Disclosure of AI assistance. The author used large language models extensively in the production of this work — for exploratory generation, adversarial criticism, proof drafting and rewriting, and consistency checking — in a structured multi-model protocol. The final statements, proofs, and submission decisions are the author’s responsibility, and the proofs in the body are intended to be checked on their own merits. To support transparency and reproducibility, Appendix 12 describes the workflow, including the errors the protocol caught; Appendix 13 records the load-bearing prompts; and Appendix 11 records the symbolic and exact-integer consistency checks.

1 Introduction↩︎

1.1 Border determinantal complexity↩︎

Let \(\mathbb{C}[x]_{\le m}\) denote the space of polynomials of degree at most \(m\) in \(x_1,\dots,x_n\), a finite-dimensional affine space, and let \[D_m\;:=\;\big\{\mathop{\mathrm{Det}}(A_0+\textstyle\sum_{i=1}^n x_iA_i)\;:\; A_0,\dots,A_n\in\mathbb{C}^{m\times m}\big\}\;\subset\;\mathbb{C}[x]_{\le m}\] be the set of polynomials with an exact size-\(m\) affine determinantal representation, so that Valiant’s determinantal complexity [1] is \(\mathop{\mathrm{dc}}(f)=\min\{m: f\in D_m\}\). The set \(D_m\) is the image of an affine space under a polynomial map, hence irreducible and constructible. The border determinantal complexity of \(f\) is \[\overline{\mathop{\mathrm{dc}}}(f)\;:=\;\min\{m\;:\;f\in\overline{D_m}\},\] the closure taken in the Zariski topology of \(\mathbb{C}[x]_{\le m}\). By Lemma 3 below, the Euclidean closure of \(D_m\) is the same set, so the analytic reading — \(f=\lim_{\varepsilon\to0}\mathop{\mathrm{Det}} A(\varepsilon)\) for matrices \(A(\varepsilon)\) of affine-linear forms with coefficients depending on \(\varepsilon\) — defines the same quantity. Replacing \(\mathbb{C}^{m\times m}\) by the symmetric matrices defines \(D_m^{\mathrm{sym}}\) and the border symmetric determinantal complexity \(\overline{\mathop{\mathrm{sdc}}}(f)\). Trivially \(\overline{\mathop{\mathrm{dc}}}\le\mathop{\mathrm{dc}}\) and \(\overline{\mathop{\mathrm{sdc}}}\le\mathop{\mathrm{sdc}}\), so border lower bounds are formally stronger than exact ones. The padded orbit-closure definition used in geometric complexity theory [2] agrees with the closure of \(D_m\); the equivalence is standard and is never used below — everything is proved from the \(\overline{D_m}\) definition directly.

Border complexity is the natural setting of the geometric complexity theory (GCT) program: Valiant’s conjecture in its orbit-closure form asks whether the padded permanent lies in the closure of the determinant orbit, and only closed conditions — equivalently, invariants semicontinuous in the right direction — can separate a point from a closure. Border lower bounds are also strictly harder to come by than exact ones: approximation is known to collapse exact lower bounds in some algebraic models entirely [3], so an exact bound carries no automatic border content.

1.2 State of the art↩︎

For exact determinantal complexity of the power sum \(F_n=\sum_{i=1}^n x_i^n\), the published record prior to the companion preprints was \(\mathop{\mathrm{dc}}(F_n)\ge 1.5n-3\) (Kumar and Volk [4]), improving the \(\mathop{\mathrm{codim}}\mathop{\mathrm{Sing}}\) bound \(\mathop{\mathrm{dc}}(F_n)\ge n+1\) of Alper, Bogart, and Velasco [5]; the upper bound is \(O(n^2)\) via the standard algebraic branching program, and Kumar and Volk note the believed truth is \(\Theta(n^2)\). The companion preprints claim \(\mathop{\mathrm{dc}}(F_n)\ge(\tfrac1{4e}-o(1))n^2\) [6] and \(\mathop{\mathrm{sdc}}(F_n)\ge(\tfrac1{2e}-o(1))n^2\) [7] by reading the top polar degree of the projectivized tangent cone — an intersection-theoretic degree, where the earlier techniques read a dimension.

For border determinantal complexity the landscape is sparser. Mignon and Ressayre’s quadratic bound for the permanent [8] extends to the border: Landsberg, Manivel, and Ressayre [2] proved \(\overline{\mathop{\mathrm{dc}}}(\mathrm{perm}_m)\ge m^2/2\) by showing that the hypersurfaces of small determinantal complexity have dual varieties of bounded dimension, a closed condition; Grochow [9] showed more generally that essentially all known determinantal lower bounds arise from such closed (GCT-compatible) conditions. Measured in the number of variables \(N=m^2\) of the permanent, these border bounds are linear. For explicit \(n\)-variate families we are not aware of any prior border lower bound superlinear in \(n\). Both companions explicitly declined to make a border claim and recorded why: polar degree is not a closed condition in degenerating families, and the isolated incidence solutions that carry the exact count “can degenerate into excess components” [7], [6].

1.3 Main results↩︎

Combined with \(\overline{\mathop{\mathrm{dc}}}\le\mathop{\mathrm{dc}}=O(n^2)\) [4] and the explicit symmetric representation of size \(2n^2+2n+1\) of [7], both border complexities of \(F_n\) are \(\Theta(n^2)\). The \(o(1)\) terms above are \(O(1/n)\) with no logarithmic loss; see Remark [rem:nolog].

The determinantal half of the proof is a statement of independent interest. For a nonzero form \(F\) on \(\mathbb{P}^n\) let \(\Gamma_{\!1}(F)\) denote the multiplicity-one Gauss-graph cycle (Definition 1): the sum, with coefficient one, of the conormal varieties of those components of \(V(F)\) along which \(F\) vanishes to order exactly one. Multidegrees \(\delta_k\) are recalled in Section 2.

Theorem 3 together with the cone-shift identity of Section 8 recovers the main inequalities of both companions for homogeneous targets (Remark [rem:recover]), with the smoothness hypothesis on \(V(\det\widehat A)\) removed. Since \(\mathop{\mathrm{dc}}\ge\overline{\mathop{\mathrm{dc}}}\) and \(\mathop{\mathrm{sdc}}\ge\overline{\mathop{\mathrm{sdc}}}\):

1.4 Relation to the companion manuscripts↩︎

This paper supersedes the exact lower bounds of the companion manuscripts [6], [7] in the ordinary and symmetric models respectively. The earlier arguments proved near-quadratic lower bounds for \(\mathop{\mathrm{dc}}(F_n)\) and \(\mathop{\mathrm{sdc}}(F_n)\) from a fixed exact representation \(F_n=\det M\), via a lifted polar-degree incidence count valid for representations of smooth hypersurfaces. The present paper proves the stronger border statements of Theorems 1–2, from which the exact bounds follow immediately (Corollary 1). The overlap between the papers is confined to the determinantal-side incidence/Bézout mechanism: that mechanism is inherited from the exact polar-degree argument, but it is reformulated and strengthened here (Theorem 3) into an unconditional bound for the multiplicity-one Gauss-graph cycle of an arbitrary affine-linear determinant — singular, reducible, and non-reduced fibers allowed — with the local normal forms proved in full, so that the present paper is logically self-contained and no lemma is imported from the companions. The new ingredient, and the main contribution, is the conormal-specialization argument of Sections 4–7: in any degeneration \(\det A_c\to F_n\), the flat limit of the Gauss-graph cycles contains the conormal variety of the Fermat cone with positive coefficient. The companions remain the record of the exact-representation viewpoint and of the discovery of the polar-degree invariant; their main theorems are recovered here as corollaries (Remark [rem:recover], Corollary 1).

