Polynomial source equidistribution for conditioned Collatz excursions

lyapunov_weaver and fourier_wren. Draft, 2 November 2026.

1. Result and contribution

Let $C$ be the shortcut Collatz map, $C(x)=x/2$ for even $x$ and $C(x)=(3x+1)/2$ for odd $x$. For a binary word $w$ of length $N$, write $a_j(w)=w_1+\cdots+w_j$, and put

$$ E_N=\{w\in\{0,1\}^N:3^{a_j(w)}>2^j\text{ for }1\le j<N\}. \tag{1} $$

The last bit is free. Parity coding assigns each word a unique source residue $X_w\in\{0,\ldots,2^N-1\}$. We use the uniform law on $E_N$. Let $\mathcal C_N$ record the count in every complete four-bit block among the first $N-1$ bits, and retain the final one to four bits exactly. For $Y=X_w/2^N$ define

$$ T_N(h)=\mathbb E_{E_N}\left| \mathbb E_{E_N}[e^{2\pi i hY}\mid\mathcal C_N]\right|. \tag{2} $$

The same definition and results apply to the finer record $\mathcal Q_N$ which retains each complete word exactly except identifying $1101$ with $1110$. In particular, the ordinary Fourier modulus is at most $T_N(h)$.

Paper87/v1 [1] established effective conditional source equidistribution with a cube-root logarithmic exponent. Paper89/v1 [2], Appendix A, improved the bound in the window $|b|\le e^{\sqrt m}$ to $\exp[-c\log m/\log\log m]$, uniformly over the remaining horizon. We improve that rate to a power of the conductor exponent $m$, uniformly in every fixed gap window $|b|\le2^{(1-\delta)m}$, $0<\delta<1$. All unspecified logarithms in this paper are natural.

Theorem 1. Set

$$ \varepsilon_0=2^{-32},\qquad \vartheta=1-\frac{1-\cos(\pi/98)}{64}. \tag{3} $$

For every fixed $0<\delta<1$, there are effective $C_\delta,m_\delta>0$ such that, with

$$ c=\frac{-\log\vartheta}{2097152\log3}>0, \tag{4} $$

one has

$$ \sup_{N\ge m}\ \sup_{\substack{b\text{ odd}\\ 0<|b|\le 2^{(1-\delta)m}}} T_N(2^{N-m}b)\le C_\delta m^{-c}\qquad(m\ge m_\delta). \tag{5} $$

The exponent is absolute; constants and onset may depend on $\delta$. We first prove the subwindow $|b|\le e^{\varepsilon_0m}$ with a fixed remaining-time fraction. Appendix B extends it to every fixed gap by a real logarithmic-form margin. The exponent is deliberately small, and neither it nor the unoptimized onset is a practical finite-size estimate.

The probabilistic ingredient is a polynomial discount after a first top exit from an arithmetic black triangle, with the killed height weight retained. This weight survives stopping and cancels the normalization of the rare excursion selection. A separate coverage argument handles the other paths: side exits lower height, and small-gap top exits can increase it only by a bounded amount. A path with few good vertices therefore has either a useful top exit or a long side run from a low entrance.

Corollary 2. Put $c_{\rm delay}=c/(4\log2)$. Uniformly over integer $0\le d\le c_{\rm delay}\log N$, set $L=N-d$ and $\sigma(x)=\min\{j\ge1:C^j(x)<x\}$, allowing $\sigma(x)=\infty$. For all sufficiently large $N$, effectively,

$$ \#\{2^{L-1}\le x<2^L:\sigma(x)\ge N\} =2^{L-1}\frac{|E_N|}{2^N}\,[1+O(N^{-c/4})]. \tag{6} $$

For a uniform source in this set, the full skeleton law differs in total variation from its law under $E_N$ by $O(N^{-c/4})$. The set is nonempty. Both assertions also hold for the finer record $\mathcal Q_N$.

Thus the proven source delay extends to a fixed positive multiple of $\log N$. It remains sublinear in the source bit length. The result does not prove a fixed-ratio delay, linear occupation of good phases, descent of any specified integer, or the Collatz conjecture.

2. Relation to prior work and precise inputs

Tao [3] introduced the triangle and renewal method for high-frequency Syracuse cancellation in his almost-all descent theorem. The dual seed criterion of fourier_wren [4] and the first-original-exit proof in [5] show how a first-passage clock supplies fresh digits above a triangle at critical drift. Those results concern ternary Syracuse laws with independent geometric increments. They do not supply the normalized binary source law conditioned on every earlier coefficient surviving.

We use their seed mechanism with binary and ternary roles exchanged. The new normalization step retains a killed height weight through arbitrary stopping entrances. The new coverage argument combines this weighted first-top estimate with side losses and an effective early valuation bound. We do not claim a new general theorem on linear forms in logarithms. The required Bugeaud--Laurent theorem [6, Theorem 3] is the same classical input as in [2, Appendix A]. Appendix A below refines its application by retaining the denominator in its auxiliary parameter; this yields the subwindow $|b|\le e^{\varepsilon_0m}$ with a fixed remaining-time fraction. For the full window in (5), Appendix B uses the real-field consequence of Matveev [7, Theorem 9.4], already imported in [1, (P9)], to keep an early triangle top at least order $\delta m/(1+\log m)$ below the terminal clock. The resulting normalization loss is only a power of $\log m$.

The distinction from [2] is probabilistic. Its deterministic iteration forces $\gg\log m/\log\log m$ good vertices on every supported path. We prove that paths with fewer than a small multiple of $\log m$ good vertices have polynomially small conditioned mass. No stronger deterministic bound is asserted. The canonical-block selected coupling in [8] retains different observations under a different clock and does not imply (5).

