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The construction and proofs are credited to Plasma AI. The graded whole-conclusion review covers the stated conditional forward direction, the reverse implication and the two-scale extension at every real ratio. The forward direction of the algebraic-family theorem is conditional on premise (R0) of [[research/leads/equilateral_two_scale/spectral_exclusion|the spectral account]]. The two-scale extension below uses the separate prose rigidity proof examined in that review. The continuous-factor and sampling arguments are supplied here. They adapt Lemma 4.1 and Proposition 4.2 of the OpenAI plane-coloring manuscript, read at proof depth and examined at their stated use in this argument.

Statement

A triangle here is an ordered triple $(0,a,b)$ of distinct points of $\mathbb C$, possibly collinear; its copies are $(p,\;p+ua,\;p+ub)$ with $p\in\mathbb C$ and $|u|=1$. Let $\mathcal T=\{(0,a_j,b_j):j\in J\}$ be a countable family. A two-coloring $c:\mathbb C\to\{-1,+1\}$ avoids $\mathcal T$ if no copy of any member is monochromatic; a measurable $c$ weakly avoids $\mathcal T$ if for each $j$ the set of $(p,u)\in \mathbb C\times S^1$ whose copy is monochromatic is null for area times arc length. Equivalently its measure in $B(0,n)\times S^1$ is zero for every positive integer $n$. Changing colors on a plane-null set preserves this condition: for each fixed $u$, hitting that set at any of the three vertices is a null event in $p$. Reflection is not built into this ordered-template convention; when reflected copies are required, include the conjugate template as well. This preserves countability and algebraic side lengths.

Theorem. Suppose every member of $\mathcal T$ has algebraic side lengths.

• (a) If some two-coloring of the plane avoids $\mathcal T$, then some Borel two-coloring weakly avoids $\mathcal T$, granting (R0).
• (b) If some measurable two-coloring weakly avoids $\mathcal T$, then some two-coloring avoids $\mathcal T$. This direction needs neither (R0) nor algebraic sides.

Two-scale extension. For Problem 173, the same proof applies to $\mathcal T$ consisting of equilateral sides $1,x$ for every real $x>0$. Use $L=F_x(i)$, where $F_x$ is the real closure of $\mathbb Q(x)$ in $\mathbb R$, and its unit group. Corollary 3 of the rigidity extension provides exactly the nonzero pair-correlation identity used in the proof. All the factor, sampling and threshold arguments below concern a countable dense carrier and work with $L$ in place of $E$. This yields the transfer for every real ratio. Theorem 4 of the circle account then excludes avoidance when $x\ne1$. The graded whole-conclusion review covers this composition. The recorded standing of the separate algebraic Lean premise (R0) is unchanged.

Why triangles and two colors

For $a,b,c\in[-1,1]$ put $Q(a,b,c)=1+ab+ac+bc$. Then

$$Q(a,b,c)=\tfrac12\bigl((1+a)(1+b)(1+c)+(1-a)(1-b)(1-c)\bigr)\ge0,$$

with equality exactly when one of $a,b,c$ equals $-1$ and one equals $+1$. On signs, $Q/4$ is the indicator that $a,b,c$ are equal. So a $[-1,1]$-valued field whose three pair correlations on a triangle sum to $-1$ has $Q=0$ almost surely there, and any rounding that keeps $+1$ and $-1$ produces a nonmonochromatic triple. Only pair correlations enter. For four points, or for three or more colors on a triangle, the monochromatic indicator has terms of degree above two, which the projection below does not preserve.

Ordinary conditional expectation does not preserve avoidance of general hyperedges. The uniform law on the six nonmonochromatic sign triples has all three conditional means zero over the trivial factor, and rounding $(0,0,0)$ gives a monochromatic triple. That projection changes each pair correlation from $-1/3$ to $0$, so it is not of the pair-preserving kind used here and does not test this mechanism.

