The net property is Proposition 29(ii); the \(s\) places of Definition 36 are distinct because \(p\ge s-1\). As in the proof of Theorem 35(1), \(P_z\equiv \sum _d\kappa _dP_{x_d}\pmod p\) for every point, and the centred difference \(a\ne 0\) of the fine positions of two distinct points lies in \(\Lambda _\kappa \). By Lemma 27(i) in each coordinate \(c\), the two points are at toroidal distance at least \((|a_c|-1)_+\varpi \) in coordinate \(c\), so their distance is at least \(\varpi \| (|a|-\mathbf1)_+\| _2\ge \varpi \lambda _{\mathrm{jit}}(\kappa )\), and \(\varpi N^{1/s}/N=p^{-1/s}\). Finally \(|a|\le (|a|-\mathbf1)_++\mathbf1\) coordinatewise gives \(\| (|a|-\mathbf1)_+\| _2\ge \| a\| _2-\sqrt s\).

Net property. The first rows are \(\mathbf{e}_x,\mathbf{e}_y,\mathbf{e}_z\), so \(\mathcal T_p\) is a graph form with the depths \((1,1,1)\), and we use Lemma 2: the box of shape \((1,1,1)\) has determinant \(1\) and puts one point in every cell. A box with depth \(2\) on \(a\), \(1\) on \(b\) and \(0\) on \(c\) has, up to sign, the determinant \((C_a)_{2,c}\), the entry of the second row of \(C_a\) in column \(c\); these off-diagonal entries of the second rows are \(h,h,r,1,r,-1\). The box with depth \(3\) on \(a\) has, up to sign, the \(2\times 2\) minor of the second and third rows of \(C_a\) off column \(a\); it is \(2h-h=h\), \(r-3\) and \(5r+1\) for \(a=x,y,z\). Here \(1\le r{\lt}p\). Since \(r\le p/8+\frac12\), \(0{\lt}5r+1{\lt}p\) for \(p\ge 11\), and \(5r+1=6\) for \(p=5,7\), a unit modulo \(5\) and \(7\). Finally \(r-3\equiv 0\) means \(r=3\), as \(-2\le r-3{\lt}p\), i.e. \(p/8\in (\frac52,\frac72)\); among primes this holds only for \(p=23\), where the box with depth \(3\) on \(y\) is singular. Lemma 1 concludes.

Separation: reduction. At integer scale \(p^3\) the point of cell \(n\in \{ 0,\dots ,p-1\} ^3\) is \(X(n)=p^2n+p\, g(n)+f(n)\), with the geometric digits \(g(n)\) and the third digits \(f(n)\) in \(\{ 0,\dots ,p-1\} ^3\). Let \(n\ne n'\), and let \(D=p^2E+p\, \Delta g+\Delta f\) be a representative of \(X(n')-X(n)\) modulo \(p^3\), where \(E\equiv n'-n\pmod p\) is an integer vector, \(\Delta g=g(n')-g(n)\) and \(\Delta f=f(n')-f(n)\in (-p,p)^3\); the toroidal distance is the least \(|D|\) over these representatives. If some \(|E_c|\ge 3\), then \(|D_c|\ge 3p^2-p(p-1)-(p-1){\gt}2p^2\). So let \(E\in \{ -2,\dots ,2\} ^3\setminus \{ 0\} \), and put \(S=E_y+E_z\) and \(T=E_y-E_z\) (so \(|S|\le 4\), and \(|S|\le 3\) if \(T\) is odd). As \(C_y\) and \(C_z\) have the same second row, \(g_y(n)=g_z(n)\) for every cell, so \(\Delta g=(\xi ,\eta ,\eta )\) with integers \(\xi ,\eta \in (-p,p)\): the \(y\)- and \(z\)-digits wrap together. Since \(\xi \equiv hS\), \(2h\equiv -1\), \(\eta \equiv rE_x-T\) and \(8r=p+\theta \), there are integers \(i\), \(j\) with

Reducing modulo \(2\) and modulo \(8\), with \(p\) odd and \(\theta \equiv -p\pmod8\), gives \(j\equiv S\pmod2\) and \(i\equiv E_x\pmod8\). As \(|\xi |{\lt}p\) and \(|S|{\lt}p\), \(|j|\le 2\), and \(j=\pm 2\) only if \(\pm S{\gt}0\). If \(|T|\le 1\), then \(|\theta E_x-8T|\le 14{\lt}p\), so likewise \(|i|\le 8\), \(i=\pm 8\) only if \(\mp (\theta E_x-8T){\gt}0\), and \(i=0\) if \(E_x=T=0\).

In cell units \(D/p^2=V+\Delta f/p^2\), with \(|\Delta f|{\lt}\sqrt3\, p\) and \(V=E+(\xi ,\eta ,\eta )/p\). So \(|D|/p^2{\gt}|V|-\sqrt3/p\), and it suffices to show \(|V|^2\ge \frac{33}{32}-\frac{41}{16p}\). With the half-integer \(\alpha =E_x+\frac j2\) and the quarter-integer \(\beta =S+\frac i4\) (the limits of \(V_x\) and \(V_y+V_z\) as \(p\to \infty \)),

The terms of order \(1/p\) are the drift (Section 11): the deviation of the geometric digits from their limit values. Each numerical estimate below increases with \(p\) and is evaluated at \(p=41\).

The cases. (a) \(|T|\ge 2\): \(|V|^2\ge T^2/2\ge 2\).

(b) \(T=0\), \(E_x=0\): then \(S\ne 0\) is even and \(i=0\), so \(V_y=V_z=E_y\) and \(|V|^2\ge S^2/2\ge 2\).

(c) \(|T|=1\), \(E_x=0\): \(j\) is odd, so \(|\alpha |=\frac12\) and \(|V_x|\ge \frac12-\frac3{2p}\); \(i\in \{ 0,\pm 8\} \), so \(\beta \) is an odd integer and \(|V_y+V_z|\ge 1-\frac2p\). Hence \(|V|^2\ge (\frac12-\frac3{2p})^2+\frac12(1-\frac2p)^2+\frac12{\gt}1.16\).

(d) \(|E_x|=2\). If \(|T|=1\), then \(j=\pm 1\), \(|\alpha |\ge \frac32\) and \(|V|^2\ge (\frac32-\frac3{2p})^2{\gt}2\). If \(T=0\), then \(S\) and hence \(j\) are even, so \(\alpha \) is an integer, and \(\alpha \ne 0\) (which would need \(j=\mp 4\)); so \(|V_x|\ge 1-\frac2p\); and \(4\beta =4S+i\equiv E_x\equiv 2\pmod4\), so \(|\beta |\ge \frac12\) and \(|V_y+V_z|\ge \frac12-\frac3{2p}\). Hence \(|V|^2\ge (1-\frac2p)^2+\frac12(\frac12-\frac3{2p})^2=\frac98-\frac{19}{4p}+\frac{41}{8p^2}\), which is at least \(\frac{33}{32}-\frac{41}{16p}\) for \(p\ge 24\).

