==== TABLE OF CONTENTS ====
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//adjugate.tex
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//conductor.tex
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//consequences.tex
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//dimension.tex
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//introduction.tex
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//main.tex
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//models.tex
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//polar.tex
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//proof-flow.tex
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//selector-wiring.tex
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//selector.tex
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//transfer.tex
A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//references.bib
===========================

==== BEGIN A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//adjugate.tex ====
\section{The adjugate scheme in characteristic two}
\label{sec:adjugate}

We now study the coefficient formula \eqref{eq:selector-coefficient} over
\[
 K=\overline{\mathbb F}_2.
\]
In this characteristic the permanent equals the determinant.  This lets
us associate a plane curve and a finite adjugate scheme to every matrix
\(X\).  Vanishing of the gradient will force this finite scheme to have
large length.

Fix an integer \(k\ge2\), pairwise distinct elements
\(\theta_1,\ldots,\theta_k\in K\), and
\(D=\operatorname{diag}(\theta_1,\ldots,\theta_k)\).  Set
\begin{equation}
 \label{eq:adjugate-parameters}
 q=\left\lfloor\frac{k-1}{3}\right\rfloor,\qquad
 r=k-2q,\qquad c=r-1,\qquad T=\binom{q+2}{2}.
\end{equation}
Thus
\begin{equation}
 \label{eq:adjugate-degrees}
 2q+c=k-1,\qquad 0\le q\le c.
\end{equation}
For a polynomial, brackets denote extraction of the indicated monomial
coefficient.  We use the homogeneous degree-\(r\) polynomial
\begin{equation}
 \label{eq:adjugate-coefficient-polynomial}
 G_D(X)=[s^qt^q]\per_k(X+sI_k+tD)
       =[s^qt^q]\det(X+sI_k+tD).
\end{equation}
Statements about its gradient below refer to the common zero locus of
all its \(k^2\) entrywise partial derivatives; no equation \(G_D=0\) is
included in that definition.

For each fixed \(X\in\Mat_k(K)\), put
\begin{equation}
 \label{eq:adjugate-curve}
 A=tD+uX,\qquad M=sI_k+A,\qquad
 F_X(s,t,u)=\det M,\qquad C_X=V(F_X)\subset\PP^2_K.
\end{equation}
Here \(F_X\) has degree \(k\), with coefficient \(1\) at \(s^k\).
We abbreviate \(C_X\) to \(C\) when \(X\) is fixed.

\begin{lemma}[Projection of the auxiliary curve]
 \label{lem:adjugate-projection}
For every \(X\in\Mat_k(K)\), the curve \(C_X\) is reduced and the map
\[
 \pi:C_X\longrightarrow\PP^1_K,\qquad [s:t:u]\longmapsto[t:u]
\]
is finite flat of degree \(k\) and is \'{e}tale over a nonempty open
subset of the base.  Every irreducible component of \(C_X\) meets the
line \(u=0\), and at each point of that intersection \(M\) has rank
\(k-1\).  Moreover, \(\pi^*\cO_{\PP^1}(1)=\cO_C(1)\), and
\begin{equation}
 \label{eq:adjugate-splitting}
 \pi_*\cO_C=\bigoplus_{j=0}^{k-1}\cO_{\PP^1}(-j),
 \qquad
 \mathcal E:=\pi_*\cO_C(k-1)
       =\bigoplus_{j=0}^{k-1}\cO_{\PP^1}(k-1-j).
\end{equation}
\end{lemma}

\begin{proof}
The restriction
\begin{equation}
 \label{eq:adjugate-fixed-fiber}
 F_X(s,t,0)=\prod_{i=1}^k(s+t\theta_i)
\end{equation}
is squarefree.  If a positive-degree irreducible homogeneous factor
\(H\) occurred at least twice in \(F_X\), its restriction to \(u=0\)
could not vanish identically: otherwise \(u\) would divide \(F_X\),
contrary to \eqref{eq:adjugate-fixed-fiber}.  That restriction is a
nonzero homogeneous polynomial of the same positive degree as \(H\).
Its square would divide \eqref{eq:adjugate-fixed-fiber}, a contradiction.
Thus \(C\) is reduced.  The restriction of each irreducible factor is
likewise a positive-degree binary form, so it has a projective zero
over \(K\).  Hence every component meets \(u=0\).

The points on that line are
\[
 P_i=[-\theta_i:1:0]\qquad(1\le i\le k).
\]
At \(P_i\), the matrix \(M\) is diagonal with exactly one zero entry.
It has rank \(k-1\), and its \(i\)-th diagonal adjugate entry is
\(\prod_{j\ne i}(\theta_j-\theta_i)\ne0\).