Continuous factors and measurable representatives

Let a standard probability space carry a measure-preserving action of the countable dense additive subgroup $E\subset\mathbb C$, with Koopman operators $T_z$. Define $H_c$ as in (R0). It is closed, since $\|T_zv-v\|_2\le2\|v-w\|_2+\|T_zw-w\|_2$, and it contains constants. Conjugation and Lipschitz composition preserve $H_c$. Products of two bounded members belong to $H_c$, by the product inequality in $L^2$.

Let $\mathcal F_c$ consist of the measurable sets $A$ with $\mathbf1_A\in H_c$, modulo completion. Complements and finite unions stay in $\mathcal F_c$ by the preceding properties; increasing unions do by $L^2$ convergence. Thus it is a sigma-algebra. For a real $h\in H_c$, the Lipschitz functions $\min(1,\max(0,n(h-t)))$ converge in $L^2$ to $\mathbf1_{\{h>t\}}$. Every real member of $H_c$ is therefore $\mathcal F_c$-measurable, and the same follows for complex members by real and imaginary parts. Conversely the simple $\mathcal F_c$-measurable functions belong to $H_c$ and are dense in $L^2(\mathcal F_c)$. Hence $H_c=L^2(\mathcal F_c)$ and its orthogonal projection is conditional expectation. In particular it sends real functions valued in $[-1,1]$ to functions with the same bounds.

For $y\in\mathbb C$ and $z_n\in E$ tending to $y$, the sequence $T_{z_n}h$ is Cauchy in $L^2$ for $h\in H_c$, because the norm of its difference at $n,m$ is $\|T_{z_n-z_m}h-h\|_2$. Its limit defines $T_yh$, independently of the approximating sequence. The group law, unitarity and strong continuity extend by limits. Realness and interval bounds are closed $L^2$ conditions, so $T_y$ preserves them. No pointwise action of $\mathbb C$ on the probability space is asserted.

For a real $g\in H_c$ with $-1\le g\le1$ there is a jointly Borel field $G(y,\eta)\in[-1,1]$ representing $T_yg$ for every fixed $y$, after choosing a bounded Borel representative of $g$. To see this, choose Borel maps $q_n:\mathbb C\to E$ with countable range and uniformly small $|q_n(y)-y|$, so that $\|T_{q_n(y)}g-T_yg\|_2\le2^{-n}$ for every $y$. Such maps exist by density of $E$ and uniform continuity of the orbit of $g$. Realize $G_n(y,\eta)=T_{q_n(y)}g(\eta)$ by the original countable action. The maps $G_n$ are jointly Borel and bounded. For each fixed $y$ the sum of the $L^1$ norms of $G_{n+1}(y,\cdot)-G_n(y,\cdot)$ is finite, so the sequence converges almost surely. Define $G$ to be this limit where it exists and $0$ otherwise. It is jointly Borel, and its section at every fixed $y$ represents the $L^2$ limit $T_yg$. The exceptional subset of the probability space may depend on $y$; Tonelli, not a common pointwise action, handles it.

Proof of (a)

Rotate each member so that $a_j>0$; with algebraic sides, $a_j,b_j\in E$, the field of algebraic numbers, and $K$ denotes its algebraic unit vectors. Let $\Omega\subset\{\pm1\}^E$ consist of the labelings with no monochromatic triple $(p,p+ua_j,p+ub_j)$, $p\in E$, $u\in K$, $j\in J$. It is closed, contains the restriction of the given coloring, and is invariant under $\eta\mapsto\eta(k\,\cdot+z)$. Amenability of $E\rtimes K$ gives an invariant probability $\mu$ on $\Omega$, as in the manuscript's Proposition 4.2. Define $T_z$, $V_k$, $f(\eta)=\eta(0)$ and the projection $P$ onto the continuous vectors $H_c$ as in the proof of Theorem 3 of the spectral account. Put $g=Pf$.

