Independent gradients and tangent directions

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

This lesson extracts local geometry from the limiting measures of Limiting measures [OpenAI-HL, Section 4]. A fixed pair of windows $A,B$ gives every label a position $p$ in the open unit square, and every other rank coordinate becomes, in the limit, a monotone function $f_C$ of $p$, differentiable at almost every $p$. Two consequences are proved. First, on a positive part of edge mass, every window active at a label contains two windows whose gradients at $p$ are independent; this uses the crossing measures and the critical-image lemma of Monotone functions of two variables. Second, lines of a fixed direction $t$ of the sweep have, after zooming in at $p$, limiting directions confined to an arc $D_p(t)$, and arcs for different times are disjoint when edge activity separates the times. Three times with three different single directions are found; the next lessons turn them into a Euclidean normalization.

We keep the subsequence of configurations of Limiting measures and pass to further subsequences when needed; every statement about "late configurations" refers to the current subsequence.

1. A pair of windows

By Lemma 3.1 of Limiting measures, for $\lambda$-almost every $z$ the measure $h_z$ has no atoms, so its support in $(0,1)$ is infinite, and two dyadic windows whose closures are disjoint and lie in $(0,1)$ both have positive $h_z$-mass. There are countably many pairs of windows, so we can fix such windows $K,K'$ for which $E=\{z:h_z(K)>0,\ h_z(K')>0\}$ has $\lambda(E)>0$.

Lemma 4.1 of Limiting measures, applied to $K'$, and $\lambda_{K'}=h_z(K')\lambda$ with $h_z(K')>0$ on $E$, show that $\lambda|_E$ vanishes on every set that is null for all crossing measures $\sigma_{CD}$ with $C,D\subseteq K'$ disjoint. Let $\Sigma=\sum c_{CD}\sigma_{CD}$ over these countably many pairs, with positive weights and finite total mass, and let $g_{CD}$ be a density of $\sigma_{CD}$ with respect to $\Sigma$ [Fremlin, 232E]. Since $\sum c_{CD}g_{CD}=1$ $\Sigma$-almost everywhere, the sets $\{g_{CD}>0\}$ cover $Z$ up to a $\Sigma$-null, hence $\lambda|_E$-null, set; so one of them, $\{g_{AB}>0\}$, meets $E$ in a set of positive $\lambda$-measure. Replace $E$ by this intersection, and name the windows so that $A$ precedes $B$. If $F\subseteq E$ is $\sigma_{AB}$-null, then $\int_Fg_{AB}\,d\Sigma=0$, so $\Sigma(F)=0$ and $\lambda(F)=0$. Thus $\lambda|_E\ \text{vanishes on }\sigma_{AB}\text{-null sets.}$ For a label $z$ put $p(z)=(z_A,z_B)\in[0,1]^2$. By Lemma 1.1(c) of Limiting measures, the image of $\sigma_{AB}$ under $p$ is at most Lebesgue measure, so the image of $\lambda|_E$ under $p$ vanishes on Lebesgue-null sets: it is absolutely continuous. In particular $p(z)\in(0,1)^2$ for $\lambda$-almost every $z\in E$.

2. Limit functions

Every window $C\subseteq K$ is disjoint from $A$ and $B$, and since $K$ and $K'$ are disjoint, $C$ either precedes both or follows both. By Proposition 3.3 of Rank coordinates, in each configuration $F_C^{(n)}=r_C\circ(r_A,r_B)^{-1}:(0,1)^2\to(0,1)$ is monotone in each coordinate with signs $(+,-)$ if $C$ precedes $A$ and $(-,+)$ if $B$ precedes $C$. Reflecting a coordinate where it decreases, we treat it as nondecreasing in both; write $u<w$ when $u_1<w_1$ and $u_2<w_2$ in the reflected coordinates. Let $G$ be the points of $(0,1)^2$ with rational coordinates. A diagonal subsequence makes $F_C^{(n)}(q)$ converge, to $\phi_C(q)$, for all $q\in G$ and all the countably many $C$. Define $\begin{gathered} f_C(p)=\sup\{\phi_C(q):q\in G,\ q<p\},\\ \bar f_C(p)=\inf\{\phi_C(q):q\in G,\ q>p\}. \end{gathered}$ Both are nondecreasing and $f_C\le\bar f_C$; $f_C$ is Borel, as a countable supremum of Borel functions. Call $p$ regular for $C$ if $f_C(p)=\bar f_C(p)$.