The only possible base point of the projection is \([1:0:0]\), which
does not belong to \(C\) because \(F_X(1,0,0)=1\).
On the base chart \(t\ne0\), use \(v=u/t\) and \(y=s/t\).  The inverse
image has coordinate algebra
\[
 K[v,y]\big/\det(yI_k+D+vX).
\]
Its defining polynomial is monic of degree \(k\) in \(y\); consequently
this algebra is free over \(K[v]\), with basis
\(1,y,\ldots,y^{k-1}\).  The chart \(u\ne0\) gives the same conclusion,
so \(\pi\) is finite flat of degree \(k\).
At \(v=0\) the monic polynomial has the distinct roots
\(-\theta_1,\ldots,-\theta_k\).
Its resultant with its \(y\)-derivative is nonzero there, and hence
nonzero on an open subset of the base.  Over that open subset the
finite flat map is \'{e}tale.  In particular its generic algebra is
a product of finite separable field extensions, also when \(k\) is even.

The sections \(t,u\) have no common zero on \(C\), so they identify
\(\pi^*\cO_{\PP^1}(1)\) with \(\cO_C(1)\).
The two monic-polynomial bases satisfy
\[
 (s/u)^j=(t/u)^j(s/t)^j\qquad(0\le j0$ and $m_0$ such that, for every integer
$m\ge m_0$,
\[
 \bdc(\per_m)\ge c m^3.
\]
One may take $c=1/(5\,529\,600e)$ and $m_0=1408$.
\end{theorem}

The constants are not optimized. Proposition~\ref{prop:model-equivalence}
identifies this coefficientwise model with the projective determinant-orbit
closure formulation, including equal permanent and determinant orders.

The same numerical lower bound holds for exact determinantal complexity.
There is also a consequence for algebraic branching programs. An
affine-linear algebraic branching program is a finite directed acyclic
graph with a source and a sink and an affine-linear form on each edge; it
computes the sum of products of edge labels along all source-to-sink paths.
Over $\C$, computing $\per_m$ requires $\Omega(m^3)$ vertices and
$\Omega(m^3)$ edges. The same bounds hold for coefficientwise limits when
the relevant vertex or edge budget remains fixed along the approximation,
even if the graph varies. Section~\ref{sec:consequences} gives the precise
statements and determinant reductions.

\subsection{Earlier bounds and geometric methods}

The early matrix-order lower bounds were linear in the permanent order $m$.
For the variable-or-constant projection model, von zur Gathen published a
bound due to Babai and Seress with leading term $\sqrt{2}\,m$ and a loss of
$6\sqrt m$ \cite[Theorem~4.4]{vonZurGathen1987}. For arbitrary affine
representations over $\C$, Meshulam's final theorem
\cite[pp.~270--271]{Meshulam1989} and Cai's dimension argument
\cite[Theorem~1.5 and Section~2.2]{Cai1990} give the scale
$\sqrt{2}\,m-O(1)$. Their arguments compare the dimension of the affine
matrix image with the minimum rank that an exact permanent representation
must have.

Mignon and Ressayre raised the exact lower bound to
$\dc(\per_m)\ge m^2/2$ in characteristic zero
\cite{MignonRessayre2004}. Their proof compares Hessians at a zero:
affine pullback cannot increase Hessian rank, the Hessian of an order-$n$
determinant has rank at most $2n$ at a noninvertible matrix, and, for
$m\ge2$, the permanent has a zero where its Hessian has full rank $m^2$.
Cai, Chen and Li extended the quadratic scale to positive characteristic
different from two by constructing sparse zeros that avoid the large
factorial factors in the earlier Hessian calculation
\cite[Theorem~2.3]{CaiChenLi2010}.

On the upper side, Grenet's construction realizes $\per_m$ as a determinant
of order $2^m-1$, using a branching program indexed by the subsets of
$\{1,\ldots,m\}$ \cite[unnumbered Theorem, p.~3]{Grenet2012}.
Its determinant entries are individual variables and constants, so it is
also an upper bound in the unrestricted exact model.

Mulmuley and Sohoni's geometric complexity theory program studies
permanent--determinant separation through orbit closures of determinants
and padded permanents, seeking equations or representation-theoretic
obstructions to containment
\cite{MulmuleySohoni2001,MulmuleySohoni2008}.
Landsberg, Manivel and Ressayre proved the unrestricted complex border
bound $\bdc(\per_m)\ge m^2/2$ by studying hypersurfaces with degenerate
duals and the effect of padding
\cite[Theorem~1.1.1]{LandsbergManivelRessayre2010}.
Theorem~\ref{thm:main} raises the lower-bound exponent from two to three
in that same model. It remains a fixed polynomial bound. Whether the
permanent has superpolynomial determinantal complexity, or superpolynomial
border determinantal complexity, remains open.