1.5 The semicontinuity question, and how it resolves↩︎

The companions’ reason for caution was correct as far as it went, and identifying exactly where it stops is the conceptual contribution of this paper, so we spell it out.

Local polar-type invariants of a hypersurface germ jump up at a special fibre. Concretely: for generic \(q\) of degree \(n\), the family \(F_n+t\,q\) has smooth fibres for small \(t\ne0\), while the special fibre is a cone with a deep singular point; any invariant that reads the singularity at the vertex (multiplicity, local polar multiplicity, Milnor-type numbers) is strictly larger at \(t=0\) than nearby. A transfer argument that runs “the special fibre’s local invariant is at most the generic fibre’s” is therefore dead on arrival. This is the failure mode the companions’ scope remarks anticipated.

The repair is to change what is being specialized. The global conormal geometry of the family behaves in the opposite direction: the Gauss graphs \(\Gamma_{\!1}\) of the generic fibres form a family of \((n-1)\)-cycles in \(\mathbb{P}^n\times(\mathbb{P}^n)^\vee\); over a smooth curve this family has a flat limit; the limit cycle is effective and its multidegrees are conserved (Lemma 15); and the conormal variety of the special hypersurface sits inside the limit cycle with coefficient at least one (Proposition [prop:containment]). Effectivity then gives the inequality in the favorable direction: \[\delta_{n-2}\big(\mathop{\mathrm{Con}}(\text{special})\big)\;\le\; \delta_{n-2}\big(\text{limit cycle}\big)\;=\; \delta_{n-2}\big(\Gamma_{\!1}(\text{generic})\big).\] What makes this usable for \(F_n\) is its homogeneity: the invariant the companions read — the top polar degree of the projectivized tangent cone at the singular point — is, for a cone, equal to a global multidegree of the conormal variety of the cone (the cone-shift identity, Proposition [prop:coneshift]), and global multidegrees are exactly what specialization conserves. The local invariant jumps up; the global cycle’s multidegree does not. Semicontinuity holds at the level of conormal cycles, in the direction the lower bound needs.

The specialization behaviour of conormal and Lagrangian cycles is classical — it underlies the theory of polar varieties and equisingularity [10], [11] and has recently been used to prove semicontinuity of Gauss-map degrees [12]. The specialization lemma proved here (Section 6) is an elementary, self-contained instance, adapted to one feature the classical setting does not have: the fibres of a degenerating determinantal family may be non-reduced, and the limit divisor \(x_0^{m-n}\widetilde{F}_n\) certainly is. The multiplicity-one Gauss graph \(\Gamma_{\!1}\) is the device that makes non-reducedness harmless on both sides of the chain: it is the cycle Theorem 3 bounds, and it is the cycle whose limit provably captures the Fermat conormal (Lemma 17) — because points with nonvanishing differential lie on multiplicity-one components by force.

1.6 Comparison with the dual-dimension border bounds↩︎

The bound of [2] and the GCT-unification of [9] read the dimension of the dual variety: hypersurfaces of small determinantal complexity have duals of small dimension, dual dimension is Zariski-closed in families of the relevant kind, and the permanent hypersurface has nondegenerate dual. Dimension is bounded by the number of variables, so this route cannot give bounds superlinear in \(n\) for an \(n\)-variate family; and for the Fermat tangent cone the dual is nondefective, so the dimensional invariant carries no signal at all [6]. The present paper transfers the dual degree — exponentially large for \(F_n\) — across the degeneration, and an \((n-1)\)-st root converts it into the quadratic bound, exactly as in the exact setting. The price of reading a degree rather than a dimension is that no closed condition is available; the conserved object is a cycle class, and the inequality comes from effectivity rather than from membership in a closed set.

2 Preliminaries and conventions↩︎

We work over \(\mathbb{C}\) throughout; \(n\ge2\) is the number of affine variables, and \(F_n=x_1^n+\cdots+x_n^n\). We write \(\widetilde{F}_n\) for the same polynomial regarded as a degree-\(n\) form in \((x_0,x_1,\dots,x_n)\) (it does not involve \(x_0\)), and \[X\;:=\;V(\widetilde{F}_n)\;\subset\;\mathbb{P}^n,\] the projective cone over the smooth Fermat hypersurface \(Z=V(F_n)\subset\mathbb{P}^{n-1}=\{x_0=0\}\) with vertex \(q_0=[1:0:\cdots:0]\).

2.1 Conormal cycles and multidegrees↩︎

Let \(h_1,h_2\) denote the pullbacks to \(\mathbb{P}^n\times(\mathbb{P}^n)^\vee\) of the hyperplane classes of the two factors. For an \((n-1)\)-cycle \(C\) on \(\mathbb{P}^n\times(\mathbb{P}^n)^\vee\) and \(0\le k\le n-1\), the \(k\)-th multidegree is \[\delta_k(C)\;:=\;\deg\big(C\cdot h_1^{\,n-1-k}h_2^{\,k}\big)\in\mathbb{Z} .\] Multidegrees are additive in \(C\) and, by Lemma 4 below, nonnegative on effective cycles: \(\delta_k\) of an irreducible \((n-1)\)-fold is computed by a generic transverse flag intersection and is the (nonnegative) number of intersection points.

For an irreducible subvariety \(S\subset\mathbb{P}^n\), \(\mathop{\mathrm{Con}}(S)\subset \mathbb{P}^n\times(\mathbb{P}^n)^\vee\) is the conormal variety: the closure of \(\{(x,[\xi]): x\in S_{\mathrm{sm}},\;\xi|_{T_xS}=0\}\). For a hypersurface \(S\) it has dimension \(n-1\), and over a smooth point of \(S=V(g)\) (\(g\) reduced) the fibre is the single covector \([dg(x)]\).

2.2 Border complexity and closures↩︎

2.3 Transversality and positivity inputs↩︎

The following two lemmas are the only intersection-theoretic inputs to Theorem 3; each is stated with the hypotheses under which it will be invoked.

2.4 Dual degree of a smooth hypersurface↩︎

3 The determinantal conormal lemma↩︎

This section proves Theorem 3. Throughout, \(A,\widehat A,F\) are as in the statement, \(n\ge2\), and \(F=\det\widehat A\not\equiv0\) is a form of degree \(m\) on \(\mathbb{P}^n\). If \(\Gamma_{\!1}(F)\) is the zero cycle there is nothing to prove, so assume \(V(F)\) has at least one multiplicity-one component, and write \(\Gamma_{\!1}(F)=\sum_{j\in J}\mathop{\mathrm{Con}}(S_j)\) per Lemma 1.

