Masks, lists and one draw per edge

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The realization theorem of The model operator and finite tests is proved by drawing, independently on every edge of a block $(i,k)$, one pair $(W,z)$ of a matrix weight and a voltage from a finite distribution whose expectation is close to $h_i(t)h_k(t)p(t)$ in every fiber. This lesson builds that distribution in three approximations. The projection field $p(t)$ is replaced by compactly supported coefficient templates at magnitudes on a logarithmic grid, multiplied by smooth scalar masks of the central parameter. The masks are realized by finitely many central powers, through their Fourier series. The second-factor templates, which carry the matrices $R_\gamma$, are replaced by finite lists, so that only finitely many matrix weights occur: one list for each template used below the layer $I=H_0$, and one common list from $I$ on, obtained by factoring through the congruence subgroup $\Lambda_q$. The lesson ends with the size of the resulting alphabet and the ranges of the voltage coordinates, which the girth argument of the next lesson needs [OpenAI-KK, Sections 9.4 to 9.6].

Templates on a logarithmic grid (Section 1).
Masks and their Fourier series (Section 2).
Finite lists on the second factor (Section 3).
One draw per edge (Section 4).
The alphabet and the coordinates of the voltages (Section 5).

The notation of the four preceding lessons is in force. In particular $L=\bar E_{\mathbb Z}$, $q=d^{A_*}$, $\delta=q^{-1}$, $N=q^{n_{\mathrm f}}$, $I=H_0$, $w_i=q^{-1}2^{-\kappa|i|}$, $M=\lfloor d^{1/32}\rfloor$, and $W\rho(g)$ stands for $\rho(g)\otimes W$ on $\ell^2(F)\otimes\mathbb C^N$, with fibers given by (1.1) of the previous lesson. A target accuracy $\varepsilon>0$ is fixed; fixed constants may depend on it, and all statements hold for large $\ell$.

1. Templates on a logarithmic grid

Let $J^\flat_\pm$ be smooth compactly supported functions given by Lemma 4.1 of Schwartz idempotents and lattice projections for a fixed $\varepsilon_0>0$, and let $\varrho\ge1$ be such that both supports lie in the box $\{|a_w|\le\varrho\}$ of the coordinates of $\bar E$. For a sign $\epsilon$, a magnitude $0<\alpha\le1$ and a family of operators $U'(\gamma)$, $\gamma\in L$, put $L_\epsilon(\alpha;U')=\sum_{\gamma\in L}\alpha^{n_{\mathrm f}}J^\flat_\epsilon\bigl(\Delta_{\alpha^{1/m}}\gamma\bigr)\,U'(\gamma) .$ The lattice $\Delta_{\alpha^{1/m}}L$ has mesh at most one and cell volume $\alpha^{n_{\mathrm f}}$ (Lemma 2.1 of An affine nilpotent group), so the coefficient $\ell^1$ norm of $L_\epsilon(\alpha;\cdot)$ is at most $\|J^\flat_\epsilon\|_\infty$ times the volume of the $1$-neighbourhood of the box, a fixed constant $C_\flat$.

Fix a spacing $s_0>0$, and a smooth $\psi^0\ge0$ supported in $(-s_0,s_0)$ with $\sum_{j\in\mathbb Z}\psi^0(v-js_0)=1$ for all $v\in\mathbb R$; it exists, as in the proof of Lemma 5.1 of The magnetic factor and the weighted trace. For $\nu,\omega\in\mathbb Z$ and $0<t<\delta$ put $\begin{aligned} \psi_\nu(t)&=\psi^0\Bigl(\log_2\frac t\delta-\nu s_0\Bigr),\\ a_\nu&=\delta\,2^{\nu s_0},\\ \theta_\omega(t)&=\psi^0\Bigl(\log_2\frac{\delta-t}\delta-\omega s_0\Bigr),\\ b_\omega&=\delta\,2^{\omega s_0}, \end{aligned}$ and $\psi_\nu=\theta_\omega=0$ off $(0,\delta)$. Then $\sum_\nu\psi_\nu=\sum_\omega\theta_\omega=1$ on $(0,\delta)$, and on the supports $|\log(a_\nu/t)|\le s_0$ and $|\log(b_\omega/(\delta-t))|\le s_0$. Only $\nu,\omega\le0$ occur, so the magnitudes are at most $\delta$. For an adjacent pair $(i,k)$ let $\mathcal T_{ik}$ be the set of pairs $(\nu,\omega)$ such that $\psi_\nu\theta_\omega$ does not vanish identically on the support of $h_ih_k$, and put $f_{ik\nu\omega}=h_ih_k\,\psi_\nu\theta_\omega .$ Since $h$ is positive on $[-0.7,0.7]$, at $b(t)=(i+k)/2$ both $h_i$ and $h_k$ are positive, so $\mathcal T_{ik}$ is not empty.