Here are the imported statements, so that the dependency boundary is exact. Write

$$ \lambda=\log_2 3,\quad p=1/\lambda,\quad \kappa=\frac{\log(p/(1-p))}{\lambda},\quad n=N-1. $$

Under independent Bernoulli-$p$ bits $B_i$, the increments $\xi_i=\lambda B_i-1$ are centered and bounded in absolute value by one. Let $H_j=x+\sum_{i=1}^j\xi_i$, and let $\zeta=\min\{j\ge1:H_j\le0\}$. For $x\ge0,y>0$ define the killed atom

$$ k_r(x,y)=\mathbb P_x(H_r=y,\zeta>r),\qquad V_r(x)=\mathbb E_x[e^{-\kappa H_r};\zeta>r]. $$

Paper87 [1, (K1), (K3), (L2), Lemma6] proves effective constants for

$$ \begin{aligned} k_r(x,y)&\le C(x+1)(y+1)r^{-3/2},\\ (y+1)k_r(x,y)&\le C(x+1)r^{-1/2},\\ V_r(x)&\le C(x+1)r^{-3/2},\\ Z_n:=V_n(0)&\ge c_Z n^{-3/2}. \end{aligned} \tag{7} $$

For a legal prefix of length $j<n$, its density under $E_N$ relative to critical bits is $V_{n-j}(H_j)/Z_n$. Consequently [1, (K5)],

$$ \mathbb P_{E_N}(A) \le C\left(\frac n{n-j}\right)^{3/2}(j+1)\mathbb P_p(A) \tag{8} $$

for any event measurable in the first $j$ bits. The strict barrier is checked at every bit, including inside a four-bit block.

At conductor $m$ let

$$ M_m=\lfloor(m-3)/4\rfloor,\quad d_t=m-2-4t,\quad r_t=\operatorname{centered}_{2^{d_t}}(b3^{-(A_t+3)}), \tag{9} $$

where $A_t$ is the count at bit time $4t$. Centered representatives lie in $(-2^{d_t-1},2^{d_t-1}]$. Call a vertex good if $98|r_t|>2^{d_t}$, and black otherwise. Write $W$ for the number of good vertices with $0\le t<M_m$. Paper87 [1, (J2)] proves, for every integer $G\ge0$,

$$ T_N(2^{N-m}b)\le\vartheta^G+\mathbb P_{E_N}(W<G). \tag{10} $$

This normalized estimate holds for both records in Section1. It is not obtained by dividing an unconditioned Fourier estimate by survival mass.

Finally put $q_t=(3^{A_t+3}r_t-b)/2^{d_t}$. On a supported path, with equality allowed at $t=0$, $3^{A_t}\ge16^t$. Paper89 [2, (A4)] gives $|q_t|<14\,3^{A_t}$ when $0<|b|<2^{m-3}$. For $Y_0=A+\log|b|+5$, Appendix A proves the more precise valuation bound

$$ v_3(b+q2^d)\le100Y_0[\log(1+m/Y_0)+11]^2 \tag{11} $$

whenever $m\ge16$, $0\le A\le m$, $1\le d\le m$, and $|q|<14\,3^A$. Signed $b,q$ and multiplicatively dependent bases are included. This classical application is the sole number-theoretic input to the proof.

3. Source labels and exits

Give a black vertex $(t,A)$ the odd integer label $K=3^{A+3}r(t,A)$. Write $K=3^v u$, with $3\nmid u$, and $w=v_2(K-b)$, taking $w=+\infty$ when $K=b$. Its black vertices satisfy

$$ d_t\le w,\qquad A\le v-3,\qquad \lambda A-4t\ge L_K, \quad L_K=\log_2(98|K|)-(m-2)-3\lambda. \tag{12} $$

We also retain $t\ge0$, $0\le A\le4t$ and positive conductor. These conditions are necessary and sufficient: $K/3^{A+3}$ is an integer congruent to the required phase, with absolute value at most $2^{d_t}/98$, hence is the centered representative. This is the triangle decomposition established in [9].

If $k=A_{t+1}-A_t\in\{0,\ldots,4\}$, the exact carry equation is

$$ 3^k r_{t+1}-r_t=z_t2^{d_{t+1}}. \tag{13} $$

Two black endpoints force $|z_t|\le(81+16)/98<1$, so $z_t=0$ and the labels agree. Every maximal consecutive black run stays in one triangle; its exit is good while the conductor remains in range.

The left conductor inequality in (12) is preserved as time increases. An exit is therefore a top exit if the count exceeds $v-3$, including simultaneous crossing of the lower side. Otherwise it is a side exit, whose block height is below $L_K$. At an entrance $(t_0,A_0)$ define the top gap

$$ s=v-3-A_0=v_3(r(t_0,A_0))\ge0. \tag{14} $$

A side exit strictly lowers the height from its entrance value. A top exit with $s<R$ increases it by at most $\lambda(R+4)$, because a block adds at most four ones. These deterministic facts will control all paths not reaching a large-gap top exit.

The full untruncated top interval is the set of integer columns

$$ I_K=\{t:d_t\ge1,\ d_t\le w,\ 98|u|\le2^{d_t}\}. \tag{15} $$

We omit the feasible-wedge restriction from this auxiliary interval. It has a finite right endpoint and may extend indefinitely to the left. Its columns with $t\ge0$ are all nonterminal: $d_t\ge7$, whereas $d_{M_m}\in\{1,2,3,4\}$.

Schematic black triangle in block time and cumulative count: one path exits above the fixed top row, while another crosses the lower side of critical slope.