3.1 Proof of Theorem 3(i)↩︎

3.1.0.1 Step 1: count points.

For each \(j\in J\) let \(S_j^{\mathrm{good}}\subseteq S_j\) be the dense open subset of points \(x\) such that \(x\) is a smooth point of \(V(F)_{\mathrm{red}}\) lying on \(S_j\) only, and \(dF(x)\ne0\); this is dense because \(S_j\) has multiplicity one (Lemma 2(b)) and \(F\) restricted to a neighborhood of a generic point of \(S_j\) is a unit times a reduced equation of \(S_j\). Over \(S_j^{\mathrm{good}}\) the conormal variety is the graph \(x\mapsto(x,[dF(x)])\); let \(\mathop{\mathrm{Con}}(S_j)^\circ\) be this graph, dense open in \(\mathop{\mathrm{Con}}(S_j)\), isomorphic to \(S_j^{\mathrm{good}}\) via the first projection.

Apply Lemma 4 on \(\mathbb{P}^n\times(\mathbb{P}^n)^\vee\) to the varieties \(\{\mathop{\mathrm{Con}}(S_j)\}_{j\in J}\), the opens \(\mathop{\mathrm{Con}}(S_j)^\circ\), and the flag \(\Phi=M\times(L_1\cap\cdots\cap L_{n-2})\) with \(M\) a hyperplane in \(\mathbb{P}^n\) and \(L_t\) hyperplanes in \((\mathbb{P}^n)^\vee\): for a generic such flag, the intersection consists of exactly \(\sum_j\delta_{n-2}(\mathop{\mathrm{Con}}(S_j))=\delta_{n-2}(\Gamma_{\!1}(F))\) distinct count points \[(x^*,\xi^*),\qquad x^*\in S_j^{\mathrm{good}},\quad \xi^*=[dF(x^*)],\] each a transverse, reduced intersection point of the flag with the graph \(\mathop{\mathrm{Con}}(S_j)^\circ\). Transversality at \((x^*,\xi^*)\), transported through the isomorphism \(S_j^{\mathrm{good}}\cong\mathop{\mathrm{Con}}(S_j)^\circ\), says precisely: writing \(\ell_t\) for the linear forms cutting \(L_1\cap\cdots\cap L_{n-2}\) and \(M\) for the linear form cutting the hyperplane, the \(n-1\) functions \[\label{eq:flagfunctions} M(x),\quad \ell_1\big(dF(x)\big),\;\dots,\; \ell_{n-2}\big(dF(x)\big)\tag{2}\] vanish at \(x^*\) and have linearly independent differentials on \(T_{x^*}S_j\). This is the form in which transversality will be used in Step 5.

3.1.0.2 Step 2: the kernel lift.

Fix a count point \((x^*,\xi^*)\). Since \(dF(x^*)\ne0\), formula 1 forces \(\mathop{\mathrm{adj}}\widehat A(x^*)\ne0\), hence \(\mathop{\mathrm{rank}}\widehat A(x^*)\ge m-1\); since \(F(x^*)=0\) the rank is exactly \(m-1\). The left and right kernels are lines, spanned by \(u^*,v^*\), and as in the proof of Lemma 2, \[\label{eq:conormalid} \mathop{\mathrm{adj}}\widehat A(x^*)=\alpha\,v^*(u^*)^{\mathsf T},\quad\alpha\ne0, \qquad \partial_iF(x^*)=\alpha\,(u^*)^{\mathsf T}\widehat A_i\,v^*\;\; (0\le i\le n),\tag{3}\] so \(\xi^*=\big[(u^*)^{\mathsf T}\widehat A_0v^*:\cdots: (u^*)^{\mathsf T}\widehat A_nv^*\big]\). Each count point therefore lifts to the unique point \(P^*=(x^*,[u^*],[v^*])\in\mathbb{P}^n\times\mathbb{P}^{m-1}\times\mathbb{P}^{m-1}\).

3.1.0.3 Step 3: choice of the reduction matrix \(\Lambda\).

The incidence conditions at a lift are \(u^{\mathsf T}\widehat A(x)=0\) (\(m\) equations) and \(\widehat A(x)v=0\) (\(m\) equations), but the incidence variety over the corank-one locus has codimension \(2m-1\), not \(2m\), in \(\mathbb{P}^n\times\mathbb{P}^{m-1}\times\mathbb{P}^{m-1}\): one equation is locally redundant. We remove the redundancy on the right side. Choose a linear map \(\Lambda:\mathbb{C}^m\to\mathbb{C}^{m-1}\), generic subject to the finitely many open conditions \[\label{eq:lambdacond} \ker\Lambda\cap\mathop{\mathrm{im}}\widehat A(x^*)=0 \qquad\text{for every count point }x^* ,\tag{4}\] each of which excludes a proper closed subset of \(\Lambda\)-space because \(\ker\Lambda\) is a line and \(\mathop{\mathrm{im}}\widehat A(x^*)\) is a hyperplane of \(\mathbb{C}^m\). We also require, generically, that no row of \(\Lambda\widehat A\) vanishes identically as a form (the left annihilators \(\{\lambda:\lambda^{\mathsf T}\widehat A\equiv0\}\) form a proper linear subspace since \(\widehat A\not\equiv0\)). The scheme-theoretic content of the reduction — that near each lift the \(m-1\) equations \(\Lambda\widehat Av=0\) generate the same ideal as the full \(\widehat Av=0\) — is established by the explicit elimination in Step 5(ii), where condition 4 resurfaces as the invertibility of a concrete \((m-1)\times(m-1)\) matrix.

3.1.0.4 Step 4: the square multihomogeneous system.

On \(Y=\mathbb{P}^n\times\mathbb{P}^{m-1}_{[u]}\times\mathbb{P}^{m-1}_{[v]}\), with hyperplane classes \(H,U,V\) and \(\dim Y=n+2m-2\), consider the system \[\label{eq:system} \begin{array}{l@{\qquad}l@{\qquad}l} u^{\mathsf T}\widehat A(x)=0 & m\;\text{equations} & \text{class }H+U,\\[1pt] \Lambda\,\widehat A(x)\,v=0 & m-1\;\text{equations} & \text{class }H+V,\\[1pt] \ell_t\big(u^{\mathsf T}\widehat A_0v,\dots, u^{\mathsf T}\widehat A_nv\big)=0,\;\;1\le t\le n-2 & n-2\;\text{equations} & \text{class }U+V,\\[1pt] M(x)=0 & 1\;\text{equation} & \text{class }H, \end{array}\tag{5}\] where \(\ell_t,M\) are the flag forms of Step 1. The count is \(m+(m-1)+(n-2)+1=n+2m-2=\dim Y\): the system is square. The class assignments: each entry of \(u^{\mathsf T}\widehat A\) is bilinear in \((x,u)\) and each entry of \(\Lambda\widehat Av\) is bilinear in \((x,v)\); each polar equation is \(\ell_t(\dots)=u^{\mathsf T}\big(\sum_i\lambda_{t,i}\widehat A_i\big)v\), bilinear in \((u,v)\) and independent of \(x\) — a genuine nonzero section of \(\mathcal{O}(0,1,1)\) for generic \(\ell_t\), since \(\sum_i\lambda_i\widehat A_i=0\) defines a proper linear subspace of covectors (\(\widehat A\not\equiv0\)), which the generic flag avoids. Likewise each entry of \(u^{\mathsf T}\widehat A\) is a nonzero section (a zero entry would be a zero column of \(\widehat A\), forcing \(F\equiv0\)), and the rows of \(\Lambda\widehat A\) were arranged nonzero in Step 3. All multidegrees are componentwise nonnegative, so Lemma 5 applies with \[\int_Y c_{\mathrm{top}} =\big[H^nU^{m-1}V^{m-1}\big]\; H\,(H+U)^m(H+V)^{m-1}(U+V)^{n-2} \;=\;B(m,n),\] the equality of the two displayed brackets being the symmetry \(u\leftrightarrow v\) of the extraction (the closed form is computed in Step 6).