Landsberg and Ressayre later proved exponential optima for exact
representations with symmetry requirements. Each specified projective
symmetry of the input must lift to a projective symmetry of the determinant
that fixes the pencil's constant matrix, preserves the image of its linear
part, and induces the given input action there. Over $\C$, for $m\ge3$,
requiring this for left multiplication by invertible diagonal and
permutation matrices gives minimum order $2^m-1$. Lifting the full
projective symmetry group of the permanent gives minimum order
$\binom{2m}{m}-1$ in the same range
\cite[Theorems~2.1 and~2.8; Propositions~2.9 and~2.10]{LandsbergRessayre2017}.
These hypotheses specify exact representation classes; the border model
of Theorem~\ref{thm:main} allows arbitrary affine determinant
approximations of fixed order.

Singularities also give geometric lower bounds for exact representations.
For a homogeneous polynomial of degree greater than two whose affine
singular locus has codimension greater than four, Alper, Bogart and Velasco
prove
$\dc(f)\ge\operatorname{codim}\operatorname{Sing}(f)+1$
\cite[Theorem~1.2]{AlperBogartVelasco2017}.
For the permanent, matrices with two zero columns show that this
codimension is at most $2m$, so this obstruction alone has only a linear
scale \cite[Remark~1.5]{AlperBogartVelasco2017}.
Our construction instead obtains a coefficient polynomial from a permanent
and finds a linear subspace on which its zero hypersurface is smooth. Its
degree is linear and the subspace dimension is quadratic in an auxiliary
matrix size.

For a smooth complex form of degree $r\ge2$ in $d\ge2$ variables, the classical
top polar degree is $r(r-1)^{d-2}$
\cite[Corollary~3.7]{Piene1978}.
The left and right kernel equations for the determinant matrix and the
Schur-complement reduction used below also appear in Sheshadri's June 2026
preprint \cite[Theorem~3 and Section~3.1]{SheshadriBorder2026}, as part of
its proposed conormal approach to border bounds for diagonal power sums.
Kumar and Volk's September 2026 preprint uses such kernel equations and a
multihomogeneous count to obtain a quadratic \emph{exact} lower bound for
$\sum_{i=1}^n x_i^n$
\cite[Theorem~1.1 and Section~3]{KumarVolk2026}.
Theorem~\ref{thm:smooth-border} proves the version needed here, including
persistence of finitely many simple polar intersections in one sufficiently
close determinant under an arbitrary coefficientwise approximation. Its
lower bound is proportional to the product $(r-1)(d-1)$.

\subsection{The coefficient construction and the proof}

To use this product at cubic scale, we seek an affine restriction of a
permanent of size proportional to $k$ that has degree proportional to $k$
in quadratically many variables and has a smooth projective zero set.
Let $k\ge1$ and $0\le q\le\lfloor k/2\rfloor$ be integers. With
indeterminates $\theta_1,\ldots,\theta_k$, put
$D=\diag(\theta_1,\ldots,\theta_k)$ and define
\[
 G_D(X)=[s^qt^q]\per_k(X+sI+tD)
 \in\mathbb Z[\theta_1,\ldots,\theta_k][X_{ij}].
\]
This polynomial is homogeneous of degree $r=k-2q$ in the $k^2$ entries
of $X$. We use the same notation after specializing the parameters in a
field. For the main construction we assume $k\ge128$ and take
$q=\lfloor(k-1)/3\rfloor$; then $r$ is linear in $k$ and
$T=\binom{q+2}{2}$ is quadratic in $k$.

The integral formula has two uses. Over $\C$, two count gates and a
cancellation gadget realize $G_D$, up to a fixed nonzero scalar, as an
affine restriction of a permanent of size $11k-2q$
(Proposition~\ref{prop:selector}). This holds for every complex choice of
the parameters. The scalar vanishes in characteristic two, so the geometric
argument specializes the integral coefficient formula itself. We will use
the complex projection at the complex parameter choice that geometry
eventually supplies.

