(i) \(N=p^{k+1}\varpi \), so \(X_c(w)-X_c(v)\equiv x:=a_c\varpi +(\theta '-\theta )\pmod N\) with \(|\theta '-\theta |{\lt}\varpi \). If \(|x|\le N/2\), the distance is \(|x|\ge |a_c|\varpi -(\varpi -1)\). Otherwise it is \(N-|x|\ge N-(|a_c|+1)\varpi +1{\gt}(p^{k+1}-|a_c|-1)\varpi \ge (|a_c|-1)\varpi \), as \(|a_c|\le p^{k+1}/2\). So \(\delta N\ge (M-1)\varpi \), and \(\varpi N^{1/s}/N=p^{m/s-k-1}=p^{-\ell /s}\). (ii) Write \(P_c(w)=P_c(v)+a_c+\varepsilon p^{k+1}\) with \(\varepsilon \in \{ -1,0,1\} \). The difference of the floors is \(\lfloor (P_c(v)+a_c)/p^j\rfloor -\lfloor P_c(v)/p^j\rfloor +\varepsilon p^{k+1-j}\). As \(|a_c|{\lt}p\le p^j\), the first part lies in \(\{ 0,\operatorname {sgn}a_c\} \), and it is nonzero only if a multiple of \(p^j\), hence of \(p^{j-1}\), lies between \(P_c(v)\) and \(P_c(v)+a_c\). Neither part uses that \(p\) is odd or prime; \(p\ge 2\) suffices.

All three follow from Lemma 2 for suitable boxes. For (N1) take depth \(k+1\) on the skeleton coordinates of a set \(I\), depth \(k\) on the full coordinates of a set \(J\) with \(|J|=|I|\), and \(k_c\) elsewhere: the over coordinates are those of \(I\), by one digit (their skeleton digits), the under coordinates those of \(J\), by one level (the level \(k\)), and the minor is \(\det A[I,J]\). (N2) is the box with depth \(k+1\) on \(z_i\) and \(k-1\) on \(z_j\); (N3) the box with depth \(k+1\) on \(I\) and \(k_e-j\) on the coordinate \(e\).

(i) Restriction. Let \(m'=s(k+1)\) and \(V\in \mathbb {F}_p[t]_{{\lt}m'}\). The first digit of the reading at a finite skeleton place \(a_z\) vanishes iff \((t-a_z)\mid V\), and at \(\infty \) iff \(\deg V{\lt}m'-1\). So the points of the \(m'\)-frame whose skeleton coordinates have first digit \(0\) are given by \(V=\prod _{z\text{ finite}}(t-a_z)\, V'\) with \(\deg V'{\lt}m\), and their readings are those of \(V'\) times a unit power series, shifted by one digit at a skeleton place. Hence this point set, with the skeleton coordinates rescaled by \(p\), is the \(m\)-frame up to per-coordinate lower-triangular scrambles, i.e. up to the moves of Lemma 2; each new elementary box is an old one with one more leading digit on each skeleton coordinate. In the graph form of the \(m'\)-frame with depths \((k+1,\dots ,k+1)\) the restriction is the vanishing of level \(0\) of the skeleton cells, and the old first offset row of \(z\) becomes its skeleton digit. The move of Proposition 13 is lower-triangular in the levels and only scales level \(0\), so it commutes with deleting the level-\(0\) columns of the skeleton blocks; this gives the normal form, with carry depth \(n\) at level \(k_d-1-n\). The proofs of Propositions 18 and 14 use only this normal form and the distinctness, with unit differences, of the \(y_{cd}\) on the rows involved, so they apply to the \(\ell _d\) skeleton rows of a column.

(ii) For each full \(x_d\), the entry of \(S_z\) on \(x_{d,k}\) is the box minor with depth \(k+1\) on \(z\) and \(k\) on \(x_d\), hence a unit. Subtract multiples of level \(k\) from the levels \(l{\lt}k\) (move (c) with \(U_{x_d}\) lower-triangular) so that \(S_z\) vanishes there, scale level \(k\) to \(\kappa _d\), and set all diagonal blocks to \(0\) (move (a)). By Lemma 2 every box minor stays a unit, and Lemma 1 applies. This is the recipe of Definition 55(\(\mathcal B^2\)) in \(s\) dimensions; it needs no normal form.