Lemma 1.1 (templates; [OpenAI-KK, Section 9.4]). For every $\varepsilon_1>0$ there are fixed $\varepsilon_0$ and $s_0$ such that, for large $\ell$:

$|\mathcal T_{ik}|\le\mathcal N$ for every adjacent pair $(i,k)$, with $\mathcal N=(2\kappa/s_0+3)^2$;
for $0<t<\delta$ and every adjacent pair $(i,k)$, $\begin{aligned} &\Bigl\|h_i(t)h_k(t)p(t)\\ &\quad-\sum_{(\nu,\omega)\in\mathcal T_{ik}}f_{ik\nu\omega}(t)\\ &\qquad\cdot L_+(a_\nu;u_t)\otimes L_-(b_\omega;V_t)\Bigr\|\\ &\quad\le\varepsilon_1 . \end{aligned}$

Proof. (1) On the support of $h_i$, $|b(t)-i|<1$, and $\log_2(t/\delta)=-\log_2(1+2^{-\kappa b})$ is a function of $b$ with derivative between $0$ and $\kappa$. So it ranges over an interval of length at most $2\kappa$, which meets the supports of at most $2\kappa/s_0+3$ functions $\psi_\nu$; the same holds for $\log_2((\delta-t)/\delta)=-\log_2(1+2^{\kappa b})$ and the $\theta_\omega$.

(2) By Proposition 3.1 of the lesson on Schwartz idempotents, with $\theta=t$, the element $K_t=\sum_\gamma t^{n_{\mathrm f}}J_+(\Delta_{t^{1/m}}\gamma)u_t(\gamma)$ satisfies $\|p_t-K_t\|\le Ct^{1/m}\le C\delta^{1/m}$ and $\|K_t\|\le C$. For $t$ in the support of $\psi_\nu$, Lemma 4.1 of that lesson gives coefficient $\ell^1$ distance at most $\varepsilon_0$ for the passage from $J_+$ to $J^\flat_+$, and at most $Cs_0$ for the passage from the magnitude $t$ to $a_\nu$; only coefficients change, the unitaries $u_t(\gamma)$ stay the same. So $\|K_t-L_+(a_\nu;u_t)\|\le\varepsilon_0+Cs_0$. With $\theta=t-\delta<0$, the same holds for $K_{t-\delta}$ and $J_-$ at magnitude $\delta-t$, evaluated on $V_t$; by Lemma 3.1 of The magnetic factor and the weighted trace, $\|p_{t-\delta}[V_t]-K_{t-\delta}[V_t]\|=\|p_{t-\delta}-K_{t-\delta}\|$, at most $C\delta^{1/m}$, and for $t$ in the support of $\theta_\omega$, $\|K_{t-\delta}[V_t]-L_-(b_\omega;V_t)\|\le\varepsilon_0+Cs_0$. Writing the difference of two tensor products as a sum of differences in one factor, $\begin{aligned} &\bigl\|p(t)-L_+(a_\nu;u_t)\otimes L_-(b_\omega;V_t)\bigr\|\\ &\quad\le C\bigl(\delta^{1/m}+\varepsilon_0+s_0\bigr) \end{aligned}$ whenever $\psi_\nu(t)\theta_\omega(t)\ne0$. The numbers $\psi_\nu(t)\theta_\omega(t)\ge0$ sum to one, and those with $h_i(t)h_k(t)\psi_\nu(t)\theta_\omega(t)\ne0$ belong to $\mathcal T_{ik}$, so the left side of (2) is at most $h_i(t)h_k(t)\,C(\delta^{1/m}+\varepsilon_0+s_0)$. Choose $\varepsilon_0$ and $s_0$ with $C(\varepsilon_0+s_0)\le\varepsilon_1/2$, and $\ell$ so large that $C\delta^{1/m}\le\varepsilon_1/2$. $\square$

The fiber of the operator $\sum_{\gamma}a_\nu^{n_{\mathrm f}}J^\flat_+(\Delta_{a_\nu^{1/m}}\gamma)\rho(\sigma_X(\gamma))$ is $L_+(a_\nu;u_t)\otimes1$, by (1.1) of the previous lesson, and that of the lifted template $\begin{gathered} \mathcal T(b)\\ =\sum_{v\in L}b^{n_{\mathrm f}}J^\flat_-\bigl(\Delta_{b^{1/m}}v\bigr)\\ \cdot R_v\,\rho(\sigma_Y(v)) \end{gathered}\tag{1.1}$ is $1\otimes L_-(b;V_t)$, because $R_v$ attached to $\sigma_Y(v)$ turns $u_t(v)$ into $V_t(v)=u_t(v)\otimes R_v$.

2. Masks and their Fourier series

For a smooth function $f$ on $\mathbb T$ put $\hat f(n)=\int_{\mathbb T}f(t)\mathbf e(-nt)\,dt$. By (1.1) of the previous lesson, the operator $\sum_n\hat f(n)\rho(z_0^n)$ acts in the fiber at $t$ by the scalar $\sum_n\hat f(n)\mathbf e(nt)$.