(e) \(|E_x|=1\), \(T=0\): \(\alpha \in \mathbb {Z}\), \(i\in \{ E_x,-7E_x\} \), and \(4\beta =4S+i\equiv E_x\pmod8\), so \(|\beta |\in \{ \frac14,\frac74,\frac94,\dots \} \); moreover \(|V_x-\alpha |\le \frac2p\) and \(|V_y+V_z-\beta |\le \frac3{4p}\). If \(\alpha =0\), then \(j=-2E_x\), so \(E_xS\le -2\) and \(E_x\beta =E_xS+E_xi/4\le -\frac74\); then \(|V|^2\ge \frac12(\frac74-\frac3{4p})^2{\gt}1.4\). If \(|\alpha |\ge 2\), or \(|\alpha |=1\) and \(|\beta |\ge \frac74\), then also \(|V|^2{\gt}1.4\). There remains \(|\alpha |=1\), \(|\beta |=\frac14\). Since \(S=\beta -\frac i4\) is an even integer, \(\beta =\frac{E_x}4\) and \(S\in \{ 0,2E_x\} \), and if \(S=2E_x\) then \(\alpha =E_x\) (as \(|j|\le 2\)). With \((a-b)^2\ge a^2-2ab\),

as \(\alpha S\in \{ 0,2\} \) and \(\theta \ge -3\).

(f) \(|E_x|=1\), \(|T|=1\): \(j=\pm 1\), so \(\alpha \in \{ \pm \frac12,\pm \frac32\} \); \(i\in \{ E_x,-7E_x\} \), and \(4\beta =4S+i\equiv 4+E_x\pmod8\), so \(|\beta |\in \{ \frac34,\frac54,\frac{11}4,\dots \} \); moreover \(|V_x-\alpha |\le \frac3{2p}\) and \(|V_y+V_z-\beta |\le \frac{11}{4p}\), as \(|\theta E_x-8T|\le 11\). If \(|\alpha |=\frac32\), then \(|V|^2\ge (\frac32-\frac3{2p})^2{\gt}2\). If \(|\beta |\ge \frac54\), then \(|V|^2\ge (\frac12-\frac3{2p})^2+\frac12(\frac54-\frac{11}{4p})^2+\frac12{\gt}1.41\). There remains \(|\alpha |=\frac12\), \(|\beta |=\frac34\). Then \(\alpha =\frac{E_x}2\) and, from \(4\beta \equiv 4+E_x\pmod8\), \(\beta =-\frac34E_x\); and \(S=\beta -\frac i4=\pm E_x\). As in (e),

as \(E_xS,E_xT\in \{ \pm 1\} \) and \(\theta \le 3\).

So \(|V|^2\ge \frac{33}{32}-\frac{41}{16p}\) in every case, which proves (2). The bound increases with \(p\); it is \(1.0002\ldots \) at \(p=197\) and below \(1\) at \(p=193\). For \(p\ge 83\) we have \(\frac{33}{32}-\frac{41}{16p}\ge 1\), and \(\sqrt a-\sqrt{a-x}=x/(\sqrt a+\sqrt{a-x})\le x/2\) gives the bound \(\sqrt{66}/8-(\frac{41}{32}+\sqrt3)/p\), with \(\frac{41}{32}+\sqrt3{\lt}3.014\).

The pair. For \(n'=(p-1)\mathbf1\equiv -\mathbf1\) one has \(g(n')=(1,p-r,p-r)\) and, as the entries of the third rows sum to \(3\), \(4\), \(6\), \(f(n')=(p-3,p-4,p-6)\); also \(g(0)=f(0)=0\). The representative with \(E=-\mathbf1\) is \(D_*=-\bigl(p^2-2p+3,\, (r-1)p+4,\, (r-1)p+6\bigr)\). Since \(|D_{*,c}|{\lt}p^2\le p^3/2\) in every coordinate, this representative is the toroidal one, so the toroidal distance is \(d_*\) and \(\delta N^{1/3}\le |D_*|/p^2=d_*/p^2\). With \(r=(p+\theta )/8\),

and for \(|\theta |\le 3\) the three coefficients exceed \(\frac14\), \(-12.3\) and \(24\), so the difference is positive for \(p\ge 53\). With (2) this gives the limit.

(1) The first rows are \(\mathbf{e}_x,\mathbf{e}_y,\mathbf{e}_z\), so \(\mathcal W_p\) is a graph form with the depths \((1,1,1)\), and we use Lemma 2: the box of shape \((1,1,1)\) has determinant \(1\) and puts one point in every cell. A box with depth \(2\) on \(a\), \(1\) on \(b\) and \(0\) on \(c\) has, up to sign, the determinant \((C_a)_{2,c}\); these off-diagonal entries of the second rows are \(-\rho ,2\rho ,-\rho ,2\rho ,h-r,h-r\). The box with depth \(3\) on \(a\) has, up to sign, the \(2\times 2\) minor of the second and third rows of \(C_a\) off column \(a\): \(\det \left(\begin{smallmatrix} -\rho & 2\rho \\ 1 & 2 \end{smallmatrix}\right)=-4\rho \), \(\det \left(\begin{smallmatrix} -\rho & 2\rho \\ 1 & 1 \end{smallmatrix}\right)=-3\rho \) and \(\det \left(\begin{smallmatrix} h-r & h-r \\ 0 & 1 \end{smallmatrix}\right)=h-r\) for \(a=x,y,z\). As \(1\le \rho \le \frac p{24}+1{\lt}p\), \(1\le h-r{\lt}p\) and \(p\ge 5\), all these are units, and Lemma 1 concludes.

(2) Reduction. At integer scale \(p^3\) the point of cell \(n\) is \(X(n)=p^2n+p\, g(n)+f(n)\), with the second digits \(g(n)=(2\zeta (n)\bmod p,\, 2\zeta (n)\bmod p,\, \zeta (n))\) and the third digits \(f(n)\). As in the proof of Proposition 106, a representative of \(X(n')-X(n)\) is \(D=p^2E+p\, \Delta g+\Delta f\) with \(E\equiv n'-n\), \(|\Delta f_c|{\lt}p\), and only \(E\in \{ -2,\dots ,2\} ^3\setminus \{ 0\} \) can give \(|D|\le 2p^2\); then \(|D|/p^2{\gt}|V|-\sqrt3/p\) with \(V=E+\Delta g/p\), and by Lemma 108 \(\Delta g=(2\Delta \zeta -p\omega ,\, 2\Delta \zeta -p\omega ,\, \Delta \zeta )\). It suffices to show \(|V|^2\ge \frac{281}{256}-\frac{345}{128p}\).

Parametrization. Put \(S=E_x+E_y\), \(Q=S-2E_z\) and \(\varepsilon =\rho /(2p)\). As \(2(h-r)=p-\rho \), \(2u\cdot E=pS-\rho Q\), so \(\Delta \zeta =p(\nu -Q\varepsilon )\) with \(\nu \in \frac S2+\mathbb {Z}\), and

With \(\theta =48r-p\), an odd integer in \([-47,-1]\) (here \(\theta \) is not that of Proposition 106), we have \(\rho =(p+\theta )/24+1\), so \(\varepsilon =\frac1{48}+\frac{\theta +24}{48p}\) and \(|\varepsilon -\frac1{48}|\le \frac{23}{48p}\); for \(p\ge 53\), \(\varepsilon \le \frac{19}{636}{\lt}\frac3{100}\). The drift of the layer is this deviation of \(\varepsilon \) from its limit.

