A finite obstruction and the route to its proof

For a rational prime \(p\equiv1\pmod4\), choose conjugate Gaussian prime factors \(\pi_p,\overline{\pi_p}\), each of norm \(p\). For a finite set \(\mathcal P\) of such rational primes, put \[\mathcal A(\mathcal P)=\{z\in\mathbb Z[i]: z\not\equiv0\pmod{\pi_p},\quad z\not\equiv0\pmod{\overline{\pi_p}}\quad(p\in\mathcal P)\}.\] Changing the chosen associates does not change this set. It is periodic under translations by \(Q\mathbb Z^2\), where \(Q=\prod_{p\in\mathcal P}p\).

Section 2 first proves that this finite obstruction implies Theorem 1. The remaining sections construct it. We fix an arbitrary deterministic infinite self-avoiding lattice walk with steps at most \(D\). Randomness enters only through sampled times and auxiliary choices of one Gaussian factor over each split prime. The aim is to show that such a walk cannot avoid all the selected zero classes.

The geometric sampling operation in Section 4 chooses a difference uniformly from the distinct differences of a long walk segment and then chooses one of its endpoints. There are many distinct differences, and a random collection of signed prime factors usually separates them. These facts force the sampled endpoint to carry more joint entropy in its residues than the earlier position did. The proof uses a torus-degree argument for the differences, followed by determinant divisibility and concentration for the factor choices. Disjoint neighborhoods in the Boolean cube avoid a separate count of all lattice directions.

The next task is coverage: for most factors of norm \(p\), the terminal residue must have probability at least \(\tau/p\) outside fewer than \(p^{1-\beta}\) exceptional residues. Joint entropy alone, at the accuracy available early in the construction, does not give this conclusion. Section 5 supplies the additional argument before the large schedule of parameters is introduced. Repeated continuations from an exact later checkpoint give one displacement vector shared by every residue coordinate. If coverage failed, this vector would put the true residue on a short list with appreciable probability. Its entropy cost is too small to create that deficit in many coordinates at once.

Section 6 arranges the geometric operations in a common schedule with decreasing scales. It first establishes the entropy and displacement estimates, then applies the coverage transfer backward through the checkpoints. Every prime batch uses the same terminal time law, denoted by \(t_*\). An independent uniform offset makes this law nearly invariant under the time shifts needed later.

Finally, assume that the walk avoids the chosen zero classes. Section 7 uses short increment words—finite lists of consecutive increments—to test candidate starting residues. Coverage and self-avoidance make these tests informative. Each prime batch near \(T\) incurs conditional information at least \(c/\log T\) per increment, where \(c>0\) depends only on \(D\). Section 8 adds these costs using nested residue vectors and dyadic word lengths at the common time law. Block subadditivity bounds their sum by the entropy available in one bounded increment, whereas the selected batches make the lower bounds diverge. This proves the finite sieve theorem. The use of approximate translation invariance and block subadditivity to bound accumulated information is methodologically related to Tao’s entropy-decrement argument [12]. Here the signed residue batches, coverage transfer, and costs forced by zero avoidance are established in full within this paper.

Split primes

The ring \(\mathbb Z[i]\) is Euclidean and hence has unique factorization. For each rational prime \(p\equiv1\pmod4\), its two conjugate Gaussian prime factors are nonassociate, have norm \(p\), and have residue fields of size \(p\); see [2]. We refer to the choice of one of these two factors as a sign. Signs of distinct rational primes will be independent and fair.