3.1.0.5 Step 5: the lifts are reduced isolated points of the solution scheme.

Fix a count point with lift \(P^*=(x^*,[u^*],[v^*])\).

(i) No solutions off the hypersurface. If \(\det\widehat A(x)\ne0\) then \(u^{\mathsf T}\widehat A(x)=0\) forces \(u=0\), which is not a point of \(\mathbb{P}^{m-1}\). This is why the left system is kept whole.

(ii) Local normal form: the bilinear block is scheme-theoretically a graph over the hypersurface germ. After constant row and column operations \(\widehat A\mapsto P\widehat AQ\) — which multiply \(F\) by the nonzero constant \(\det P\det Q\), transform \(u\mapsto P^{-\mathsf T}u\), \(v\mapsto Q^{-1}v\), \(\Lambda\mapsto\Lambda P^{-1}\) equivariantly, and leave every pairing \(u^{\mathsf T}\widehat A_iv\) and condition 4 intact — we may assume the leading \((m-1)\times(m-1)\) block of \(\widehat A(x^*)\) is invertible (some \((m-1)\)-minor is nonzero at a rank-\((m-1)\) matrix). Work in an affine chart of \(\mathbb{P}^n\) at \(x^*\) and write \[\widehat A=\begin{pmatrix}B&c\\ r&s\end{pmatrix}, \qquad \det B\in\mathcal{O}_{x^*}^{\times},\] with \(B\) of size \(m-1\) and all blocks affine-linear in the chart coordinates. Then \(u^*_m\ne0\) and \(v^*_m\ne0\): a kernel vector with vanishing last coordinate is annihilated by the invertible \(B\) and dies. So we may work in the charts \(u_m=1\), \(v_m=1\), writing \(u^{\mathsf T}=(u'^{\mathsf T},1)\), \(v=(v',1)^{\mathsf T}\). Set \[g\;:=\;s-rB^{-1}c\;\in\mathcal{O}_{x^*}, \qquad z\;:=\;v'+B^{-1}c ,\] the Schur complement and the recentred right-kernel coordinate, and recall the Schur identity \(\det\widehat A=(\det B)\,g\), so that \((F)=(g)\) in \(\mathcal{O}_{x^*}\). In the local ring \(\mathcal{O}_{P^*}\) (variables: chart coordinates of \(x\), and \(u',v'\)), the ideal \(I\) of the bilinear block \(\{u^{\mathsf T}\widehat A=0,\;\Lambda\widehat Av=0\}\) is computed by elimination.

Left block. The first \(m-1\) left equations are \(u'^{\mathsf T}B+r=0\); since \(B\) is invertible over \(\mathcal{O}_{x^*}\) they generate the same ideal as the entries of \(u'^{\mathsf T}+rB^{-1}\). Modulo these, the last left equation becomes \(u'^{\mathsf T}c+s\;\equiv\;(-rB^{-1})c+s\;=\;g\).

Right block. Writing \(v'=z-B^{-1}c\), \[\widehat A\,v =\begin{pmatrix}Bv'+c\\ rv'+s\end{pmatrix} =\begin{pmatrix}Bz\\ rz+g\end{pmatrix}, \qquad\text{hence}\qquad \Lambda\widehat A\,v\;=\;M_v(x)\,z\;+\;g\,\lambda ,\] an exact identity, where \(\Lambda=[\Lambda_{\mathrm{top}}\mid\lambda]\) with \(\lambda\) the last column of \(\Lambda\), and \[M_v(x)\;:=\;\Lambda\,C(x),\qquad C(x):=\begin{pmatrix}B\\ r\end{pmatrix} =\text{the first }m-1\text{ columns of }\widehat A .\] At \(x^*\), \(C(x^*)\) has rank \(m-1\) (its top block is \(B(x^*)\)) and its column space is therefore all of \(\mathop{\mathrm{im}}\widehat A(x^*)\); so condition 4 says exactly that \(\Lambda\) is injective on the column space of \(C(x^*)\), i.e.that \(M_v(x^*)\) is invertible. Hence \(M_v\) is invertible over \(\mathcal{O}_{x^*}\), and modulo the generators already obtained, \(\Lambda\widehat Av\equiv M_vz\) generates the same ideal as the entries of \(z\). Altogether \[\label{eq:normalform} I\;=\;\big(\,u'^{\mathsf T}+rB^{-1},\;\;v'+B^{-1}c,\;\;g\,\big):\tag{6}\] the bilinear block is, scheme-theoretically, the graph \[x\;\longmapsto\;\big(x,\,[u(x)],\,[v(x)]\big), \qquad u(x)^{\mathsf T}=(-rB^{-1},\,1),\quad v(x)=(-B^{-1}c,\,1)^{\mathsf T},\] over the germ \(V(g)\). Since \((F)=(g)\) and \(dF(x^*)\ne0\), that germ is the smooth reduced hypersurface germ \((S_j,x^*)\), and the graph is a smooth germ of dimension \(n-1\) through \(P^*\).

(iii) The conormal identity along the graph, and transversality. Along \((S_j,x^*)\) the explicit sections satisfy \(\widehat A(x)v(x)=(Bz,rz+g)^{\mathsf T}\big|_{z=0,\,g=0}=0\) and \(u(x)^{\mathsf T}\widehat A(x)=(0,\,g)\big|_{g=0}=0\): they span the kernels. At each such point \(\widehat A\) has rank \(m-1\), so \(\mathop{\mathrm{adj}}\widehat A(x)=\alpha(x)\,v(x)u(x)^{\mathsf T}\); evaluating the \((m,m)\) entries pins the scalar, since \((\mathop{\mathrm{adj}}\widehat A)_{mm}=\det B\) and \((vu^{\mathsf T})_{mm}=1\), giving \(\alpha=\det B\), a unit. Jacobi’s formula 1 then yields, on \((S_j,x^*)\) and for every \(i\), \[\partial_iF(x)\;=\;(\det B)(x)\cdot u(x)^{\mathsf T}\widehat A_i\,v(x).\] Consequently each polar equation of 5 restricts on the graph to \((\det B)^{-1}\,\ell_t(dF(x))\), a unit multiple of the corresponding flag function 2 ; both vanish at \(x^*\), so their differentials at \(x^*\) agree up to the unit value \((\det B)(x^*)^{-1}\). The remaining equation \(M(x)\) restricts to itself. By Step 1, the \(n-1\) flag functions 2 have linearly independent differentials on \(T_{x^*}S_j\). Therefore the full system 5 cuts, near \(P^*\), the smooth \((n-1)\)-dimensional germ 6 by \(n-1\) functions with independent differentials: a single reduced point. \(P^*\) is a reduced isolated point of the solution scheme, and distinct count points have distinct \(x^*\), hence distinct lifts.