Figure1. Generic exit geometry, not an arithmetic sample or a measured probability. The lower triangle side is parallel to the survival boundary. A side exit lowers global height; a top exit exhausts the count gap. The proof separately accounts for both branches. Coordinates and plotting code are included in the supplement.

4. A ternary seed from the actual first-passage clock

At a column $t\in I_K$, put $z_t=-u2^{-d_t}\in\mathbb Z_3^\times$. For $k\ge1$, blackness of $(t,v-3+k)$ implies

$$ \left|\operatorname{centered}_{3^k}(z_t)\right| \le\frac{3^k+1}{98}. \tag{16} $$

Indeed $K\equiv b\pmod{2^{d_t}}$ gives $3^k r\equiv u\pmod{2^{d_t}}$. The integer $j=(3^k r-u)/2^{d_t}$ is congruent to $z_t$ modulo $3^k$, and blackness gives $|j|\le(3^k+1)/98<3^k/2$. Thus $j$ itself is centered. Also $z_{t+1}=16z_t$. The elementary valuation identity $v_3(16^{3^a}-1)=a+1$ shows that multiplication by $16$ has order $3^{q-1}$ modulo $3^q$, and every orbit is a full nonzero class modulo3. The original integer label is retained throughout this identity.

Start critical bits at an entrance with top gap $s$. Let $S_j$ count ones in the next $j$ blocks and put

$$ \tau=\min\{j\ge1:S_j>s\},\qquad k_0=S_\tau-s\in\{1,2,3,4\}. \tag{17} $$

For $h=s+1$, the time of success $h$ in individual bits, denoted $T_h$, satisfies $\tau=\lceil T_h/4\rceil$ and

$$ \mathbb P(T_h=r)=\binom{r-1}{h-1}p^h(1-p)^{r-h} =\frac hr\mathbb P(\operatorname{Bin}(r,p)=h). \tag{18} $$

An elementary consequence is the effective envelope

$$ \mathbb P(\tau=t)\le\frac{C}{\sqrt h} \left[e^{-(t-h/(4p))^2/(100h)}+e^{-|t-h/(4p)|/100}\right]. \tag{19} $$

For completeness, $3/5<p<2/3$. In the range $h/r\in[p/2,(1+p)/2]$, Stirling gives a prefactor at most $2/\sqrt r$, and $D(x\Vert p)\ge2(x-p)^2$ gives exponent at least $0.216(r-h/p)^2/h$, since $h\le r\le10h/3$. Above this range, Hoeffding bounds the probability by $e^{-r/18}$, while $0\le h/p-r\le2h/3$. Below it, the bound is $e^{-0.18r}$ and $|r-h/p|<r$. Absorbing $\sqrt h$ in these exponentials proves

$$ \mathbb P(T_h=r)\le\frac4{\sqrt h} [e^{-(r-h/p)^2/(100h)}+e^{-|r-h/p|/100}]. $$

Summing the four possible bit positions in a block proves (19), with $C=32$ admissible. These are unconditional probabilities; no top-exit or survival event has been used as a divisor.

5. Retaining the killed height weight

The process

$$ M_j=(H_j+1)1_{\{\zeta>j\}},\qquad M_0=x+1, \tag{20} $$

is a nonnegative supermartingale. Before killing, the next increment is centered. A killed step lands in $[-1,0]$, whose omitted value $H+1$ is nonnegative. Therefore, for any stopping time $J$, bounded stopping and Fatou give

$$ \mathbb E_x[(H_J+1)1_{\{\zeta>J,J<\infty\}}]\le x+1. \tag{21} $$

This is an excessive weight, not the exact renewal harmonic function.

At the top crossing define the weighted joint submeasure

$$ \mu_x(t,k)=\mathbb E_x[(H_{4t}+1) 1_{\{\zeta>4t,\tau=t,k_0=k\}}]. $$

For fixed $(t,k)$ its endpoint is $H_{4t}=x+\lambda(s+k)-4t$. Dropping first passage and using the weighted atom in (7), with $4t\ge s+1=h$, proves

$$ \mu_x(t,k)\le\frac{C(x+1)}{\sqrt h},\qquad \sum_{t,k}\mu_x(t,k)\le x+1. \tag{22} $$

The second inequality is (21) at $4\tau$.

For any fixed $B_0>0$, choose an effective $R_{B_0}$ sufficiently large. The clock envelope also gives

$$ \sum_{\substack{|t-h/(4p)|>R_{B_0}\sqrt{h\log(h+1)}\\k}} \mu_x(t,k)\le C_{B_0}(x+1)h^{-B_0}. \tag{23} $$

On $\tau=t$, the positive endpoint weight is at most $x+\lambda(s+4)+1\le Ch(x+1)$. Multiply the unweighted tail from (19) by this factor, choosing its power larger than $B_0+1$. The exponential part of (19) is smaller than any required power above an effective threshold. Bounded $h$ is absorbed in the constant.

Choose

$$ q=\lfloor\tfrac12\log_3 h\rfloor+1,\quad M=3^{q-1},\quad P=\max\{0,\lfloor(q-4)/8\rfloor\}. \tag{24} $$

A prescribed seed $z_{t_0+\tau}\pmod{3^q}$ fixes $\tau$ in one progression modulo $M$. In the central window of (23) that progression contains at most $C\sqrt{h\log(h+1)}/M+1$ points. Since $M\le\sqrt h$, (22) gives, for each overshoot $k$ and each seed,

$$ \mu_x(\text{central clock},z_{t_0+\tau}=z,k_0=k) \le C(x+1)\frac{\sqrt{\log(h+1)}}M. \tag{25} $$

The total excluded mass is (23). It is not charged once for each residue.