We use the following consequences repeatedly. If \(\pi\mid v\) and \(\overline\pi\mid v\), then \(p\mid v\), meaning that \(p\) divides both integer coordinates of \(v\). If \(a\in\mathbb Z\), then \(\pi\mid a\) if and only if \(p\mid a\). Every nonzero multiple of \(\pi\) has length at least \(\sqrt p\). Finally, multiplication by a Gaussian integer \(\alpha\) acts on \(\mathbb Z^2\) by an integer matrix of determinant \(N(\alpha)\). Thus if \(\alpha\mid v,w\), then \[\det(v,w)\in N(\alpha)\mathbb Z.\]

For \(T>1\), let \[\mathcal B_T=\{p\text{ rational prime}:T\le p\le2T,\ p\equiv1\pmod4\}, \qquad k_T=|\mathcal B_T|, \qquad L_T^*=\frac1{k_T}\sum_{p\in\mathcal B_T}\log p.\] All uses have \(T\) sufficiently large that \(k_T>0\). For one fixed batch we abbreviate \(k_T,L_T^*\) by \(k,L_*\). The prime number theorem in arithmetic progressions for the fixed modulus \(4\) gives [11] \[ k_T=(1/2+o(1))\frac{T}{\log T}, \qquad \log T\le L_T^*\le\log(2T). \tag{1}\] Indeed Selberg’s weighted asymptotic \(\sum_{p\le x,\ p\equiv1(4)}\log p\sim x/2\), subtracted at \(T\) and \(2T\), gives a sum over \(T<p\le2T\). Adding the possible endpoint term at \(p=T\) changes it by at most \(\log T\). The weighted sum over \(\mathcal B_T\) therefore equals \((1/2+o(1))T\) and lies between \(k_T\log T\) and \(k_T\log(2T)\), whose ratio tends to \(1\). Only fixed-modulus, fixed-ratio intervals are involved. In particular, all constants implicit in (1) hold for every sufficiently large \(T\).

Entropy conventions

For a finite variable \(X\) with law \(p\), \(H(X)=-\sum_xp(x)\log p(x)\), with \(0\log0=0\). We use conditional entropy and conditional mutual information in their usual finite-alphabet forms: \[I(X;Y\mid Z)=H(X\mid Z)-H(X\mid Y,Z).\] Standard identities and inequalities may be found in Cover and Thomas [3]. When residue maps are selected randomly, we first fix those maps and compute entropy in the walk-sampling experiment; only afterward do we average over the maps. An expectation outside an entropy in this usage is an average of entropies, not the entropy of a mixture that includes the map choices.

For laws \(p,q\) on a finite set, put \(\mathop{\mathrm{TV}}(p,q)=\frac12\sum_x|p(x)-q(x)|\). Relative entropy is \(D_{\mathrm{KL}}(p\Vert q)=\sum_xp(x)\log(p(x)/q(x))\), with the usual value \(+\infty\) if \(q\) vanishes where \(p>0\). In particular, on an alphabet of size \(a\), \[D_{\mathrm{KL}}(p\Vert\mathrm{Unif})=\log a-H(p), \qquad D_{\mathrm{KL}}(p\Vert q)\ge2\mathop{\mathrm{TV}}(p,q)^2.\] The second inequality is Pinsker’s inequality in natural logarithms [3].

For completeness, let \(P,Q\) be finite probability laws and put \(B=\{x:P(x)>Q(x)\}\), \(u=P(B)\), and \(v=Q(B)\). The log-sum inequality, from convexity of \(t\log t\), gives \[D_{\mathrm{KL}}(P\Vert Q) \ge D_{\mathrm{KL}}(\operatorname{Ber}(u)\Vert \operatorname{Ber}(v)).\] Here \(\operatorname{Ber}(u)\) is the Bernoulli law with success probability \(u\). As a function of \(u\), the expression on the right has value and first derivative zero at \(u=v\), and second derivative \(1/[u(1-u)]\ge4\). It is therefore at least \(2(u-v)^2=2\mathop{\mathrm{TV}}(P,Q)^2\), with boundary cases by continuity.

We shall also use \(h(u)=-u\log u-(1-u)\log(1-u)\) for binary entropy, with its continuous values at \(0\) and \(1\).