Which \((\nu ,\omega )\) occur. Let \(p\ge 53\). Then \(|Q|\le 8\) and \(4\rho \le \frac p6+4{\lt}h\). By Lemma 108: if \(\nu =0\), then \(\Delta \zeta =-Q\rho /2\), \(|\Delta \zeta |\le h\), and \(\omega \in \{ 0,-\operatorname {sgn}Q\} \) (only \(\omega =0\) if \(Q=0\)); if \(\nu =\pm \frac12\), then \(\Delta \zeta =\frac12(\pm p-Q\rho )\) has the sign of \(\nu \), \(\omega =\operatorname {sgn}\nu \) occurs, and \(\omega =0\) occurs if and only if \(\nu Q{\gt}0\); if \(\nu =\pm 1\), the step occurs only if \(\nu Q{\gt}0\), and then \(|\Delta \zeta |{\gt}h\) and \(\omega =\operatorname {sgn}\nu \); and \(|\nu |\ge \frac32\) gives \(|\Delta \zeta |\ge \frac32p-4\rho {\gt}p\), which does not occur. None of these rules depends on \(p\) or on its class.

Long options. \(A\) lies in the lattice \(\Lambda _{\mathcal W}=\{ A:A_x,A_y\in \mathbb {Z},\ 2A_z\in \mathbb {Z},\ A_x+A_y\equiv 2A_z\pmod2\} \), as \(A_x+A_y\equiv S\equiv 2\nu \equiv 2A_z\). By the rules \(|2\omega -2\nu |\le 2\), and \(Q=(A_x+A_y-2A_z)-2\nu +2\omega \), so \(|Q|\le \sqrt6\, |A|+2\). Hence \(|V|\ge |A|-3|Q|\varepsilon \ge (1-\frac{9\sqrt6}{100})|A|-\frac9{50}{\gt}1.05\) if \(|A|^2\ge 3\), and \(1.05^2{\gt}\frac{281}{256}\).

Short options. Since \(2A_z\in \mathbb {Z}\), \(|A|^2\) is an integer or an integer plus \(\frac14\), and \(|A|^2=\frac94\) would need \(A=(\pm 1,\pm 1,\pm \frac12)\) or \(\pm \frac32\mathbf{e}_z\), which violate the parity condition; so the vectors of \(\Lambda _{\mathcal W}\) with \(0{\lt}|A|^2{\lt}3\) are those with \(|A|^2\le 2\): \(\pm \mathbf{e}_z\), \((\pm 1,0,\pm \frac12)\), \((0,\pm 1,\pm \frac12)\) and \((\pm 1,\pm 1,0)\). For each of them \(E=A-(2\nu -\omega )(1,1,0)-\nu \mathbf{e}_z\), with \(\nu \equiv A_z\pmod1\) and \(|\nu |\le 1\), so at most three values of \(\nu \), and checking the rules for these values and their \(\omega \) leaves the options of Table tab:wcontacts; \(A=0\) does not occur (it would need \(E=\omega (1,1,0)\) with \(\nu =0\), \(\omega =-\operatorname {sgn}Q=-\operatorname {sgn}(2\omega )\), or \(E=-\nu \mathbf1\) with \(|\nu |=1\) and \(Q=0\)). The squared length \(q(\varepsilon )=|V|^2\) of each row is a convex quadratic in \(\varepsilon \), so it lies above its tangent line at \(\frac1{48}\), and as \(|\varepsilon -\frac1{48}|\le \frac{23}{48p}\), \(q(\varepsilon )\ge q(\frac1{48})-|q'(\frac1{48})|\cdot \frac{23}{48p}=:T(p)\). For the diagonal contact (a), \(T(p)=\frac{281}{256}-\frac{345}{128p}\), as \(\frac{45}8\cdot \frac{23}{48}=\frac{345}{128}\). For the step (b) along \(\mathbf{e}_z\), \(T(p)=\frac{211}{192}-\frac{253}{96p}\), which is larger, as \(\frac{211}{192}{\gt}\frac{281}{256}\) and \(\frac{253}{96}{\lt}\frac{345}{128}\). For the rows (c)–(g), \(q(\frac1{48}){\gt}\frac{281}{256}\), so \(p\, (T(p)-\frac{281}{256})=p\, (q(\frac1{48})-\frac{281}{256})-\frac{23}{48}|q'(\frac1{48})|\) is affine in \(p\) with positive slope, and at \(p=53\) it is positive (for row (c), \(53\cdot \frac5{96}-\frac{23}{48}\cdot \frac{37}8{\gt}0\), and the other rows have more room); so these options have \(|V|^2{\gt}\frac{281}{256}\). This proves (2). The constant \(\frac{345}{128}\) is the slope of the diagonal contact (a) times the largest drift \(\frac{23}{48p}\), which is attained for \(p\equiv 1\pmod{48}\); exactly, with \(x=\varepsilon -\frac1{48}=\frac{\theta +24}{48p}\),

The bound increases with \(p\); it is \(1.0024\ldots \) at \(p=67\), below \(1\) at \(p=61\), \(1.0164\ldots \) at \(p=97\) and below \(\sqrt{66}/8=1.01550\ldots \) at \(p=89\).

(3) Here \(\zeta (1,0,0)=h-r\) and \(\zeta (2,0,p-1)=p-2-4r\), so the second digits of the two points are \((p-1-2r,\, p-1-2r,\, h-r)\) and \((p-4-8r,\, p-4-8r,\, p-2-4r)\) (as \(p\ge 8r+4\)), and the third digits are \((0,1,0)\) and \((p-2,1,0)\). Hence \(D=(p^2-(6r+2)p-2,\, -(6r+3)p,\, -(h+3r+2)p)\) is a representative of the difference with \(|D_c|{\lt}p^3/2\), i.e. the toroidal one. As \(6r+3=3\rho \) and \(h+3r+2=\frac12(p+3\rho )\), \(D/p^2=(1-6\varepsilon ,-6\varepsilon ,-\frac12-3\varepsilon )+(\frac1p-\frac2{p^2},0,0)\): the contact (a) with the step \(E=(1,0,-1)\), plus a jitter. So \(d_*/p^2\le |V_{\mathrm a}|+\frac1p\), where by the display in (2), with \(|x|\le \frac{23}{48p}\), \(|V_{\mathrm a}|^2\le \frac{281}{256}+\frac{345}{128p}+\frac{4761}{256p^2}{\lt}\frac{281}{256}+\frac{3.05}p\) for \(p\ge 53\); with \(\sqrt{a+x}\le \sqrt a+x/(2\sqrt a)\) this gives \(d_*/p^2{\lt}\sqrt{281}/16+2.46/p\), slightly more than (3) states. Together with (2), \(\delta N^{1/3}\to \sqrt{281}/16\).

(4) By (2), \(\delta N^{1/3}{\gt}1\) for every prime \(p\ge 67\) and \({\gt}\sqrt{66}/8\) for every prime \(p\ge 97\). The remaining primes are settled by the exact minimum distances (Appendix app:verify): at every prime \(17\le p\le 61\), \(\delta N^{1/3}\ge 1.0009\ldots \) (least at \(p=47\)), and at every prime \(53\le p\le 89\), \(\delta N^{1/3}\ge 1.0257\ldots \) (least at \(p=89\)).