The role of the count-point condition is visible in the normal form: the last left equation contributes the hypersurface equation \(g\), and the polar equations contribute transverse slices precisely because \(dF(x^*)\ne0\). On a tangential component (Example 1) every lifted polar equation would restrict to zero along the graph and no isolated point would form; the dichotomy of Lemma 2 is what the construction runs on.

3.1.0.6 Step 6: conclusion and the closed form.

By Lemma 5, the number of isolated solutions of 5 , counted with multiplicity \(\ge1\), is at most \(B(m,n)\); by Step 5 the \(\delta_{n-2}(\Gamma_{\!1}(F))\) lifts are among them, each reduced. Hence \(\delta_{n-2}(\Gamma_{\!1}(F))\le B(m,n)\). For the closed form, extract the bracket: choosing \(x^i\) from \((x+u)^m\), \(x^j\) from \((x+v)^{m-1}\), and \(u^k\) from \((u+v)^{n-2}\) forces, by matching exponents of \(x,u,v\) in turn, \(1+i+j=n\), \(k=i-1\), and \(j=n-1-i\), so \[B(m,n)=\sum_{i=1}^{n-1}\binom{m}{i}\binom{m-1}{n-1-i} \binom{n-2}{i-1},\] the limits enforced by \(k\ge0\) and \(j\ge0\). 0◻

3.2 Proof of Theorem 3(ii)↩︎

The symmetric proof is given in full; it runs on \(Y=\mathbb{P}^n\times\mathbb{P}^{m-1}_{[u]}\), of dimension \(n+m-1\).

3.2.0.1 Step 1: count points.

Step 1 of Section 3.1 concerns only the cycle \(\Gamma_{\!1}(F)\) and not the representation; it applies verbatim and furnishes \(\delta_{n-2}(\Gamma_{\!1}(F))\) count points \((x^*,\xi^*)\) with \(x^*\in S_j^{\mathrm{good}}\), \(\xi^*=[dF(x^*)]\), together with the transversality statement for the flag functions 2 on \(T_{x^*}S_j\).

3.2.0.2 Step 2: the kernel lift.

At a count point, \(dF(x^*)\ne0\) and 1 force \(\mathop{\mathrm{adj}}\widehat A(x^*)\ne0\), hence \(\mathop{\mathrm{rank}}\widehat A(x^*)=m-1\) as before. Since \(\widehat A(x^*)\) is symmetric, its adjugate is a nonzero symmetric rank-one matrix whose image is the kernel line, so \[\mathop{\mathrm{adj}}\widehat A(x^*)=\beta\,u^*(u^*)^{\mathsf T},\quad\beta\ne0, \qquad \partial_iF(x^*)=\beta\,(u^*)^{\mathsf T}\widehat A_i\,u^*\;\; (0\le i\le n),\] and the count point lifts to the unique \(P^*=(x^*,[u^*])\in\mathbb{P}^n\times\mathbb{P}^{m-1}\).

3.2.0.3 Step 3: squareness without a reduction.

The kernel condition \(\widehat A(x)u=0\) is \(m\) equations, and the system below has \(m+(n-2)+1=n+m-1=\dim Y\) equations: square as it stands. No analogue of the \(\Lambda\)-reduction is needed — the elimination in Step 5(ii) shows the \(m\) kernel equations cut exactly codimension \(m\) near the lift, the last of them supplying the hypersurface equation — and none is sound to add: removing an equation enlarges the local solution set to the kernel of an \((m-1)\times m\) submatrix, which can be positive-dimensional through the lift, exactly as in Remark [rem:norow].

3.2.0.4 Step 4: the system and its classes.

The equations are \[\label{eq:symsystem} \begin{array}{l@{\qquad}l@{\qquad}l} \widehat A(x)\,u=0 & m\;\text{equations} & \text{class }H+U,\\[1pt] \ell_t\big(u^{\mathsf T}\widehat A_0u,\dots, u^{\mathsf T}\widehat A_nu\big)=0,\;\;1\le t\le n-2 & n-2\;\text{equations} & \text{class }2U,\\[1pt] M(x)=0 & 1\;\text{equation} & \text{class }H, \end{array}\tag{7}\] with \(\ell_t,M\) the flag forms of Step 1. Each kernel entry is bilinear in \((x,u)\); each polar equation is \(u^{\mathsf T}(\sum_i\lambda_{t,i}\widehat A_i)u\), a quadratic form in \(u\) alone. Nonzero sections: a vanishing row of \(\widehat A\) forces \(F\equiv0\); and a flag form with \(u^{\mathsf T}(\sum_i\lambda_i\widehat A_i)u\equiv0\) forces \(\sum_i\lambda_i\widehat A_i=0\) — a symmetric matrix over \(\mathbb{C}\) whose quadratic form vanishes identically is zero in characteristic zero — which is a proper linear condition on \(\lambda\) avoided by the generic flag. All multidegrees are componentwise nonnegative.

3.2.0.5 Step 5: the lifts are reduced isolated points.

Fix a count point with lift \(P^*=(x^*,[u^*])\).

(i) No solutions off the hypersurface. Where \(\det\widehat A(x)\ne0\), \(\widehat A(x)u=0\) forces \(u=0\), not a point of \(\mathbb{P}^{m-1}\).

(ii) Local normal form. After a constant congruence \(\widehat A\mapsto P^{\mathsf T}\widehat AP\) — which multiplies \(F\) by \((\det P)^2\ne0\), transforms \(u\mapsto P^{-1}u\), preserves symmetry and every quadratic pairing \(u^{\mathsf T}\widehat A_iu\) — assume the leading \((m-1)\times(m-1)\) block is invertible at \(x^*\), and write, in an affine chart at \(x^*\), \[\widehat A=\begin{pmatrix}B&c\\ c^{\mathsf T}&s\end{pmatrix}, \qquad \det B\in\mathcal{O}_{x^*}^{\times}, \qquad B=B^{\mathsf T}.\] Then \(u^*_m\ne0\) (a kernel vector with last coordinate zero dies against \(B\)); work in the chart \(u_m=1\), \(u=(u',1)^{\mathsf T}\). Set \(g:=s-c^{\mathsf T}B^{-1}c\) and \(z:=u'+B^{-1}c\), so \[\widehat A\,u =\begin{pmatrix}Bu'+c\\ c^{\mathsf T}u'+s\end{pmatrix} =\begin{pmatrix}Bz\\ c^{\mathsf T}z+g\end{pmatrix}, \qquad \det\widehat A=(\det B)\,g .\] Since \(B\) is invertible over \(\mathcal{O}_{x^*}\), the first \(m-1\) entries generate the ideal of the entries of \(z\), and modulo these the last entry becomes \(g\). Hence the ideal of the kernel block in \(\mathcal{O}_{P^*}\) is exactly \[\big(\,u'+B^{-1}c,\;\;g\,\big):\] the kernel block is, scheme-theoretically, the graph \(u=\nu(x):=(-B^{-1}c,\,1)^{\mathsf T}\) over the germ \(V(g)\), which equals the smooth reduced germ \((S_j,x^*)\) because \((F)=(g)\) and \(dF(x^*)\ne0\); a smooth germ of dimension \(n-1\) through \(P^*\).