Lemma 2.1 (masks; [OpenAI-KK, Section 9.4]). For every adjacent pair $(i,k)$ and $(\nu,\omega)\in\mathcal T_{ik}$, the mask $f=f_{ik\nu\omega}$ is smooth on $\mathbb T$, supported in an interval of length at most $Cw_i$, and for every $j\ge0$ $\begin{gathered} \|f^{(j)}\|_\infty\le C_jw_i^{-j},\\ \|f^{(j)}\|_1\le C_jw_i^{1-j},\\ |\hat f(n)|\le C_jw_i(1+|n|w_i)^{-j} . \end{gathered}$ Consequently $\sum_n|\hat f(n)|\le C$, and for $\Theta\ge1$ and $K_i=\lceil\Theta/w_i\rceil$, the truncation $f^{\le}(t)=\sum_{|n|\le K_i}\hat f(n)\mathbf e(nt)$ satisfies $\sup_t|f(t)-f^{\le}(t)|\le C\Theta^{-1}$. The constants depend on $\kappa$, $h$, $\psi^0$ and $s_0$, not on $\ell$, $i$, $k$, $\nu$, $\omega$.

Proof. On the support of $h_i$, $t=\delta\frac{2^{\kappa b}}{1+2^{\kappa b}}$ and $\delta-t=\frac\delta{1+2^{\kappa b}}$ with $|b-i|<1$, so both are at least $c_\kappa w_i$: one of them is at least $\delta/2\ge w_i/2$, and the other is comparable to $\delta2^{-\kappa|b|}$. The $j$-th derivative of $\psi_\nu$ is a combination of derivatives of $\psi^0$ with products of derivatives of $\log_2t$, so it is at most $C_jt^{-j}$; likewise $|\theta_\omega^{(j)}|\le C_j(\delta-t)^{-j}$. With $\|h_i^{(j)}\|_\infty\le C_jw_i^{-j}$ (Lemma 5.1 of The magnetic factor and the weighted trace), $w_k\ge2^{-\kappa}w_i$ for $|k-i|\le1$, and the Leibniz rule, $\|f^{(j)}\|_\infty\le C_jw_i^{-j}$. The support of $h_i$ is a compact subset of $(0,\delta)$ of length at most $Cw_i$, which gives smoothness on $\mathbb T$ and the $L^1$ bounds.

$|\hat f(n)|\le\|f\|_1\le Cw_i$, and $j$ integrations by parts give $|\hat f(n)|\le\|f^{(j)}\|_1(2\pi|n|)^{-j}$, at most $C_jw_i^{1-j}|n|^{-j}$. If $|n|w_i\le1$ the first bound gives the claim, otherwise the second, since then $|n|w_i\ge\frac12(1+|n|w_i)$. With $j=2$, $\begin{aligned} &\sum_nw_i(1+|n|w_i)^{-2}\\ &\quad\le w_i+2\int_0^\infty w_i(1+sw_i)^{-2}\,ds\le3 , \end{aligned}$ and, the summand being decreasing in $|n|$, $\begin{aligned} &\sum_{|n|>K_i}w_i(1+|n|w_i)^{-2}\\ &\quad\le2\int_{K_i}^\infty w_i(1+sw_i)^{-2}\,ds\\ &\quad=\frac2{1+K_iw_i}\le\frac2\Theta . \end{aligned}$ The uniform bound for $f-f^{\le}$ is this tail, times $C_2$. $\square$

The truncated masks need not vanish off the supports of the masks; only the uniform closeness is used.

3. Finite lists on the second factor

The scalar masks vary through infinitely many layers, but the alphabet of matrix weights must be finite. Below the layer $I$ only $O(1+\log q)$ second-factor templates occur, each replaced by its own list; from $I$ on, one common list serves all layers.

Lemma 3.1 (lists below $I$; [OpenAI-KK, Section 9.5]). Let $\mathcal W_<$ be the set of magnitudes $b_\omega$ with $(\nu,\omega)\in\mathcal T_{ik}$ for some adjacent pair with $i<I$. Then $|\mathcal W_<|\le C(1+\log q)$. For every fixed $\varepsilon_2>0$ and large $\ell$, there are for each $b\in\mathcal W_<$ elements $v_{b,1},\ldots,v_{b,M}\in L$ and scalars $c_{b,1},\ldots,c_{b,M}$ with $|c_{b,h}|\le C$ and $\begin{aligned} &\Bigl\|\frac1M\sum_{h=1}^Mc_{b,h}R_{v_{b,h}}\rho\bigl(\sigma_Y(v_{b,h})\bigr)-\mathcal T(b)\Bigr\|\\ &\quad\le\varepsilon_2 . \end{aligned}$

Proof. For $i<I$ and $t$ in the support of $h_i$, $b(t)<i+1\le I$, so $\delta\ge\delta-t\ge\delta2^{-\kappa(I+1)}/2$. On the support of $\theta_\omega$, $b_\omega$ is within the factor $e^{s_0}$ of $\delta-t$, so $\log_2b_\omega^{-1}$ lies in the interval from $\log_2q-C$ to $\log_2q+\kappa(I+1)+C$, which contains at most $(\kappa I+C)/s_0$ grid points. Since $I=H_0\le h_*\log_2d$ and $\log_2q=A_*\log_2d$, this is at most $C(1+\log q)$.

