Let \(e=\mathbf{e}_a+\varepsilon \mathbf{e}_b\) be the step of the proposition, from the cell \(n\) to the cell \(n+e\). Write \(\rho '_b\) and \(\rho ''_b\) for the second offset digit of \(b\) at \(n\) and at \(n+e\); along \(e\) it changes by a constant \(\sigma '_b\) modulo \(q\). Count the cells \(n\) with the following properties:

1. the last digits of \(n_a\) and \(n_b\) cause no carry along \(e\);

2. \(r_a+\sigma _a\ge q\), so that \(a\) wraps (\(\sigma _a\) values of \(r_a\));

3. \(r_b\) does not wrap (\(q-|\sigma _b|\) values);

4. \(\rho '_b\) lies in a nonempty set \(\mathcal V_b\) of digit values chosen so that \(\varepsilon (\rho ''_b-\rho '_b)\le 0\);

5. \(r_u+\sigma _u\in [0,q)\) for every coordinate \(u\notin \{ a,b\} \).

For one \(u\) at a time, the digits involved form a box with depths \(k+1\) on \(a\), \(k+2\) on \(b\), \(k+1\) on \(u\) and \(0\) elsewhere, of total \(3k+4\le sk\), so they are jointly uniform; the conditions (a)–(d) have positive density and the fraction of cells violating (e) at \(u\) is \(|\sigma _u|/q\), so some cell satisfies all of them, because \(\sum _u|\sigma _u|{\lt}q\). For such a cell, \(|\Delta _a|{\lt}(\sigma _a+1)/q\), \(|\Delta _u|{\lt}(|\sigma _u|+1)/q\) and \(\varepsilon \Delta _b{\lt}1-|\sigma _b|/q+1/q^2\). The second part of the proposition (the case \(\varepsilon \sigma _b{\gt}0\)) uses, in the same way, the boxes with depths \(k+1\) on \(a\), \(b\) and \(u\) and \(0\) elsewhere (total \(3k+3\le sk\)), with \(r_a\) and \(r_b\) both wrapping, so that \(|\Delta _a|{\lt}(\sigma _a+1)/q\) and \(|\Delta _b|{\lt}(|\sigma _b|+1)/q\). The pair lies inside the cube, so the bound holds for the toroidal and the Euclidean distance.

Let \(K=K_{\mathrm{pair}}{\lt}\infty \). Every diagonal entry is \(-\alpha _a\) with \(1\le \alpha _a\le K\). Every pair has a small entry: if \(|x|=|\tilde M_{ab}|{\gt}K\), put \(\varepsilon =\operatorname {sgn}x\); then \(\sigma _a=-\alpha _a+|x|{\gt}0\), so \(b\) is backward, \(\varepsilon \sigma _b=\varepsilon y-\alpha _b\le 0\) for \(y=\tilde M_{ba}\), and if it is negative the definition gives \(\alpha _b-\varepsilon y\le K\); so \(|y|\le \max (\alpha _b,K-\alpha _b)\le K\). Orient \(c\to d\) when \(|\tilde M_{cd}|\le K\), and let \(\mathrm{out}(c)\) be the number of \(d\ne c\) with \(c\to d\). Every pair carries at least one arc, so \(\sum _c\mathrm{out}(c)\ge \binom s2\), and by convexity \(\sum _c\binom {\mathrm{out}(c)}2\ge s\binom {(s-1)/2}2=s(s-1)(s-3)/8\). The left side counts the triples \((c,\{ d,d'\} )\) with \(c\to d\) and \(c\to d'\), so averaging over the \(\binom s2\) pairs \(\{ d,d'\} \), some two columns \(d,d'\) have at least \((s-3)/4\) common rows \(c\notin \{ d,d'\} \) with \(|\tilde M_{cd}|,|\tilde M_{cd'}|\le K\). By \(t=0\) and Lemma 2, the box with depths \(k+1\) on two such rows \(c,c'\), \(k-1\) on \(d,d'\) and \(k\) elsewhere makes every \(2\times 2\) minor \(\tilde M_{cd}\tilde M_{c'd'}-\tilde M_{cd'}\tilde M_{c'd}\) a unit, hence nonzero, and the \(1\times 1\) boxes make the entries nonzero. So these rows give pairwise non-proportional vectors in \(([-K,K]\setminus 0)^2\), of which there are at most \(2K^2\) directions: \((s-3)/4\le 2K^2\).