(iii) By (i), the only constraints are these. On a full column \(d\) (\(\ell _d=2\)): at carry depth \(0\), \(A_{id}=\lambda _i\mu _dw_{id}\) (a generalized Cauchy column); at carry depth \(1\) Proposition 18 only asks that the interpolating polynomial have a unit leading coefficient, and for two rows this is the determinant \(\det [A_{\cdot d},\beta _{\cdot d}]\), which is (N3). On a skeleton column (\(\ell _d=1\)): \(g_{ij}\ne 0\), which is (N2). Larger carry depths are set to \(0\) by Proposition 14, for every \(k\), and the own terms by move (a). It remains to realize the ratios \(A_{2d}/A_{1d}\). Put \(z_1\) at \(\infty \) and \(z_2\) at a finite place \(a_2\); then \(w_{1d}=1\) and \(w_{2d}=1/(a_2-b_d)\), with \(b_d\) the place of \(x_d\), and the condition is \(b_d=a_2-\Lambda A_{1d}/A_{2d}\) with \(\Lambda =\lambda _2/\lambda _1\). Take \(\Lambda =1\), \(\lambda _1=1\) and \(\mu _d=A_{1d}\). The \(b_d\) are finite, differ from \(a_2\) because \(A_{1d}\ne 0\), and are pairwise distinct because the \(2\times 2\) minors of \(A\) are nonzero (N1). The net property follows as in (ii).

For skeleton digits of this form, (N1)–(N3) say: every entry and every \(2\times 2\) minor of \(A\) is nonzero (N1, which contains (N3) for \(j=1\) on a full coordinate); \(g_{12}\ne 0\) and \(g_{21}\ne 0\) (N2, which is also (N3) for \(j=1\) on a skeleton coordinate); and \(\det [A_{\cdot d},\beta _{\cdot d}]\ne 0\) for every full coordinate \(x_d\) ((N3) for \(j=2\); for \(\ell =2\) no skeleton coordinate lies outside a set \(I\) of size \(2\)). These conditions involve only the coefficients, not \(k\). If: Proposition 29(iii). Only if: let \(k\ge 1\), and let a graph form with the given skeleton digits be reachable from the frame for some choice of the places. All box minors of \(\Phi ^F\) are units (Lemma 11, over \(\mathbb {F}_p\)), and the moves multiply every box minor by a unit (Lemma 2), so the graph form is a \((0,m,s)\)-net over \(\mathbb {F}_p\) (Lemma 1). Lemma 28 applies to every such net and gives (N1)–(N3).

