A precise limit of fixed-alphabet root-word focusing

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Scope

The source reassessment checked Karagiannis's Theorems 1--3, Prömel--Voigt's definitions of parameter words and Theorems C.3/C.7, and Voigt's product canonization Theorems 2--3 in their full reading copies, after the digests. These are genuine Ramsey structure theorems. They do not assert that evaluating a variable word at root labels preserves an additive relation of the alphabet.

The theorem below identifies an exact obstruction to that transfer. It does not color all signed relations on roots, and does not exclude fixed-height adaptive focusing, increasing alphabets, or non-coordinatewise encodings.

Theorem

Fix an alphabet $A$ of size $m\geq2$, and a finite set of coordinates $J$. For each $j$, choose a map $a\mapsto x_{j,a}\in\mu_{n_j}$, where the odd integers $n_j$ are pairwise coprime. The local maps need not be injective. Evaluate words by

$$\pi((a_j)_{j\in J})=\prod_{j\in J}x_{j,a_j}.$$

There is a coloring of $\pi(A^J)$, with a number of colors depending only on $m$, for which the set of distinct values of every monochromatic parameter-subspace image is rationally independent.

A parameter subspace fixes some coordinates and partitions the others into nonempty variable blocks; each block takes one common letter, independently of the other blocks. This is the point-subspace conclusion to which a word-based relation-forcing argument would have to apply. The conclusion is uniform in finite word length; it does not assert independence of a union of subspace images at different lengths or domains.

Proof

Put $L=m-1$, and let $P_j$ be the equality partition of $A$ induced by the values $x_{j,a}$. For every partition $P$ of $A$ and every class $C\in P$ except one fixed omitted class, record

$$\#\{j\in J:P_j=P,\ a_j\in C\}\pmod L. \tag{1}$$

There are at most $L^{\sum_P(|P|-1)}$ records, with the one-color interpretation for $L=1$. These records depend only on the evaluated root: the product of the conductor groups is direct, so the local root values are determined by their product, and a local root value determines its equality class.

Let $B$ be one variable block in a monochromatic parameter subspace. For any nontrivial partition type $P$, choose a recorded class and letters respectively inside and outside it. Comparing their substitutions in that block in the corresponding record (1) gives

$$t_P:=\#\{j\in B:P_j=P\}\equiv0\pmod L. \tag{2}$$

Thus a nontrivial type that occurs in the block occurs at least $m-1$ times. For $m=2$ this just says that an occurring type occurs at least once, as required.

Consider the distinct diagonal outputs $y_a=\prod_{j\in B}x_{j,a}$. Fix a target output and representatives for its at most $m-1$ competitors. Each competitor is separated from the target by some partition type. Assign each competitor a different coordinate of a separating type; (2) supplies enough copies, even if all competitors require the same type. At an assigned coordinate choose a rational linear functional that kills the competitor's local root but not the target's. Such a functional exists: distinct odd-order roots are never rationally proportional. At unused coordinates choose a rational functional nonzero on the target. Tensoring the functionals isolates its diagonal output.

Multiplication identifies $\bigotimes_{j\in B}\mathbb Q(\zeta_{n_j})$ with their compositum: the usual cyclotomic degree $\varphi(\prod_jn_j)=\prod_j\varphi(n_j)$ equals the tensor dimension, and the multiplication map is surjective. Hence this is a separating rational functional on the actual root values. All distinct diagonal outputs are rationally independent.

Different variable blocks use disjoint conductor factors. The products of their independent output sets remain independent by the same tensor-field identification. Fixed coordinates multiply the set by one nonzero scalar, which does not change independence. This proves the theorem.

Consequence for the live route

Fixed-alphabet coordinatewise word focusing can guarantee a monochromatic structure whose evaluated values are all independent. It therefore cannot, without an additional encoding ingredient, transfer an alphabet relation to the required monochromatic determinant support.

Karagiannis's increasing-alphabet theorem lies outside this obstruction, but its statement does not bound variable multiplicities or the locations of the chosen words. In ordinary multiplicative evaluation a variable word takes the form $c a^r$; repetition can annihilate the finite conductor of a root alphabet. In conductor-disjoint evaluation the theorem above exhibits the different problem of becoming independent. A useful repair must prove a relation-preserving, noncollapsing encoding for the structures the Ramsey theorem actually returns. No such repair is supplied here.

This finite theorem has no computational input. It is a scoped obstruction, not a proof or disproof of finite dissociated colorability of the full root group. The fixed-height premise in the main argument remains unproved. The finite theorem and its scope received independent internal review.