For $b\in\mathcal W_<$ let $\mathfrak Q_b\subset L$ be the coordinate rectangle with radius $\lceil\varrho b^{-w/m}\rceil$ in every coordinate of weight $w$, and $n_b=|\mathfrak Q_b|$. It contains every $v$ with $J^\flat_-(\Delta_{b^{1/m}}v)\ne0$, and since the weights of the $2n_{\mathrm f}$ coordinates of $L$ add up to $mn_{\mathrm f}$, $\begin{aligned} n_b&\le\prod_w(2\varrho+3)\,b^{-w/m}\\ &=(2\varrho+3)^{2n_{\mathrm f}}\,b^{-n_{\mathrm f}} . \end{aligned}$ For $v$ uniform in $\mathfrak Q_b$, the random operator $n_bb^{n_{\mathrm f}}J^\flat_-(\Delta_{b^{1/m}}v)R_v\rho(\sigma_Y(v))$ has expectation $\mathcal T(b)$ and matrix part of norm at most $(2\varrho+3)^{2n_{\mathrm f}}\|J^\flat_-\|_\infty$. Its group coordinates are at most $(\varrho+1)b^{-1}\le Cq2^{\kappa I}\le q^a$ for a fixed $a\ge n_{\mathrm f}$, because $2^{\kappa I}\le d^{\kappa h_*}=q^{\kappa h_*/A_*}$. Lemma 5.1 of the previous lesson, applied to these at most $C(1+\log q)$ distributions, gives the lists. $\square$

From $I$ on, let $\Lambda=\Lambda_q$, of index $Q_q=q^{2n_{\mathrm f}}$ in $L$ (Section 2 of The magnetic factor and the weighted trace), and $\mathcal V_q$ the set of elements of $L$ with all coordinates in $\{0,\ldots,q-1\}$. Every $\gamma\in L$ is uniquely $\gamma=yv$ with $y\in\Lambda$, $v\in\mathcal V_q$ (Exercise 6.1), and $R_y=1$ for $y\in\Lambda$. Put $\begin{gathered} C_q=\frac1{Q_q}\sum_{v\in\mathcal V_q}R_v\,\rho(\sigma_Y(v)),\\ \mathcal S(b)=Q_q\sum_{y\in\Lambda}b^{n_{\mathrm f}}J^\flat_-\bigl(\Delta_{b^{1/m}}y\bigr)\rho(\sigma_Y(y)) . \end{gathered}$ By Lemma 5.1 of the previous lesson, applied to the uniform distribution on $\mathcal V_q$, there are $v_1,\ldots,v_M\in\mathcal V_q$ with $\|C_q^M-C_q\|\le\varepsilon_2$, where $C_q^M=\frac1M\sum_hR_{v_h}\rho(\sigma_Y(v_h))$. This one list serves all layers $i\ge I$.

Lemma 3.2 (the subgroup factorization; [OpenAI-KK, Section 9.5]). Let $\mathcal W_\ge$ be the set of magnitudes $b_\omega$ with $(\nu,\omega)\in\mathcal T_{ik}$ for some adjacent pair with $i\ge I$. For large $\ell$:

for every such pair and $b=b_\omega$, $b^{-1/m}/q\ge2^{2E_0H_i} ;\tag{3.1}$
the coefficient $\ell^1$ norms of the $\mathcal S(b)$, $b\in\mathcal W_\ge$, are at most a fixed constant, and every fiber of $\mathcal S(b)C_q$ and of $\mathcal S(b)C_q^M$ has norm at most a fixed constant;
$\|\mathcal S(b)C_q^M-\mathcal S(b)C_q\|\le C\varepsilon_2$;
for $t$ in the support of $\theta_\omega$, the fiber of $\mathcal S(b_\omega)C_q$ at $t$ is within $\varepsilon_2$ of $1\otimes L_-(b_\omega;V_t)$, uniformly in $\omega$ and $t$.

Proof. (1) For $i\ge I$ and $t$ in the support of $h_i$, $b(t)>i-1\ge0$, so $\delta-t\le\delta2^{-\kappa b}\le2^\kappa q^{-1}2^{-\kappa i}$, and $b_\omega\le e^{s_0}(\delta-t)$. Hence $\begin{aligned} \log_2\frac{b^{-1/m}}q&\ge\frac\kappa mi\\ &\quad-\Bigl(1-\frac1m\Bigr)A_*\log_2d-C . \end{aligned}$ Since $i\ge I=H_0>\log_2d$ and $H_i=H_0+i\le2i$, the right side minus $2E_0H_i$ is at least $(\kappa/m-A_*-4E_0)i-C$, which is positive for large $\ell$ by the choice $\kappa/m>10(E_0+A_*+C_\bullet+1)$.

