Two-factor products of coefficient-two circuits have two colors

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After the prime staircase retired density-only descent, a further actual-color attempt examined the full products of its coefficient-two circuits, not just their mandatory subsets. Up to rational basis changes and column signs these factors are

$$C_d=\{e_1,\ldots,e_d,w\},\qquad w=\tfrac12\sum_a e_a.$$

The fixed ratio two would flatten two factors by the coefficient-growth criterion if the requisite full dependent supports had one product color, but no mechanism forces that condition. Proper subalphabets of a factor are independent, so arbitrary subalphabet focusing does not preserve its circuit.

The two-factor check

An exact two-factor check gives a useful small-case boundary without a computation. For $d,e\ge4$, the product $C_d\otimes C_e$ has two rationally independent color classes. Color the leaf grid by surjective row and column types in $\mathbb Z/4$, red when column type minus row type is in $\{0,1\}$, and blue otherwise. Each color's spanning bipartite graph is a blowup of a connected eight-cycle. Color each of the two wildcard wings with both colors and color the center arbitrarily. A rational relation has coordinate equations

$$4k_{ab}+2y_a+2z_b+t=0.$$

For a monochromatic relation, the opposite-color grid is connected and has $k_{ab}=0$, forcing constant row coefficients $y_a=\alpha$ and column coefficients $z_b=\beta$. An opposite-color vertex in each wing gives $\alpha=\beta=0$, hence $t=0$ and all $k_{ab}=0$. The disjoint-wing ternary relation $\sum_a e_a\otimes w-\sum_b w\otimes e_b=0$ gives the matching lower bound two. Thus all finite-height coloring numbers equal two in this two-factor case. This algebraic check makes no assertion for three or arbitrarily many factors.

Consequence

The full-product attempt therefore still needs actual-color control of all leaves or a coefficient-safe substitute. It is not an amplification theorem. The required negative solution still needs either unbounded $\chi_h(G)$ at one fixed finite height or a valid actual-color substitute; the coefficient-growth inequality supplies a broader sufficient flattening target, and the staircase shows why raw short-relation rank cannot force it. This page is author-recorded; a separate reviewer checked the two-factor algebra.