As \(u\ne 0\), \(\zeta \) takes every value, and \(\zeta (n')\equiv \zeta (n)+\sigma \); so \(\Delta \zeta =\sigma \) for \(\zeta (n)\le p-1-\sigma \) and \(\Delta \zeta =\sigma -p\) otherwise, and both occur if \(\sigma \ne 0\). Reversing the pair exchanges \((\Delta \zeta ,\omega )\) and \((-\Delta \zeta ,-\omega )\), so let \(\Delta \zeta =d{\gt}0\) and \(\zeta =\zeta (n)\in [0,p-1-d]\). As \(2\zeta \ge p\) if and only if \(\zeta \ge h+1\), \(\omega \in \{ 0,1\} \), and \(\omega =1\) if and only if \(\zeta \le h{\lt}\zeta +d\); such \(\zeta \) form the interval \([\max (0,h+1-d),\min (h,p-1-d)]\), which is not empty, as \(1\le d\le p-1\) and \(h+2\le p\). And \(\omega =0\) if and only if \(\zeta +d\le h\), possible if and only if \(d\le h\), or \(\zeta \ge h+1\), possible if and only if \(h+1\le p-1-d\), i.e. \(d\le h-1\).

Call a step \(E\) with a pair \((\xi ,\omega )\) as in the definition of \(F\) an option of \(\psi \), with the vector \(V=E+\xi \mathbf w-\omega (1,1,0)\). Every option has \(|\xi |{\lt}1\) and \(|2\xi -\omega |{\lt}1\), so \(|V|{\gt}2\) if some \(|E_c|\ge 3\). The step \(\mathbf{e}_z\) gives \(V=\mathbf{e}_z\) if \(\psi _z\equiv 0\), and otherwise, with \(\xi =\{ \psi _z\} -1\) and \(\omega =-1\), it gives \(V=(2\xi +1,2\xi +1,1+\xi )\), whose entries lie in \((-1,1)\). Hence \(F\le 3\), and only the steps \(E\in \{ -2,\dots ,2\} ^3\setminus \{ 0\} \) matter.

(1) At integer scale \(p^3\) the point of cell \(n\) is \(p^2n+p\, g(n)+f(n)\) with the third digits \(f(n)\). As in the proof of Proposition 107(2), the representatives of the difference of the points of the cells \(n\) and \(n'\) are \(D=p^2E+p\, \Delta g+\Delta f\) with \(E\equiv n'-n\pmod p\) and \(|\Delta f_c|{\lt}p\); without the third digits they are \(p^2V\) with \(V=E+\Delta g/p\), and \(\bigl||D|/p^2-|V|\bigr|\le |\Delta f|/p^2{\lt}\sqrt3/p\). Comparing a pair that attains \(\delta \) with one that attains \(\sqrt{F_p(u)}\) gives \(|\delta N^{1/3}-\sqrt{F_p(u)}|{\lt}\sqrt3/p\). By Lemma 108 (which holds for every \(u\); part (1) uses neither \(u\ne 0\) nor the net property), \(\Delta g=(2\Delta \zeta -p\omega ,\, 2\Delta \zeta -p\omega ,\, \Delta \zeta )\), and the pairs \((\Delta \zeta ,\omega )\) that occur for the step \(E\) are those of the lemma. With \(\xi =\Delta \zeta /p\) they are exactly the pairs of the definition of \(F\) for \(\psi =u/p\): \(\Delta \zeta \in \{ \sigma ,\sigma -p\} \) (only \(0\) if \(\sigma =0\)) means \(\xi \in (-1,1)\) and \(\xi \equiv u\cdot E/p\pmod1\), and, as \(p\) is odd, \(|\Delta \zeta |\le h\) means \(|\xi |{\lt}\frac12\). Then \(V=E+\xi \mathbf w-\omega (1,1,0)\). As \(p\ge 5\), every step \(E\in \{ -2,\dots ,2\} ^3\setminus \{ 0\} \) joins two distinct cells, so \(F_p(u)=F(u/p)\).

(2) Let \(\psi =(\frac12-\varepsilon ,\frac12-\varepsilon ,2\varepsilon )\) with \(0{\lt}\varepsilon \le \frac3{100}\). With \(S=E_x+E_y\) and \(Q=S-2E_z\), \(\psi \cdot E=\frac S2-Q\varepsilon \), so \(\xi =\nu -Q\varepsilon \) with \(\nu \in \frac S2+\mathbb {Z}\), and \(V=A-Q\varepsilon \mathbf w\) with \(A=E+(2\nu -\omega )(1,1,0)+\nu \mathbf{e}_z\). This is the parametrization of the proof of Proposition 107(2), where \(u/p\) is of this form with \(\varepsilon =\rho /(2p)\). As \(|Q\varepsilon |\le 8\varepsilon {\lt}\frac12\), the pairs that occur follow the rules stated there: for \(\nu =0\), \(|\xi |{\lt}\frac12\) and \(\omega \in \{ 0,-\operatorname {sgn}Q\} \); for \(\nu =\pm \frac12\), \(\xi \) has the sign of \(\nu \), \(\omega =\operatorname {sgn}\nu \) occurs, and \(\omega =0\) occurs if and only if \(\nu Q{\gt}0\); \(\nu =\pm 1\) occurs only if \(\nu Q{\gt}0\), and then \(\omega =\operatorname {sgn}\nu \); and \(|\nu |\ge \frac32\) does not occur. The rest of that proof uses only \(\varepsilon \le \frac3{100}\): the options with \(|A|^2\ge 3\) have \(|V|{\gt}1.05\), and the others are the rows of Table tab:wcontacts. On \((0,\frac3{100}]\) the rows (c), (d) and (f) decrease, to \(1.1081\), \(1.1681\) and \(1.5524\), and the rows (e) and (g) are at least \(\frac54\); all of these, and \(1.05^2=1.1025\), exceed \(\mu ^{*2}\), while \(\min (q_a(\varepsilon ),q_b(\varepsilon ))\le \mu ^{*2}\), as \(q_a\) decreases and \(q_b\) increases through \(\varepsilon ^*\). The rows (a) and (b) occur, since \((\nu ,\omega )=(\frac12,1)\) and \((0,0)\) are always allowed. So \(F(\psi )=\min (q_a(\varepsilon ),q_b(\varepsilon ))\).

(3) Let \(\psi \in (\mathbb {R}/\mathbb {Z})^3\) with \(F(\psi )\ge \mu ^{*2}\), and represent \(\psi _x,\psi _y,\psi _z\) by numbers in \([0,1)\). We use only options with \(\omega =\operatorname {sgn}\xi \), which always occur, and options with \(\omega =0\) and \(|\xi |{\lt}\frac12\). The numbers \(\frac{246}{225}=1.0933\ldots \), \(\frac{17}{16}\) and \(\frac9{25}\) are below \(\mu ^{*2}=1.0983\ldots \).