(2) In coordinates $\Lambda$ is the set of points of $L$ with all coordinates divisible by $q$. With $\beta=b^{1/m}$, $\Delta_\beta\Lambda$ is a rectangular lattice with spacing $q\beta^w\le q\beta\le1$ in weight $w$, by (3.1), and cell volume $Q_qb^{n_{\mathrm f}}$. So the coefficient $\ell^1$ norm $Q_qb^{n_{\mathrm f}}\sum_{y\in\Lambda}|J^\flat_-(\Delta_\beta y)|$ of $\mathcal S(b)$ is bounded, as in Section 1. The operators $C_q$ and $C_q^M$ are averages of unitaries, and the fiber norms of a product are at most the products of the coefficient $\ell^1$ norms.

(3) $\|\mathcal S(b)\|\le C$ and $\|C_q^M-C_q\|\le\varepsilon_2$.

(4) The factor $Q_q$ cancels, and $\begin{aligned} \mathcal S(b)C_q&=\sum_{y\in\Lambda,\,v\in\mathcal V_q}b^{n_{\mathrm f}}J^\flat_-(\Delta_\beta y)\\ &\qquad\cdot R_v\,\rho(\sigma_Y(y))\rho(\sigma_Y(v)) . \end{aligned}$ By (1.1) of the previous lesson and $R_y=1$, the term acts in the fiber at $t$ by $1\otimes V_t(y)V_t(v)$, and by Lemma 3.1 of the lesson on the magnetic factor, $V_t(y)V_t(v)=\mathbf e(\tau c(y,v))V_t(yv)$ with $\tau=t-\delta$. Since $(y,v)\mapsto yv$ is a bijection onto $L$, the fiber is $1\otimes\sum_\gamma d_\gamma V_t(\gamma)$ with $d_{yv}=b^{n_{\mathrm f}}J^\flat_-(\Delta_\beta y)\mathbf e(\tau c(y,v))$, to be compared with the coefficients $b^{n_{\mathrm f}}J^\flat_-(\Delta_\beta(yv))$ of $L_-(b;V_t)$. By Lemma 2.1 of An affine nilpotent group, $\Delta_\beta(yv)=(\Delta_\beta y)(\Delta_\beta v)$ and $\tau c(y,v)=(\tau/b)\,c(\Delta_\beta y,\Delta_\beta v)$, with $|\tau|/b\le e^{s_0}\le3$ on the support of $\theta_\omega$. The coordinates of $\Delta_\beta v$, $v\in\mathcal V_q$, are at most $q\beta\le2^{-2E_0H_i}=:\vartheta$. On the compact $1$-neighbourhood $K_1$ of the support of $J^\flat_-$, the group law and the cocycle are uniformly continuous, $c(a,1)=0$, and $J^\flat_-$ is Lipschitz; so for $\Delta_\beta y\in K_1$, $\bigl|d_{yv}-b^{n_{\mathrm f}}J^\flat_-(\Delta_\beta(yv))\bigr|\le b^{n_{\mathrm f}}\,\varpi(\vartheta),$ with $\varpi(\vartheta)\to0$ as $\vartheta\to0$. For $\Delta_\beta y\notin K_1$ and small $\vartheta$, both coefficients vanish, since $\Delta_\beta(yv)$ is then outside the support. The nonzero differences occur at $\gamma$ with $\Delta_\beta\gamma$ in a fixed compact set, whose number of points times $b^{n_{\mathrm f}}$ is bounded. So the coefficient $\ell^1$ distance, and with it the fiber distance, is at most $C\varpi(\vartheta)$, which is below $\varepsilon_2$ for large $\ell$, uniformly. $\square$

4. One draw per edge

For an adjacent pair $(i,k)$, fix $\varrho$ as in Section 1 and define a random pair $(W,z)\in M_N(\mathbb C)\times F$ by the following choices, independent given $(\nu,\omega)$.