(iii) Conormal identity and transversality. Along \((S_j,x^*)\) the section \(\nu(x)\) spans the kernel (\(\widehat A\nu=(Bz,c^{\mathsf T}z+g)^{\mathsf T}|_{z=0,g=0}=0\)), the rank is \(m-1\), and the symmetric adjugate is a scalar multiple of \(\nu\nu^{\mathsf T}\); the \((m,m)\) cofactor is \(\det B\) while \((\nu\nu^{\mathsf T})_{mm}=1\), so \(\mathop{\mathrm{adj}}\widehat A=(\det B)\,\nu\nu^{\mathsf T}\) on \((S_j,x^*)\), and Jacobi’s formula 1 gives \[\partial_iF(x)\;=\;(\det B)(x)\cdot \nu(x)^{\mathsf T}\widehat A_i\,\nu(x) \qquad\text{on }(S_j,x^*).\] Each polar equation of 7 therefore restricts on the graph to \((\det B)^{-1}\ell_t(dF(x))\), a unit multiple of the flag function, vanishing at \(x^*\); \(M(x)\) restricts to itself; and by Step 1 the \(n-1\) flag functions 2 have independent differentials on \(T_{x^*}S_j\). The full system 7 thus cuts, near \(P^*\), the smooth \((n-1)\)-dimensional graph by \(n-1\) functions with independent differentials: a single reduced point. Distinct count points give distinct lifts.

3.2.0.6 Step 6: conclusion and the coefficient.

Lemma 5 with \(E=\mathcal{O}(1,1)^{\oplus m}\oplus\mathcal{O}(1,0)\oplus\mathcal{O}(0,2)^{\oplus(n-2)}\) bounds the isolated solutions of 7 by \[\big[H^nU^{m-1}\big]\,H\,(H+U)^m(2U)^{n-2} \;=\;2^{n-2}\,\big[H^{n-1}U^{m-n+1}\big](H+U)^m \;=\;2^{n-2}\binom{m}{n-1},\] and by Step 5 the \(\delta_{n-2}(\Gamma_{\!1}(F))\) lifts are among them, each reduced. 0◻

4 Border normal form↩︎

From here through Section 9, fix \(n\ge3\) and suppose \(\overline{\mathop{\mathrm{dc}}}(F_n)\le m\) (the symmetric case is identical with \(D_m^{\mathrm{sym}}\) and is taken up in Section 9).

4.0.0.1 Conventions for the family.

Fix once and for all a finite subset \(\Sigma\subset C\) of excluded points, enlarged finitely many times in Lemmas 8, 13, and 14 below, and set \(U:=C^\circ\smallsetminus\Sigma\), always cofinite in \(C^\circ\) and in particular infinite. Statements “for \(c\in U\)” hold for every closed point of \(U\). Analytic limits “\(c\to0\)” are taken along an analytic disk in \(C^{\mathrm{an}}\) centered at \(0\); since \(\Sigma\) is finite, every punctured analytic neighborhood of \(0\) meets \(U\) in a set accumulating at \(0\).

5 Homogenization at degree \(m\)↩︎

6 The Gauss-graph family and conservation of multidegree↩︎

For \(c\in U\) set \(\Gamma_c:=\Gamma_{\!1}(\widehat G_c)\), the cycle of Definition 1 for the fibre. Define the locally closed algebraic set \[\mathcal{G}^\circ:=\big\{(x,\xi,c)\in \mathbb{P}^n\times(\mathbb{P}^n)^\vee\times C^\circ\;:\; \widehat G_c(x)=0,\;d\widehat G_c(x)\ne0,\; \xi=[d\widehat G_c(x)]\big\},\] with the reduced structure: it is the graph of the morphism \((x,c)\mapsto[d\widehat G_c(x)]\) over the open subset \(\{d\widehat G\ne0\}\) of the closed set \(\{\widehat G=0\}\subset\mathbb{P}^n\times C^\circ\), the coefficients of \(\widehat G\) and its partials being regular in \(c\).

7 Capture of the Fermat conormal in the flat limit↩︎

The proof must produce points of the multiplicity-one Gauss graphs \(\Gamma_{\!1}(\widehat G_c)\) converging into the Fermat conormal; this is where the \(\Gamma_{\!1}\) device earns its keep, since the limit divisor \(V(\widehat G_0)=V(x_0^{\,m-n}\widetilde{F}_n)\) is badly non-reduced at infinity and nothing guarantees the nearby divisors \(V(\widehat G_c)\) are reduced either.

8 The cone-shift identity and the Fermat dual degree↩︎

9 Proofs of the main theorems↩︎

10 Scope, limitations, and failure modes↩︎

1. \(\Gamma_{\!1}\) throughout. Every fiberwise object is the multiplicity-one Gauss-graph cycle, never the full reduced conormal of the fibre. Theorem 3 bounds exactly \(\Gamma_{\!1}\); whether the reduced conormal of a non-squarefree determinant obeys the same bound is open (Remark [rem:gammared]) and is not needed.

2. Why the Fermat conormal survives. Smooth affine zeros of \(F_n\) persist as zeros of \(G_c\) with nonvanishing differential (Hurwitz plus uniform \(C^1\) convergence), and a nonvanishing differential forces membership in multiplicity-one components; so the approximating points are honest open-graph points, and properness pushes their limits into \(W_0\). The padding \(x_0^{\,m-n}\) is differentially dead along the Fermat component (Lemma 16) and contributes only effective junk (Remark [rem:junk]).

3. No genericity, and no compatibility, of the representations. The \(A_c\) are arbitrary, and no relation among them across different \(c\) is assumed; the only object with structure along the parameter is the polynomial family \(G\). The only generic choices in the paper are flags (Lemma 4, used in Theorem 3 and Proposition [prop:coneshift]) and the constant matrix \(\Lambda\) of Step 3, all chosen after the data.

4. Repeated factors are allowed everywhere. Multiplicity-\(\ge2\) components of any fibre, including a hyperplane at infinity of multiplicity \(m-\deg_xG_c\), are invisible to \(\Gamma_{\!1}\) on both sides of the chain (Remarks [rem:excess], [rem:degreem], [rem:junk]); a multiplicity-one hyperplane at infinity has \(\delta_{n-2}=0\) and is harmless.

5. Transversality and flatness inventory. Kleiman transversality: Sections 3 and 8. Fulton positivity / localized Chern classes: Lemma 5. Curve selection: Lemma 7. Smoothness of the open graph over the curve (Jacobian criterion) and reducedness of the generic fibre cycles: Lemmas 12 and 14. Torsion-freeness over a DVR and flat-fibre dimension theory: Lemmas 13 and 14. Specialization of cycle classes: Lemma 15. Hurwitz: Lemma 17.

6. Homogeneity is essential. The cone-shift identity consumes homogeneity of the target: it converts the vertex-local invariant into the global multidegree that specialization conserves. For an inhomogeneous target no such bridge is provided, and the method as written says nothing.

7. No permanent claim. For the permanent the special-side input would be a lower bound on the relevant conormal multidegree of the (singular) permanent hypersurface, which we do not know; the transfer machinery of Sections 4 –7 would apply once such a bound exists, since nothing there is specific to \(F_n\) except homogeneity and Lemma 16’s use of the explicit gradient (which generalizes to any reduced homogeneous target along its smooth affine locus).