The step \(\mathbf{e}_z\). If \(\psi _z=0\), then \(V=\mathbf{e}_z\) and \(|V|^2=1\). Otherwise \(\xi =\psi _z-1\), \(\omega =-1\) gives \(V=(2\psi _z-1,2\psi _z-1,\psi _z)\) with \(|V|^2=9\psi _z^2-8\psi _z+2\), a convex function with the values \(\frac{246}{225}\) at \(\psi _z=\frac2{15}\) and \(\frac{17}{16}\) at \(\psi _z=\frac34\); so \(\psi _z\notin [\frac2{15},\frac34]\). If \(\psi _z{\gt}\frac34\), then \(\xi =\psi _z-1\), \(\omega =0\) gives \(|V|^2=8(1-\psi _z)^2+\psi _z^2\le \frac{17}{16}\). So \(0{\lt}\psi _z{\lt}\frac2{15}\), and \(\xi =\psi _z\), \(\omega =0\) gives \(V=(2\psi _z,2\psi _z,1+\psi _z)\) with \(|V|^2=q_b(\psi _z/2)\); as \(q_b\) increases, \(\psi _z\ge 2\varepsilon ^*\).

The steps \(\mathbf{e}_x\) and \(\mathbf{e}_y\). If \(\psi _x=0\), then \(|V|^2=1\). Otherwise \(\xi =\psi _x\), \(\omega =1\) gives \(V=(2\psi _x,2\psi _x-1,\psi _x)\) with \(|V|^2=9\psi _x^2-4\psi _x+1\), a convex function with the values \(1\) at \(\psi _x=0\) and \(\frac{246}{225}\) at \(\psi _x=\frac7{15}\); so \(\psi _x{\gt}\frac7{15}\). If \(\psi _x{\gt}\frac12\), then \(\xi =\psi _x-1\), \(\omega =0\) gives \(V=(1-2t,-2t,-t)\) with \(t=1-\psi _x\), and \(|V|^2=9t^2-4t+1\) is the same function, so \(t{\gt}\frac7{15}\). Hence \(\alpha =\frac12-\psi _x\in (-\frac1{30},\frac1{30})\), and likewise \(\beta =\frac12-\psi _y\in (-\frac1{30},\frac1{30})\). Put \(\gamma =\psi _z\).

The step \(\mathbf1\) (the lock). Here \(\psi \cdot \mathbf1\equiv \gamma -\alpha -\beta =:\tilde\sigma \in (-\frac1{15},\frac15)\). If \(\tilde\sigma {\gt}0\), then \(\xi =\tilde\sigma -1\), \(\omega =-1\) gives \(V=(2\tilde\sigma ,2\tilde\sigma ,\tilde\sigma )\) with \(|V|^2=9\tilde\sigma ^2{\lt}\frac9{25}\). So \(\alpha +\beta \ge \gamma \).

The diagonal steps \((1,0,-1)\) and \((0,1,-1)\). For \(E=(1,0,-1)\), \(\psi \cdot E\equiv \psi _x-\psi _z=\frac12-\lambda \) with \(\lambda =\alpha +\gamma \in (2\varepsilon ^*-\frac1{30},\frac16)\subset (0,\frac16)\), and \(\xi =\frac12-\lambda \), \(\omega =1\) gives \(V=(1-2\lambda ,-2\lambda ,-\frac12-\lambda )\) with \(|V|^2=\frac54-3\lambda +9\lambda ^2=q_a(\lambda /3)\): row (a) of Table tab:wcontacts with \(\varepsilon =\lambda /3\). Likewise \(E=(0,1,-1)\) gives \(q_a(\lambda '/3)\) with \(\lambda '=\beta +\gamma \). As \(q_a\) decreases on \([0,\frac1{18}]\) and \(q_a(\varepsilon ^*)=\mu ^{*2}\le F(\psi )\), we get \(\lambda \le 3\varepsilon ^*\) and \(\lambda '\le 3\varepsilon ^*\). But \(\lambda +\lambda '=\alpha +\beta +2\gamma \ge 3\gamma \ge 6\varepsilon ^*\), by the lock and the step \(\mathbf{e}_z\). So equality holds throughout: \(\lambda =\lambda '=3\varepsilon ^*\), \(\alpha +\beta =\gamma \) and \(\gamma =2\varepsilon ^*\), hence \(\alpha =\beta =\varepsilon ^*\), i.e. \(\psi =\psi ^*\).

Hence \(F(\psi )\ge \mu ^{*2}\) only at \(\psi =\psi ^*\), where \(F(\psi ^*)=\mu ^{*2}\) by (2): \(F\le \mu ^{*2}\) everywhere, with equality only at \(\psi ^*\). As \(\varepsilon ^*\) is irrational, \(u/p\ne \psi ^*\), so \(F_p(u)=F(u/p){\lt}\mu ^{*2}\), and by (1) \(\delta N^{1/3}{\lt}\sqrt{F_p(u)}+\sqrt3/p{\lt}\mu ^*+\sqrt3/p\).

By Lemma 19(P) over \(\mathbb K\), with \(\pi _{ij}=\tilde M_{ij}\tilde M_{ji}\), the numbers \(\chi ^2=\pi _{ab}\pi _{cd}/(\pi _{ac}\pi _{bd})\) and \(\psi ^2=\pi _{ab}\pi _{cd}/(\pi _{ad}\pi _{bc})\) satisfy \((\chi ^2-\psi ^2-1)^2=4\psi ^2\). (1) The entries of a layer of Cauchy form are units, so for odd \(p\) the \(h_{ij}\equiv 2\tilde M_{ij}\) are units modulo \(p\) (for \(p=2\) the hypothesis fails, the \(h_{ij}\) being odd), and \(4\pi _{ij}\equiv h_{ij}h_{ji}\). Let \(A_1=|h_{ab}h_{ba}h_{cd}h_{dc}|\), \(A_2=|h_{ac}h_{ca}h_{bd}h_{db}|\), \(A_3=|h_{ad}h_{da}h_{bc}h_{cb}|\), odd integers at most \(\Omega ^4\). Then \(\chi ^2\equiv \varepsilon _2A_1/A_2\) and \(\psi ^2\equiv \varepsilon _3A_1/A_3\) with signs \(\varepsilon _2,\varepsilon _3\in \{ \pm 1\} \), and multiplying the last relation by \(A_2^2A_3^2\) shows that \(p\) divides

The bracket is a sum of three odd numbers, hence odd, so \(Z\) is odd and in particular nonzero, and \(|Z|\le (3\Omega ^8)^2+4\Omega ^{16}=13\Omega ^{16}\). (2) Over \(\mathbb {Q}\) the same computation with \(h_{ij}=2\tilde M_{ij}\), odd integers by assumption, gives \(Z=0\) exactly, contradicting that \(Z\) is odd.

