证明
Lemma 22 ([The left nullspace]). The left nullspace of the limiting matrix in…Lemma 21 ([The limiting derivative columns]). The following derivative columns…观点
None of the four forward offsets is zero. The sum of two forward offsets lies between 2 and 8 , and is therefore nonzero modulo 11 . Thus a path of one or two forward steps cannot return to its initial block. Constant input steps preserve each block. The path expansions for the output and for its Schur expressions consequently show that the prescribed constant and linear terms, with zero higher coefficients, solve all the Schur equations through degree two. Suppose a partial substitution solves them in degrees less than ℓ≥3 . Add a correction V B on a reverse edge j←i , where B is a word of length ℓ-1 , and add arbitrary degree- ℓ loop corrections. At the block-diagonal constant term, an off-diagonal input variation gives an output variation in the same off-diagonal block. Its compression on any fixed root block is zero. Thus the reverse correction changes no Schur equation through degree ℓ-1 . Its contribution in degree ℓ must pair it with a single prescribed linear arrow. There is a unique forward edge that returns along this reverse edge: the reverse offsets 10,9,8,7 correspond respectively to 1,2,3,4 . If that edge has coefficient U and color s_c , the contributions are c|c|c block, word, selected response\ i, s_cB, m_i(U,V); j, Bs_c, m_j(V,U). This is exactly the two-arrow Schur calculation defining the paired responses. A previously present coefficient of degree at least two cannot interact with the new degree- ℓ-1 coefficient in degree ℓ . Two new corrections have total degree at least 2(ℓ-1)>ℓ . The degree- ℓ variation is therefore linear. Loop corrections give D_i independently at every vertex and every word of length ℓ . We compute the annihilator of this finite correction map. Let f_ i,ω ∈(E^a)^* be a collection of functionals annihilating its image. The loop columns first imply that f_ i,ω annihilates L_i . Write its induced functional on H_i uniquely as f_ i,ω =p_i(c_ i,ω ), c_ i,ω ∈G. Here and below we use the same notation for the induced functional and its pullback to E^a . By and , annihilation of the reverse columns is equivalent to c_ i,s_cB =c_ j,Bs_c every edge i←j of color c. The two offsets of either color are consecutive integers, say a,a+1 . Applying to both edges out of j gives c_ j+a,s_cB =c_ j+a+1,s_cB . As j ranges over ℤ/11ℤ , this makes the value independent of the vertex. Equation then identifies its value with that on the rotated word Bs_c . Iterating rotations shows that the annihilator consists exactly of one element of G per cyclic orbit of words. The converse follows from the same paired-annihilator identity. Equation says that every such annihilator kills the degree- ℓ error. Finite-dimensional linear duality therefore puts the negative error in the image of the correction map. Choose a correction solving all selected equations. Lower degrees remain solved, so transfer and the active partition imply that every nonselected equation vanishes in degree ℓ as well. Continue inductively. A coefficient of degree n can change only at steps n and n+1 , so the resulting free series is well-defined. Every fixed Schur coefficient is eventually zero and remains zero. The construction also covers g=0 , when the annihilator is zero. For a periodic word the orbit consists of its distinct rotations; no multiplicity is inserted into the annihilator calculation or into .