Lemma 2.1. (a) If $p$ is regular for $C$ and $p_n\to p$, then $F_C^{(n)}(p_n)\to f_C(p)$.

(b) Every point where $f_C$ is continuous is regular for $C$. Almost every point of $(0,1)^2$ is regular, and $f_C$ is differentiable at almost every point.

Proof. (a) For grid points $q^-<p<q^+$, eventually $q^-<p_n<q^+$, so $F_C^{(n)}(q^-)\le F_C^{(n)}(p_n)\le F_C^{(n)}(q^+)$; in the limit, $\begin{aligned} \phi_C(q^-)&\le\liminf F_C^{(n)}(p_n)\\ &\le\limsup F_C^{(n)}(p_n)\le\phi_C(q^+). \end{aligned}$ Taking the supremum over $q^-$ and the infimum over $q^+$ gives $f_C(p)\le\liminf$ and $\limsup\le\bar f_C(p)=f_C(p)$.

(b) If $p<p''$, a grid point $q$ with $p<q<p''$ has $\bar f_C(p)\le\phi_C(q)\le f_C(p'')$. Letting $p''\to p$ at a continuity point gives $\bar f_C(p)\le f_C(p)$. By Theorem 2.2 of Monotone functions of two variables, $f_C$, which is bounded, Borel and monotone in each variable, is differentiable, hence continuous, almost everywhere. $\square$

Lemma 2.2 (coordinate identities). Let $z$ lie in the support of $\lambda$, with $p=p(z)\in(0,1)^2$ regular for $C$. Then $z_C=f_C(p)$.

Proof. For every $k$, the ball of radius $1/k$ about $z$ has positive $\lambda$-mass, so by Lemma 2.1(a) of Measures on compact metric spaces it contains the label of a left endpoint of an edge in every late configuration. Choosing diagonally, there are vertices $q_n$ of configuration $n$ whose labels $z^{(n)}$ tend to $z$. Then $p(z^{(n)})=(r_A,r_B)(q_n)\to p$ and $z^{(n)}_C=r_C(q_n)=F_C^{(n)}(p(z^{(n)}))$, which tends to $f_C(p)$ by Lemma 2.1(a); it also tends to $z_C$. $\square$

The points $p$ that fail to be regular for some $C$, or where some $f_C$ is not differentiable, form a Lebesgue-null set, a countable union of null sets. Its preimage under $p$ is $\lambda|_E$-null, by absolute continuity. Removing it, and the $\lambda$-null complement of the support of $\lambda$, from $E$, we may assume that for every $z\in E$: $p=p(z)\in(0,1)^2$, every $f_C$ with $C\subseteq K$ is differentiable at $p$, and $z_C=f_C(p)$.

3. Independent gradients

Lemma 3.1. After removing a further $\lambda$-null set from $E$, every $z\in E$ has the following property: if $J\subseteq K$ is a window with $h_z(J)>0$, then some disjoint windows $C,D\subseteq J$ satisfy $\det\begin{pmatrix}\nabla f_C(p)\\\nabla f_D(p)\end{pmatrix}\ne0,\qquad p=p(z).$