Suppose \(K_{\mathrm{pair}}\le 1\). Then every diagonal entry is \(-1\), and the definition of \(K_{\mathrm{pair}}\) forces the pair rule: each pair \((\tilde M_{ab},\tilde M_{ba})\) is in \(\{ \pm 1\} ^2\), or one entry \(x\) has \(|x|\ge 2\) and the other is \(\operatorname {sgn}x\) (the step \(e_a+\varepsilon e_b\) has \(\sigma _a=-1+\varepsilon x\), \(\sigma _b=y-\varepsilon \) for \((x,y)=(\tilde M_{ab},\tilde M_{ba})\); if \(|x|\ge 2\) and \(\varepsilon =\operatorname {sgn}x\), \(a\) is forward, so \(b\) is backward with \(|\sigma _b|\le 1\) whenever \(\varepsilon \sigma _b{\lt}0\), which with \(y\ne 0\) forces \(y=\operatorname {sgn}x\)). So pairs in one class of Lemma 19(C) have \(\pi =-1\), and pairs across classes have an integer \(\pi =P\ge 1\). A class of size \(\ge 4\) contradicts (H) (\(1=1+1\)), so for \(s=5\) the classes are \(X=\{ x_1,x_2,x_3\} \) and \(Y=\{ y_1,y_2\} \). For \(\{ x_i,x_j,y_1,y_2\} \), (H) gives \(1=1/u+1/v\) with integers \(u^2=P_{i1}P_{j2}\), \(v^2=P_{i2}P_{j1}\); squaring shows \(u,v\in \mathbb {Q}\), hence integers, hence \(u=v=2\); so \(P_{i1}P_{j2}=4\) for all \(i\ne j\), which forces \(P_{i1}=\alpha \) for all \(i\). Then the three pairings of \(\{ x_1,x_2,x_3,y_1\} \) all have magnitude \(\alpha ^{-1/2}\), and \(1=2\) is false. For \(s=6\) a class of size \(\ge 4\) again contradicts (H), so both classes have size \(3\), and any five of the coordinates form classes of sizes \(3\) and \(2\), to which the case \(s=5\) applies; for \(s\ge 7\) some class has size \(\ge 4\). Since \(K_{\mathrm{pair}}\) is an integer, \(K_{\mathrm{pair}}\ge 2\).

One-axis steps give \(0\le \alpha ^{(j)}_c\le K\) for all \(c\) and \(j\). Suppose \(u=M^{(j_1)}[a][b]\) and \(v=M^{(j_2)}[b][a]\) have \(|u|,|v|{\gt}K\). Take the step along \(a\) and \(b\) with carry depths \(j_a=j_2\), \(j_b=j_1\) and \(e_ae_b=\varepsilon :=\operatorname {sgn}u\). Then \(e_a\sigma _a=-\alpha ^{(j_2)}_a+|u|{\gt}0\), so \(a\) is forward, and \(e_b\sigma _b=\varepsilon v-\alpha ^{(j_1)}_b\). If \(\operatorname {sgn}v=\operatorname {sgn}u\), then \(b\) is forward too and the cost is \(+\infty \); otherwise \(b\) is the only backward coordinate, of cost \(\alpha ^{(j_1)}_b+|v|+1{\gt}K\). Both contradict the definition of \(K\).

Let \(\nu =s-2\) and \(\lambda \) the common value. By Proposition 18 at carry depth \(0\), \(M^{(0)}[c][d]=\lambda '_cg_{d,0}w_{cd}\) with \(g_{d,0}\ne 0\); put \(\mu _d=\lambda g_{d,0}\). In column \(0\) the finite rows \(c=1,\dots ,\nu \) give the integers \(\mu _0/c\), so \(\operatorname {lcm}(1,\dots ,\nu )\) divides \(\mu _0\) and \(|M^{(0)}[1][0]|=|\mu _0|\ge \operatorname {lcm}(1,\dots ,\nu )\). In column \(1\) the rows \(c\in \{ 0,2,\dots ,\nu \} \) give the integers \(\mu _1/(c-1)\), \(c-1\in \{ -1,1,\dots ,\nu -1\} \), so \(|M^{(0)}[0][1]|=|\mu _1|\ge \operatorname {lcm}(1,\dots ,\nu -1)\). Lemma 91 for the pair \(\{ 0,1\} \) gives \(K(F)\ge \operatorname {lcm}(1,\dots ,\nu -1)\).