Let $K=\overline{\mathbb F}_2$, choose pairwise distinct
$\bar\theta_1,\ldots,\bar\theta_k\in K$, and write
$\bar D=\diag(\bar\theta_1,\ldots,\bar\theta_k)$. Since permanent
and determinant agree over $K$, every matrix $X$ defines a plane curve
\[
 C_X=V\bigl(\det(sI+t\bar D+uX)\bigr)\subset\PP^2_K.
\]
Its fiber on $u=0$ consists of the distinct points
$[-\bar\theta_i:1:0]$. This fixed fiber makes the curve reduced and its
projection to $[t:u]$ finite and generically separable, and it controls the adjugate
of $sI+t\bar D+uX$ on the curve. The full adjugate defines a finite
subscheme $Z\subset C_X$. If all entrywise derivatives of $G_{\bar D}$
vanish at $X$, the coefficient $[s^qt^qu^{r-1}]$ of every adjugate entry
vanishes. We prove that these
entries span all degree-$(k-1)$ forms vanishing on $Z$, so the same
coefficient vanishes on every such form. For a degree-$q$ monomial
$s^it^ju^\ell$, multiplication by $s^{q-i}t^{q-j}u^{r-1-\ell}$ lets this
functional recover its coefficient in any degree-$q$ form. These exponents
are nonnegative because $r-1\ge q$. Thus restriction of degree-$q$ forms to $Z$ is
injective, giving the length bound in Proposition~\ref{prop:adjugate-length}:
\[
 K[s,t,u]_q\hookrightarrow H^0(Z,\cO_Z(q)),
 \qquad \length Z\ge\dim_K K[s,t,u]_q=T.
\]

The length of $Z$ is bounded above by the \emph{normalization defect}
$\delta(C_X)$, the length of the quotient between the normalization's
structure sheaf and the curve's structure sheaf. The comparison follows
the evaluation-ideal argument of Fantechi, G\"ottsche and van Straten
over $\C$ \cite[Proposition~C.2 and Corollary~C.3]{FGvS1999};
Section~\ref{sec:conductor} proves the needed conductor inclusion and
colength identity for our generically separable pencils in arbitrary
characteristic.
Thus every critical matrix lies over a curve with $\delta(C_X)\ge T$.

Applying the arbitrary-characteristic Severi dimension theorem of Christ,
He and Tyomkin \cite[Lemma~2.6, Proposition~2.7, and Remark~2.8]{CHT2023}
to the integral components bounds the locus of curves with this defect.
Their result builds on Tyomkin's dimension bound for toric pairs
\cite[Theorem~1.2(1)]{Tyomkin2013}. The sum
$R_X=(1,\ldots,1)\adj(sI+t\bar D+uX)$ of the adjugate rows controls
the matrices over one fixed curve: its zero subscheme has length
$k(k-1)/2$, and the common
boundary frame makes that subscheme determine the matrix uniquely.
A Hilbert-scheme estimate and an algebraic incidence then bound the
dimension of each matrix fiber. Combining the curve and matrix bounds gives
\[
 \dim\{X:\nabla G_{\bar D}(X)=0\}\le k^2+k-T,
 \qquad
 \codim\{X:\nabla G_{\bar D}(X)=0\}\ge T-k.
\]

The projective zero scheme of the raw partial derivatives forms a proper
family over the integral parameter space. Specialization transfers the
dimension bound to a nonempty open set of complex parameters. Over $\C$,
Euler's identity identifies these gradient zeros with the hypersurface's
singular scheme, and a general linear section avoids that scheme and meets
the smooth locus transversely. Proposition~\ref{prop:smooth-projection}
then supplies, for $k\ge128$ and the stated choice of $q$, a nonzero form
$G$ of degree $r$ in $d=\lfloor k^2/100\rfloor$ variables whose
projective zero set is smooth. For each $m\ge11k-2q$, it also supplies
a fixed affine map $\Phi_m$ satisfying $G=\per_m\circ\Phi_m$.

Figure~\ref{fig:proof-flow} shows how the two uses of the coefficient
formula meet and how the polar bound converts their output into the cubic
estimate.

\input{figures/proof-flow}

Sections~\ref{sec:selector}--\ref{sec:transfer} establish this
construction. Section~\ref{sec:polar} then proves the smooth-form bound
stated in Section~\ref{sec:models}, using selected simple polar
intersections and the left and right kernels of one nearby determinant.
Section~\ref{sec:consequences} combines the two results and gives the
exact, branching-program, and smooth-initial-form consequences.

\FloatBarrier
==== END A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//introduction.tex ====