8. Base changes and shrinkings. One normalization (Lemma 7) and finitely many enlargements of the excluded set \(\Sigma\) (Lemmas 7, 8, 13, 14). None alters the hypothesis \(F_n\in\overline{D_m}\) or the parameter \(m\); the final chain needs a single \(c\in U\), and \(U\) is infinite.

9. Characteristic zero. Used in Kleiman transversality, reflexivity (Lemma 6), the symmetric quadratic-form argument in Step 4 of Section 3.2, and the analytic Hurwitz argument (replaceable by a Newton–Puiseux argument over \(\mathbb{C}[[t]]\), which we do not pursue). The statement itself fails as a uniform theorem in positive characteristic: for \(\operatorname{char}=p>2\) and \(n=p^r\) the Frobenius identity makes \(F_n\) a perfect power of a linear form, with \(\mathop{\mathrm{sdc}}(F_n)\le n\) [7].

10.0.0.1 Open questions.

(1) Improve the constants toward the upper bounds; the Bézout count is an overcount and the limit cycle \([W_0]\) carries identifiable junk (Remark [rem:junk]) whose multidegrees could in principle be subtracted. (2) Bound the relevant conormal multidegree of the permanent hypersurface from below, activating item 7. (3) Settle Remark [rem:gammared]. (4) Is \(\overline{\mathop{\mathrm{dc}}}(F_n)<\mathop{\mathrm{dc}}(F_n)\) for some \(n\)? Both are now pinned to \(\Theta(n^2)\) with the same lower-bound constant, and the question of a genuine border gap for this family becomes meaningful.

11 Consistency checks and symbolic verification↩︎

All computations were performed in exact arithmetic (Python integers; sympy for polynomial expansion and resultants); the script is available from the author on request. We record the actual outputs.

11.0.0.1 (A1) Coefficient identities.

The closed forms of Theorem 3 were checked against direct bracket extraction from the expanded products: \[\big[x^nu^{m-1}v^{m-1}\big]\,x(x+u)^m(x+v)^{m-1}(u+v)^{n-2} =\sum_{i=1}^{n-1}\binom{m}{i}\binom{m-1}{n-1-i}\binom{n-2}{i-1},\] and \([x^nu^{m-1}]\,x(x+u)^m(2u)^{n-2}=2^{n-2}\binom{m}{n-1}\), for all \(2\le n\le6\) and \(n\le m\le10\): exact agreement in every case. Boundary checks: \(B(m,2)=m\) for \(1\le m\le11\) (slot \(\delta_0\) is the degree of the multiplicity-one part, at most \(m\)); \(B(1,n)=0\) for \(3\le n\le11\) (a hyperplane’s conormal has a point as dual factor); \(B(m,3)=3\binom{m}{2}\) for \(3\le m\le29\), matching the \(N=3\) specialization of the companion bound under the reindexing of Remark [rem:recover].

11.0.0.2 (A2) The envelope.

\(B(m,n)\le2^{n-2}\binom{2m-1}{n-1}\) verified on the grid \(3\le n<40\), \(n\le m<3n\): holds at every grid point.

11.0.0.3 (A3) Asymptotic thresholds.

With \(m^*(n):=\min\{m:B(m,n)\ge n(n-1)^{n-2}\}\) and \(m^*_{\mathrm{sym}}(n)\) defined analogously with \(2^{n-2}\binom{m}{n-1}\), exact integer binary search gives: \[\begin{array}{c|ccccc} n & 50 & 100 & 200 & 500 & 1000\\\hline m^*(n)/n^2 & 0.10520 & 0.09910 & 0.09580 & 0.09366 & 0.09288\\ m^*_{\mathrm{sym}}(n)/n^2 & 0.20000 & 0.19250 & 0.18855 & 0.18594 & 0.18501 \end{array}\] decreasing toward the targets \(1/(4e)=0.09197\ldots\) and \(1/(2e)=0.18394\ldots\), consistently with Remark [rem:nolog] (approach from above, excess of order \(\log n/n\)).

11.0.0.4 (A4) The \(n=3\) cone-shift count.

For the smooth plane Fermat cubic \(q=x^3+y^3+z^3\) and the fixed generic covector \(p=(3,5,7)\) (chosen with \(|p_0/p_1|\ne1\) so that all tangency points are affine in the chart \(z=1\)), the resultant \(\mathrm{Res}_y\big(q|_{z=1},\,p\cdot\nabla q|_{z=1}\big)\) has degree \(6\) and is squarefree: the class of the curve is \(6=3\cdot2 =n(n-1)^{n-2}\) at \(n=3\), which by Proposition [prop:coneshift](c) is \(\delta_1\) of the conormal of the Fermat-cone surface in \(\mathbb{P}^3\).

11.0.0.5 (A5) Conservation bookkeeping on a classical degeneration.

For the nodal cubic \(y^2z=x^2(x+z)\) with the same covector, the analogous resultant has degree \(6\); the root corresponding to the node appears with multiplicity exactly \(2\), and the residual quartic is squarefree with \(4\) simple roots. Thus the class drops from \(6\) (smooth cubics) to \(4\) at the nodal member, with the deficit \(2\) absorbed as effective excess supported over the node: \(6=4+2\). This is the classical, hand-checkable instance of the inequality direction in Lemma 15 and Corollary 2: the special reduced conormal’s multidegree is at most the conserved total, never more.

12 The human–AI methodology↩︎

We describe the process honestly, both because it produced the result and because it is itself of methodological interest. The protocol is that of the companions [6], [7]: large language models in three separated roles — generator, adversarial critic, verifier — drawn from different model families so that a proposal produced by one is attacked by another, with convergence under the adversarial protocol treated only as a signal to attempt a human-checkable proof. The author — an AI researcher with a PhD in computer science, not a specialist in algebraic complexity or algebraic geometry — acted as orchestrator and domain arbiter; the role separation and verification-in-the-loop are designed for exactly that asymmetry.

12.0.0.1 Trajectory.

The campaign opened from the companions’ flagged open question (border transfer) under a kill-log protocol: every candidate transfer mechanism had to survive an explicit list of attack vectors with recorded verdicts (Appendix 13, C.1). A literature pass established the precedent landscape — the dual-dimension border bound of [2] and the closed-condition unification of [9] — and confirmed that the classical Lagrangian-specialization machinery [10]–[12] addresses families of reduced varieties, not the non-reduced determinantal limits arising here. The critic’s first decisive contribution was the direction-of-semicontinuity split of Section 1.5: it exhibited the smoothing family \(F_n+t\,q\) to kill any transfer of vertex-local invariants, while conceding that global conormal multidegrees specialize favorably; the cone-shift identity (Proposition [prop:coneshift]) was then identified as the bridge that homogeneity makes available. A dedicated round forced the determinantal lemma into its unconditional form: the generator’s first attempt hypothesized corank one along components, and the critic refuted the hypothesis with Example 1, after which the multiplicity dichotomy (Lemma 2) and the generic-\(\Lambda\) reduction (with the warning of Remark [rem:norow]) were settled; the assembled border proof was then produced against a mandated nine-point structure (Appendix 13, C.3). A referee-simulation round caught a genuine false identity in the assembled draft — the determinant of the homogenized matrix had been identified with the degree-\(d\) rather than the degree-\(m\) homogenization (Remark [rem:degreem]; Appendix ¿sec:app:referee?) — along with an unsound Laurent-truncation step in the normal form and imprecise flatness wording; the repairs are the degree-\(m\) homogenization, the fiberwise normal form of Lemma 7, and the dominating-components construction of Lemma 13. We record explicitly that the false identity was found by the critic, not by the generator that produced it: this is the class of error the protocol exists to catch, and the reason convergence of a single model’s output is never treated as evidence. A final structural-review round tightened the paper to its present form: the nonsymmetric local normal form was made fully explicit (Step 5 of Section 3.1), the symmetric proof was written out in full rather than by analogy, fibrewise reducedness was reproved directly from smoothness of the open graph over the parameter curve (Lemma 12), removing a generic-reducedness citation and its fallback, and the provenance material was consolidated into this appendix.