A rational lies in \(Q_K\) if and only if its numerator and denominator in lowest terms are at most \(K\) in absolute value. Put \(\varphi (c)=w_{cd}/w_{cd'}\). One of \(d,d'\) is \(\infty \). As \(\varphi \mapsto 1/\varphi \) and \(Q_K=Q_K^{-1}\), we may take \(d'=\infty \); then \(c\) is finite and \(\varphi (c)=1/t\) with \(t=c-d\in [-\nu ,\nu ]\setminus 0\), and the condition says \(tQ'/P'\in Q_K\) for \(\varrho ^{-1}=P'/Q'\) in lowest terms. With \(e=\gcd (t,P')\) the lowest terms of \(tQ'/P'\) are \((tQ'/e)/(P'/e)\), so \(t=et'\) with \(e\mid P'\) and \(|t'|\le K\): at most \(2K\operatorname {div}(P')\) values. Some \(c_0\in \mathcal C\) gives \(\varrho ^{-1}\in t_0Q_K\), so \(|P'|\le \nu K\). \(d,d'\) finite. Row \(\infty \) has \(\varphi =1\) and contributes at most one element. For finite \(c\) put \(u=c-d\) and \(g=d-d'\ne 0\), so \(\varphi (c)=(u+g)/u\); with \(h=\gcd (u,g)\), \(u=hu'\), \(g=hg'\), this is \((u'+g')/u'\) in lowest terms. Let \(\varrho =P/Q\) in lowest terms, \(A=\gcd (Q,u')\) and \(B=\gcd (P,u'+g')\). Then \(\varphi (c)/\varrho =\bigl(\frac QA\cdot \frac{u'+g'}B\bigr)\big/\bigl(\frac PB\cdot \frac{u'}A\bigr)\) is in lowest terms (each pair of factors is coprime, by \(\gcd (P,Q)=\gcd (u',u'+g')=1\) and \(\gcd (Q/A,u'/A)=\gcd (P/B,(u'+g')/B)=1\)), so \(u'+g'=B\beta \) with \(1\le |\beta |\le K\). For fixed \((h,A,B)\), \(\beta \) determines \(u'=B\beta -g'\), hence \(c=d+hu'\): at most \(2K\operatorname {div}(g)\operatorname {div}(Q)\operatorname {div}(P)\) places, with \(|g|\le \nu \) and, as above, \(|P|,|Q|\le \nu K\).

By Lemma 91 at carry depth \(0\) every pair \(\{ c,d\} \) has an entry \(|M^{(0)}[c][d]|\le K\); orient \(c\to d\) for such an entry. The convexity count in the proof of Theorem 89 gives columns \(d\ne d'\) and a set \(\mathcal C\) of at least \((s-3)/4\) rows \(c\notin \{ d,d'\} \) with \(1\le |M^{(0)}[c][d]|,|M^{(0)}[c][d']|\le K\) (entries at carry depth \(0\) are nonzero integers). By Proposition 18, \(M^{(0)}[c][d]=\lambda '_cg_{d,0}w_{cd}\), so \(w_{cd}/w_{cd'}=(g_{d',0}/g_{d,0})\, M^{(0)}[c][d]/M^{(0)}[c][d']\) lies in \(\varrho \, Q_K\) with \(\varrho =g_{d',0}/g_{d,0}\), and Lemma 93 with \(n=s-2\) bounds \(|\mathcal C|\). For the image of the places under \(g\in PGL_2(\mathbb Q)\), in normalized homogeneous coordinates every \(D_{cd}\), hence every \(w_{cd}\), is multiplied by a factor depending only on \(c\) times one depending only on \(d\); the first cancels in \(w_{cd}/w_{cd'}\) and the second changes only \(\varrho \).