Sampling a deterministic walk

Fix \(D\ge1\) and an infinite deterministic sequence \((z_t)_{t\ge0}\) of distinct points of \(\mathbb Z[i]\), with \(|z_{t+1}-z_t|\le D\). A random time is a finitely supported nonnegative integer-valued variable. All later sampling procedures produce such times. The walk itself is never assigned a probability law.

Recall the increment bound \[K_D=|\{u\in\mathbb Z[i]:|u|\le D\}|.\] This includes the zero vector; although a self-avoiding walk never uses it as an increment, this harmless upper bound simplifies notation.

Continuity and short lists

The next lemma adapts the error-indicator proof of Fano’s inequality to lists; see [3].

How geometry produces many differences

Separating differences with randomly chosen factors

Many distinct differences give high projected entropy only if reduction separates them. The next lemma excludes collisions by ruling out nonzero multiples in a rectangle containing differences of differences. Recall that a sign choice over a split rational prime means selecting one of its two conjugate Gaussian factors, with equal probability.

From differences to endpoint entropy

The two geometric estimates now give a sampling operation that can be iterated. We first specify how entropy is averaged over residue maps; the same convention will be used throughout the sampling schedule.

Fix a batch \(\mathcal P\) of \(k\) distinct primes \(p\equiv1\pmod4\) in \([T,2T]\), and write \[L_*=\frac1k\sum_{p\in\mathcal P}\log p.\] A signed subset \(\mathcal S\) consists of a subset of distinct primes from \(\mathcal P\), with one of the two conjugate Gaussian factors chosen for each. Thus its size counts rational primes, and it never contains both factors of the same prime. For a finitely supported point variable \(Z\in\mathbb Z[i]\), define \[F_{\mathcal S}(Z)=(Z\bmod\pi)_{\pi\in\mathcal S}, \qquad f_Z(s)=\mathbb E_{\mathcal S}H(F_{\mathcal S}(Z)), \quad 0\le s\le k,\] where the \(s\) underlying primes are chosen uniformly without replacement and their signs independently and fairly. All index choices are independent of the point variables. Entropy is computed for each fixed signed subset before averaging.

We have \[f_Z(0)=0,\qquad 0\le f_Z(s)\le sL_*.\] The function \(f_Z\) is concave on the integer interval. Indeed, choose a uniform random ordering of the batch and independent signs. The difference \(f_Z(s+1)-f_Z(s)\) is the average conditional entropy of the next coordinate given the first \(s\). Exchangeability of the unexposed coordinates and the fact that conditioning decreases entropy show that these differences decrease with \(s\). This is the usual entropy submodularity argument [3].

Let \(Z=z_\tau\), where \(\tau\) is any finitely supported random time. For parameters \(U>0\) and \(0<g<10^{-2}\), put \(n=\lfloor e^{gU}\rfloor\). For an exact time \(t\), write \[E_t=\{z_t,z_{t+1},\ldots,z_{t+n}\}.\] Conditional on \(\tau=t\), choose \(h\) uniformly from the set \(E_t-E_t\), including zero, and then choose one ordered pair \((X,Y)\in E_t^2\) with \(X-Y=h\) by a fixed deterministic rule. Thus differences, not ordered pairs, receive equal weight. Finally let \(V\) equal \(X\) or \(Y\), each with probability \(1/2\), using an independent coin. Self-avoidance identifies each endpoint with a unique walk time in \([t,t+n]\), so this defines a forward transition of the time law. The transition advances time by at most \(n\) and uses no prime indices.

The right side retains one half of the starting entropy fraction and adds one half of the maximal fraction. When successive transitions match the next input coordinate count to the preceding output count, repetition therefore drives the entropy deficit downward. Section 6 will arrange this matching, starting even from a deterministic time. The error \(O_D(g)\) is uniform; choosing the scale large enough permits \(g\) to decrease as well.

One shared conditioning cost

We first isolate why a common data vector is useful. If one separately conditioned on data chosen for each coordinate, its entropy cost would be paid once per coordinate. A single vector can instead be paid for once in the joint entropy, then distributed over the coordinates.