The box matrix is the Hermite matrix of Lemma 9 for the \(2s\) points with the multiplicities \(\alpha _P\), \(\sum _P\alpha _P=\sum _cd_c=m\) (no point at \(\infty \)); this gives the value, a unit modulo \(p\) if the points are distinct modulo \(p\). Conversely, a coincidence modulo \(p\) puts two equal rows into some box: \(V(v_c)\) and \(V(v_d)\) for \(v_c\equiv v_d\) (the box \((k,\dots ,k)\)); \(V(u_c)\) and \(V(v_d)\) for \(u_c\equiv v_d\), \(c=d\) allowed (depth \(k+1\) on \(c\), at least \(1\) on \(d\)); \(V(u_c)\) and \(V(u_d)\) for \(u_c\equiv u_d\), \(c\ne d\) (depths \(k+1\) on \(c\) and \(d\), \(k-1\) on two other coordinates, \(k\) elsewhere; here \(s\ge 4\) is used). Then Lemma 1.

As \(\deg V{\lt}sk=\deg \Pi \), partial fractions give \(V=\sum _d\Pi _dN_d\) with \(\deg N_d{\lt}k\); as \(V\equiv \Pi _dN_d\pmod{(t-v_d)^k}\) and \(\Pi _d(v_d)\) is a unit, the cell of \(d\) determines \(N_d\) and conversely. The first offset digit of \(c\) is \(V(u_c)\). For the column (level \(j\) of the cell of \(d\), all other cells \(0\)) we have \(V=\Pi _dN_d\) with \(N_d\equiv (t-v_d)^j/\Pi _d\pmod{(t-v_d)^k}\), i.e. \(N_d=\sum _{j\le i{\lt}k}\varphi _{i-j}(t-v_d)^i\) with \(\varphi (x)=\sum _n\varphi _nx^n=1/\Pi _d(v_d+x)\). Hence, with \(\vartheta =\vartheta _{cd}\), \(V(u_c)=\Pi _d(u_c)\sum _{j\le i{\lt}k}\varphi _{i-j}\vartheta ^i\). Weighting the level \(j=k-1-n\) by \(x^n\) gives \(\Pi _d(u_c)\, \vartheta ^{k-1}\varphi (x)\sum _{r\ge 0}(x/\vartheta )^r=\Pi _d(u_c)\vartheta ^k\varphi (x)/(\vartheta -x)\) modulo \(x^k\), and \(\Pi _d(u_c)\vartheta ^k=\Pi (u_c)=\Theta _c^k\).

The normal form is Lemma 115 times \(\Theta _c^{-k}\) and \(\Pi _d(v_d+x)\), as in Proposition 13, with unit diagonal entries of \(L_c\) and \(U_d\). The proofs of Propositions 14 and 18 use only that the \(w_{cd}\) are units and that the \(y_{cd}\) of a column are distinct with unit differences (the matrix \((w_{cd}y_{cd}^n)_{c\ne d,\, n\le s-2}\) is then a unit diagonal times an invertible Vandermonde matrix), and the proof of Corollary 16 uses only the unit box minors of the frame, here Lemma 114.

By Proposition 116, \(M^{(0)}[c][d]=\frac12\Lambda _c\vartheta _{cd}^{-1}\) and \(M^{(j)}[c][d]=\frac12\Lambda _c\vartheta _{cd}^{-1}(1+2\vartheta _{cd}^{-j})\) for \(1\le j\le s-2\). For \(1\le i\le s-1\), \(\Lambda _c/\vartheta _{cd}^{\, i}\) is a product of odd integers, hence an odd integer; so the first term lies in \(\frac12+\mathbb {Z}\) and the second in \(\mathbb {Z}\). The leading coefficients are \(1\) and \(2\), and \(\mu _d=1\). For \(H_s\) the differences \(u_c-u_{c'}\), \(v_d-v_{d'}\) and \(\vartheta _{cd}=2(c-d)-1\) have absolute value at most \(2s-1\).

(1) In cell units, coordinate \(c\) of the point of cell \(n\) is \(n_c+(r_c+f_c)/p\) with \(r_c\) the first offset digit and \(f_c\in [0,1)\). Let \(n\ne n'\) and let \(E\) be a torus representative of \(n'-n\) (modulo \(p^k\) in each coordinate). If some \(|E_c|\ge 3\), then the two points differ by \(|D_c|{\gt}|E_c|-1\ge 2\) cells in coordinate \(c\), with \(D\) their difference in cell units, since \((r'_c+f'_c)-(r_c+f_c)\in (-p,p)\). Otherwise adding \(E_d\) to the cell integer \(n_d\) changes its level \(k-1\) by \(E_d\) and the levels \(k-1-i\), \(1\le i\le j_d\), by \(\operatorname {sgn}(E_d)\) (carries or borrows; a carry is at most \(1\) since \(|E_d|\le 2{\lt}p\), and the torus wrap is the full carry); since row \(0\) vanishes below level \(k-L\), capping \(j_d\) at \(L-1\) is exact. So \(\Delta r_c=r'_c-r_c\equiv \sigma _c\pmod p\) with \(\Delta r_c\in (-p,p)\). As \(F_0\in (\frac12+\mathbb {Z})^{s\times s}\) and \(F_i\) is integral, \(\sigma _c\in \Sigma E/2+\mathbb {Z}\). If \(\Sigma E\) is odd, \(\sigma _c\pm \frac p2\) are the integers congruent to \(\sigma _c\) (with \(\frac12\equiv (p+1)/2\)), and as \(|\sigma _c|{\lt}p/2\) they are the only ones in \((-p,p)\); if \(\Sigma E\) is even, \(\sigma _c\) is an integer with \(|\sigma _c|{\lt}p\), and the integers in \((-p,p)\) congruent to it are \(\sigma _c\) and \(\sigma _c-p\operatorname {sgn}\sigma _c\) (only \(0\) if \(\sigma _c=0\)). So \(\Delta r_c\in O_c\), and \(p|D_c|=|pE_c+\Delta r_c+\Delta f_c|{\gt}|pE_c+\Delta r_c|-1\), i.e. \(p|D_c|\ge b_c\).

(2) First \(|\sigma _c|\le 2\alpha _c+2A_c\le B\), since the own term is \(-E_c\alpha _c\) and the term of an axis \(d\ne c\) is \(E_dM^{(j_d)}[c][d]\) (\(|E_d|=1\)) or \(\operatorname {sgn}(E_d)(M^{(0)}+M^{(j_d)})[c][d]\) (\(|E_d|=2\)); so (1) applies. Let \(e=|E|_\infty \le 2\) and \(|E_c|=e\). Then \(E_c\sigma _c\le -e^2\alpha _c+e\cdot eA_c\le 0\): every coordinate of maximal size is backward. If \(\Sigma E\) is odd, every coordinate has \(|pE_c+t|\ge p|E_c\pm \frac12|-|\sigma _c|\ge p/2-|\sigma _c|\) for \(t\in O_c\), so \(|D_c|\ge \frac12-\epsilon _p\) for all \(s\) coordinates. If \(\Sigma E\) is even, a backward coordinate \(c\) with \(|E_c|=e\) has \(\operatorname {sgn}\sigma _c\ne \operatorname {sgn}E_c\), so both options give \(|pE_c+t|\ge pe-|\sigma _c|\) and \(|D_c|\ge e-\epsilon _p\). For \(e=2\) this is at least \(2-\epsilon _p\ge \sqrt2(1-\epsilon _p)\); for \(e=1\) an even \(\Sigma E\) needs at least two nonzero coordinates, each of maximal size, so \(|D|\ge \sqrt2(1-\epsilon _p)\).