(L) If \(0{\lt}|u|_\infty \le \rho -1\), then \(|z-\sum _d\rho ^dx_d|\le (\rho -1)(1+\rho +\dots +\rho ^{s-1})=\rho ^s-1{\lt}p\), so \(z=\rho (x_1+\rho x_2+\cdots )\) exactly. Then \(\rho \mid z\) and \(|z|{\lt}\rho \) give \(z=0\), then \(|x_1|{\lt}\rho \) and \(\rho \mid x_1\) give \(x_1=0\), and so on: \(u=0\). The vector \(x=e_1\), \(z=\rho \) shows equality, and \(\rho =\lfloor p^{1/s}\rfloor \) attains the pigeonhole bound. (L1) Let \(u=(x,z_1,\dots ,z_\ell )\ne 0\) and \(B=|u|_\infty \). If \(B(\rho +1)\ge p\), then \(B\ge p/(\rho +1)\ge \rho ^\ell \). Otherwise \(|z_1-\rho x|\le B(1+\rho ){\lt}p\) gives \(z_1=\rho x\), and for \(i{\lt}\ell \), \(z_{i+1}-\rho z_i\equiv \rho ^{i+1}x-\rho ^{i+1}x=0\) with \(|z_{i+1}-\rho z_i|{\lt}p\) gives \(z_{i+1}=\rho z_i\); so \(z_\ell =\rho ^\ell x\). As \(u\ne 0\), \(x\ne 0\), so \(B\ge |z_\ell |\ge \rho ^\ell \); the vector \((1,\rho ,\dots ,\rho ^\ell )\) shows equality. (T) If \(0{\lt}|u|_\infty \le \rho -2\), then \(|z_1-\rho x_1-x_2|\le (\rho -2)(\rho +2){\lt}p\), so \(z_1=\rho x_1+x_2\), and likewise \(z_2=x_1+\rho x_2\). If \(x=0\) then \(z=0\). Otherwise, say \(|x_1|\ge |x_2|\), and \(|z_1|\ge (\rho -1)|x_1|\ge \rho -1\), a contradiction. (G) Let \(0{\lt}|u|_\infty \le B\). Since \(|z_j-A_jx|\le B(1+\| A_j\| _1){\lt}p\), the relations are exact. Then \(Z=z_1+iz_2\) is divisible by \(\zeta _1\) in \(\mathbb {Z}[i]\), and \(|Z|\le \sqrt2B{\lt}|\zeta _1|\), so \(Z=0\). Dividing by \(\zeta _1\) shows that \(\zeta _2\) divides the rational integer \(x_1\). For a rational integer \(n\), \(n/\zeta _2=n(\nu _2-i)/(\nu _2^2+1)\), so \(\zeta _2\mid n\) iff \(\nu _2^2+1\mid n\); as \(|x_1|\le B{\lt}\nu _2^2+1\), \(x_1=0\). Continuing with \(\zeta _3,\dots ,\zeta _{r-1}\) gives \(x_1=\dots =x_{r-2}=0\) and then \(x_{r-1}+ix_r=0\), so \(u=0\).

(a) Columns \(r-1\) and \(r\) of \(A\) are \((\operatorname {Re}\Omega _{r-1},\operatorname {Im}\Omega _{r-1})\) and \((-\operatorname {Im}\Omega _{r-1},\operatorname {Re}\Omega _{r-1})\), so \(\| A_j\| _1\ge |\Omega _{r-1}|=\prod _j(\nu _j^2+1)^{1/2}\ge \sqrt5\, (\nu _1^2+1)^{1/2}\), as \(r\ge 3\) and \(\nu _l\ge 2\). Hence \(p{\gt}B\sqrt5\, (\nu _1^2+1)^{1/2}\). Since \(\nu _1\le \sqrt2B+1\), we have \(\nu _1^2+1\le 2B^2+2\sqrt2B+2\le 5B^2\) for \(B\ge 2\), and \(\nu _1^2+1=5=5B^2\) for \(B=1\). So \(p{\gt}\nu _1^2+1\ge \nu _j^2+1\). (b) The arguments of all \(\Omega _l\) lie in \((0,\pi /2)\), so the entries are nonzero integers, of absolute value \({\lt}p\) by the hypothesis. The \(2\times 2\) minor of the columns \(l{\lt}l'\le r-1\) is \(\operatorname {Im}(\overline{\Omega _l}\, \Omega _{l'})=|\Omega _l|^2\operatorname {Im}(\zeta _{l+1}\cdots \zeta _{l'})\), and that of the columns \(l\le r-1\) and \(r\) is \(|\Omega _l|^2\operatorname {Re}(\zeta _{l+1}\cdots \zeta _{r-1})\). Here \(|\Omega _l|^2\) is a product of the factors \(\nu _j^2+1{\lt}p\), so it is prime to \(p\). The second factor is a nonzero integer of absolute value at most \(|\zeta _2\cdots \zeta _{r-1}|{\lt}|\Omega _{r-1}|{\lt}p\). (c) The \(\nu _j\) are nondecreasing in \(B\). At \(B=g^*+1\) they are \((6,3)\), \((6,3,3)\), \((8,3,3,3)\), \((8,3,3,3,3)\) for \(s=5,6,7,8\), with arctangent sums below \(0.487\), \(0.809\), \(1.090\), \(1.412\), all \({\lt}\pi /2\). For every \(B\), \(\nu _j{\gt}\sqrt B\) (Definition 32), so the sum is \({\lt}(r-1)/\sqrt B\).