==== BEGIN A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//main.tex ====
\documentclass[11pt]{article}
\def\DoNotLoadEpstopdf{}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{microtype}
\usepackage{booktabs,array}
\usepackage{placeins}
\usepackage{etoolbox}
\usepackage{xcolor}
\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
\usepackage[figure]{hypcap}
\usepackage[nameinlink,capitalize,noabbrev]{cleveref}
\makeatletter
\newcommand\chResultRefstepcounter[2][]{%
  \let\chSavedCrefRefstepcounter\cref@old@refstepcounter
  \let\cref@old@refstepcounter\Hy@theorem@refstepcounter
  \refstepcounter[#1]{#2}%
  \let\cref@old@refstepcounter\chSavedCrefRefstepcounter
}
\AddToHook{begindocument/end}{%
  \patchcmd{\cref@thmnoarg}
    {\refstepcounter}{\Hy@theorem@refstepcounter}
    {}{\PackageError{ch418}{Could not patch the plain theorem counter path}{%
      Recheck the amsthm, hyperref, and cleveref versions.}}%
  \patchcmd{\cref@thmoptarg}
    {\refstepcounter}{\chResultRefstepcounter}
    {}{\PackageError{ch418}{Could not patch the shared theorem counter path}{%
      Recheck the amsthm, hyperref, and cleveref versions.}}%
}
\makeatother
\AddToHook{cmd/thebibliography/after}{%
  \addcontentsline{toc}{section}{\refname}%
}
\pdfinfoomitdate=1
\pdftrailerid{}
\pdfsuppressptexinfo=15
\hypersetup{pdftitle={A cubic lower bound for border determinantal complexity of the permanent},pdfauthor={OpenAI}}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\C}{\mathbb C}
\newcommand{\PP}{\mathbb P}
\newcommand{\cO}{\mathcal O}
\DeclareMathOperator{\Mat}{Mat}
\DeclareMathOperator{\per}{per}
\DeclareMathOperator{\dc}{dc}
\DeclareMathOperator{\bdc}{\overline{dc}}
\DeclareMathOperator{\adj}{adj}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\length}{length}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Hilb}{Hilb}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Gr}{Gr}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\Sing}{Sing}
\DeclareMathOperator{\Supp}{Supp}
\DeclareMathOperator{\codim}{codim}
\newcommand{\dd}{\mathrm d}
\title{A cubic lower bound for border determinantal complexity of the permanent}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove that the complex border determinantal complexity of the $m\times m$
permanent is $\Omega(m^3)$, allowing arbitrary affine-linear determinant
representations and coefficientwise limits. It also gives cubic lower bounds
for exact determinantal complexity and for the numbers of vertices and edges
in affine-linear algebraic branching programs, including coefficientwise
limits with a fixed vertex or edge budget.
\end{abstract}
\tableofcontents
\input{sections/introduction}
\input{sections/models}
\input{sections/selector}
\input{sections/adjugate}
\input{sections/conductor}
\input{sections/dimension}
\input{sections/transfer}
\input{sections/polar}
\input{sections/consequences}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//main.tex ====

==== BEGIN A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//models.tex ====
\section{Border models and smooth homogeneous forms}
\label{sec:models}

We first identify the two border models precisely. All closures in this
section are over $\C$. In a finite-dimensional polynomial space, coefficientwise
convergence means ordinary convergence of the coefficients of every monomial.
The image of a polynomial map is constructible
\cite[Theorem~29.23.3, Tag~054K]{StacksProject}, and a complex constructible
set has the same Euclidean and Zariski closures. For the latter fact, decompose
the set into finitely many locally closed subsets: each contains a Zariski
open dense subset of its closure, which is also Euclidean dense. These facts
apply both to the polynomial map taking an affine matrix to its determinant
and to the action map defining a linear-group orbit. Thus coefficientwise
sequences can be used for either closure below.

\begin{lemma}[Affine substitution]
\label{lem:affine-substitution}
Let $f$ be a complex polynomial, let $A$ be an affine map into its variable
space, and let $a\ne0$ be a complex scalar. Then
\[
 \bdc(a f\circ A)\le \bdc(f).
\]
\end{lemma}
\begin{proof}
Fix a size $n$ and determinants $Q_j=\det M_j$ converging coefficientwise to
$f$, with all entries of $M_j$ affine-linear. The entries of $M_j\circ A$
are still affine-linear. Substitution by the fixed map $A$ is a continuous
linear map on spaces of polynomials of degree at most $n$, so
$Q_j\circ A\to f\circ A$. Multiplying one row of each substituted matrix
by $a$ gives the assertion. This argument imposes no bound on the matrix
entries in the approximating sequence.
\end{proof}

For $n\ge m$, let $W_n=\Mat_n(\C)$ and regard $\det_n$ as a degree-$n$
form on $W_n$. Choose $m^2$ independent linear coordinates $x_{ij}$ on $W_n$.
If $n>m$, choose one more independent coordinate $z$ and set
\[
 P_{n,m}=z^{n-m}\per_m(x).
\]
If $n=m$, set $P_{m,m}=\per_m(x)$; no padding coordinate is needed.
Let $\mathcal D_n$ be the projective Zariski closure of the orbit of
$[\det_n]$ under invertible linear substitutions on $W_n$.

\begin{proposition}[Equivalence of the two border models]
\label{prop:model-equivalence}
For every $n\ge m\ge1$,
\[
 \bdc(\per_m)\le n
 \quad\Longleftrightarrow\quad
 [P_{n,m}]\in\mathcal D_n.
\]
\end{proposition}

\begin{proof}
Every linear map $L:W_n\to W_n$ is a limit of invertible ones. Consequently
$\det_n\circ L$ belongs to the affine closure of the determinant orbit.
This orbit is invariant under nonzero scalar multiplication: multiplying
one row of a matrix variable by $a\ne0$ multiplies its determinant by $a$
and is an invertible linear substitution. Its affine closure is therefore
the cone over $\mathcal D_n$. In particular a nonzero form belongs to this
affine closure exactly when its projective class belongs to $\mathcal D_n$.