Choose $(\nu,\omega)$ uniformly in $\mathcal T_{ik}$ and $n$ uniformly in $\{-K_i,\ldots,K_i\}$. Record the scalar $c_1=|\mathcal T_{ik}|(2K_i+1)\hat f_{ik\nu\omega}(n)$ and the element $z_0^n$.
Choose $x$ uniformly in the rectangle $\mathfrak P_\nu\subset L$ with radius $\lceil\varrho a_\nu^{-w/m}\rceil$ in every coordinate of weight $w$. Record $c_2=|\mathfrak P_\nu|\,a_\nu^{n_{\mathrm f}}J^\flat_+(\Delta_{a_\nu^{1/m}}x)$ and $\sigma_X(x)$.
If $i<I$, choose $h$ uniformly in $\{1,\ldots,M\}$; record $c_3=c_{b_\omega,h}$, the matrix $R=R_{v_{b_\omega,h}}$ and $g_3=\sigma_Y(v_{b_\omega,h})$. If $i\ge I$, choose $y$ uniformly in the set $\mathfrak Q^\Lambda_\omega$ of $y\in\Lambda$ with all coordinates of weight $w$ at most $\varrho b_\omega^{-w/m}$, and independently $h$ uniformly in $\{1,\ldots,M\}$; record $c_3=|\mathfrak Q^\Lambda_\omega|\,Q_qb_\omega^{n_{\mathrm f}}J^\flat_-(\Delta_{b_\omega^{1/m}}y)$, the matrix $R=R_{v_h}$ and $g_3=\sigma_Y(y)\sigma_Y(v_h)$.

Put $z=z_0^n\sigma_X(x)g_3$, let $c^\flat$ be a nearest point to $c_1c_2c_3$ in a fixed finite set $G_\varepsilon$ of complex numbers within $\varepsilon_2$ of every point of the disk of radius $C_\varepsilon+1$, and $W=c^\flat R$. The voltage $z$ is kept even when $W=0$.

The scalars are bounded: $|c_1|\le\mathcal N(2\Theta/w_i+3)\,Cw_i\le C$ by Lemma 2.1; $|c_2|\le C$ as in the proof of Lemma 3.1; $|c_3|\le C$ by Lemma 3.1 for $i<I$, and for $i\ge I$ because, by (3.1), $|\mathfrak Q^\Lambda_\omega|\le\prod_w(2\varrho+3)b_\omega^{-w/m}/q$, which is $(2\varrho+3)^{2n_{\mathrm f}}(Q_qb_\omega^{n_{\mathrm f}})^{-1}$. This gives the constant $C_\varepsilon$.

Proposition 4.1 (the expectation of one draw; [OpenAI-KK, Section 9.6]). For every $\varepsilon>0$ there are fixed choices of $\varepsilon_1$, $\varepsilon_0$, $s_0$, $\Theta$, $\varepsilon_2$ and $G_\varepsilon$ such that for large $\ell$ and every adjacent pair $(i,k)$, $\begin{gathered} \operatorname*{ess\,sup}_t\bigl\|\mathbb E[W\rho(z)](t)\\ -h_i(t)h_k(t)p(t)\bigr\|\le\varepsilon , \end{gathered}\tag{4.1}$ where $(t)$ denotes the fiber at $t$, and $h_ih_kp(t)=0$ for $t\notin(0,\delta)$.

Proof. Choose $\varepsilon_1=\varepsilon/4$; Lemma 1.1 fixes $\varepsilon_0$, $s_0$ and $\mathcal N$. Then choose $\Theta$ and $\varepsilon_2$ so small, in terms of $\mathcal N$ and the fixed constants, that the bound below is at most $\varepsilon$.

Let $X=c_1c_2c_3R\rho(z)$. Since $R$ and $\rho(z)$ are unitary, rounding changes $W\rho(z)$ by at most $\varepsilon_2$ in norm. As $\rho$ is a homomorphism, $R$ commutes with the $\rho(g)$, and the choices are independent given $(\nu,\omega)$, $\mathbb EX=\sum_{(\nu,\omega)\in\mathcal T_{ik}}F_{\nu\omega}\,\mathcal A_\nu\,\mathcal B_\omega ,$ where $F_{\nu\omega}=\sum_{|n|\le K_i}\hat f_{ik\nu\omega}(n)\rho(z_0^n)$; $\mathcal A_\nu=\sum_xa_\nu^{n_{\mathrm f}}J^\flat_+(\Delta_{a_\nu^{1/m}}x)\rho(\sigma_X(x))$, because $\mathfrak P_\nu$ contains every $x$ with a nonzero coefficient; and $\mathcal B_\omega$ is the list average of Lemma 3.1 if $i<I$, and $\mathcal S(b_\omega)C_q^M$ if $i\ge I$. In the fiber at $t$, $F_{\nu\omega}$ acts by $f^{\le}(t)$ and $\mathcal A_\nu$ by $L_+(a_\nu;u_t)\otimes1$.