Proof. Fix $J\subseteq K$, and let $B_J$ be the set of $z\in E$ with $h_z(J)>0$ at which the determinant vanishes for all disjoint $C,D\subseteq J$. For each such pair, Lemma 3.1 of Monotone functions of two variables, applied to $F=(f_C,f_D)$ on $(0,1)^2$, gives a Borel null set $N_{CD}$ containing the values $F(p)$ at the points where $F$ is differentiable with $\det DF(p)=0$. For $z\in B_J$, $(z_C,z_D)=(f_C(p),f_D(p))\in N_{CD}$, by Lemma 2.2. By Lemma 1.1(c) of Limiting measures, the image of $\sigma_{CD}$ under $z\mapsto(z_C,z_D)$ is at most Lebesgue measure, so $\sigma_{CD}(B_J)=0$. This holds for all disjoint $C,D\subseteq J$, so Lemma 4.1 of Limiting measures gives $\lambda_J(B_J)=0$, that is, $\int_{B_J}h_z(J)\,d\lambda=0$. Since $h_z(J)>0$ on $B_J$, $\lambda(B_J)=0$. Remove $B_J$ for all countably many $J$. $\square$

4. First-order expansions

Fix $z\in E$ and $p=p(z)$.

Lemma 4.1. Let $\mathcal C$ be a finite set of windows in $K$, $\rho<\infty$ and $\epsilon>0$. There is $h_0>0$ such that for every $h\in(0,h_0)$ there is $n_0(h)$ with $\begin{aligned} &\bigl|F_C^{(n)}(p+hw)-f_C(p)\\ &\quad-h\nabla f_C(p)\cdot w\bigr|\le\epsilon h \end{aligned}$ for all $C\in\mathcal C$, $|w|\le\rho$ and $n\ge n_0(h)$.

Proof. Let $\eta>0$ be small. By differentiability, for $h<h_0$ the function $f_C(p+hw)$ differs from $f_C(p)+h\nabla f_C(p)\cdot w$ by at most $\eta h$ for $|w|\le\rho+1$ and all $C\in\mathcal C$. Fix such $h$. Regular points are dense, so there is a finite set of regular points $q$ in the disk of radius $(\rho+1)h$ about $p$ such that every point $x=p+hw$, $|w|\le\rho$, has points of the set in each of the four closed quadrants with vertex $x$, within distance $\eta h/(1+\max_C|\nabla f_C(p)|)$ of $x$. For each $C$, two of them satisfy $q^-\le x\le q^+$ in the order of the reflected chart of $C$. By Lemma 2.1(a) with constant sequences, $F_C^{(n)}(q)\to f_C(q)$ at these finitely many points; take $n_0(h)$ with all errors at most $\eta h$. Monotonicity squeezes $F_C^{(n)}(p+hw)$ between $F_C^{(n)}(q^\pm)$, which are within $\eta h+\eta h+\eta h$ of $f_C(p)+h\nabla f_C(p)\cdot w$. Choose $\eta=\epsilon/3$. $\square$

A fast diagonal is a pair of sequences $h_j\downarrow0$ and indices $n_j\to\infty$ such that for every finite $\mathcal C$, every $\rho$ and every $\epsilon>0$, the inequality of Lemma 4.1 holds with $h=h_j$, $n=n_j$ for all large $j$. Fast diagonals exist: enumerate the windows of $K$ as $C_1,C_2,\dots$, and for each $j$ choose $h_j<h_0$ and $n_j\ge n_0(h_j)$ for $\mathcal C=\{C_1,\dots,C_j\}$, $\rho=j$ and $\epsilon=1/j$. Subsequences of fast diagonals are fast diagonals. For a point $X$ of configuration $n_j$, its zoom coordinate is $w=((r_A,r_B)(X)-p)/h_j$.

5. Tangent directions

For $t$ in the interior $K^\circ$ of $K$, the direction $\varsigma(t)$ of Rank coordinates lies, in late configurations, outside the spans of $A$ and $B$, on the same side of both, because $t$ and $K'$ are separated by a positive gap. By Exercise 7.3 of Rank coordinates, along a line of direction $t$, oriented by increasing first coordinate, both $r_A$ and $r_B$ increase strictly if the span of $K$ lies above those of $A,B$, and both decrease strictly otherwise. Let $Q$ be the corresponding closed quadrant, $\{w:w_1,w_2\ge0\}$ or $\{w:w_1,w_2\le0\}$; forward displacements along lines of direction $t$ have rank displacements in $Q$.