(3) Since \(|2F_0[c][d]|\le 2C{\lt}p\) is odd, every entry of \(F_0\) is a unit modulo \(p\), and \(C{\lt}p/2-1\). First take the step \(E=\mathbf{e}_a\) from a cell in which the last digit of \(n_a\) is not \(p-1\) (no carry) and \(r_a\) wraps, i.e. \(\Delta r_a=F_0[a][a]-\frac p2\); such cells exist, because \(r_a\) depends on the last digit of \(n_d\), \(d\ne a\), with the unit coefficient \(F_0[a][d]\). Then \(|D_a|\le \frac12+(C+1)/p\), and every other coordinate has \(\Delta r_c=F_0[c][a]\pm \frac p2\) and \(|D_c|\le \frac12+(C+1)/p\); so \(\delta N^{1/s}\le \sqrt s(\frac12+(C+1)/p)\). Next take \(E=\mathbf{e}_a-\mathbf{e}_b\), with the last digit of \(n_a\) not \(p-1\) and that of \(n_b\) not \(0\) (no carries); then \(\sigma _c=\beta _c:=F_0[c][a]-F_0[c][b]\) is an integer with \(|\beta _c|\le 2C\). Fix these two digits. For every \(c\), \(r_c\) depends with a unit coefficient on the last digit of some \(n_g\), \(g\notin \{ a,b\} \) (as \(s\ge 3\)), so it is uniformly distributed on the remaining cells, and the fraction of them on which \(r_c+\beta _c\notin [0,p)\) is \(|\beta _c|/p\). As \(\sum _c|\beta _c|\le 2sC{\lt}p\), some cell has no wrap; there \(|D_c-E_c|\le (|\beta _c|+1)/p\le (2C+1)/p\) for every \(c\), so \(|D|\le \sqrt2+\sqrt s(2C+1)/p\).

Let \(H_s\) be the design of Lemma 117 on \(u_c=2c\), \(v_d=2d+1\), let \(\mathcal H^s_{k,p}=\mathcal P^{u,v,H_s}_{k,p}\), \(B_s=\max _c(2\alpha _c+2A_c)\) and \(C_s=\max _c\alpha _c\), which is the largest entry of \(F_0\) in absolute value, since \(|F_0[c][d]|\le A_c\le \alpha _c\). For a prime \(p\ge 2s\) and every \(k\ge 1\), Lemma 117 and Proposition 116 make \(\mathcal H^s_{k,p}\) a digital \((0,sk,s)\)-net with one point in every cell, whose row \(0\) is \(H_s\) truncated to \(L=\min (k,s-1)\) levels and \(0\) below. It satisfies (Half) and the dominance hypothesis of Proposition 118(2) (with \(A_c\) over all \(j\le s-2\ge L-1\)), so Proposition 118(2) gives the lower bound (as \(p\ge 2B_s+2{\gt}2B_s\); for \(p\le 2B_s+2\) it would not be positive) and Proposition 118(3) the upper bound, both independent of \(k\). Finally \(|\vartheta _{cd}|\le 2s-1\), so \(|\Lambda _c|\le (2s-1)^{(s-1)^2}\), \(|M^{(j)}[c][d]|\le \frac32|\Lambda _c|\) (absolute values, as \(\Lambda _c\) is negative for some \(c\) when \(s\) is even), \(\alpha _c\le A_c+1\) and \(B_s\le 6(s-1)\max _c|\Lambda _c|+2=e^{O(s^2\log s)}\).

In cell units coordinate \(c\) of the point of cell \(n\) is \(n_c+r_c/p+(d_c+g_c)/p^2\), with \(r_c\) and \(d_c\) the first and second offset digits and \(g_c\in [0,1)\). For two distinct points and a torus representative \(E\) of their cell difference with some \(|E_c|\ge 3\), coordinate \(c\) differs by more than \(2\), as in the proof of Proposition 118(1). Otherwise, as there (at \(k=1\) there are no carries), \(\Delta r_c\in O_c\). As \(d=D_1n\bmod p\) is linear in the cell, \(\Delta d_c\equiv v_c\pmod p\) with \(\Delta d_c\in (-p,p)\) and \(|v_c|{\lt}p\), so \(\Delta d_c\in U_c\); and \(|\Delta g_c|{\lt}1\). So \(p^2|D_c|=|p^2E_c+p\Delta r_c+\Delta d_c+\Delta g_c|\ge b_c\). The comparison with the toroidal distance is that of the proof of Proposition 118(1).

(1) The generator matrix of coordinate \(c\) has the rows \(\mathbf{e}_c\) and row \(c\) of \(F_0^*\), \(D_1\), \(D_2\), \(D_3\). By Lemma 2 its box minors are the minors of the stacked offset rows with disjoint row and column sets; each of the \(126\) of them, computed in exact rational arithmetic, is a nonzero rational whose numerator and denominator have no prime factors other than \(2\), \(3\) and \(5\). Lemma 1 concludes.

(2) Over \(E\in \{ -2,\dots ,2\} ^5\), \(\max |2(F_0^*E)_c|=22\) and \(\max |(D_1E)_c|=24\), so Lemma 120 applies for \(p\ge 29\). Evaluated exactly, \(\mathrm{LB}_2(p){\gt}1\) for every prime \(37\le p{\lt}200\) (the least value is \(1.0127\ldots \), at \(p=37\); and \(\mathrm{LB}_2(31)=0.9918\ldots \)), and \(\mathrm{LB}_2^*(200)^2=1.1877\ldots {\gt}1\), so \(\mathrm{LB}_2(p)\ge \mathrm{LB}_2^*(p)\ge \mathrm{LB}_2^*(200){\gt}1\) for every \(p\ge 200\). For the limit, with the row-sum norm \(\| F_0^*\| _\infty =\max _c\sum _d|F_0^*[c][d]|=\frac{11}2\) every option of Proposition 118(1) is at least \(p\, \ell _c(E)-(2\| F_0^*\| _\infty +1)\), where

\(\omega \) over \(\{ 0,\operatorname {sgn}\beta _c\} \) (over \(\{ 0\} \) if \(\beta _c=0\)). So \(\delta N^{1/5}\ge \lambda -\sqrt5(2\| F_0^*\| _\infty +1)/p\) with \(\lambda =\min _E|\ell (E)|\). If \(\Sigma E\) is odd, every coordinate gives at least \(\frac12\), with equality exactly on the binding steps; if \(\Sigma E\) is even, \(|\ell (E)|^2\) is an integer, and a finite check over the \(5^5-1\) vectors shows that it is at least \(2\). So \(\lambda =\sqrt5/2\), and Proposition 118(3) with \(C=\frac32\) gives \(\delta N^{1/5}\le \sqrt5/2+4\sqrt5/p\) for \(p{\gt}32\).

