Two scoped limits of sparse direction and full-profile focusing

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These results constrain two proposed Ramsey implementations. They do not exclude color-adaptive cancellation between several mixed trades, and are not a coloring theorem for the whole line family.

1. Bounded reciprocal mass forces bounded actual colors

For signed linear direction matchings of sizes $d_i$, put $S=\sum_i1/d_i$. Then

$$\chi_1\leq\chi_{\mathbb Q}\leq\lceil1+S\rceil. \tag{1}$$

For any column subset $B$, normalize its columns to unit length and let $G_B$ be their Gram matrix. A size-$d_i$ column meets at most $d_i$ disjoint members of direction $j$; each squared correlation is at most $1/(d_id_j)$. Hence

$$\operatorname{tr}G_B=|B|,\qquad \operatorname{tr}(G_B^2)\leq(1+S)|B|.$$

Cauchy--Schwarz on its nonzero eigenvalues gives $\operatorname{rank}_{\mathbb Q}B\geq|B|/(1+S)$. Edmonds, Theorem 1 now proves (1). Both the digest and exact source statement were checked.

Thus assigning domains so large that $S$ remains bounded cannot produce an unbounded-color direction family. The established extraction phase condition permits divergent $S$; (1) does not rule out that regime.

2. Initial-plane profiles need not homogenize on a later column

In each field, flag, or polynomial line model, partition rows into initial affine $p$-planes. Every later column meets each such plane at most once. Later sizes are $p^k$, $k\geq2$, with at most $p^k$ directions at each size.

Claim. The initial planes can be labeled with sixteen labels so that no later column meets planes all bearing the same label.

For a finite stage, color planes independently and uniformly with sixteen labels. The event for a column of size $D$ has probability $16^{1-D}$. Assign it local-lemma weight $x_D=2^{-D}$. A plane contains $p^2$ points and so meets at most $p^2M_E$ columns among $M_E$ directions of size $E$. Thus the sum of neighboring event weights for a $D$-column is at most

$$Dp^2\sum_{k\geq2}p^k2^{-p^k}\leq(8/7)D. \tag{2}$$

Successive summands have ratio $p\,2^{-(p-1)p^k}\leq1/8$. The first summand, after multiplication by $p^2$, is at most one since $p^4\leq2^{p^2}$. This proves (2). Since all $x_D\leq1/16$, the dependency product is at least

$$\exp(-(16/15)(8/7)D)=\exp(-128D/105).$$

For $D\geq4$,

$$\frac{D-1}{D}\log16-\log2\geq2\log2>4/3>128/105.$$

Therefore

$$16^{1-D}\leq2^{-D}\exp(-128D/105),$$

which verifies the asymmetric local-lemma hypothesis. The relevant independent-variable statement was checked after its digest in Moser--Tardos, Theorems 1.1--1.2. Compactness gives the same labeling on a nested infinite model.

Realization by proper colors on the initial columns

In each initial $p$-plane, split every parallel class into two nonempty parts. Each binary part is rationally independent: a relation in the full affine plane is constant within each parallel class, whereas the part omits a column from every class. Pair this binary color with the plane's sixteen-valued label. Across the disjoint initial planes, this gives thirty-two proper initial-column colors.

Any later column encounters distinct plane labels and hence distinct complete initial-plane color profiles. Consequently a Ramsey argument that requires identical full old-plane profiles along one entire later column cannot be universal.

This coloring is only on the initial columns. It does not color the later ones or split their mixed relations; several later columns can cancel outside a chosen profile class. No general color-adaptive lifting theorem is refuted. The finite dependency estimates and scope are author-recorded. No computational search was run.