The target $G(t)$, the sum over $\mathcal T_{ik}$ of $f(t)\,L_+(a_\nu;u_t)\otimes L_-(b_\omega;V_t)$, is within $\varepsilon_1$ of $h_ih_kp(t)$ for $0<t<\delta$ by Lemma 1.1, and equal to it, namely zero, for other $t$. For each pair, the fiber difference is $\begin{aligned} &(f^{\le}-f)(t)\,(L_+\otimes1)\mathcal B_\omega(t)\\ &\quad+f(t)\,(L_+\otimes1)\\ &\qquad\cdot\bigl(\mathcal B_\omega(t)-1\otimes L_-(b_\omega;V_t)\bigr) . \end{aligned}$ The first term is at most $C\Theta^{-1}\cdot C_\flat\cdot C$, since the fibers of $\mathcal B_\omega$ are bounded (Lemma 3.1 with $\|\mathcal T(b)\|\le C_\flat$, and Lemma 3.2(2)). The second term vanishes off the support of $\theta_\omega$, and on it is at most $C_\flat$ times $\varepsilon_2$ (Lemma 3.1, with the fiber of $\mathcal T(b)$ from Section 1) or $\varepsilon_2+C\varepsilon_2$ (Lemma 3.2(3) and (4)). Summing over at most $\mathcal N$ pairs, $\begin{aligned} &\bigl\|\mathbb E[W\rho(z)](t)-h_ih_kp(t)\bigr\|\\ &\quad\le\varepsilon_2+\mathcal NC\bigl(\Theta^{-1}+\varepsilon_2\bigr)+\frac\varepsilon4 \end{aligned}$ for almost every $t$, which is at most $\varepsilon$ for the choices made. $\square$

5. The alphabet and the coordinates of the voltages

Lemma 5.1 (alphabet and coordinates; [OpenAI-KK, Section 9.6]). For large $\ell$:

the matrices $W$ that occur form a set of at most $C(1+\log q)M$ elements, and the pairs of a label of $Y$ and such a matrix number $o(dD)$;
there is a fixed $C$ such that for every adjacent pair $(i,k)$ every possible $z$, and $z^{-1}$, has all coordinates at most $2^{CH_i}$ in absolute value;
if $i\le2I$, then $a_\nu^{-1/m}\ge2^{2E_0H_i}$ for every $(\nu,\omega)\in\mathcal T_{ik}$, and the superdiagonal coordinates $(b_x-1)_{uv}$, $u<v$, of $x$ in step 2 are independent and uniform on integer intervals with at least $2^{E_0H_i}$ points;
if $i\ge I$, the numbers $(b_y-1)_{uv}/q$, $u<v$, for $y$ in step 3 are independent and uniform on integer intervals with at least $2^{E_0H_i}$ points.

Proof. (1) The matrices are $c^\flat R$ with $c^\flat\in G_\varepsilon$, a fixed finite set, and $R$ in the lists of Lemma 3.1, at most $C(1+\log q)$ lists of $M$ entries, or in the one list of $C_q^M$. With $M\le d^{1/32}$ and $\log q=A_*\log d$, there are at most $Cd^{1/32}\log d$ matrices. The labels number $m_\ell\le27K^2d\,2^{r+2}=O(\ell^2d^{11/4})$ by Lemma 4.1 of Gated permutations and the two towers, since $2^r=d^{7/4}$ and $K=320\ell$. The product is $O(d^{89/32}\ell^3)=o(d^3)=o(dD)$.

(2) By Lemma 2.1, $|n|\le K_i\le\Theta q2^{\kappa|i|}+1$. On the support of $h_i$, $t$ and $\delta-t$ are at least $c_\kappa w_i$, so $a_\nu^{-1},b_\omega^{-1}\le Cq2^{\kappa|i|}$, and the coordinates of $x$, $v_{b,h}$ and $y$ are at most $(\varrho+1)Cq2^{\kappa|i|}$; those of $v_h\in\mathcal V_q$ are at most $q$. Since $\log_2(q2^{\kappa|i|})=A_*\log_2d+\kappa|i|$ is at most $(A_*+\kappa)H_i$, every factor of $z$ has coordinates at most $2^{C'H_i}$, and the fixed polynomial laws of multiplication and inversion give the bound $2^{CH_i}$ for $z$ and $z^{-1}$.

(3) On the support of $h_i$, $t<\delta$, so $a_\nu\le e^{s_0}\delta$ and $\log_2a_\nu^{-1}\ge\log_2q-2$. If $0\le i\le2I$, then $H_i\le3H_0\le3h_*\log_2d$, and $A_*/m>10E_0h_*$ gives $\begin{aligned} \frac1m\log_2a_\nu^{-1}&\ge10E_0h_*\log_2d-2\\ &\ge\tfrac{10}3E_0H_i-2\ge2E_0H_i . \end{aligned}$ If $i<0$, then $b(t)<i+1\le0$ and $t\le\delta2^{\kappa b}\le2^\kappa q^{-1}2^{-\kappa|i|}$, so $\log_2a_\nu^{-1}\ge\log_2q+\kappa|i|-\kappa-2$, and with $\kappa/m>10E_0$ and $H_i=H_0+|i|\le h_*\log_2d+|i|$, $\begin{aligned} \frac1m\log_2a_\nu^{-1}&\ge10E_0\bigl(h_*\log_2d+|i|\bigr)-C\\ &\ge2E_0H_i . \end{aligned}$ The coordinate $(b_x-1)_{uv}$ has weight $v-u\ge1$, so its interval has radius at least $\varrho a_\nu^{-1/m}\ge2^{2E_0H_i}$; a uniform point of a rectangle has independent uniform coordinates.