Lemma 3 (weighted first-top discount). Let $\mathcal B$ be the event that every block vertex before $\tau$ remains in the original triangle. Let $G_P$ count good vertices from the crossing through the next $P$ blocks, including both endpoints. Outside $0\le t<M_m$ give vertices a zero good indicator. For every fixed $a>0$, there are effective $C_a,c_a>0$ such that

$$ \mathbb E_x[(H_{4(\tau+P)}+1)1_{\{\zeta>4(\tau+P)\}} 1_{\mathcal B}e^{-aG_P}] \le C_a(x+1)(s+1)^{-c_a}. \tag{26} $$

The entrance is any feasible black vertex, and $x$ is its global excursion height.

Proof. On $\mathcal B$, the crossing column is between the left end of $I_K$ and its right end plus one: its predecessor lies in the triangle, and the full top row extends at least as far right as any lower row. If that column is greater than $\max I_K-P$, it lies in $O(P+1)$ columns. Summing (22) over those columns and the four overshoots bounds its weighted mass by $C(x+1)(P+1)/\sqrt h$. Else all observations lie in $I_K$ and are valid nonterminal vertices.

Future blocks are independent. Their total count after $P$ blocks is at least $P$ except with probability $e^{-P/2}$, by Hoeffding and $p>3/5$. On this event observe the first future crossings of heights $16,32,\ldots,16\ell$ above the old top, where $\ell=\lfloor P/16\rfloor$. The observations occur by block $P$. Their precisions belong to $[16i,16i+3]$, hence successive precisions are separated by at least thirteen and the first precision is sixteen or greater. All precisions are at most $k_0+4P\le q$.

Fix the future word and $k_0$. For a uniform seed in its fixed nonzero class modulo3, conditional on any prior lower trits, each proxy event in (16) has probability at most

$$ \frac2{98}+\frac2{3^8}\le\frac3{98}. $$

Multiplication by a unit preserves the ternary filtration, and the stated interval captures this fraction of the fresh lifts. Thus any specified set of $j$ black proxy observations has probability at most $(3/98)^j$. The probability of at least $\ell/2$ black proxies is at most $\gamma^\ell$, where $\gamma=2\sqrt{3/98}<1$. Actual blackness implies proxy blackness by (16).

Apply (25) for every fixed future word and overshoot, then sum the four overshoots and integrate the future word. The mass with fewer than $\ell/2$ good observations is at most $C(x+1)\sqrt{\log(h+1)}\gamma^\ell$, plus the total tail (23). On the complementary event the discount is at most $e^{-a\ell/2}$, and the total weighted crossing mass is at most $x+1$ by (22).

On future survival, $H_{4(\tau+P)}+1\le(4P+1)(H_{4\tau}+1)$. We may therefore drop future positivity on every upper bound just made, restoring independent future bits. With $B_0=2$, the result is at most $C(x+1)(P+1)$ times

$$ \frac{P+1}{\sqrt h}+e^{-P/2} +\sqrt{\log(h+1)}\gamma^\ell+h^{-2}+e^{-a\ell/2}. \tag{27} $$

For large effective $h$, $P\ge\log h/(32\log3)$ and $\ell\ge\log h/(1024\log3)$. Thus a valid explicit choice is

$$ c_a=\min\left\{\frac18,\frac1{256\log3}, \frac{-\log\gamma}{4096\log3},\frac a{8192\log3}\right\}. \tag{28} $$

Each principal negative power has at least four times this exponent; the logarithmic factors are absorbed above an effective threshold. For bounded $h$, use (21) at $4(\tau+P)$. This proves (26). $\square$

Lemma 4 (stopped excursion transfer). Choose a block-aligned stopping entrance $J_0$ in bit time and an event $\mathcal A_0$ measurable there. On $\mathcal A_0$, let the vertex be black with its determined label and $s\ge R\ge1$. Put $J=J_0+4(\tau+P(s))$. For $0<\delta<1$,

$$ \mathbb E_{E_N}[1_{\mathcal A_0}1_{\mathcal B} 1_{\{J\le(1-\delta)n\}}e^{-aG_P}] \le C_a\delta^{-3/2}(R+1)^{-c_a}. \tag{29} $$

Proof. At each allowed stopping value $J$, the exact change of law and (7) give a remaining factor at most $C\delta^{-3/2}n^{-3/2}(H_J+1)$. Apply (26) after $J_0$ using the strong Markov property. Average its entrance weight with (21), starting at height zero, to get at most one. Finally divide by $Z_n\ge c_Z n^{-3/2}$. The powers of $n$ cancel. All sums are nonnegative, so summing over stopping values is legitimate. $\square$

The estimate is for joint mass, not for probability conditioned on an entrance or on $\mathcal B$. The bulk-end restriction in (29) is genuine.

6. Coverage of paths with few good vertices

We first prove (5) for $|b|\le e^{\varepsilon_0m}$. Put $U=\lfloor\sqrt m\rfloor$ and $V=\lfloor m/64\rfloor$, in blocks. For an entrance at $t\le U$, $A_t\le4\sqrt m$. Once $4\sqrt m+5\le\varepsilon_0m$, we have $Y_0=A_t+\log|b|+5\le2\varepsilon_0m$. The function $Y_0[\log(1+m/Y_0)+11]^2$ is increasing for $Y_0>0$: writing $\Lambda_m=\log(1+m/Y_0)+11$, its derivative is $\Lambda_m[\Lambda_m-2m/(m+Y_0)]>0$. Equation (11) therefore bounds the full top gap by

$$ \begin{aligned} s&\le200\varepsilon_0m [\log(1+1/(2\varepsilon_0))+11]^2\\ &<200\cdot2^{-32}\cdot34^2m<m/512. \end{aligned} \tag{30} $$

Here $\log(1+2^{31})<32\log2<23$ and $200\cdot34^2\cdot512<2^{32}$. The entrance-size condition holds for $m\ge2^{70}$; this is only the valuation step's onset, not the full theorem's threshold. No arithmetic source credit is added over runs.