Let \(\Lambda\) be a lattice, that is, a finitely generated free abelian group, and let \(k\ge1\) be an integer. For \(1\le j\le k\) and \(\sigma\in\{+,-\}\) let \[\phi_{j,\sigma}:\Lambda\longrightarrow G_{j,\sigma}\] be homomorphisms to finite abelian groups of order \(p_j\). Assume \(T\le p_j\le2T\) and \(T\ge2\), and put \[\overline L=\frac1k\sum_{j=1}^k\log p_j,\qquad \mathop{\mathrm{av}}_c u_c=\frac1{2k}\sum_{j=1}^k\sum_{\sigma\in\{+,-\}}u_{j,\sigma}.\] Here \(c=(j,\sigma)\) denotes one signed coordinate; write \(p_c=p_j\), \(G_c=G_{j,\sigma}\), and \(\phi_c=\phi_{j,\sigma}\). For an integer \(1\le s\le k\), choose \(s\) distinct indices uniformly and one independent fair sign at each. Denote this signed subset by \(S\), and write \(F_S(Y)=(\phi_c(Y))_{c\in S}\) for a finite \(\Lambda\)-valued variable \(Y\). The choice of \(S\) is auxiliary and independent of every variable whose entropy is measured. In particular, \(\mathbb E_S H(F_S(Y))\) means the average of the entropies for fixed \(S\).

We will contradict (8) with the short-list bound of Lemma 6. Its success event may depend on both the residue and the data; independence of these variables is unnecessary.

A transfer from later coverage

We now mix a family of displacement laws. Each conditional law will have a probability floor outside a small exceptional set. High joint entropy of the starting position, together with a small cost for shared data, will strengthen the exceptional-set bound for the mixture.

Let \(A\) be a finite variable, let \(y\) map its alphabet to \(\Lambda\), and put \(Y=y(A)\). For every value \(a\) of \(A\), specify a finitely supported displacement law \(\nu_a\) on \(\Lambda\). Conditional on \(A\), sample \(h_1,\ldots,h_n\) independently with law \(\nu_A\), where \(n\ge1\) is an integer, and define \[\mathcal H=(h_1,\ldots,h_n),\qquad Z=Y+h_1.\] In the walk application, \(A\) will record an exact intermediate time, \(Y\) its position, and \(\nu_a\) the displacement law from that time to the terminal position. Here \(A\) is simply the finite mixing variable; its influence on both \(Y\) and \(\mathcal H\) is retained. The coordinate subset \(S\) used to measure entropy is independent of this whole experiment.

For a variable \(X\) in an alphabet of size \(p\), say explicitly that it has \((\tau,\beta)\) coverage when \[ \#\{u:\mathbb P(X=u)<\tau/p\}<p^{1-\beta}. \tag{9}\] The strict inequality concerns the number of exceptional residues; no claim is made that this set is empty.

The transfer below lowers the probability floor from \(\tau'\) to \(\tau\), but improves the exceptional-set bound from \(p^{1-\beta'}\) to \(p^{1-\beta}\), with \(\beta\ge\beta'\). The exceptional sets of the conditional laws may vary with \(a\), so mixing them does not directly give this improved bound. The theorem asks for three inputs: coverage for each conditional law, a joint entropy budget that can absorb the common data, and enough independent trials to make the candidate lists small.

The theorem has charged the repeated displacements once, through \(H(\mathcal H)\). Its independence assumption is only conditional on the exact checkpoint \(A\). In the next section the schedule will make the joint entropy error and this single data cost small enough to transfer coverage at every step of a backward induction.

Choice of constants and scales

Fix \(D\ge1\), and set \[\beta_0=\frac1{200},\qquad e_0=\frac{\beta_0}{50}.\] We will first choose a fixed iteration count \(l_0\) and a sufficiently small constant \(0<g_0<10^{-2}\), depending only on \(D,e_0\). They determine a constant \(c_0>0\); only then do we choose a sufficiently large fixed \(B>2\). The precise requirements are specified below. All these constants are fixed before the asymptotic parameter \(M\), and are independent of the walk and starting time.