Say that a window $C\subseteq K$ lies strictly before $t$ if its closure lies in $[0,t)$, and strictly after $t$ if its closure lies in $(t,1]$. For $t\in K^\circ$ let $D_p(t)$ be the set of unit vectors $d\in Q$ with $\nabla f_C(p)\cdot d\ge0$ for every window $C\subseteq K$ lying strictly before $t$, and $\nabla f_C(p)\cdot d\le0$ for every $C\subseteq K$ lying strictly after $t$. It is the intersection of the unit circle with a closed convex cone inside $Q$, so a closed arc of length at most $\pi/2$, or empty.

Lemma 5.1 (directions of lines). Let $(h_j,n_j)$ be a fast diagonal and $t\in K^\circ$. Let $X_j,Y_j$ lie on one line of direction $t$ in configuration $n_j$, with $Y_j$ forward of $X_j$, and suppose their zoom coordinates converge to $w\ne w'$. Then $(w'-w)/|w'-w|\in D_p(t)$. Moreover $D_p(t)$ is not empty.

Proof. The rank displacement from $X_j$ to $Y_j$ lies in $Q$, so $w'-w\in Q$. Let $C\subseteq K$ lie strictly before $t$. In late configurations the span of $C$ lies below $\varsigma(t)$, so $r_C$ increases along the line (Exercise 7.3 of Rank coordinates), and $F_C^{(n_j)}$ at the rank point of $Y_j$ is at least its value at that of $X_j$. By the fast diagonal, these values are $f_C(p)+h_j\nabla f_C(p)\cdot w_j+o(h_j)$ and the same with $w'_j$, where $w_j\to w$ and $w'_j\to w'$ are the zoom coordinates; so $\nabla f_C(p)\cdot(w'-w)\ge0$. For $C$ strictly after $t$ the inequality is reversed.

For non-emptiness, let $X_j$ be the point of configuration $n_j$ with $(r_A,r_B)(X_j)=p$, and follow the forward ray of direction $t$ from $X_j$. The preimage of the closed disk of radius $h_j$ about $p$ is compact, by Proposition 3.2 of Rank coordinates, so the ray leaves it; let $Y_j$ be its first point with zoom coordinate of length one. Along a subsequence these zoom coordinates converge to some $w'$ of length one, and the first part gives $w'\in D_p(t)$. $\square$

Lemma 5.2 (separated directions). Let $s<t$ lie in $K^\circ$ with $h_z((s,t))>0$. Then $D_p(s)\cap D_p(t)=\varnothing$. There are times $\tau_1<\tau_2<\tau_3$ in $K^\circ$ for which $D_p(\tau_i)$ are single unit vectors $d_i$, no two of them parallel.

Proof. Choose a window $J$ whose closure lies in $(s,t)$ with $h_z(J)>0$; one exists, since $(s,t)$ is a countable union of such windows. By Lemma 3.1 there are disjoint $C,D\subseteq J$ with independent gradients. Both lie strictly after $s$ and strictly before $t$. A vector in $D_p(s)\cap D_p(t)$ would have nonpositive and nonnegative scalar products with both gradients, hence zero products, and would be zero.

Let $T$ be the support of $h_z$ in $K^\circ$, with the endpoints of the components of $K^\circ\setminus T$ removed; only countably many points are removed, and $h_z(T)=h_z(K)>0$ since $h_z$ has no atoms. Two points $s<t$ of $T$ have $h_z((s,t))>0$: otherwise $(s,t)$ would lie in a component of the complement of the support, with $s$ and $t$ among its endpoints. So the arcs $D_p(t)$, $t\in T$, are pairwise disjoint and nonempty. Disjoint arcs of positive length contain different rational angles, so only countably many $t\in T$ have $D_p(t)$ of positive length; removing them leaves a set of positive $h_z$-measure, in particular three points $\tau_1<\tau_2<\tau_3$. Their directions lie in the quadrant $Q$ and are distinct, so no two are parallel. $\square$

