Paper V: Nonlinear interactions in residue coordinates

E12 Nonlinear interactions in residue coordinates

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Linear transfer matrices do not directly describe nonlinear products. Residue coordinates still obey exact conservation laws because integer addresses add modulo the packet modulus.

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Theorem E12.1: Residue law for Cauchy products

Let

\[ F(z)=\sum_{n\geq0}f_nz^n, \qquad G(z)=\sum_{n\geq0}g_nz^n \]

be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo \(N\):

\[ F=\sum_{r=0}^{N-1}F_r, \qquad G=\sum_{s=0}^{N-1}G_s. \]

If \(H=FG\), then its residue packet \(H_t\) is

\[ \boxed{ H_t = \sum_{r+s\equiv t\!\!\pmod N}F_rG_s. } \]

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Proof of Theorem E12.1

Every monomial in \(F_rG_s\) has degree congruent to \(r+s\) modulo \(N\). Conversely, every coefficient of degree congruent to \(t\) is formed from pairs whose residues sum to \(t\). Grouping the Cauchy product by this residue gives the formula.

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Theorem E12.2: Shifted pointwise monomial law

Let

\[ y(n)=\prod_{j=1}^{m}x(n-a_j) \]

and write

\[ a_j=Nc_j+b_j, \qquad 0\leq b_j<N. \]

For output residue \(r\), define

\[ r_j\equiv r-b_j\pmod N, \qquad 0\leq r_j<N, \]

and let

\[ \kappa_{r,b_j} = \begin{cases} 1,&r<b_j,\\ 0,&r\geq b_j. \end{cases} \]

Then the output channel is

\[ \boxed{ y_r(q) = \prod_{j=1}^{m} x_{r_j} \bigl(q-c_j-\kappa_{r,b_j}\bigr). } \]

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Proof of Theorem E12.2

For \(n=Nq+r\),

\[ n-a_j =N(q-c_j)+r-b_j. \]

If \(r\geq b_j\), this address has residue \(r-b_j\) and coarse index \(q-c_j\). If \(r<b_j\), add \(N\) to the residue and subtract one from the coarse index. Multiplying the resulting channel values gives the formula.

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Corollary E12.3: Polynomial shift filters

Every finite polynomial in shifted samples has an exact finite expression in residue coordinates. Each monomial selects a deterministic list of input residues and coarse shifts, and the complete output is their finite sum.

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Proof of Corollary E12.3

Apply Theorem E12.2 to each monomial and add the results.

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Example E12.4: Quadratic nearest neighbor interaction

For

\[ y(n)=x(n)x(n-1) \]

and modulus \(N\),

\[ y_r(q) = \begin{cases} x_r(q)x_{r-1}(q),&r\geq1,\\ x_0(q)x_{N-1}(q-1),&r=0. \end{cases} \]

The carry at residue zero is the nonlinear analogue of a transfer crossing a coarse cell boundary.

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