(3) Proposition 118(1) with \(k=1\) and \(F_0=F_0^*\) applies for \(p{\gt}22\). Evaluated exactly, \(\mathrm{LB}{\gt}1\) for every prime \(53\le p{\lt}200\) (\(\mathrm{LB}(47)=0.9906\ldots \), \(\mathrm{LB}(53)=1.0049\ldots \)), and the nondecreasing bound \(\mathrm{LB}^*\) after Proposition 118 has \(\mathrm{LB}^*(200)^2=1.1689\ldots {\gt}1\). (The monotone bounds already exceed \(1\) from \(p=50\) for \(\mathrm{LB}_2^*\) and from \(p=63\) for \(\mathrm{LB}^*\), so the exact scans up to \(200\) in (2) and (3) are more than needed.)

Diagonal. \(\mathrm{dr}(E)\) contains \(\alpha _c\) exactly for the coordinates with \(E_c\ne 0\), since \(-E_c\beta _c=\alpha _c-E_c\sum _{d\ne c}F_0[c][d]E_d\), and \(\beta _c\) does not involve \(\alpha _c\) when \(E_c=0\). So \(\mathrm{dr}(F_0)\) does not increase when every \(\alpha _c\) is lowered to \(m_c\), which keeps (B2) and (MIN); we assume \(\alpha =m\).

Minor tax. At most one row of \(F_0\) has all its off-diagonal entries in \(\{ \pm \frac12\} \): if rows \(a\ne b\) did, the three ratios \(F_0[a][c]/F_0[b][c]\), \(c\notin \{ a,b\} \), would lie in \(\{ \pm 1\} \), two would be equal, and the \(2\times 2\) minor on those two columns would vanish. So at most one \(m_c\) is \(\frac12\): at most one row is light, with all off-diagonal entries \(\pm \frac12\), and the others are heavy, and \(\sum _c\alpha _c\ge 4\cdot \frac32+\frac12=\frac{13}2\).

Balance on the full steps. A full step has all \(E_c=\pm 1\). For \(E\in \{ \pm 1\} ^5\) (binding, as \(5\) is odd), \(\mathrm{dr}(E)=-E^{\! \top }F_0E=\sum _c\alpha _c+\phi (E)\) with \(\phi (E)=\sum _{c{\lt}d}\gamma _{cd}E_cE_d\) and \(\gamma _{cd}=-(F_0[c][d]+F_0[d][c])\in \mathbb {Z}\). The function \(\phi \) has mean \(0\) over \(\{ \pm 1\} ^5\), and it vanishes identically if and only if the off-diagonal part of \(F_0\) is skew-symmetric; otherwise \(\max \phi \ge 1\), as \(\phi \) is integer-valued. So \(\mathrm{dr}(F_0)\ge \sum _c\alpha _c\), and \(\mathrm{dr}(F_0)\ge \sum _c\alpha _c+1\) if the off-diagonal part is not skew.

Skew layers. If some entry has \(|F_0[c][d]|\ge \frac52\), it occurs in the rows \(c\) and \(d\), and with the minor tax \(\sum \alpha \ge \frac52+\frac52+\frac32+\frac32+\frac12=\frac{17}2\). Otherwise all off-diagonal entries are \(\pm \frac12\) or \(\pm \frac32\); let \(G\) be the graph on the five coordinates with the edges \(\{ c,d\} \) where \(|F_0[c][d]|=\frac32\). For two coordinates \(x,y\) and the set \(S\) of the other three, the binding steps with \(E_x=E_y=0\) and \(E_S\in \{ \pm 1\} ^3\) have, as the skew terms inside \(S\) cancel, \(\mathrm{dr}(E)=\sum _{c\in S}\alpha _c+|u\cdot E_S|+|w\cdot E_S|\) with \(u=F_0[x,S]\), \(w=F_0[y,S]\). The step with \(E_S=\operatorname {sgn}u\) (coordinatewise) has \(u\cdot E_S=\| u\| _1\) and \(w\cdot E_S\in \frac12+\mathbb {Z}\), as \(w\) has three entries in \(\frac12+\mathbb {Z}\); so

(i) If there is a light row \(\ell \), then \(\sum \alpha =\frac{13}2\), the other four rows have degree at least \(1\) in \(G\), and \(\ell \) is isolated. \(G\) is not a perfect matching on the four heavy rows: for an edge \(\{ a,b\} \) of such a matching, the rows \(a\) and \(b\) would be \(\pm \frac12\) on the three columns outside \(\{ a,b\} \), which (MIN) excludes as in the minor tax. So some heavy \(h\) has degree at least \(2\), and the pair \(\{ x,y\} =\{ h,\ell \} \) gives \(\mathrm{dr}(F_0)\ge \frac92+(\frac32+2+\frac12)=\frac{17}2\). (ii) If there is no light row, \(\sum \alpha =\frac{15}2\) and every degree is at least \(1\); as \(5\) is odd, \(G\) is not a perfect matching, so some \(x\) has degree at least \(2\), and with \(y\) outside \(x\) and its neighbours if \(\deg x=2\) (any \(y\ne x\) otherwise) we get \(\deg _S(x)\ge 2\) and again \(\mathrm{dr}(F_0)\ge \frac92+4=\frac{17}2\).

Layers that are not skew. If some entry has absolute value at least \(\frac52\), then \(\sum \alpha \ge \frac52+3\cdot \frac32+\frac12=\frac{15}2\), and balance gives \(\mathrm{dr}\ge \frac{17}2\); the same holds if \(\sum \alpha \ge \frac{15}2\). There remain the layers with entries in \(\{ \pm \frac12,\pm \frac32\} \), one light row and four heavy rows with \(\alpha =\frac32\). Conjugation by permutation matrices and by diagonal sign matrices preserves the binding steps, \(\mathrm{dr}\), (B2) and (MIN), so we may take the light row to be row \(4\), equal to \((\frac12,\frac12,\frac12,\frac12,-\frac12)\). Each heavy row then has \(240\) possibilities, and an exhaustive enumeration with the (MIN) conditions finds \(1{,}004{,}544\) layers satisfying (MIN), none with \(\mathrm{dr}\le \frac{15}2\). Finally \(\mathrm{dr}(F_0^*)=\frac{17}2\) is a finite check over the \(61\) binding steps up to sign.

The design \(H_5^*\) satisfies (Half); its geometric layer \(F_0\) has the row scalars \(\lambda '=\frac12(-9,-3,-3,3,9)\) and the column scalars \(\mu =(5,5,5,-5,1)\). The primes of the point differences, of the units \(\lambda '_c\mu _d\) and of the leading coefficients of the column polynomials \(g_{d,j}\) (exact interpolation) are \(2,3,5,7\) and \(97,101\), the last two only in leading coefficients at carry depth \(j=3\); Proposition 116 gives the net statement. For the separation, \(\max |\sigma _c|=253\), so Proposition 118(1) applies for \(p{\gt}506\); the monotone bound \(\mathrm{LB}^*\), minimized over the number of design levels \(L=\min (k,4)\in \{ 1,2,3,4\} \) (that is, over \(k\)), is \(1.026529\ldots \) at \(p=1511\), the same value as \(\mathrm{LB}\) (attained at \(L=4\), i.e. \(k\ge 4\)). As \(\mathrm{LB}^*\) is nondecreasing in \(p\) for each \(L\), \(\delta N^{1/5}\ge \mathrm{LB}^*(p)\ge \mathrm{LB}^*(1511)=1.026529\ldots {\gt}1.0265\) for every prime \(p\ge 1511\) and every \(k\).