Suppose first that $\bdc(\per_m)\le n$. Identity blocks let us take every
approximating matrix to have size exactly $n$. Write
\[
 Q_j(x)=\det\left(A_{j,0}+\sum_{a,b=1}^m x_{ab}A_{j,ab}\right)
 \longrightarrow\per_m(x).
\]
For $n>m$, homogenize to degree $n$ using the independent coordinate $z$:
\[
 Q_j^{\mathrm{hom}}(x,z)=\det\left(zA_{j,0}+\sum_{a,b=1}^m x_{ab}A_{j,ab}\right)
 \longrightarrow z^{n-m}\per_m(x).
\]
Homogenization here is a fixed linear map of coefficient spaces. Each
displayed determinant is a linear substitution of $\det_n$ on $W_n$,
so its limit belongs to the same closed cone. The coordinates fit because
$m^2+1\le n^2$ when $n>m$.

For $n=m$, instead take the degree-$m$ homogeneous component of each $Q_j$.
It is precisely
\[
 \det\left(\sum_{a,b=1}^m x_{ab}A_{j,ab}\right),
\]
and these components converge to $\per_m$. They again are linear
substitutions of $\det_m$. This proves the forward implication without
introducing an extra variable in the case $n=m$.

Conversely, the cone observation and equality of the two closures give
determinant-orbit representatives converging coefficientwise to $P_{n,m}$.
When $n>m$, set $z=1$ and all unused coordinates to zero; when $n=m$, just
use the $m^2$ coordinates $x_{ab}$. Each resulting polynomial is an affine
determinant of size $n$, and the limit is $\per_m$. This proves the converse.
Finally, a determinant of size less than $m$ has degree less than $m$, as
does every polynomial in its coefficientwise closure. Thus
$\bdc(\per_m)\ge m$, and the stated range $n\ge m$ covers every size
that can satisfy the affine condition.
\end{proof}

The quantitative obstruction used in the proof depends on the degree and
the number of variables of a smooth homogeneous polynomial. Here smoothness
means that the projective zero set has no point at which all first partial
derivatives vanish.

\begin{theorem}[Lower bound for smooth homogeneous forms]
\label{thm:smooth-border}
Let $G\in\C[x_1,\ldots,x_d]$ be a nonzero homogeneous polynomial of degree
$r\ge2$, where $d\ge2$. If its zero set in $\PP^{d-1}$ is smooth, then
\[
 \bdc(G)\ge \frac{(r-1)(d-1)}{4e}.
\]
\end{theorem}

The complete proof is in Section~\ref{sec:polar}. For a constant vector
$w\in\C^d$, write
\[
 \partial_wG=\sum_{j=1}^d w_j\frac{\partial G}{\partial x_j}
\]
for the first directional derivative, or \emph{first polar}, of $G$.
If $G$ is a coefficientwise limit of affine determinants of size $n$, the proof
chooses $d-2$ first polars whose common intersection with the hypersurface
consists of $r(r-1)^{d-2}$ simple points. It then shows that these selected
points persist for one sufficiently close determinant and satisfy
\begin{equation}
 r(r-1)^{d-2}
 \le 2^{d-2}\binom{2n-1}{d-1}
 \le \left(\frac{4en}{d-1}\right)^{d-1}.
 \label{eq:smooth-count-synopsis}
\end{equation}
The binomial coefficient is understood to be zero when $d-1>2n-1$;
in that case the positive left side already rules out the approximation.
The first inequality comes from lifting the selected points to the right
and left kernel lines of the determinant matrix and counting the resulting
bilinear equations. Taking the $(d-1)$st root gives the theorem. The proof
establishes persistence and simplicity of those selected lifts using only
coefficientwise closeness of the determinant polynomials, so it allows
unbounded coefficients in the approximating matrices.

This comparison specifies our construction's target. A nonzero homogeneous
form of degree proportional to $k$ in quadratically many variables, with
a smooth projective zero set, forces cubic determinant size. We now produce
such a form as a fixed affine restriction of a permanent whose size is
linear in $k$.
==== END A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//models.tex ====

==== BEGIN A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//polar.tex ====
\section{Counting polars of a smooth hypersurface}\label{sec:polar}

We now prove Theorem~\ref{thm:smooth-border}. A smooth degree-$r$
hypersurface in $\PP^{d-1}$ supplies $r(r-1)^{d-2}$ simple intersections
with first polars. These points persist for one sufficiently close affine
determinant. Right and left kernel variables then express each selected
intersection by bilinear equations, whose multidegrees give the numerical
comparison stated in Section~\ref{sec:models}. The determinant kernel-incidence
count and its Schur-complement reduction
build on \cite[Theorem~3 and Section~3.1]{SheshadriBorder2026}. We give the
argument needed here for finitely many simple polar intersections,
including unrestricted affine limits and possible excess components of
the incidence.