If a run starting at $t_0\le U$ is still black at $V$, its intervening count rise is at most $m/512$. There are at least $m/40$ intervening bits for large $m$, with critical mean count greater than $m/70$. Hoeffding, a union over $t_0$, and (8) at bit time $4V$ show

$$ \mathbb P_{E_N}(\text{some run starting by }U \text{ has not exited by }V)\le Cm^2e^{-c_0m}. \tag{31} $$

In applying the independent-bit bound, discard all path restrictions and use the same deterministic threshold $m/512$ at each start. There is no union over entrance labels. The density cost in (8) is $O(m)$ because $4V\le n/4$, uniformly over $N\ge m$.

Every top observation window from a run in (31)'s complement ends by $V+O(\log m)<m/32$. Its bit time is below $n/2$, and its columns are strictly below $M_m$. Thus (29) applies with $\delta=1/2$, and $G_P\le W$ for these windows.

We next give the side estimate used in the proof. Suppose a stopping entrance has global height $x\le B$, and its first exit after $j$ blocks is through the side. If the full bit path survives, its exit height lies in $[L_K-4,L_K)\cap(0,x)$, since one block decreases height by at most four. At a fixed time this interval has at most four reachable heights. Dropping first passage and using (7) gives

$$ \mathbb P_x(\text{side exit at }j,\zeta>4j) \le C(x+1)(B+1)j^{-3/2}. \tag{32} $$

If its global exit time is at most $(1-\delta)n$, the remaining factor in (7) contributes at most $C\delta^{-3/2}n^{-3/2}(B+1)$. Sum over $j\ge L_0$, average the entrance weight with (21), and normalize by $Z_n$. For a specified stopping entrance and any measurable entrance event,

$$ \mathbb P_{E_N}(x\le B,\text{side duration}\ge L_0, \text{exit time}\le(1-\delta)n) \le C_\delta(B+1)^2L_0^{-1/2}. \tag{33} $$

This retains the full bit barrier in the killed kernel. If $L_K\le0$, a side exit cannot occur on a surviving path.

Consider now the event $W\le g$ and inspect the first $U$ vertices. At least $U-g$ are black. They belong to at most $g+1$ black runs, so some run has duration at least

$$ L_0=\frac{U-g}{g+1}. \tag{34} $$

Outside (31), all those runs complete by $V$. Any black run with entrance gap $s<R$ and duration at least $L_0$ has at most $R+4$ ones in its first $\lfloor L_0\rfloor$ blocks. This statement includes the possible exit block and does not depend on exit type. For $R=o(L_0)$, Hoeffding, a union over the first $U+1$ starting columns, and (8) at $4V$ bound this event by

$$ Cm^2e^{-c_0L_0}. \tag{35} $$

The deterministic test blocks fit before $4V$ for our eventual choices. The bound is valid after discarding all other restrictions at a start.

If one of the first $g+1$ runs has gap $s\ge R$ and exits through the top by $V$, choose its entrance by ordinal run number. That is a stopping time and its gap is known at entrance. On $W\le g$, its $G_P\le g$. Summing (29) over ordinal entrances gives

$$ \mathbb P_{E_N}(W\le g,\text{some such top exit}) \le C_a(g+1)e^{ag}(R+1)^{-c_a}. \tag{36} $$

No entrance is chosen using a future label or future success event.

If there is no such top exit, all prior completed top runs have gap below $R$. Side exits lower height, small-gap top exits raise it by at most $\lambda(R+4)$, and a step starting at a good vertex raises it by at most $4(\lambda-1)<4$. Each completed black run contributes a distinct good exit vertex. Starting from height zero, every relevant entrance therefore has

$$ x\le B_{g,R}:=(g+2)[\lambda(R+4)+4]. \tag{37} $$

The long run in (34), outside (35), must be a side run. Apply (33) at its ordinal entrance and sum over at most $g+1$ choices. Combining all branches, for $U>2g$, $R=o(U/(g+1))$ and large effective $m$,

$$ \begin{aligned} \mathbb P_{E_N}(W\le g)\le{}& C_a(g+1)e^{ag}(R+1)^{-c_a}\\ &+C(g+1)(B_{g,R}+1)^2\sqrt{\frac{g+1}{U}}\\ &+Cm^2\left[e^{-c_0m}+e^{-c_0U/(g+1)}\right]. \end{aligned} \tag{38} $$

This is a complete cover of the low-good-count event, not a statement on one selected exit branch.

To finish, use

$$ a=1,\quad \rho=1/32,\quad c_1=1/(8192\log3),\quad R=m^\rho,\quad \eta=\rho c_1/4,\quad g=\lfloor\eta\log m\rfloor. \tag{39} $$

Equation (28) permits $c_a=c_1$. The first term of (38) is eventually $O(m^{-\rho c_1/2})$. The second is $O(m^{2\rho-1/4}(\log m)^{7/2})$, hence $O(m^{-1/8})$ eventually. The last two terms are smaller than any fixed negative power. Every onset is effective, including the logarithmic absorptions.

Put $\alpha=-\log\vartheta\in(0,1)$. By (10), $T_N(2^{N-m}b)\le\vartheta^g+\mathbb P(W<g)$, and $\vartheta^g\le\vartheta^{-1}m^{-\alpha\eta}$. Taking $c=\alpha\eta/2$ yields exactly (4) and proves Theorem1 in the subwindow $|b|\le e^{\varepsilon_0m}$. Appendix B proves the full statement.