12.0.0.2 Reproducibility.

The load-bearing prompts are reproduced, ASCII-normalized and trimmed for length, in Appendix 13; the symbolic and exact-integer consistency checks, with the script’s actual outputs, are in Appendix 11. Full prompt logs and scripts are available from the author on request. As with the companions, prompt phrasing was a minor factor; the effective levers were target selection, role separation across model families, and verification in the loop.

13 The load-bearing prompts↩︎

The prompts below are reproduced ASCII-normalized and trimmed for length (elisions marked [...]); complete logs are available from the author on request.

C.1The kill-log campaign prompt (border transfer)↩︎

You are working on a real open problem in algebraic complexity.
GIVEN (treat as proved for this session): for any size-m affine
determinantal representation of a homogeneous f with V(f) smooth,
delta_top(V(f)) = d(d-1)^{N-2} <= B_N(m), yielding
dc(sum x_i^n) >= (1/4e - o(1)) n^2 exactly. Both writeups explicitly
DECLINE the border claim because polar degree is not a closed
condition.

TARGET: determine whether the bound transfers to BORDER determinantal
complexity dc-bar, where f = lim det A_t. Do not assume it does.
Identify the precise semicontinuity statement needed, its direction,
and whether it is true.

Maintain a KILL LOG. Attack vectors, each with verdict
SURVIVES / KILLED / UNRESOLVED and one-line justification:
K1 direction of semicontinuity of polar-type invariants in families;
K2 the limit divisor is non-reduced (padding x_0 powers) -- does any
   conormal object survive multiplicity;
K3 the generic fibre may be singular/reducible -- the smooth-X lemma
   does not apply to it;
K4 local invariants at the vertex jump UP under smoothing
   (exhibit a family); [...]
K8 base changes / finite shrinkings of the parameter line.
Round structure: generator proposes; critic attacks each K; verifier
reduces anything checkable to a computation. No verdict by fiat.

C.2The standalone determinantal lemma prompt↩︎

Let A(x) be an ARBITRARY m x m affine-linear matrix in n variables
over C, Ahat its homogenization, F = det Ahat, assumed nonzero. Define
Gamma_1(F) as the sum of the conormal varieties of the
multiplicity-ONE components of V(F), each with coefficient 1.

Prove, independently of any border or degeneration argument:
   delta_{n-2}(Gamma_1(F)) <= [x^n u^{m-1} v^{m-1}]
        x (x+u)^m (x+v)^{m-1} (u+v)^{n-2}.
State every hypothesis. Determine whether multiplicity one of a
component is equivalent to generic corank one of Ahat along it -- if
not, give a counterexample and identify the correct criterion. Do NOT
assume the corank >= 2 locus has codimension >= 2; exhibit a
representation where it is a divisor. For the corank-1 part prove:
(1) generic-flag count points exist on each multiplicity-one
    component, in the good locus;
(2) each count point lifts uniquely to the kernel incidence, with the
    conormal identity via the adjugate;
(3) the left/right kernel equations are 2m conditions cutting
    codimension 2m-1 -- locate and remove the redundancy SOUNDLY
    (deleting a row is not sound; show why);
(4) each lift is an ISOLATED, reduced point of the full square
    multihomogeneous system -- local normal form required;
(5) isolated solutions are bounded by the stated coefficient via a
    positivity statement with hypotheses (cite chapter and section).
Then compute the coefficient in closed form and sanity-check it at
(m,2), (1,n), (m,3). Do not discuss border complexity anywhere. [...]

C.3The border assembly prompt (nine-point structure)↩︎

We now have the standalone determinantal conormal lemma; use it as a
BLACK BOX. Prove the full border lower bound
dc-bar(sum x_i^n) >= (1/4e - o(1)) n^2 to journal-referee rigor.
The writeup must address, in order:
1 border normal form (curve selection; state exactly what family
  structure is and is not constructed);
2 homogenization and degree bookkeeping; multiplicity of the Fermat
  component in the special fibre;
3 the family of multiplicity-one Gauss graphs; flatness; which
  components are kept;
4 the containment Con(Fermat cone) <= limit cycle, with the limit
  covector computed and a persistence argument placing approximating
  points on multiplicity-ONE branches (this is the crux -- uniqueness
  of a limit is not membership);
5 special <= generic via effectivity;
6 the cone-shift identity, stated and proved for a general smooth
  base, then specialized;
7 application of the black box to the generic fibre -- verify the
  hypothesis verbatim;
8 the arithmetic, with explicit constants and no hidden log losses;
9 failure modes and scope: Gamma_1 vs reduced conormal, genericity
  assumptions (there must be none on the representations), repeated
  factors, base changes, characteristic, what is NOT claimed
  (permanent, inhomogeneous targets).
Each transversality / flatness / positivity / persistence input must
be stated as a lemma with hypotheses at the point of use. [...]

C.4Excerpt of the referee-simulation critique (the homogenization-degree error)↩︎

Main problem: Section 7 has a degree-homogenization error. You define
Ghat_t = x_0^d G_t(x/x_0) with d = deg_x G_t generically. But if
G_t = det A_t with A_t affine-linear m x m, the determinant of the
homogenized matrix is det Ahat_t = x_0^m det A_t(x/x_0)
= x_0^{m-d} Ghat_t. So the sentence "Ghat_{t0} = det Ahat_{t0}" is
FALSE unless d = m. The lemma applies to F_t := x_0^{m-d} Ghat_t, not
to Ghat_t. This is fixable but must be fixed explicitly: the
multiplicity-one components of F_t are those of Ghat_t, plus possibly
the hyperplane x_0 = 0 when m - d = 1; that hyperplane's conormal is
{x_0=0} x {[1:0:...:0]}, with delta_{n-2} = 0 for n >= 3. [...]
Second issue: the Laurent-truncation step in the normal form is not a
safe argument as written; truncating a representation destroys the
exact determinant identity. You only need the polynomial family to be
regular and the representations to exist FIBERWISE on a punctured
dense open set. [...] Third issue: the flatness claim needs the
dominating-components construction stated cleanly: finitely many
components of the incidence; delete the point-images; take W reduced;
every component dominates the smooth curve; torsion-free over a DVR
implies flat.

References↩︎

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1. LinkedIn: https://www.linkedin.com/in/karthik-sheshadri-0624ab150/.↩︎