Canonical cyclic lifts of one circuit have two actual colors

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The canonical cyclic lift flattens any integer relation. Unlike the earlier designated-template splitting bound, the theorem below colors all ternary relations of specified covers of a single rational circuit. It does not bound arbitrary covers or simultaneous constructions using several interacting circuits.

1. The canonical polynomial matrix

Let $A\in\{0,1\}^{X\times V}$ have distinct nonzero columns forming a rational circuit. Put $n=|V|$ and choose its primitive relation

$$\ker_{\mathbb Q}A=\mathbb Q a,\qquad a\in\mathbb Z^V,\quad \gcd_v|a_v|=1.$$

All $a_v\ne0$. Write $L_v=|a_v|$ and

$$p_x=\sum_{a_v>0}A_{xv}a_v =\sum_{a_v<0}A_{xv}|a_v|.$$

For each row, tile $[0,p_x)$ separately by its positive and negative incident lengths, in any fixed orders. Let $g_{xv}\geq0$ be the tile start and $G=\max_{A_{xv}=1}g_{xv}$. Set

$$A(z)_{xv}=A_{xv}z^{g_{xv}},\qquad h_v(z)=\operatorname{sgn}(a_v)S_{L_v}(z),\qquad S_L(z)=1+\cdots+z^{L-1}.$$

The two tilings of every row give the exact identity

$$A(z)h(z)=0\quad\hbox{over }\mathbb Z[z]. \tag{1}$$

For any degree $m>\max_xp_x$, use incidence shifts $b\mapsto b+g_{xv}$ modulo $m$. The coefficients of $h$, assigned to sheets, form a nonzero ternary relation in the resulting cover. All used sheets are distinct within a type because $L_v\leq\max_xp_x<m$.

2. Full dissociation of the two parity classes

Theorem. Suppose $\max_vL_v\geq2$. For every odd prime $P$ with

$$P-1>nG,\qquad 2P>\max_xp_x, \tag{2}$$

the degree-$2P$ canonical cover is properly dissociated-colored by $c(v,b)=b\bmod2$. In particular its actual coloring number is exactly two.

Proof. Since $A(1)=A$ has rank $n-1$, while (1) supplies a kernel,

$$\ker_{\mathbb Q(z)}A(z)=\mathbb Q(z)h(z).$$

The vectors $h(z)$ and $h(-z)$ are not proportional. To verify this,

$$\frac{S_L(-z)}{S_L(z)} =\frac{1-z}{1+z} \begin{cases} 1,&L\ \text{even},\\ (1+z^L)/(1-z^L),&L\ \text{odd}. \end{cases} \tag{3}$$

An even and odd length have unequal ratios. Two distinct odd lengths also have unequal ratios (compare their first nonconstant terms after canceling the common factor). Thus proportionality requires either all lengths even, or all the same odd length. Primitivity excludes the first case and reduces the second to $L_v=1$ for all $v$, contrary to the hypothesis.

Therefore the stacked matrix

$$M(z)=\begin{bmatrix}A(z)\\ A(-z)\end{bmatrix}$$

has full column rank over $\mathbb Q(z)$. Choose a nonzero $n$-row minor $D(z)$. Its polynomial degree is at most $nG$.

Suppose $f_v(b)\in\{0,\pm1\}$ is a relation supported on sheet parity $\varepsilon$, and put $F_v(z)=\sum_{b=0}^{2P-1}f_v(b)z^b$. At every $2P$-th root of unity,

$$A(z)F(z)=0,\qquad F(-z)=(-1)^\varepsilon F(z).$$

Applying the first identity at $-z$ also gives $M(z)F(z)=0$. Every $2P$-th root other than $1,-1$ has order $P$ or $2P$ and degree $P-1$ over $\mathbb Q$. For completeness, irreducibility of $1+z+\cdots+z^{P-1}$ follows by substituting $z+1$ and applying Eisenstein at $P$; the order-$2P$ case is obtained by $z\mapsto-z$. Equation (2) thus ensures $D(z)\ne0$ at all these roots. Consequently $F(z)=0$ there.

Write $F_v(z)=z^\varepsilon Q_v(z^2)$, where $\deg Q_v<P$. It vanishes whenever $z^2$ is a nontrivial $P$-th root. Fourier inversion on $\mathbb Z/P\mathbb Z$ shows that its coefficient sequence is constant:

$$f_v(\varepsilon+2j)=\lambda_v\in\{0,\pm1\} \quad(0\leq j<P). \tag{4}$$

Sum the covering relation over all sheets. It gives $A\lambda=0$, so $\lambda=q a$ for rational $q$. Since $a$ is primitive and $\lambda$ integral, Bezout's identity gives $q\in\mathbb Z$. The bounds $|\lambda_v|\leq1$ and $\max|a_v|\geq2$ force $q=0$. Thus (4) vanishes and each parity class is fully dissociated. The nonzero canonical relation from Section 1 proves that one color does not suffice. $\square$

3. Fixed voltages preserve type independence at large prime degree

Here is a separate useful stabilization fact. Fix any nonnegative integer translation voltages on any finite binary base, and a rationally independent set $U$ of its column types. Some $|U|$-row minor of $A(z)_U$ is nonzero at $z=1$, and has degree at most $|U|G$. For a prime cover degree $Q$ with $Q-1>|U|G$, that minor is nonzero at every $Q$-th root. Fourier decomposition of the cyclic cover then shows that all copies of the types in $U$ are rationally independent.

For a rational circuit, partition its types into any two nonempty proper sets and apply this fact to each. For its canonical voltages, when

$$Q-1>nG,\qquad Q>\max_xp_x,$$

the degree-$Q$ cover has $\chi_1=\chi_{\mathbb Q}=2$, including when the base circuit is already ternary. The upper bound is a type coloring; the canonical relation gives the lower bound. This is a theorem for fixed voltages and growing prime degree, not for all covers of those degrees.

4. Adaptive cuts also exist, but do not defeat these colorings

Multiplying (1) by $(z-1)Y(z)$, for any binary sheet function $y$, gives the ternary relation

$$f_v(b)=\operatorname{sgn}(a_v) (y(b-L_v)-y(b)). \tag{5}$$

These are cuts of the circulant multigraph with a type-$v$ edge between $b-L_v$ and $b$, labeled by the covered column $(v,b)$. This graph is connected because $\gcd_vL_v=1$. A coloring avoids all nonzero relations (5) exactly when deleting the edges of each individual color leaves a connected graph: a constant-on-components indicator gives the reverse implication.

Thus the interval template is only one member of a genuinely adaptive family. Nevertheless Section 2 colors every ternary relation, including all these cuts, in the stated nonflat-circuit covers. A successful cover amplifier must go beyond independently chosen canonical one-circuit lifts, for example by coupling different relation spaces. No such arbitrary-color coupling theorem is established here.

The polynomial identities, Fourier argument, primitive coefficient step, and quantifier restrictions are author-recorded, checked against the supplied proof. No computation or new external paper theorem is used; the review is internal.