For a constant vector $w\in\C^N$, write
\[
 \partial_w F=\sum_{j=1}^N w_j\frac{\partial F}{\partial x_j}.
\]
We call this an ambient first polar: after restricting to an affine slice,
the direction $w$ need not be tangent to the slice. A common zero of $s$
equations in $s$ local coordinates is \emph{simple} if its full Jacobian
is invertible.

\subsection{Simple polar intersections}

The numerical count below is the classical top polar degree of a smooth
hypersurface \cite[Corollary~3.7]{Piene1978}. We give the transversality
argument to retain the individual simple points needed in the limit.
General-transversality principles of this kind are developed in
\cite[Corollary~5]{Kleiman1974}; the argument below works directly over
$\C$ with the incidence of polar directions.

\begin{lemma}\label{polar:simple-points}
Let $r,d\ge2$, and let $G\in\C[x_1,\ldots,x_d]$ be homogeneous of degree
$r$. Suppose that $Y=V(G)\subset\PP^{d-1}$ is a smooth hypersurface.
There are constant directions $w_1,\ldots,w_{d-2}\in\C^d$ for which
\begin{equation}\label{polar:projective-certificate}
 G=0,\qquad \partial_{w_i}G=0\quad(1\le i\le d-2)
\end{equation}
has exactly $\mu=r(r-1)^{d-2}$ distinct simple projective solutions.
There is a linear form $L_0$ nonzero at every one of these points.
\end{lemma}

\begin{proof}
The sections $\partial_wG$ of $\cO_Y(r-1)$ have no common zero on $Y$:
otherwise all partial derivatives of $G$ would vanish at a point of the
smooth hypersurface. We spell out why general choices give simple points.
If $d=2$, the smooth degree-$r$ hypersurface in $\PP^1$ already consists
of $r$ distinct points, and there are no polar equations. Assume $d\ge3$.

Put $\mathcal P=(\C^d)^{d-2}$ and consider
\[
 \mathcal I=\{(x,w_1,\ldots,w_{d-2})\in Y\times\mathcal P:
                  \partial_{w_i}G(x)=0\text{ for every }i\}.
\]
For each $x$, the equation in each separate $w_i$ block is a nonzero
linear equation. Thus $\mathcal I$ is the total space of a vector bundle
of rank $(d-1)(d-2)$ over $Y$. In particular it is smooth, and
$\dim\mathcal I=d(d-2)=\dim\mathcal P$. The incidence is closed in
$Y\times\mathcal P$, so its projection to $\mathcal P$ is proper because
$Y$ is projective. It is surjective: $G$ and any
$d-2$ chosen polars are $d-1$ positive-degree homogeneous equations in
$d$ variables, so the projective dimension theorem gives a common
projective zero.

Characteristic-zero generic smoothness makes the generic fiber of each
dominating component smooth. Remove the proper images of the
non-dominating components and of the locus where the projection is not
smooth. Properness makes these images closed, and generic smoothness
ensures that they omit the generic point of $\mathcal P$. The resulting
nonempty open subset has smooth zero-dimensional fibers. Each such fiber
is a finite set of reduced points. At each point the differentials of the polar
sections on $Y$ are independent. Since $dG\ne0$ in local projective
coordinates, the full system~\eqref{polar:projective-certificate} is
simple there.

This is a proper intersection of hypersurfaces of degrees
$r,r-1,\ldots,r-1$ in $\PP^{d-1}$. Projective B\'ezout therefore gives
exactly $r(r-1)^{d-2}$ points, each of multiplicity one.
Finally, the linear forms vanishing at at least one of these finitely
many points form a finite union of proper hyperplanes in $(\C^d)^*$.
Choose $L_0$ outside that union.
\end{proof}

\subsection{A count that permits excess components}

The kernel equations below need not meet properly everywhere, so we count
only their selected simple zeros. The coefficient in the next lemma is
the multihomogeneous B\'ezout number. Its standard specialization bound
controls nonsingular isolated zeros even when other components are present
\cite[Sections~3--3.2]{Wampler1996}. We give a direct proof of the bilinear
case needed here.