Let \(M\) be a positive integer tending to infinity, and put \[X_m=100^m\quad(\lceil M/2\rceil\le m\le M).\] Use every batch \(\mathcal B_T\) with \[ T=B^j,\quad j\in\mathbb Z,\qquad X_m\le\log T\le1.05X_m \quad\text{for some }\lceil M/2\rceil\le m\le M. \tag{17}\] The batches are disjoint because \(B>2\). Write \(T_+=\exp(1.05X_M)\). For this value of \(M\), the union of all these batches is the finite prime selection that will be used in Theorem 2.

Starting at time \(0\) on a fixed infinite self-avoiding \(D\)-step walk, apply the transition of Lemma 9 with scales \(U\) in decreasing order. For each \(m\), use the three bands in Table 1. In a band with parameter \(g\), start at its upper endpoint in \(\log U\), and successively subtract \(2\log(1/g)\) as long as the resulting scale stays in that band. Equivalently, successive \(U\)’s have ratio \(g^2\). All randomizations are fresh conditional on the current exact walk time. No prime subset or sign choice enters these transitions.

After all transitions in all windows have been performed, add one independent uniform offset in \(\{0,\ldots,N-1\}\), where \(N=\lceil T_+^{10}\rceil\). Denote the resulting random time by \(t_*\). Thus all batches will be measured at the same time law.

A checkpoint is a stage just before a specified transition, or the terminal stage. It also denotes the random walk time at that stage. Conditioning on an exact start at a checkpoint means fixing that walk time and then running the remaining schedule with fresh randomness. Such conditional experiments are defined at every nonnegative integer start, whether or not it has positive probability in the experiment starting from time \(0\). The term “exact start” specifies a new continuation experiment, not conditioning on an event that may have probability zero.

Iterating and preserving joint entropy

For a fixed batch write \(f_Z(s)\) as in Section 4. Every use of Lemma 9 below has \(q\le k_T\). For each entropy estimate we begin at a scale where \(q\le k_T\); all subsequent scales in that block are smaller. The other limiting conditions hold uniformly: \[\log U\ge cX_{\lceil M/2\rceil},\qquad \log(1/g)=O(M^{20}),\qquad \log\log T=O(M).\] In particular \(g^CU/\log T\to\infty\) for every fixed \(C>0\).

Suppose \(l\) successive transitions in one band end just before its next scale \(V\), and that the initial subset size is at most \(k_T\). The output size of one transition and the input size of the next are both exactly \(\lfloor g^2U/L_*\rfloor\). Writing the normalized entropy deficit as \(d_j\), Equation (4) gives \(d_{j+1}\le d_j/2+C_Dg\). Since \(d_0\le1\), iteration gives \(d_l\le2^{-l}+2C_Dg\), and hence \[ \frac{f_Z(\lfloor V/L_*\rfloor)} {\lfloor V/L_*\rfloor L_*} \ge1-2^{-l}-O_D(g) \tag{19}\] at that checkpoint. The same bound, with another \(O_D(g)\) error, holds at every later checkpoint. Indeed Lemmas 12 and 4 show that the entropy loss at this fixed subset size is \(O_D(gV)\), whereas \(\lfloor V/L_*\rfloor L_*\ge V/2\) for large \(M\). The argument is uniform for every initial law, including an exact start fixed before these \(l\) transitions. The later entropies here are marginal entropies in that experiment; we are not conditioning on the exact later time, which would determine its walk position.