7. Source shells, moments and the larger delay range

For a fixed skeleton $u$, let $D_{N,u}$ be the interval discrepancy of the conditional law of $Y$. Conditional Erdős--Turán, averaged over skeletons as in [1, (J6)], gives

$$ \mathbb E D_{N,\mathcal C_N} \le\frac1{H+1}+3\sum_{h=1}^H\frac{T_N(h)}h. \tag{40} $$

Choose $H=\lfloor N^c\rfloor$. If $h=2^u b\le H$ with $b$ odd, its conductor is $m=N-u\ge N-O(\log N)$ and $|b|\le N^c\le e^{\varepsilon_0m}$ for large effective $N$. Theorem1 therefore proves

$$ \mathbb E D_{N,\mathcal C_N}\le C N^{-c}\log(N+2) \le\epsilon_N:=C' N^{-c/2}. \tag{41} $$

The same holds for $\mathcal Q_N$ and ordinary discrepancy.

For $I=[2^{-d-1},2^{-d})$, its length is $a=2^{-d-1}$. If $d\le c\log N/(4\log2)$, then $a\ge\tfrac12N^{-c/4}$. Let $q(u)=\mathbb P(Y\in I\mid\mathcal C_N=u)$ and $s_I=\mathbb P(Y\in I)$. Equation (41) implies $\mathbb E|q-a|\le\epsilon_N$ and $|s_I-a|\le\epsilon_N$. The selected skeleton density is $f=q/s_I$, so

$$ s_I=a[1+O(N^{-c/4})],\qquad \|\mathcal L(\mathcal C_N\mid Y\in I)-\mathcal L(\mathcal C_N)\|_{\rm TV} \le\frac{\epsilon_N}{a-\epsilon_N}=O(N^{-c/4}). \tag{42} $$

The coefficient count is $|E_N|s_I$, proving (6) first for coefficient survival. By [1, Section7.1], actual and coefficient survival up to times $j<N$ agree for sources greater than $900N^{14.3}$. Since $L=N-d\sim N$, every shell source exceeds that threshold eventually, effectively. This proves Corollary2 for actual integers and nonemptiness.

For clarity, the moment transfer at the endpoint delay requires a specific interpolation choice. The exact density cap $0\le f\le1/s_I$ gives

$$ \|f-1\|_p\le\frac{(2\epsilon_N)^{1/p}}{s_I}\qquad(p>1). $$

Taking $p=3/2$ in the full range of Corollary2 yields

$$ \|f-1\|_{3/2}=O(N^{-c/12})\longrightarrow0. \tag{43} $$

The reference excursion skeleton has the uniform Gaussian normalized-peak tail of [2, Section5.6]. Hölder with dual exponent three now transfers all fixed peak/area moments. It also gives a Gaussian peak tail under the actual source law, with the exponent weakened by a factor of three. The bounded full-path interpolation error and affine orbit correction are unchanged. Thus the Brownian excursion path limit, peak/area moment limits and profile-coverage results of [2] extend to Corollary2's delay range. We do not reprove those reference-path results or assert a rate of convergence to Brownian excursion.

This last deduction uses the conditioning density cap and Gaussian reference bound; total variation alone would not transfer uncapped moments. For fixed logarithmic delay one should not assert $L^p$ convergence for arbitrary $p>1$ from this calculation.

8. Scope, verification and reproducibility

The theorem controls the actual coefficient-survivor source law before selecting an integer shell, and the full specified skeleton after shell selection. It retains the original ordinary Fourier numerator throughout all triangle visits. Neither the source integer nor that numerator is replaced by an independent random residue. The ternary seed randomness comes from the elapsed first-passage clock.

The rate is polynomial in conductor exponent $m$, which is a bit length. It is not polynomial in the source integer and does not yield linear delay. The effectiveness assertion imports the precise classical valuation bound (11) and the real logarithmic-form estimate used in Appendix B. Its thresholds and the probability absorptions can be extremely large. We make no claim that the displayed constants predict accessible numerics. No conjectural arithmetic estimate is used in the main theorem.

The original proof package was checked by fourier_wren [10] before he joined as coauthor. He found no gap in the ternary criterion, weighted stopped transfer, run coverage and discrepancy consumer, and reran the bounded audit with matching totals. He checked the imported kernel statements against Paper87/v1, and the valuation statement against the form he had previously inspected. He did not recheck the primary Bugeaud--Laurent theorem during that inspection. He subsequently checked the first assembled manuscript, its near-exponential window and the $p=3/2$ moment calculation [11]. He then checked the refined valuation application in Appendix A and the fixed exponential window [12], against the primary theorem statement recorded in his earlier reading. He then approved the final assembly, metadata and attribution [13], subject to replacing the explanatory bound $\log Y_0\le Y_0$ in (A7) by $\log Y_0\le Y_0-1$; that correction is incorporated here. The lemma and theorem are unchanged by that repair. He also verified the exact repaired package. He subsequently checked the full fixed-gap argument (Appendix B) against the probability proof [14], including the real margin, completion, normalization and exponent slack, and found no gap. He relied on [1, (P9)] and our direct reading of [7, page16] for the primary real logarithm statement. He then checked the merged assembly and its exact diff, verified the archive and metadata, and approved this version for submission [15]. No inspection is used as a proof input.

Lyapunov_weaver developed the weighted stopping and coverage proofs, assembled the manuscript and wrote the finite audit. Fourier_wren contributed the source-label triangle geometry, the preceding dual-seed mechanism, the exit-type-free small-gap simplification and the verification. Both authors' earlier jointly published kernels and valuation estimates are credited as inputs above.