Net. Proposition 29(ii): \(p{\gt}2^s\ge s-1\), and the prescribed coefficients \(\rho ^d\) are units since \(\rho ^{s-1}{\lt}p\).

(1) Since \(x_{d,k}\equiv P_{x_d}\) and \(P_z\equiv S_z\pmod p\), every point satisfies \(P_z\equiv \sum _d\rho ^dP_{x_d}\pmod p\). This congruence passes to the differences modulo \(p^{k+1}\), so every difference \(a\ne 0\) lies in \(\Lambda _A\) of Lemma 31(L), and \(|a|_\infty \ge \rho \). Lemma 27(i) gives \(\delta N^{1/s}\ge (\rho -1)p^{-1/s}\). There are no faults. With \(\rho =\lfloor p^{1/s}\rfloor \) we have \(p{\lt}(\rho +1)^s\), so the bound exceeds \((\rho -1)/(\rho +1)\), which is \(\ge 1/3\) for \(\rho \ge 2\) (\(p{\gt}2^s\)) and \(\ge 1/2\) for \(\rho \ge 3\) (\(p{\gt}3^s\)).

(2) There are \(N=p^{s(k+1)-1}\) points with distinct \(P\), all in \(\Lambda _A\), and \(\Lambda _A\) has \(p^{s(k+1)}/p=N\) classes modulo \(p^{k+1}\mathbb {Z}^s\). So the occupied fine cells are exactly the cells of \(\Lambda _A\). Optimality is Lemma 31(L).

(3) Let \(Q\in \{ 0,\dots ,p^{k+1}-1\} ^s\) be the fine cell of a point \(y\in [0,1)^s\). We look for a cell \(a=Q-v\in \Lambda _A\) in the same range with \(|v|_\infty \le \mu :=\lceil p/\rho ^{s-1}\rceil \). Prescribe the sign \(\epsilon _c\) of \(v_c\): \(-1\) if \(Q_c{\lt}p^{k+1}/2\), else \(+1\). Since \(\mu \le p/4+1\le p^{k+1}/2\), any \(v\) with these signs and \(|v|_\infty \le \mu \) keeps \(a\) in range. With \(T\equiv Q_z-\sum _d\rho ^dQ_{x_d}\pmod p\), the vector \(a\) lies in \(\Lambda _A\) iff \(v_z-\sum _d\rho ^dv_{x_d}\equiv T\). Choose the representative \(T'\) of \(T\) in \([0,p)\) or in \((-p,0]\), and expand \(T'=\sum _{i=0}^{s-1}t_i\rho ^i\) in base \(\rho \) with signed digits: \(t_0,\dots ,t_{s-2}\) are taken one at a time from the window \([0,\rho -1]\) or \([-(\rho -1),0]\), according to the sign of \(T'\), and \(t_{s-1}\) is the rest. Put \(v_z=t_0\) and \(v_{x_i}=-t_i\); the windows give the required signs of \(v_z,v_{x_1},\dots ,v_{x_{s-2}}\). The lower digits contribute less than \(\rho ^{s-1}\) in absolute value, so \(t_{s-1}\) has the sign of \(T'\) (which we choose) and \(|t_{s-1}|{\lt}p/\rho ^{s-1}+1\). So \(|v|_\infty \le \mu \), since also \(\rho -1{\lt}p/\rho ^{s-1}\). The point in the cell \(a\) differs from \(y\) by less than \((\mu +1)p^{-(k+1)}\) in every coordinate, so

For \(\rho =\lfloor p^{1/s}\rfloor \), \(\mu {\lt}\rho (1+1/\rho )^s+1\), so both bounds are bounded in \(p\) for fixed \(s\), and \(h/\delta \le \sqrt s\, (1+o(1))\) as \(p\to \infty \).