Choose a small absolute \(a_0>0\) such that \(a_0T/L_*\le k_T\) for large \(T\), using (1). For a batch \(T\) in window \(m\), begin an entropy estimate at the first top-band scale \(U\le a_0T\). Its size lies between \(g_0^2a_0T\) and \(a_0T\). Choose a fixed integer \(l_0\) large enough that \(2^{-l_0}<e_0/3\), and take \(g_0\) small enough to make the total error in (19) and its preservation less than \(2e_0/3\). These choices can be made before \(B\) and \(M\). Run \(l_0\) transitions and let \(q_T=\lfloor V/L_*\rfloor\) be the recorded size. All these scales lie in the top band for large \(M\). At every later checkpoint, for its point variable \(Z\), \[ f_Z(q_T)\ge(1-e_0)q_TL_*, \qquad q_TL_*\ge c_0T \tag{20}\] with \(c_0>0\) depending only on the fixed choices. For example any sufficiently small constant below \(a_0g_0^{2(l_0+1)}/2\) works in the second inequality.

By (1), \(k_{T'}\log(2T')\le C T'\). Since the batches are a subsequence of the geometric progression \(B^j\), we can now choose \(B>2\) so large that for every used batch \[ \sum_{T'<T}k_{T'}\log(2T')\le e_0c_0T. \tag{21}\] In fact the left side is at most \(CT/(B-1)\). This completes the fixed-constant choices in the order \(D,e_0\), then \(l_0,g_0,c_0\), then \(B\), and finally \(M\). The top band has now supplied a batch entropy of order \(T\) that survives the remaining schedule, and the geometric spacing makes all smaller residue alphabets inexpensive compared with it. We next build the more accurate conditional estimates needed for coverage.

Intermediate checkpoints and almost uniform marginals

These checkpoints will transmit coverage backward from an accurate one-coordinate base case. Their spacing must allow joint entropy to grow while keeping subsequent displacement data inexpensive. For each batch \(T\), put \[ \beta_i=2^{-i}\beta_0,\qquad r_0=\tfrac14,\qquad r_{i+1}=r_i-2\beta_i. \tag{22}\] Stop at the least \(J=J(T)\) with \(\beta_J\log T\le M^{20}\). Since \(\beta_0\log T>M^{20}\) for sufficiently large \(M\), we have \(J\ge1\); minimality then gives \[ J=O(M),\qquad M^{20}/2<\beta_J\log T\le M^{20},\qquad 0.23<r_i\le0.25. \tag{23}\] The last assertion follows from \(r_i=1/4-4\beta_0(1-2^{-i})\). Let \(A_i\) be the checkpoint just before the first scheduled operation with \(U\le T^{r_i}\). All these checkpoints lie in the middle band for this \(m\), since \[0.23X_m<r_i\log T\le0.2625X_m.\] Its scale is in \((g_1^2T^{r_i},T^{r_i}]\). Let \(S_i\) be a deterministic upper bound for the remaining time shift from \(A_i\) supplied by Lemma 12, and define \[\mathcal D_i=\{v\in\mathbb Z[i]:|v|\le DS_i\}.\] This set does not depend on the exact start. From every exact start \(A_i=a\), every possible displacement to any later checkpoint, including \(z_{t_*}-z_a\), belongs to \(\mathcal D_i\). Since the scale at \(A_i\) is at most \(T^{r_i}\), lattice-point counting gives \[ \log|\mathcal D_i|\le C_D T^{r_i}. \tag{24}\] Consequently, uniformly in the starting law, for any later checkpoint \(A\) we have \[ H(z_A-z_{A_i})\le C_D T^{r_i}. \tag{25}\] The common support bound is stronger than a separate conditional entropy bound at each exact start. It will control the unconditional entropy of displacement data when the next checkpoint is random.

Fix an exact start at \(A_i\), where \(i<J\), and let \(Y=z_{A_{i+1}}\). There is a deterministic subset size \(s_i\), depending on the batch and the schedule but not on that start, such that \[ f_Y(s_i)\ge(1-e^{-cM^3})s_iL_*, \qquad s_iL_*\ge\exp(r_i\log T-O(M^6)) \tag{26}\] for a fixed \(c>0\). To see this, take \(\lceil M^3\rceil\) iterations starting at \(A_i\). The first log-scale differs from \(r_i\log T\) by less than \(2M^3\), and these iterations consume \(O(M^6)\) log-scale. The starting subset size is at most \(k_T\), since \(U\le T^{r_i}\le T^{1/4}=o(T)\) and \(k_TL_*\asymp T\). They fit before \(A_{i+1}\), because \[(r_i-r_{i+1})\log T=2\beta_i\log T>2M^{20}.\] Equation (19), preservation, and \(g_1=e^{-M^3}\) prove the first bound, and the recorded scale proves the second.