6. One triple for all labels

The times of Lemma 5.2 depend on $z$. A common normalization of the configurations needs one fixed triple. For every $z\in E$, since $h_z$ has no atoms and $h_z(K)>0$, there are rational times $t_1<t_2<t_3$ in $K^\circ$ with $h_z((t_1,t_2))>0$ and $h_z((t_2,t_3))>0$. There are countably many rational triples, so one triple serves on a subset of $E$ of positive $\lambda$-measure; then a window $K_0$ with closure in $(t_1,t_2)$ has $h_z(K_0)>0$ on a subset of positive measure. Restrict $E$ to it. We have reached:

Proposition 6.1. There are windows $A,B$ and $K_0$, times $t_1<t_2<t_3$ in $K^\circ$ with $K_0$ inside $(t_1,t_2)$, and a Borel set $E\subseteq Z$ with $\lambda_{K_0}(E)>0$, such that the image of $\lambda|_E$ under $p$ is absolutely continuous, and for every $z\in E$ and $p=p(z)$: every $f_C$, $C\subseteq K$, is differentiable at $p$ with $z_C=f_C(p)$; Lemma 3.1 holds at $z$; $D_p(t_1),D_p(t_2),D_p(t_3)$ are pairwise disjoint closed arcs; and three times $\tau_1<\tau_2<\tau_3$ in $K^\circ$ have single, pairwise non-parallel directions $D_p(\tau_i)=\{d_i\}$.

The disjointness of the three arcs follows from Lemma 5.2, since $h_z((t_1,t_2))$, $h_z((t_2,t_3))$ and $h_z((t_1,t_3))$ are positive. $\lambda_{K_0}(E)=\int_Eh_z(K_0)\,d\lambda>0$.

7. Exercises

7.1 (easy). Show that the intersection of a closed convex cone contained in a closed quadrant with the unit circle is a closed arc of length at most $\pi/2$, or empty.

7.2 (medium). In Lemma 2.1, give an example of nondecreasing functions $F^{(n)}$ on $(0,1)$, converging at rational points, for which $F^{(n)}(p_n)$ does not converge to $f(p)$ at a point $p$ where $f$ jumps. Why does this not harm the argument?

7.3 (medium). Why must $C$ and $D$ in Lemma 3.1 be windows inside $J$, rather than arbitrary windows of $K$?

8. Solutions

7.1. In polar angle, a closed convex cone of angle less than $\pi$ is the set of vectors with angle in a closed interval; inside a quadrant this interval has length at most $\pi/2$.

7.2. Let $F^{(n)}=\mathbf 1_{[1/2+1/n,1)}$ on $(0,1)$. At every rational $q$, $F^{(n)}(q)\to\mathbf 1_{q>1/2}$, so $f(1/2)=0$ and $\bar f(1/2)=1$: the point $1/2$ is not regular. Along $p_n=1/2+2/n\to1/2$, $F^{(n)}(p_n)=1$, which does not tend to $f(1/2)$. The lemma is applied only at regular points, which fill the square up to a null set, and the measures involved do not charge null sets of the square.

7.3. The conclusion is used to separate the directions $D_p(s)$ and $D_p(t)$, and this needs windows lying strictly between $s$ and $t$. Windows elsewhere in $K$ give constraints of the same sign at both times.

References

[Fremlin] D. H. Fremlin, Measure Theory, Volume 2, 232E (Radon–Nikodym theorem); the text of the core course Measure and Integration. https://www1.essex.ac.uk/maths/people/fremlin/mt.htm
[OpenAI-HL] OpenAI, A power saving for planar halving lines, OpenAI Math Release preprint, 25 September 2026, Section 4. https://github.com/openai/math/tree/main/preprints/A-power-saving-for-planar-halving-lines-September-25-2026