(N1). For \(s=4\) the entries \(\rho \), \(1\) are units, and \(\det A=\rho ^2-1=(\rho -1)(\rho +1)\) with \(0{\lt}\rho -1{\lt}\rho +1{\lt}p\), since \(\rho ^2{\lt}p+4\) and \(p\ge 5\). For \(s\ge 5\) use Lemma 33: in (2), \(B(p)\ge g^*+1\) exactly for \(p{\gt}132\), \(440\), \(2170\), \(6095\) (\(s=5,\dots ,8\)), which gives the primes \(137\), \(443\), \(2179\), \(6101\). These four numbers are \(\Gamma (g^*+1)\) for \(\Gamma (B)=B(1+\max _j\| A_j(B)\| _1)\), and \(\Gamma \) is increasing on \(B\ge g^*+1\) (below), so some \(B\ge g^*+1\) has \(\Gamma (B){\lt}p\) iff \(\Gamma (g^*+1){\lt}p\). In (3) the hypothesis is that of Lemma 33(b), and the closed-form sufficient condition is that of Lemma 33(c). Net. Proposition 29(iii).

Monotonicity of \(\Gamma \). Let \(5\le s\le 8\) and \(g^*+1\le B{\lt}B'\). The \(\nu _j\) do not decrease from \(B\) to \(B'\), and \(\sum _j\arctan (1/\nu _j){\lt}\pi /2\) at \(B\) by Lemma 33(c). The absolute values of the entries of \(A(B)\) are the real and imaginary parts of \(\Omega _1,\dots ,\Omega _{r-1}\) (proof of Lemma 33(a)). Raise the \(\nu _j\) one at a time from their values at \(B\) to those at \(B'\). If \(\nu _j\) is a factor of \(\Omega _l=P\, (\nu _j+i)\), the other factors have arctangents no larger than at \(B\), so \(P=a+ib\) with \(\arg P\in [0,\pi /2)\), i.e. \(a{\gt}0\), \(b\ge 0\); and \(P(\nu _j+i)=(a\nu _j-b)+i(b\nu _j+a)\), so raising \(\nu _j\) by \(\Delta \ge 0\) raises the real part by \(a\Delta \) and the imaginary part by \(b\Delta \). All parts stay positive, so every \(|A_{jd}|\) is nondecreasing, \(\| A_j(B')\| _1\ge \| A_j(B)\| _1\), and \(\Gamma (B'){\gt}\Gamma (B)\). (The thresholds are also verified in Lean, by a different route: \(\Gamma \) is evaluated exactly for \(g^*+1\le B{\lt}B_1\), \(B_1=7,7,9,9\), and \((B\max _j\| A_j(B)\| _1)^2\ge 2B^s\), from \(\| A_j\| _1\ge |\Omega _{r-1}|\), covers \(B\ge B_1\).)

Separation. Let \(a\ne 0\) be a difference and \(M=|a|_\infty \). If \(M\ge p-1\), then \(M\ge \lambda _\infty (\Lambda _A)-g^*\) below holds, since \(\lambda _\infty (\Lambda _A)\le \lfloor p^{2/s}\rfloor \le p-1\). Otherwise let \(\xi _c=\xi _{c,1}\) be the floor changes of Lemma 27(ii); they give the changes of the level-\((k-1)\) digits modulo \(p\). Put

