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The zoom at a rank point

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

Rank coordinates are compact labels, but a small disk in the rank square could correspond to a hugely distorted region of the plane. This lesson recovers Euclidean control at the points of the good set \(E\) of Independent gradients and tangent directions [OpenAI-HL, Section 5]. Zooming into the rank chart at such a point, the three sweep directions with single tangent directions become three families of lines whose images have limiting directions, and the theorem of Three transverse foliations shows that the zoomed maps converge to a nonsingular linear map. A comparison with one normalization common to all points then gives the output used in A spatial measure: at every point of \(E\), the inverse rank map distorts circles of small radius by a bounded factor, with the radius fixed before the configuration.

We keep the notation of the previous lessons: the windows \(A,B\), the chart \((r_A,r_B)\), the set \(E\) and the times \(t_1<t_2<t_3\) and \(\tau_1<\tau_2<\tau_3\) of Proposition 6.1 there, and the direction \(\varsigma(t)\) of lines of direction \(t\).

1. A linear normalization

Lemma 1.1. Let \(d_1,d_2,d_3\) be pairwise non-parallel vectors of the plane and \(e_1,e_2,e_3\) also. There is an invertible linear map \(B\) with \(Bd_i\in\mathbb Re_i\) for \(i=1,2,3\).

Proof. Write \(d_3=\alpha d_1+\beta d_2\) and \(e_3=\alpha'e_1+\beta'e_2\); all four coefficients are nonzero, since no two of the vectors are parallel. Let \(B\) map \(\alpha d_1\) to \(\alpha'e_1\) and \(\beta d_2\) to \(\beta'e_2\). Then \(Bd_3=e_3\). \(\square\)

In each late configuration the slopes \(\varsigma(t_1),\varsigma(t_2),\varsigma(t_3)\) are distinct. Fix the target lines \(\mathbb Rv_1,\mathbb Rv_2,\mathbb Rv_3\), with \(v_1=(1,0)\), \(v_2=(0,1)\), \(v_3=(1,1)\), and let \(T_n\) be a linear map, given by Lemma 1.1, sending the lines of slopes \(\varsigma(t_i)\) to them. The common normalization of configuration \(n\) is \[ L_n=T_n\circ(r_A,r_B)^{-1}:(0,1)^2\longrightarrow\mathbb R^2 , \] a homeomorphism onto the plane. It does not depend on any label. For a closed disk \(\bar B(p,h)\subseteq(0,1)^2\) put \[ \begin{gathered} U_n(p,h)=\max_{|u-p|=h}|L_n(u)-L_n(p)|,\\ V_n(p,h)=\min_{|u-p|=h}|L_n(u)-L_n(p)|. \end{gathered} \]

2. The zoom converges to a linear map

Fix \(z\in E\) and \(p=p(z)\), with the times \(\tau_i\) and unit directions \(d_i\), \(D_p(\tau_i)=\{d_i\}\). By Lemma 1.1 fix an invertible \(B\) with \(Bd_i\in\mathbb Rv_i\). Let \((h_j,n_j)\) be a fast diagonal (Section 4 of Independent gradients and tangent directions).

In configuration \(n_j\), let \(X_j\) be the point with \((r_A,r_B)(X_j)=p\), and let \(L_j^\tau\) be a linear map sending the lines of slopes \(\varsigma(\tau_i)\) to \(\mathbb Rv_i\). For a scale \(c>0\) put \(Y(x)=X_j+c\,(L_j^\tau)^{-1}x\) and consider \[ F_j(x)=\frac1{h_j}B\bigl((r_A,r_B)(Y(x))-p\bigr). \] On the ray \(\{cv_1\}\) the norm \(|F_j(v_1)|\), as a function of \(c\), is continuous, zero at \(c=0\), and larger than \(1\) for large \(c\): as \(c\to\infty\) the point leaves every compact set of the plane, so its rank image approaches the boundary of the square, at distance from \(p\) larger than \(h_j\|B^{-1}\|\) once \(h_j\) is small. Choose \(c=c_j\) with \(|F_j(v_1)|=1\).

Proposition 2.1. Along a subsequence, \(F_j\to\sigma\,\mathrm{id}\) and \(F_j^{-1}\to\sigma\,\mathrm{id}\) uniformly on compact sets, for some \(\sigma\in\{-1,1\}\).