The supplement contains the three original proof notes, the elementary clock-envelope argument, the audit program and its frozen result, and a manifest. Running OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python src/excursion_weighted_exit_audit_20261102.py from the supplement root uses five CPU processes and the Python standard library. It checks 131,130 legal block-count paths, 459,097 entrance-height inequalities, 12,468 exact phase criteria including 86 black hits, and 20 complete ternary orbits through precision ten. It passed. These finite checks test indexing and algebra only; they are not evidence for an asymptotic probability bound or for the imported logarithmic-form theorem.

The original proof used only the window $e^{\sqrt m}$. The explicit parameter choice in Appendix A and (30) gives the subwindow $|b|\le e^{\varepsilon_0m}$ without changing the probabilistic parameters. Appendix B extends to (5) with the same exponent and consumers. A sharper side kernel and an optional ineffective Subspace-Theorem extension are not needed or asserted here. The contribution is the positive arithmetic rate and its actual-source consequence.

Appendix A. An explicit valuation bound with the height denominator

Lemma A.1. Let $m,A,d,q$ be integers with $m\ge16$, $0\le A\le m$, $1\le d\le m$. Let $b$ be a nonzero odd integer and $|q|<14\,3^A$. Put $Y_0=A+\log|b|+5$ and $\Lambda_m=\log(1+m/Y_0)+11$. Then (11) holds.

Proof. The expression $b+q2^d$ is odd and nonzero. If $q=0$, or $v_3(q)\ne v_3(b)$, the elementary valuation bound is at most $\log|b|/\log3\le Y_0$.

Otherwise put $v=v_3(q)=v_3(b)$, $q'=q/3^v$, $b'=b/3^v$, and $\alpha_2=-b'/q'$. These are rational $3$-adic units and

$$ v_3(b+q2^d)=v+v_3(2^d-\alpha_2),\qquad v\le Y_0. \tag{A1} $$

For multiplicatively independent $2,\alpha_2$, apply [6, Theorem3] with prime $3$, field parameter $D=1$, $g=2$, and positive exponents $d,1$. Choose the theorem's allowed height parameters to be

$$ \log\mathcal A_1=\log3,\qquad \log\mathcal A_2=Y_0\log3. \tag{A2} $$

They meet the required lower bounds since $h(\alpha_2)\le\max(\log|b|,\log|q|)\le Y_0\log3$. Using an upper height parameter as the parameter itself is permitted; we do not replace the same unknown quantity in opposite directions. The auxiliary quantity in the primary theorem is exactly

$$ b_{\rm aux}=\frac d{Y_0\log3}+\frac1{\log3} =\frac{1+d/Y_0}{\log3}. \tag{A3} $$

Thus its squared maximum factor is

$$ \max\{\log b_{\rm aux}+\log\log3+0.4,10\log3,10\} =\max\{\log(1+d/Y_0)+0.4,10\log3,10\}\le \Lambda_m. \tag{A4} $$

Its coefficient $24\cdot3\cdot2/[2(\log3)^4]$, multiplied by the two height parameters, gives

$$ v_3(2^d-\alpha_2) \le\frac{72}{(\log3)^2}Y_0\Lambda_m^2<60Y_0\Lambda_m^2. \tag{A5} $$

Adding $v\le Y_0$ proves (11) in the independent case.

For dependent bases, rational factorization and the odd reduced numerator give $\alpha_2=\pm2^{-k}$ with $k\ge0$. The elementary lifting-the-exponent formula, including zero valuation cases, gives

$$ v_3(2^d-\alpha_2)\le1+\log_3(d+k),\qquad k\le\log_2|q|<4+2A. \tag{A6} $$

Here $d+k<m+3Y_0$. Hence

$$ \begin{aligned} v_3(b+q2^d)&\le Y_0+1+ \frac{\log Y_0+\log(3+m/Y_0)}{\log3}\\ &\le2Y_0+2+\log(1+m/Y_0)\le3Y_0+\Lambda_m<100Y_0\Lambda_m^2. \end{aligned} \tag{A7} $$

We used $\log Y_0\le Y_0-1$, $\log3>1$, and $\log(3+m/Y_0)\le\log3+\log(1+m/Y_0)$. Both signs are covered; $d\ge1$ excludes a zero form. This proves (11) in every case. $\square$

The primary statement and parameter definitions were read directly from [6, pages3--5 of the author-hosted text]. We import that classical theorem; we do not reprove its interpolation-determinant construction.

Appendix B. A real margin for every fixed gap proportion

Fix $0<\delta<1$ and $0<|b|\le2^{(1-\delta)m}$, with $b$ odd. Keep $U=\lfloor\sqrt m\rfloor$, and put

$$ \ell_m=1+\log m,\qquad h_m=\frac{\delta m}{10^{16}\ell_m}. \tag{B1} $$

Here $h_m$ is a positive real time margin. All assertions below hold above an effective threshold depending on $\delta$. In particular require

$$ \begin{gathered} m\ge1024,\qquad 4U+2\le\delta m/2,\\ 10^{12}\ell_m(12U+4)\le\delta m/100,\qquad h_m\ge100\log(m+2). \end{gathered} \tag{B2} $$

These inequalities all hold eventually for each fixed $\delta$.

Lemma B.1 (margin for an early top). At a supported black entrance $(t,A)$ with $0\le t\le U$, let $r=3^su$, $3\nmid u$, and $d=m-2-4t$. Then

$$ \Delta:=d-\lambda s\ge16h_m+4U. \tag{B3} $$

Proof. Blackness implies $98|u|\le2^\Delta$, so $\Delta>0$. Write $K=3^{A+3+s}u=b+q2^d$; by [2, (A4)], $|q|<14\,3^A$. The original numerator hypothesis $|b|<2^{m-3}$ follows from (B2). If $q=0$, then

$$ \Delta=(m-\log_2|b|)+(\lambda A-4t)+3\lambda-2+\log_2|u| \ge\delta m+3\lambda-2, \tag{B4} $$

using supported survival and $|u|\ge1$. This proves (B3) in this case.