By the skeleton digits, the change of \(S_{z_i}\) is \(A_ia_x\) plus the contributions of the level-\((k-1)\) changes \(\xi \); subtracting these, as in \(u\), leaves a vector of \(\Lambda _A\). So \(u\in \Lambda _A\), and \(|u|_\infty \le M+g^*\) with \(g^*=1+\lceil r/2\rceil \). If \(u\ne 0\), then \(M\ge \lambda _\infty (\Lambda _A)-g^*\). If \(u=0\), then \(a_x=0\), so every \(\xi _{x_d}=0\), and \(a_{z_1}=\xi _{z_2}\), \(a_{z_2}=-\xi _{z_1}\). If \(\xi _{z_2}=1\) then \(a_{z_2}{\gt}0\) and \(a_{z_1}=1\), so \(\xi _{z_1}\in \{ 0,1\} \) and \(a_{z_2}=-\xi _{z_1}\le 0\), a contradiction. The case \(\xi _{z_2}=-1\) is symmetric, and \(\xi _{z_2}=0\) gives \(a_{z_1}=0\), \(\xi _{z_1}=0\), \(a_{z_2}=0\), contradicting \(a\ne 0\). Here the opposite signs of \(g_{12}\) and \(g_{21}\) give the contradiction, so no dependence of \(S_{z_i}\) on its own cell is needed. So \(M\ge \lambda _\infty (\Lambda _A)-g^*\), and Lemma 27(i) gives \(\delta N^{1/s}\ge (\lambda _\infty (\Lambda _A)-g^*-1)p^{-2/s}\). With \(\lambda _\infty \ge \rho -1\) and \(g^*=2\) (\(s=4\)), resp. \(\lambda _\infty \ge B+1\) (Lemma 31), this is the claim. For \(s=4\), \(\rho \ge 5\) iff \(p\ge 23\), and \((\rho -4)/\sqrt p\to 1\).

The moves only scale the entries of the first offset rows on the last cell level, so by Proposition 29(i) these entries form a Cauchy layer \(W_{ce}=\lambda _c\mu _e/D_{ce}\) in the places (Section 3), with \(W_{z_iz_j}=g_{ij}\) and \(W_{z_ix_d}=\kappa _i\). Lemma 19(X) for the coordinates \(z_i,z_j,z_l,x_d\) reads \(g_{ji}\kappa _l/(\kappa _jg_{li})+g_{ij}\kappa _l/(\kappa _ig_{lj})=1\); multiplying by \(g_{li}g_{lj}/\kappa _l\) gives the identity.

Suppose such a graph form exists. The moves multiply every box minor by a unit, so it is a \((0,m,s)\)-net, and \(g_{ij}\ne 0\) by (N2). Multiply the identity for \((i,j,l)=(1,2,3)\) by \(\rho ^3\): \(g_{21}g_{32}\rho +g_{12}g_{31}\rho ^2\equiv g_{31}g_{32}\pmod p\). The difference of the two sides is at most \(G^2(\rho ^2+\rho +1){\lt}p\) in absolute value, so the congruence is an equality of integers. Hence \(\rho \mid g_{31}g_{32}\). But \(0{\lt}|g_{31}g_{32}|\le G^2{\lt}\rho \), a contradiction.

Write \(\kappa =(4,7,8)\), \(\beta =(-2,-1,-1)\) and \(\eta =(1,0,0)\) for the coefficients of the \(S_{z_i}\) on \(x_k\), \(x_{k-1}\), \(x_{k-2}\), \(g_{ij}\) and \(H_{ij}\) for those of \(S_{z_i}\) on \(z_{j,k-1}\) and \(z_{j,k-2}\) (\(i\ne j\)), and \(\alpha =(0,1,3)\) for the places of \(z_1,z_2,z_3\); all faults \(g_{ij}\) are \(\pm 1\).

