Mobile anchors and triangular gluing

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Motivation and exact remaining requirement

In every proper $q$-coloring of a configuration with coloring number exactly $q$, each pair of color classes has a nonzero common signed sum: otherwise their union is dissociated and the two colors can be merged. These pairwise sums may depend on the coloring and need not be roots. They offer a possible replacement for the universal anchors lost by simplex gluing.

The constructions below control the entire rational ternary closure, not merely planted rows. For a fixed source with finite $\tau_q$, they exclude triangular padding whose buffer lengths all exceed the explicit source-dependent threshold below. They do not exclude shorter buffers, changing sources with changing thresholds, or sources with $\tau_q=\infty$. Recurrent constraint interaction is a possible way to leave this excluded class, not a proved necessary feature of every successful construction.

Arbitrary mobile words eventually separate — proved

Let $\chi(B)=q$, $U_q(B)=\{0\}$, and $T=\tau_q(B)<\infty$. Set

$$H=\max_{\text{rational lines }L}|(\Sigma(B)\cap L)\setminus\{0\}|, \qquad h=2+\lceil T\log(qH)\rceil.$$

For any nonzero rational anchor word $u_1,\ldots,u_m$, $m\geq h$, there are proper base colorings $f_i$ satisfying, for every color $c$,

$$\bigcap_{i=1}^m \{t\in\mathbb Q:t u_i\in\Sigma(f_i^{-1}(c))\}=\{0\}. \tag{1}$$

Fix a $T$-element separating family of colorings. After choosing the first coloring, at most $qH$ nonzero pairs $(c,t)$ remain. At any later block each surviving pair is omitted by at least one member of the family. A greedy choice reduces the survivor count by a factor at most $1-1/T$. After $m-1>T\log(qH)$ steps the count is less than one.

If $B\subset\mathbb Z^d$ has $N$ points and entries bounded by $M$, then $H\leq2NM$: a rational line's integer points are integer multiples of a primitive vector, and a nonzero coordinate bounds the possible multipliers of any signed sum by $NM$.

A precise triangular construction

Take independent copies of $\Gamma=\langle B\rangle_{\mathbb Z}$. Let integral gluing rows be $r_1,\ldots,r_s\in\Gamma^n$. Give each row a buffer set $P_e$ of size $m_e$, with the sets $P_e$ pairwise disjoint. Assume that on every $v\in P_e$, row $r_e$ has a primitive nonzero anchor $u_{e,v}$, while every later row $r_f$, $f>e$, vanishes. Earlier rows may overlap these buffers. Quotient by the rational span of all rows, and denote the image family by $G$. Put $L_e=\max_{v\in P_e}\lambda_B(u_{e,v})$, using primitive width.

Saturation and distinctness

Choose one primitive pivot from each buffer set. Integer functionals taking value one on those pivots give a unitriangular minor of the row matrix. Thus the rows generate a saturated lattice; the quotient group is free. In a nonzero rational row combination, its least active row has nonzero entries on every one of its buffers, since all later rows vanish there. If every $m_e\geq3$, the combination has at least three nonzero blocks. Consequently each block embeds, and no new zero, equal, or antipodal labels occur between blocks.

Uniform analytic bound

Exactly as in primitive-anchor phase gluing,

$$\rho(G)\geq\rho(B)-\frac{\pi^2}{2} \max_e\frac{L_e^2}{m_e^2}. \tag{2}$$

Independently maximize every block polynomial. Correct row constraints in reverse order, spreading a principal phase $\theta_e\in[-\pi,\pi]$ across its buffers with the separate minimizing functionals $\ell_{e,v}(u_{e,v})=1$. Later rows are unaffected because they vanish on these buffers. The disjoint buffer sets mean every polynomial block moves only once from its independent maximum. Its first-order variation vanishes, and its quadratic loss is bounded by $\pi^2L_e^2/(2m_e^2)$ times its coefficient mass. Summing gives (2), with a maximum rather than a sum over rows.

Nevertheless the quotient has no chromatic increase

If every $m_e\geq h$, then $\chi(G)=q$. Color each buffer word using (1) and color all remaining blocks properly. Suppose a nonzero monochromatic relation had block-sum vector $x=\sum_e\lambda_e r_e$. On the buffers of its least active row $e$, its sums are exactly $\lambda_e u_{e,v}$. Equation (1) forces $\lambda_e=0$, a contradiction. If $x=0$, blockwise properness already makes the relation trivial. Restriction to an embedded block gives the matching lower bound $q$.

This proof permits arbitrary mobile labels, not just a finite list of directions. Its no-increase conclusion requires both finite $T$ and $m_e\geq h(B,q)$ for every buffer. Analytic smallness alone does not imply these inequalities, so the theorem does not rule out every cheap triangular gluing or every iteration with changing sources.

A full-closure counting criterion without triangularity

For $B\subset\mathbb Z^d$ as above, let $R\subseteq(\mathbb Q^d)^n$ have dimension $r$, and suppose every nonzero vector of $R$ has at least $D$ nonzero blocks. The quotient image family is $q$-colorable whenever

$$q(2NM+1)^r(1-1/T)^D<1. \tag{3}$$

Color blocks independently from the fixed $T$-witness family. A coordinate projection onto $r$ scalar coordinates injects on $R$, so $|R\cap\Sigma(B)^n|\leq(2NM+1)^r$. Each fixed nonzero vector has probability at most $(1-1/T)^D$ of belonging to the signed spans of a fixed color in all blocks. A union bound over these vectors and the colors proves (3); zero block sums are excluded by blockwise properness. Again the count covers all induced relations, not just a selected row list.