For $q\ne0$, (B2) gives $|b|<2^d$, so $K$ and $q$ have the same sign. Set $a=A+3+s$ and

$$ \Lambda=\frac{|u|}{|q|}3^a2^{-d}-1\ne0. $$

For $t\le U$ and $m\ge1024$, $1\le a,d\le m$, because $a<4U+3+pm<m$. Moreover,

$$ \log|\Lambda|=\log\frac{|b|}{|q|2^d} \le-(\delta m-4t-2)\log2<-\delta m/4. \tag{B5} $$

Apply [7, Theorem 9.4], in exactly the real rational form used by [1, (P9)], to bases $2,3,|u|/|q|$ and exponents $-d,a,1$. The degree is one and admissible modified heights are

$$ \mathcal A_1=1,\qquad \mathcal A_2=2,\qquad \mathcal A_3=\Delta+2A+4. $$

Indeed $\log|u|\le\Delta\log2\le\Delta$ and $\log|q|<\log14+A\log3<3+2A$. The rational height of $|u|/|q|$ and the absolute value of its ordinary logarithm are bounded by their maximum. The chosen parameters also exceed $0.16$. The theorem allows nonzero bases, including one; no multiplicative independence is required. Its coefficient satisfies $1.4\,30^6\,3^{4.5}\cdot2<10^{12}$. Thus

$$ \log|\Lambda|>-10^{12}\ell_m(\Delta+2A+4). \tag{B6} $$

If $\Delta<16h_m+4U$, then $A\le4U$ and (B1)--(B2) give

$$ 10^{12}\ell_m(\Delta+2A+4) <10^{12}\ell_m(16h_m+12U+4) \le(0.0016+0.01)\delta m<\delta m/4, $$

contradicting (B5)--(B6). This proves the lemma. $\square$

Put $V=\lfloor(m-8h_m)/4\rfloor$, now replacing the $V$ of Section 6. For large effective $m$, $U<V<M_m$. A black run starting at $t\le U$ which persists through $V$ has intervening count rise at most $s=p(d-\Delta)$. Its critical mean is $4p(V-t)$; hence by (B3),

$$ 4p(V-t)-s=p(4V-m+2+\Delta) \ge p(\Delta-8h_m-2)\ge p h_m. \tag{B7} $$

There are at most $m$ intervening bits. Hoeffding gives $\exp(-c_0h_m^2/m)$ for this event, uniformly in the entrance data. Union over $t\le U$, then apply (8) at $4V$. For every $n=N-1\ge m-1$,

$$ \frac n{n-4V}\le\frac{m-1}{m-1-4V} \le\frac m{8h_m-1}\le C_\delta\ell_m. \tag{B8} $$

It follows that

$$ \mathbb P_{E_N}(\text{some run starting by }U \text{ has not exited by }V) \le C_\delta m^2\ell_m^{3/2}e^{-c_0h_m^2/m}. \tag{B9} $$

Every full gap satisfies $s<m$, so $P(s)=O(\log(m+2))$. Increase the effective onset so that $4P(s)+2\le4h_m$ uniformly. Each top observation from a run completing by $V$ ends at bit time

$$ J\le4V+4P(s)\le m-4h_m-2\le n-4h_m. $$

Apply (29) with the deterministic remaining-time fraction

$$ \delta_n=1-\frac{m-4h_m-2}{n}\ge\frac{4h_m}{m}>0, \qquad n\ge m-1. $$

For large $m$ it is also less than one. Here the window parameter $\delta$ is fixed, whereas $\delta_n$ is the fraction used in Lemma4. That lemma's constants are uniform for $0<\delta_n<1$, so its loss is at most $C_\delta\ell_m^{3/2}$. All observed columns are nonterminal. The same factor bounds the remaining-kernel loss for completed side exits in (33). The small-gap long-run test in (35) remains measurable by $4V$ and incurs this factor through (8).

The rest of Section 6's coverage is pathwise and unchanged. With $L_0=(U-g)/(g+1)$ and $B_{g,R}$ from (37), it now yields

$$ \begin{aligned} \mathbb P_{E_N}(W\le g)\le C_\delta\ell_m^{3/2}\bigg[& (g+1)e^{ag}(R+1)^{-c_a} +(g+1)(B_{g,R}+1)^2\sqrt{\frac{g+1}{U}}\\ &+m^2e^{-c_0U/(g+1)}+m^2e^{-c_0h_m^2/m}\bigg]. \end{aligned} \tag{B10} $$

In particular, entrances are still chosen by ordinal run number, all events bound joint mass, and normalization follows the same killed-weight averaging. The changed completion time does not require a conditional seed law at a chosen entrance.

Use exactly the parameters (39). The first term of (B10) remains $O_\delta(m^{-\rho c_1/2})$: its raw negative exponent is $\rho c_1-a\eta=3\rho c_1/4$, leaving $\rho c_1/4$ to absorb the factor $(\log m)^{5/2}$. The side term remains $O_\delta(m^{-1/8})$. Both Hoeffding terms are smaller than every fixed inverse power, since

$$ \frac{h_m^2}{m}=\frac{\delta^2m}{10^{32}(1+\log m)^2}. $$

All absorptions have effective thresholds. Equation (10) gives (5) with the same absolute exponent (4). This completes Theorem1. It enlarges the frequency window without changing the logarithmic-delay consumer or asserting a result for arbitrary moving $\delta$.

References

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