$. Such halving failures therefore lie in the event counted by (37) with $H=2$. Under (41), its polynomial factor is absorbed into $K2^{\eta k}$.

Every aligned block of width $2^k$ with positive left endpoint satisfies $x\ge2^k$. For the aligned block starting at zero, partition its positive part into $[2^j,2^{j+1})$, $0\le j<k$. A failure to halve by $T(k)$ is also a failure by the earlier $T(j)$, so summing the smaller-scale bounds costs a convergent geometric factor because $\eta>0$. Finally any interval of $m$ integers meets at most two aligned blocks of width $2^k$, with $k=\lceil\log_2m\rceil$. This proves the uniform halving bound. Each shortcut step is at most two steps of the ordinary map.

To obtain Theorem 2, take

$$ d=24/25,\qquad \delta=1/10000,\qquad \gamma=5\cdot10^{-7}. \tag{47} $$

The exact inequalities in Appendix A.4 show that (36), (41) and $c(1+\delta)<1$ hold. In particular $\theta<.95$, so $(d-\theta)\delta>10^{-6}$. This completes the proof.

6. Evidence, reproducibility, and limitations

The theorems are analytic. A fixed finite audit supports the conditional coding and constant calculations; it does not establish the eventual arithmetic separation. The supplement includes the original audit programs and retained outputs, with a manifest of file sizes and SHA-256 hashes. No source enumeration was enlarged or rerun in preparing this standalone paper.

The signed-numerator audit enumerates 5,515 composition source cases and 2,728 auxiliary source cases, checks second and fourth signed energies, compares three small cases with direct quadruple enumeration, and includes a negative control for dropping the conditioning cost. It also checks the exact rational fourth-moment rate. The finite composition cases verify the explicit binary/residue interface; they do not sample or prove the limiting separation threshold.

A second script checks the general-band constants at five specified values of $\varepsilon$ and verifies exact rational inequalities used in the variance, strip and ratio estimates. The proof for all $0<\varepsilon\le1/2$ is in Appendix A, not inferred from those five cases. A retained exact-arithmetic certificate records the inequalities used in the compression and halving parameters. The historical filename of that certificate refers to halving; the same inequalities serve both consumers here.

From the extracted supplement root, run

python src/signed_numerator_atom_check_20261031.py
python src/direct_moment_dimension_band_constants_20261031.py
python src/direct_moment_compression_constants_20261031.py

The first command uses Python 3.10+, a C++17 compiler and five CPU worker processes. The other two use only the Python standard library. No GPU or third-party Python package is required. Expected outputs and the finite audit domains are recorded in the supplement README and JSON results. The supplemental author-audit notes are historical proof checks, not independent reviews; their old section numbers refer to the proposed integrated draft from which this focused contribution was separated. They are not needed in place of the complete proof above and below.

The third command directly reproduces the exact comparisons in Appendix A.3--4 without enumerating sources. Original software is provided under the MIT License, and the manuscript, explanatory text and retained data under CC BY 4.0; see the supplement license file.

The explicit band does not make the Subspace-Theorem threshold or the constants in Theorem 2 effective. It gives sufficient slopes, without claiming optimality. It leaves the broader fixed-order entropy results of [1] unchanged. Exact-total estimates concern the stated critical $o(s)$ window; the source-counting reduction instead uses the proved unconditional law and pays its exact word weights.

Most importantly, upper concentration and fewer labelled images do not ensure that every supplied source has a smaller positive source in its own labelled fibre. A particular source can remain exceptional or be the least element of that fibre. Neither divergent orbits nor nontrivial positive cycles are excluded. The missing connection to a pointwise descent mechanism is not supplied by the finite audits or by an entropy interpretation of these bounds.

Appendix A. Uniform constants and exact comparisons

A.1. General moment and packing constants

Let $0<\varepsilon\le1/2$. If $m_0=\lceil2/\varepsilon\rceil$, then $3^{2m_0}=9^{m_0}\ge1+8m_0>16/\varepsilon$. Consequently $4\le q\le2m_0<4/\varepsilon+2$ for the choice in (16). Thus

$$ N<512/\varepsilon^2+4/\varepsilon+2 \le514.5/\varepsilon^2<515/\varepsilon^2. \tag{A1} $$

The bounds $.63<\alpha<.631$ follow from exact comparisons of powers of two and three. At a good total (17), the two leading margins needed for $1/2\le(\ell-1)/(T-1)\le2/3$ are bounded below by

$$ \begin{aligned} .63(1-\varepsilon/4)-\tfrac12(1-3\varepsilon/16) &\ge157/1600,\\ \tfrac23(1-5\varepsilon/16)-.631(1-\varepsilon/4) &\ge83/8000. \end{aligned} \tag{A2} $$

Both left sides are decreasing affine functions of $\varepsilon$; their displayed lower bounds occur at $\varepsilon=1/2$. Floor errors are bounded, so the margins are uniform once $P$ is large.