Finally, conditional on any exact start at \(A_J\), proceed to the bottom band for this \(m\). Take \(\lceil200M^{20}\rceil\) iterations, recording a subset size \(s\). Begin at its first scale, which is at most \(\exp(0.10X_m)=o(T)\), so again the initial subset size is at most \(k_T\). The required log-scale span is \(O(M^{40})=o(X_m)\), so the iterations and their recording scale fit in that band. Their deficit, including preservation to \(t_*\), is at most \[2^{-\lceil200M^{20}\rceil}+O_D(e^{-100M^{20}}) \le e^{-90M^{20}}.\] Concavity of \(f\) gives \(f(1)\ge f(s)/s\). Thus in every such exact-start experiment, \[ f_{z_{t_*}}(1)\ge(1-e^{-90M^{20}})L_*. \tag{27}\] Equations (26) and (27) are the two inputs to the coverage induction: high joint entropy between checkpoints and an extremely accurate one-coordinate base case. Figure 1 summarizes the two orders now in play: sampling moves forward through the schedule, while coverage will be proved by induction from \(A_J\) back to \(A_0\).

Coverage from every exact checkpoint

We now apply the earlier transfer theorem to the estimates just proved. The coordinate groups are \(\mathbb Z[i]/(\pi)\), their maps are reduction modulo the two factors over each prime of \(\mathcal B_T\), and the abstract average \(\overline L\) is exactly \(L_*\). Averages over signed primes assign equal weight to all \(2k_T\) prime-factor pairs.

Recall that \((\tau,\beta)\) coverage means that a law on an alphabet of size \(p\) assigns mass at least \(\tau/p\) outside fewer than \(p^{1-\beta}\) residues. We use the following floors and exceptional fractions at the checkpoints already constructed: \[\begin{align*} \tau_0&=\tfrac14,& \delta_i&=2^{-i-6},& \tau_{i+1}&=\tau_i+4\delta_i, \tag{28}\\ \eta_0&=\tfrac1{100},&&& \eta_{i+1}&=\eta_i\delta_i/100. \tag{29}\end{align*}\] Here \(\beta_i\), \(r_i\), and \(J\) are those of (22)–(23). In particular, \[\tau_i=\frac38-2^{-i-3}<\frac38,\qquad \eta_i=100^{-(i+1)}2^{-i(i-1)/2-6i}.\] Since \(J=O(M)\), these formulas imply, uniformly over \(0\le i\le J\), \[ \log(1/\delta_i)=O(M),\qquad \eta_i\ge\exp(-O(M^2)), \qquad \eta_i\delta_i\beta_{i+1}\ge\exp(-O(M^2)). \tag{30}\] The last estimate is needed only for \(i<J\).

For a specified exact start \(a\) at checkpoint \(A_i\), denote the terminal residue law of its continuation experiment by \[b_{i,a,\pi}(u)= \Pr\{z_{t_*}\bmod\pi=u: \text{the remaining schedule is started at }A_i=a\}.\] The colon specifies the experiment, including at starts that have probability zero under the schedule begun at time \(0\).

At \(A_0\), the conclusion gives mass at least \(1/(4p)\) outside fewer than \(p^{199/200}\) residues for at least \(99\%\) of the signed primes, from every exact start. The next section will use this coverage and the common terminal-time law to test candidate starting residues under zero avoidance. There the repeated tests will be independent conditional on residue vectors, whose laws mix exact starts; that conditioning will be treated separately.