Reachability. By Proposition 29(i), Propositions 18 and 14 hold for the skeleton rows: column \(x\) carries the three rows \(S_{z_i}\) and the levels \(k\), \(k-1\), \(k-2\) (the carry depths \(0,1,2\)), and column \(z_j\) carries the two rows \(S_{z_i}\), \(i\ne j\), and the levels \(k-1\), \(k-2\); one vector \(\lambda '\) of row scalars is shared by all columns. The coefficients of \(S_{z_i}\) on its own cell are diagonal entries, free by move (a). By Proposition 18 (with \(w=1\), \(y=\alpha _i\) on the column at \(\infty \), and \(w=y=1/(\alpha _i-\alpha _j)\) on the column \(z_j\)), the digits are reachable, with all deeper levels \(0\), if there are units \(\lambda '_i\), \(\mu _x\), \(\mu _{z_j}\) such that, modulo \(13\): (R1) \(\kappa _i=\lambda '_i\mu _x\); \(\kappa _i+\beta _i=\lambda '_i(c_0+c_1\alpha _i)\) with \(c_1\ne 0\); and \(\kappa _i+\beta _i+\eta _i=\lambda '_iG(\alpha _i)\) with \(\deg G=2\); (R2) \(g_{ij}=\lambda '_i\mu _{z_j}/(\alpha _i-\alpha _j)\), and \(g_{ij}+H_{ij}=\lambda '_i(\alpha _i-\alpha _j)^{-1}\bigl(c'_0+c'_1/(\alpha _i-\alpha _j)\bigr)\) with \(c'_1\ne 0\), for each \(j\) and the two \(i\ne j\). Take \(\lambda '=\kappa \) and \(\mu _x=1\). For (R2) at level \(k-1\) each column must give one value of \(\mu _{z_j}=g_{ij}(\alpha _i-\alpha _j)/\kappa _i\) for both \(i\ne j\): column \(z_1\) needs \(7\mu =-1\) and \(8\mu =-3\), column \(z_2\) needs \(4\mu =1\) and \(8\mu =2\), column \(z_3\) needs \(4\mu =3\) and \(7\mu =2\), and indeed \((\mu _{z_1},\mu _{z_2},\mu _{z_3})=(-2,-3,4)\) solves all six. For (R1), \((\kappa _i+\beta _i)/\kappa _i=(2/4,6/7,7/8)=(7,12,9)=c_0+c_1\alpha _i\) with \((c_0,c_1)=(7,5)\). The remaining conditions say that leading coefficients of interpolation polynomials are nonzero: three values \(\lambda '_iG(\alpha _i)\) at distinct \(\alpha _i\) always come from a \(G\) of degree at most \(2\), and since \(\det [\lambda ',\lambda '\odot (c_0+c_1\alpha ),\lambda '\odot G(\alpha )]\) is \(\lambda '_1\lambda '_2\lambda '_3\, c_1\) times the leading coefficient of \(G\) times a Vandermonde determinant, \(\deg G=2\) is equivalent to \(\det [\kappa ,\beta ,\eta ]\ne 0\); here \(\det [\kappa ,\beta ,\eta ]=7\cdot (-1)-8\cdot (-1)=1\). Likewise \(c'_1\ne 0\) in column \(z_j\) is equivalent to \(g_{ij}H_{lj}-H_{ij}g_{lj}\ne 0\) for \(\{ i,l\} =\{ 1,2,3\} \setminus \{ j\} \), and these are \(1\), \(-1\), \(1\) for \(j=1,2,3\). The places \(\infty ,0,1,3\) are distinct modulo \(13\). So Proposition 14 gives, for every \(k\ge 1\), moves of Lemma 2 that turn the graph form \(\Phi ^F\) of the frame into one with these skeleton digits (truncated for \(k=1\)). The box minors of \(\Phi ^F\) are units (Lemma 11), the moves multiply them by units, and Lemma 1 gives the net property.

Separation. Let \(a\) be the centred difference of the fine positions of two distinct points (Lemma 27). If some \(|a_c|\ge 13/2\), then \(|a|_\infty \ge 7\). Otherwise, as \(P_x\equiv x_k\) and \(P_{z_i}\equiv S_{z_i}\pmod{13}\), Lemma 27(ii) gives

with sign-consistent prefix patterns \((\xi _{c,1},\xi _{c,2})\in \{ (0,0),(\operatorname {sgn}a_c,0),(\operatorname {sgn}a_c,\operatorname {sgn}a_c)\} \) (and \(\xi =0\) for missing levels). We enumerate \(|a_x|\le 6\) and all patterns (\(4625\) combinations, with both signs for the patterns of the \(z_i\) before the sign check; rerun by the scripts of Appendix app:verify), take the centred \(a_{z_i}\) and keep the sign-consistent solutions: the least \(|a|_\infty \) of a nonzero solution is \(M_{13}=4\). Lemma 27(i) gives \(\delta N^{1/4}\ge (M_{13}-1)\, 13^{-3/4}=0.4381\ldots \).