For the allowed mode parameter, $|1-2b|\le1/3$. The definition of $q$ gives $\kappa_q(b)\le(1+\varepsilon/16)/2$. Using $\ln(1+x)\le x$ and $\ln2>1/2$,

$$ -\log_2\kappa_q(b)\ge1-\varepsilon/8. \tag{A3} $$

Since $T\le P$ and $T-1\ge(1-5\varepsilon/16)P-\rho-1$, (14) is bounded by

$$ P^q2^{\rho+1} 2^{-(1-\varepsilon/8)(1-5\varepsilon/16)P} \le8P^q2^{-(1-7\varepsilon/16)P}. \tag{A4} $$

Here $\rho<2$ and $(1-\varepsilon/8)(1-5\varepsilon/16) \ge1-7\varepsilon/16$. On the good strip each summand of $B$ is at most $3^{\ell-1}2^H$, giving (19). Its polynomial factor is absorbed by the positive gap $\varepsilon P/16$.

A.2. Maximal strip and clock estimates

For a positive geometric variable with parameter $\alpha'$ satisfying (24), write $\rho'=1/\alpha'$ and let $\psi(\lambda)=\log\mathbb E\exp(\lambda(e-\rho'))$. With $r=1-\alpha'<37/100$, for $|\lambda|\le1/32$ we have

$$ \psi''(\lambda)=\frac{re^\lambda}{(1-re^\lambda)^2} <\frac{229400}{229441}<1. \tag{A5} $$

Indeed $e^{1/32}<32/31$ gives $re^\lambda<296/775$, and the rational expression is increasing there. Since $\psi(0)=\psi'(0)=0$, it follows that $\psi(\lambda)\le\lambda^2/2$. Apply the exponential-martingale maximal inequality separately to each sign with magnitude $\lambda=\varepsilon/16\le1/32$. A deviation from the critical line by $H$ implies a centered deviation by at least $H-\ell|\rho'-\rho|$. Hence

$$ \begin{aligned} \Pr(\text{bad chunk}) &\le2\exp\{-\lambda H+\lambda\ell|\rho'-\rho| +\ell\lambda^2/2\}\\ &\le2\exp\{-7\varepsilon^2P/3072\} \le2\exp\{-\varepsilon^2P/512\}. \end{aligned} \tag{A6} $$

We used $\ell<2P/3$ and the exact comparison $1/256-1/3072-1/768=7/3072>1/512$. This controls the complete prefix strip, including its interior, not merely the terminal total.

For the clock cap, at $\lambda=1/10$ the geometric exponential moment is at most $63/53<e^{1/5}$: use $\alpha'>.63$ and $e^{1/10}<10/9$ in its explicit generating function. Chernoff then gives $\Pr(S>4s)\le e^{-s/5}$. These arguments hold uniformly in a fixed neighborhood satisfying (24).

A.3. Fourth-moment constants

With (29), good totals satisfy $.97P-\rho\le T\le.99P$. The bounds on $\alpha$ imply, with strict leading margins, that eventually $1/2\le(\ell-1)/(T-1)\le16/25$. Therefore

$$ \kappa_4(b)\le\frac{1+(7/25)^4}{2} =\frac{196513}{390625}=\kappa. \tag{A7} $$

Equation (30) says $\kappa^{.97}<2^{-.961}$. Since $T\le P$ and $T-1>.97P-3$, it gives $T^4\kappa^{T-1}\le\kappa^{-3}P^4 2^{-.961P}$. The numerator bound follows from $\rho\ell+H\le.99P$: its polynomial factor is absorbed in the gap to $.995P$.

The selected length is at most $10000(.98)(.631)P=6183.8P<.989408s$. A centered placement thus leaves more than $.005s$ on each side asymptotically, so the more conservative $s/250$ bound holds for all large $s$.

Choose $\alpha'>.63$ and $|\rho'-\rho|\le1/10000$. On $|\lambda|\le1/100$, the variance bound in (A5) improves to $3663/3844<1$, using $re^\lambda<37/99$. Set $\lambda=1/100$, use $H=P/100$ and $\ell<.62P$, and retain the drift. This gives

$$ \Pr(\text{bad chunk}) \le2\exp[-(.0001-.000031-.00000062)P] \le2\exp(-68P/10^6). \tag{A8} $$

Fewer than four good factors forces at least $9997$ independent failures before total conditioning. Their rate satisfies (31), since $68\cdot9997=679796>679000$ and $\ln2<.7$. The clock cap is unchanged. The resulting moment, exceptional, and target exponents are respectively $.961$, $.97$, and $.960$, leaving strict margins for every polynomial conditioning cost.

A.4. Compression parameters

Two exact integer comparisons are

$$ 3^{6309}<2^{10000},\qquad 10000^{10000}<2^{9500}6309^{6309}3691^{3691}. \tag{A9} $$

The first gives $\alpha>6309/10000$. Since $H_2$ decreases on $(1/2,1)$, the second gives $\theta<H_2(6309/10000)<19/20$. Thus with (47), $(d-\theta)\delta>10^{-6}>\gamma$. Moreover,

$$ \frac{\delta}{\alpha(1+\delta)} <\frac{10000}{6309\cdot10001}<\frac1{6250}. \tag{A10} $$

The last comparison is $62500000<63096309$. Finally $c=\log_2(3/2)<3/5$, since $(3/2)^5<2^3$, and $(3/5)(10001/10000)<1$. These are exactly the strict assumptions needed in (36), (41), and the scale recursion. Equations (30) and (A9) can be verified with integer exponentiation; the retained certificate records the bit lengths of their positive differences. Their verification does not make any asymptotic onset effective.

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