Bounds. The continuous-factor argument above shows that $g$ is real with $-1\le g\le1$, and extends the translations on $H_c$ to a strongly continuous unitary representation of $\mathbb C$ with $-1\le T_yg\le1$ for every $y$.

Correlations. As in Theorem 3, (R0) gives $\langle f,T_zf\rangle=\langle g,T_zg\rangle$ for all nonzero $z\in E$. Let $\kappa(y)=\langle g,T_yg\rangle$, continuous on $\mathbb C$. For $u\in K$ every $\eta\in\Omega$ satisfies $\eta(0)\eta(ua_j)+\eta(0)\eta(ub_j)+\eta(ua_j)\eta(ub_j)=-1$, and the three displacements $ua_j$, $ub_j$, $u(b_j-a_j)$ are nonzero. Integrating,

$$1+\kappa(ua_j)+\kappa(ub_j)+\kappa(u(b_j-a_j))=0\qquad(u\in K),$$

and by continuity and density of $K$ in $S^1$ this holds for every unit $u$.

Pointwise vanishing. For $p\in\mathbb C$ and unit $u$ the three functions $G_1=T_pg$, $G_2=T_{p+ua_j}g$, $G_3=T_{p+ub_j}g$ take values in $[-1,1]$, and $\int Q(G_1,G_2,G_3)\,d\mu$ is the left side above, because $\langle T_yg,T_{y'}g\rangle=\kappa(y'-y)$. Since $Q\ge0$, $Q(G_1,G_2,G_3)=0$ $\mu$-almost surely.

A measurable sample. Use the jointly Borel field constructed above. For each fixed $(p,u)$ its three sections represent the three vectors in the preceding paragraph. Tonelli then gives, for every $j\in J$ and every integer $n\ge1$,

$$\int_\Omega\int_{|p|<n}\int_{S^1} Q\bigl(G(p,\eta),G(p+ua_j,\eta),G(p+ub_j,\eta)\bigr) \,d\sigma(u)\,dp\,d\mu(\eta)=0 .$$

These are countably many conditions, so one $\eta_0$ makes every inner double integral vanish. Put $c'(y)=+1$ if $G(y,\eta_0)\ge0$ and $c'(y)=-1$ otherwise; $c'$ is Borel. For almost every $(p,u)$ and every $j$ the three values of $G(\cdot,\eta_0)$ include $+1$ and $-1$, which $c'$ keeps, so the copy is not monochromatic. Thus $c'$ weakly avoids $\mathcal T$. $\square$

Proof of (b)

Let $c$ weakly avoid $\mathcal T$, and let $A^*$ and $B^*$ be the points of density one of $\{c=1\}$ and $\{c=-1\}$. They are disjoint and their union is conull. Suppose $p$, $p+ua_j$, $p+ub_j$ all lie in $A^*$. For small $h>0$ choose $p'$ uniformly in the disk of radius $h$ about $p$ and, independently, $u'$ uniformly on an arc about $u$ of angular width $h/(1+|a_j|+|b_j|)$. Each of $p'+u'a_j$ and $p'+u'b_j$ has a density at most $1/(\pi h^2)$, supported in the disk of radius $2h$ about $p+ua_j$, respectively $p+ub_j$. Density one at the three points makes the probability that all three random points lie in $\{c=1\}$ tend to one. The law of $(p',u')$ is absolutely continuous, which contradicts weak avoidance. The same holds for $B^*$. This is the argument of the manuscript's Lemma 4.3 with three points.

Every finite set of points has a translate inside the conull set $A^*\cup B^*$, where the density classes color it with no monochromatic copy. Each avoidance condition is closed in $\{\pm1\}^{\mathbb C}$ and involves three coordinates, so product compactness gives a coloring of the whole plane avoiding $\mathcal T$. $\square$

This yields an arbitrary avoider, not a measurable completion of the density classes on the omitted null set.