Proposition

Lemma 20 ([Splitting repeated blocks]). Fix L≥1 and arbitrary scalar matrices S_1,S_2∈M_L(E) . Substitute s_c=δS_c into Z , with a commuting Taylor parameter δ , and add an order- δ^ℓ input loop Y_i in each old block i . Here Y_i is an …

Lemma 20 ([Splitting repeated blocks]). Fix L≥1 and arbitrary scalar matrices S1,S2∈ML(E) . Substitute sc=δSc into Z , with a commuting Taylor parameter δ , and add an order- δℓ input loop Yi in each old block i . Here Yi is an arbitrary pair in Mb(E)⊗ML(E) . Put Bi,h =∑_ |ω|=ℓ ei,h,ω Sω+Di,h (Yi), where Di,h is applied entrywise to the repeat-factor coefficients. One can choose the Yi so that every Bi,h , for all root labels h , has simple spectrum. For such a choice, after a finite splitting extension the leaf output has branches λi,h,j (δ) =λi,h +δℓθ_ i,h,j +O(δℓ+1 ), 1≤j≤L, each scalar of multiplicity t , where θi,h,j are the eigenvalues of Bi,h . Their spectral projections are integral in δ and reduce to πi,h ⊗Πi,h,j , where Πi,h,j are the rank-one spectral projections of Bi,h .

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Theorem 9.1 ([Cyclic compatibility of the error]). Let Z(s_1,s_2) be a pair of…Proposition 19 ([Compatible dual relations]). For any finite nonempty…Proposition
Matrix substitution is legitimate coefficientwise in δ . The leading Schur expression on the repeated (i,h) -space is I_t⊗B_ i,h : the original errors give the first sum, and an order- δ^ℓ input loop contributes its first response. All lower orders are zero. For each label h , D_ i,h is a nonzero linear functional on the input directions. Its matrix-valued extension is therefore onto M_L(E) . The discriminant of B_ i,h , as a polynomial in the loop coordinates, is consequently not identically zero. There are finitely many labels, so the product of these discriminants is nonzero. Over the infinite field in use, or a scalar extension, choose loops at which their product is nonzero. This choice is made for all labels, not merely the selected ones. For L=1 the simple-spectrum condition is automatic. Separate the old distinct roots by a formal conjugation equal to the identity at δ=0 . The separated (i,h) -block is λ_ i,h I+δ^ℓ(I_t⊗B_ i,h ) +O(δ^ ℓ+1 ). Here is an exact comparison with the fixed-root Schur expression. Write M=W-λ_ i,h I in the fixed root/complement decomposition, with blocks M_ uv . Its complementary block M_ 22 is invertible. If F is the separated root block minus λ_ i,h I , its invariant columns can be written QR , with Q=I+O(δ) , R=O(δ) , and M QR= QR F . Eliminating the lower row gives (M_ 11 -M_ 12 M_ 22 ^ -1 M_ 21 )Q =(Q-M_ 12 M_ 22 ^ -1 R)F. The left parenthesis is the fixed-root Schur expression. Both outer factors relating it to F reduce to identity at zero. Consequently it and F have the same first nonzero coefficient. Its vanishing below degree ℓ forces the same vanishing for F , and its degree- ℓ coefficient is I_t⊗B_ i,h . This proves . Within this block divide the change from λ_ i,h I by δ^ℓ . Its constant term is I_t⊗B_ i,h , with L distinct eigenvalues. A second formal conjugation, integral in δ , separates these slopes into blocks of size t . Across all old labels there are exactly mLe_b=e_ bmL such blocks. Their characteristic polynomials are pairwise coprime over the Laurent fraction field: different old roots are already distinct at order zero, and distinct slopes separate the blocks inside one old root. The minimal-degree bound at size bmL is e_ bmL . Each separated block contributes at least one to that degree, so each contributes exactly one and is scalar. The two integral conjugations also give integral spectral projections with the stated limits.
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