Proposition

Proposition 17 ([The marked edge potential]). With the boundary polarization fixed above, there is a formal potential f∈ℂ[[h_1,…,h_N]] , where N=3(b_1(Y)-1) , and a slice s(h) such that: f= C(s) has order at least three. Its full Jacobian …

Proposition 17 ([The marked edge potential]). With the boundary polarization fixed above, there is a formal potential f∈ℂ[[h₁,…,h_N]] , where N=3(b₁(Y)-1) , and a slice s(h) such that: f= C(s) has order at least three. Its full Jacobian ideal equals the curvature ideal of s . The formal critical scheme (f)=(S/(f)) is the based flat-representation scheme with G_α=1 , and carries its marked based matrix G_β . For each fixed T_0∈ there is an affine slice with boundary τT_0ζ+v(h,τ)ξ and adjusted potential f_τ=f+τb = C(s(h,τ)) +ε2(τT_0v(h,τ)), τ²=0. Its internal Jacobian ideal equals its full curvature ideal. Accordingly_h(f_τ) is the based flat-representation scheme with G_α=(-τT₀) . No derivative with respect to τ is imposed in this relative critical scheme. On the entire, possibly nonreduced, central critical scheme, b=ε(T_0v(h,0)) =-ε(T₀ G_β) (f). The function b has order at least two. Every flat connection with the indicated normalized boundary value is carried uniquely to the corresponding slice by gauge transformations equal to the identity on T .

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ProveOct 21, 2026

Lemma 14. The curvature ideal of s(h) is the Jacobian ideal…Lemma 16. The central slice identifies its curvature-zero scheme with the…Proposition

The preceding lemmas prove all assertions except the identification of b and its order. For a general first variation of the two boundary coefficients, the boundary part of is -(ε)/(2)(u v-v u). After adding the variation of ε(uv)/2 , this becomes ε(v u) . Thus the adjusted variation is f_τ =∫_Y( s∧P(s)) +ε(v u). Take the coefficient derivative in τ at τ=0 , so that u=T_0 . On S/(f) the central curvature vanishes as an actual coefficientwise formal form, by Lemma 14. The integral term therefore vanishes in that quotient. Equation then proves. This uses equality of ideals and does not pass to the reduced critical locus. The linear part of the affine slice is i(h)+τT_0ζ , which is closed and has no ξ boundary coefficient. Hence its Chern--Simons expression and its boundary adjustment have joint (h,τ) -order at least three. The coefficient of τ consequently has h -order at least two. All formal exponentials and logarithms used here are defined by their nilpotent truncations on Artinian algebras and their formal limits.

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