(4) The coordinate $(b_y-1)_{uv}$ ranges over the multiples of $q$ of absolute value at most $\varrho b_\omega^{-(v-u)/m}$, which number at least $b_\omega^{-1/m}/q\ge2^{2E_0H_i}$ by (3.1). $\square$

The ranges in (3) and (4) overlap: every layer $i$ is covered by one of them, and the layers from $I$ to $2I$ by both.

6. Exercises

6.1 (easy). Show that $|\mathcal V_q|=Q_q$, and that every $\gamma\in L$ is uniquely $yv$ with $y\in\Lambda$ and $v\in\mathcal V_q$.

6.2 (easy). Let $g$ be smooth with support in $(-1,1)$, $0<w<\frac12$, and $f(t)=g((t-\frac12)/w)$ on $\mathbb T$. Show that $\hat f(n)=w\,\hat g(nw)\,\mathbf e(-n/2)$, with the Fourier transform $\hat g(\xi)=\int g(s)\mathbf e(-\xi s)ds$, and deduce bounds of the form in Lemma 2.1.

6.3 (medium). Show that $\mathbb E[c_2\rho(\sigma_X(x))]$ in step 2 does not change if $\mathfrak P_\nu$ is replaced by a larger rectangle with the corresponding count, and that the bound $|c_2|\le C$ requires the rectangle radius to be of order $a_\nu^{-w/m}$ in weight $w$.

6.4 (medium). Derive $V_t(y)V_t(v)=\mathbf e((t-\delta)c(y,v))V_t(yv)$ for $y\in\Lambda_q$ directly from $\sigma(y)\sigma(v)=(0,1,c(y,v))\sigma(yv)$, the multiplier of $u_t$, and $R(0,1,z)=\mathbf e(-\delta z)$.

7. Solutions

6.1. Reduction modulo $q$ is a surjective homomorphism $L\to\bar E_{\mathbb Z/q\mathbb Z}$ with kernel $\Lambda$, and $\mathcal V_q$ maps bijectively onto $\bar E_{\mathbb Z/q\mathbb Z}$, a set of $q^{2n_{\mathrm f}}$ elements. For $\gamma\in L$, let $v\in\mathcal V_q$ have the same reduction; then $y=\gamma v^{-1}\in\Lambda$. If $yv=y'v'$, then $v$ and $v'$ have the same reduction, so $v=v'$ and $y=y'$.

6.2. $\begin{aligned} \hat f(n)&=\int g((t-\tfrac12)/w)\,\mathbf e(-nt)\,dt\\ &=w\int g(s)\,\mathbf e(-n(\tfrac12+ws))\,ds\\ &=w\,\mathbf e(-n/2)\hat g(nw) . \end{aligned}$ Since $|\hat g(\xi)|\le C_j(1+|\xi|)^{-j}$, $|\hat f(n)|\le C_jw(1+|n|w)^{-j}$, and $\sum_n|\hat f(n)|\le C$, as in Lemma 2.1.

6.3. The expectation over a rectangle $\mathfrak P$ of $|\mathfrak P|a^{n_{\mathrm f}}J^\flat_+(\Delta x)\rho(\sigma_X(x))$ is $\sum_{x\in\mathfrak P}a^{n_{\mathrm f}}J^\flat_+(\Delta x)\rho(\sigma_X(x))$, which is the same for every rectangle containing the support. The scalar is at most $|\mathfrak P|a^{n_{\mathrm f}}\|J^\flat_+\|_\infty$; with radii $\lceil\varrho a^{-w/m}\rceil$ the count is at most $Ca^{-n_{\mathrm f}}$, while radii larger by a factor $\lambda\ge1$ in every coordinate multiply it by about $\lambda^{2n_{\mathrm f}}$.

6.4. $V_t(y)V_t(v)=u_t(y)u_t(v)\otimes R_yR_v$. The first factor is $\mathbf e(tc(y,v))u_t(yv)$. For the second, $\begin{aligned} &R(\sigma(y))R(\sigma(v))\\ &\quad=R(\sigma(y)\sigma(v))\\ &\quad=R(0,1,c(y,v))R(\sigma(yv))\\ &\quad=\mathbf e(-\delta c(y,v))R_{yv} . \end{aligned}$ Together, $\mathbf e((t-\delta)c(y,v))V_t(yv)$; and $R_y=1$ is not needed for this identity.

References

[OpenAI-KK] OpenAI, A torsion-free counterexample to the Kadison–Kaplansky projection conjecture, OpenAI Math Release preprint, 23 September 2026, Sections 9.4 to 9.6. https://github.com/openai/math/tree/main/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026