Proof. We check the hypotheses of Theorem 2.1 of Three transverse foliations. The map \(F_j\) is a homeomorphism of the plane onto \(\Omega_j=h_j^{-1}B((0,1)^2-p)\), and these sets eventually contain every compact set, since \(p\) is interior and \(h_j\to0\). A line of type \(i\) in the domain of \(F_j\) corresponds to a line of direction \(\tau_i\) in configuration \(n_j\). Both coordinates of \(B^{-1}F_j\), which are \(r_A\) and \(r_B\) up to translation and scaling, are monotone along lines of direction \(\tau_i\), with a sign that is the same for all such lines, by Section 5 of Independent gradients and tangent directions; the orientation induced by \(v_i\) may differ from the forward orientation, but along a subsequence it is fixed for each \(i\). So the first hypothesis holds with \(P=B^{-1}\).

Let \(x_j,y_j\) lie on a line of type \(i\) with \(F_j(x_j)\to a\) and \(F_j(y_j)\to b\ne a\). The zoom coordinates of the corresponding points are \(B^{-1}F_j(x_j)\to B^{-1}a\) and \(B^{-1}F_j(y_j)\to B^{-1}b\), and they lie on one line of direction \(\tau_i\). By Lemma 5.1 there, the unit vector of \(\pm B^{-1}(b-a)\), the sign given by the forward order, lies in \(D_p(\tau_i)=\{d_i\}\). So \(b-a\in\mathbb RBd_i=\mathbb Rv_i\). This holds along every subsequence, since subsequences of fast diagonals are fast diagonals. The third hypothesis holds by the choice of \(X_j\) and \(c_j\). \(\square\)

3. Comparison with the common normalization

Let \(A_j=T_{n_j}\circ(L_j^\tau)^{-1}\), the linear map from the \(\tau\)-normalized plane to the common one. It sends the lines \(\Xi_{i,j}=L_j^\tau(\text{lines of slope }\varsigma(t_i))\) to \(\mathbb Rv_i\).

Lemma 3.1. Along a further subsequence, \(\Xi_{i,j}\) converges to a line \(\Xi_i\) with \(B^{-1}\Xi_i\) spanned by a vector of \(D_p(t_i)\). The three lines \(\Xi_1,\Xi_2,\Xi_3\) are distinct, and the angle between any two of them is at least a constant \(\theta_p>0\) depending only on \(p\) and \(B\).

Proof. Pass to a subsequence along which the lines \(\Xi_{i,j}\) converge, and let \(u_j\) be a unit vector on \(\Xi_{i,j}\), oriented forward, with \(u_j\to u\). The segment from \(0\) to \(u_j\) in the \(\tau\)-normalized plane corresponds to a segment on a line of direction \(t_i\) in configuration \(n_j\). By Proposition 2.1 its endpoints have images \(F_j(0)=0\) and \(F_j(u_j)\to\sigma u\ne0\), uniformly. Their zoom coordinates are \(0\) and \(B^{-1}F_j(u_j)\to\sigma B^{-1}u\), so by Lemma 5.1 of Independent gradients and tangent directions the unit vector of \(\sigma B^{-1}u\) lies in \(D_p(t_i)\). The arcs \(D_p(t_1),D_p(t_2),D_p(t_3)\) are pairwise disjoint compact subsets of one closed quadrant, which contains no two opposite unit vectors; so the lines they span form three pairwise disjoint compact sets of lines, at mutual angle at least some \(\theta'>0\). Applying \(B\) changes angles by a bounded factor. \(\square\)

Lemma 3.2. The matrices \(A_j/\|A_j\|\) have, along a subsequence, a limit \(A\), and \(A\) is invertible, with condition number bounded by a constant depending only on \(\theta_p\).

Proof. A subsequence converges, to \(A\) with \(\|A\|=1\). If \(A\) were singular, it would have rank one, with a kernel line and an image line. At most one of the three distinct lines \(\Xi_i\) is the kernel; for the other two, say \(\Xi_a\) and \(\Xi_b\), \(A\) maps them onto the image line. But \(A_j\Xi_{a,j}=\mathbb Rv_a\) and \(A_j\Xi_{b,j}=\mathbb Rv_b\), and passing to the limit gives \(A\Xi_a=\mathbb Rv_a\) and \(A\Xi_b=\mathbb Rv_b\), two different lines. So \(A\) is invertible. The same argument applies to every sequence of matrices of norm one mapping three lines with mutual angles at least \(\theta_p\) to the three fixed lines, so by compactness their condition numbers are bounded. \(\square\)

Proposition 3.3. Let \(K_p\) be the bound of Lemma 3.2 times the condition number of \(B\). Along every fast diagonal there is a subsequence along which, for some scalars \(\kappa_j>0\), the maps \[ w\longmapsto\kappa_j\bigl(L_{n_j}(p+h_jw)-L_{n_j}(p)\bigr) \] converge uniformly on compact sets to an invertible linear map with condition number at most \(K_p\).

Proof. Write \(X=X_j+c_j(L_j^\tau)^{-1}x\). Then \(L_{n_j}((r_A,r_B)(X))-L_{n_j}(p)\) equals \(T_{n_j}(X-X_j)=c_jA_jx\), and \((r_A,r_B)(X)=p+h_jB^{-1}F_j(x)\). So for \(w\) in a compact set, with \(x=F_j^{-1}(Bw)\), \[ \begin{aligned} &\frac1{c_j\|A_j\|}\bigl(L_{n_j}(p+h_jw)-L_{n_j}(p)\bigr)\\ &\quad=\frac{A_j}{\|A_j\|}F_j^{-1}(Bw)\longrightarrow\sigma ABw , \end{aligned} \] uniformly, by Proposition 2.1 and Lemma 3.2. \(\square\)

4. Radial comparison

Proposition 4.1 (radial comparison; OpenAI). For every \(z\in E\), with \(p=p(z)\), there is \(C_p<\infty\) with the following property. For every integer \(T\ge2\) there is \(h_{p,T}>0\) such that for every fixed \(h\in(0,h_{p,T})\), in all sufficiently late configurations \(n\), \[ \begin{gathered} V_n(p,h/T)\ge\frac{U_n(p,h)}{C_pT},\\ U_n(p,h/T)\le\frac{C_pV_n(p,h)}T . \end{gathered} \]

Proof. Take any \(C_p>K_p\). Suppose the statement fails for some \(T\). Then there are \(h_k\to0\) and, for each \(k\), configurations as late as we like in which one of the inequalities fails for \(h=h_k\). Choose such a configuration \(n_k\) so late that \((h_k,n_k)\) is a fast diagonal. By Proposition 3.3, along a subsequence and after positive rescaling, \(w\mapsto L_{n_k}(p+h_kw)-L_{n_k}(p)\) converges uniformly on the closed unit disk to an invertible linear map \(\Lambda\) of condition number \(\kappa\le K_p<C_p\). Rescaling does not change the two ratios. On the circles \(|w|=1/T\) and \(|w|=1\), the ratio \(V_{n_k}(p,h_k/T)/U_{n_k}(p,h_k)\) tends to \[ \frac{\min_{|w|=1/T}|\Lambda w|}{\max_{|w|=1}|\Lambda w|}=\frac1{T\kappa}, \] and \(U_{n_k}(p,h_k/T)/V_{n_k}(p,h_k)\) tends to \(\kappa/T\). Since \(1/(T\kappa)>1/(C_pT)\) and \(\kappa/T<C_p/T\), both inequalities hold for large \(k\), contradicting the choice of \(n_k\). \(\square\)

The order of quantifiers is essential: the radius \(h\) is fixed first, and only then is the configuration taken late. No rate uniform in \(p\) or in the configuration is claimed.

5. Exercises

5.1 (easy). Find a linear map \(B\) with \(B(1,1)\in\mathbb R(1,0)\), \(B(1,-1)\in\mathbb R(0,1)\) and \(B(1,0)\in\mathbb R(1,1)\).

5.2 (medium). Show that for an invertible linear map \(\Lambda\) of condition number \(\kappa\), \(\min_{|w|=r}|\Lambda w|/\max_{|w|=R}|\Lambda w|=r/(R\kappa)\).

5.3 (medium). Why can Proposition 4.1 not be proved with a single fast diagonal fixed in advance?

6. Solutions

5.1. \((1,0)=\frac12(1,1)+\frac12(1,-1)\). With the proof of Lemma 1.1 (\(\alpha=\beta=\frac12\), \(\alpha'=\beta'=1\)), \(B\) maps \(\frac12(1,1)\) to \((1,0)\) and \(\frac12(1,-1)\) to \((0,1)\): \(B(x,y)=(x+y,\,x-y)\), and \(B(1,0)=(1,1)\).

5.2. \(\min_{|w|=r}|\Lambda w|=r\,s_{\min}\) and \(\max_{|w|=R}|\Lambda w|=R\,s_{\max}\), with the singular values \(s_{\min},s_{\max}\) and \(\kappa=s_{\max}/s_{\min}\).

5.3. A fixed diagonal says nothing about radii \(h\) not in the sequence \((h_j)\), nor about configurations other than \(n_j\). The proposition concerns every small fixed radius and all late configurations; the proof argues by contradiction and builds a fast diagonal out of the failing pairs.

References