\begin{lemma}\label{polar:product-count}
Let $Y=\prod_{j=1}^s\PP^{a_j}$, where $a_j\ge0$ and
$M=\sum_j a_j\ge1$. For each $1\le\ell\le M$, let $F_\ell$ be a
multihomogeneous polynomial of degree one in two distinct coordinate
groups, indexed by $u_\ell,v_\ell$, and degree zero in the other groups.
Any finite collection of distinct simple common zeros of these equations
has size at most
\begin{equation}\label{polar:product-coefficient}
 [H_1^{a_1}\cdots H_s^{a_s}]
       \prod_{\ell=1}^M(H_{u_\ell}+H_{v_\ell}).
\end{equation}
Other zeros and components of any dimension are allowed.
\end{lemma}

\begin{proof}
First construct a system with the displayed number of zeros, all simple.
Give each occurrence of a coordinate group in an equation a different
label. On $\PP^{a_j}$ choose the labeled linear forms so that every
subfamily of at most $a_j+1$ coefficient vectors is independent.
In coordinates $[z_0:\cdots:z_{a_j}]$, forms
\[
 z_0+t z_1+\cdots+t^{a_j}z_{a_j}
\]
with distinct values of $t$ have this property by the Vandermonde
determinant. When $a_j=0$, use the nowhere-zero form $z_0$ on $\PP^0$.
Replace equation $\ell$ by $Q_\ell=L_\ell R_\ell$, using its two
labeled forms on factors $u_\ell$ and $v_\ell$.

At a common zero of all $Q_\ell$, let $b_j$ count the labeled forms
vanishing in factor $j$. General position implies $b_j\le a_j$.
Each of the $M$ equations requires at least one vanishing form, and no
label occurs in two equations. Hence
\[
 M\le\sum_j b_j\le\sum_j a_j=M.
\]
Exactly one labeled factor vanishes in each equation, and exactly $a_j$
forms vanish in factor $j$. Conversely, choosing one factor of each
equation, with exactly $a_j$ choices assigned to factor $j$, determines
one point of $Y$: the selected $a_j$ hyperplanes in that factor meet in
one point. No additional labeled form vanishes there, because $a_j+1$
such hyperplanes have empty intersection. Different choices therefore
give different points. Expanding the product in
\eqref{polar:product-coefficient} counts exactly these choices; denote
that coefficient by $B$.

Every comparison zero is simple. In equation $\ell$, one factor is zero
and the other is nonzero, so its differential is a nonzero multiple of
the selected linear form's differential. In each projective factor the
selected hyperplanes have independent differentials. The full Jacobian
is therefore invertible. This construction also covers $B=0$, when the
comparison system has no common zero.

It remains to compare an arbitrary system with this one. Interpolate by
\[
 F_{\ell,t}=(1-t)Q_\ell+tF_\ell\qquad(t\in\C).
\]
The common-zero incidence is closed in $Y\times\C$. On it, rank
deficiency of the differential in the $Y$ directions is a closed
algebraic condition. Indeed, it is given by maximal Jacobian minors in
each projective chart. Coordinate changes act invertibly on the
differential, and changing a trivialization multiplies an equation by a
unit. At a common zero the derivative of that unit contributes zero,
so the rank condition is independent of the chart.

Since $Y$ is projective, the parameter values having a rank-deficient
common zero form a Zariski-closed subset $\mathcal B\subset\C$.
The comparison system shows that $0\notin\mathcal B$, so $\mathcal B$
is finite. For $t\notin\mathcal B$, all zeros are isolated by the
implicit function theorem. Their set is closed in the compact space
$Y$, hence finite. The number of zeros is locally constant on
$\C\setminus\mathcal B$: the implicit function theorem preserves each
zero in a disjoint neighborhood, and compactness prevents additional
zeros from appearing outside those neighborhoods as the parameter varies.
The same argument covers an empty fiber. Because the complement of a
finite subset of $\C$ is connected, this number is always the number at
$t=0$, namely $B$.

Finally, choose any finite set of simple zeros at $t=1$. They persist in
disjoint neighborhoods for parameters sufficiently close to $1$. Such
parameters can be chosen outside $\mathcal B$, where the total count is
$B$. The selected set has at most $B$ members. This conclusion does not
assign multiplicities to, or impose conditions on, any excess components
at $t=1$.
\end{proof}

\subsection{Ambient polars of affine determinant limits}

We next apply the product count to an arbitrary affine determinant
approximation. The directions in the statement remain ambient directions;
this permits us to use a projective polar certificate on an affine chart.

\begin{proposition}\label{polar:ambient-bound}
Let $N,n,\mu\ge1$ and $h\ge0$ be integers. Suppose that
$P\in\C[x_1,\ldots,x_N]$ is a coefficientwise limit of determinants of
$n\times n$ matrices of affine-linear forms. Let
$\Psi:\C^{h+1}\to\C^N$ be an affine map and
$w_1,\ldots,w_h\in\C^N$ constant directions. If
\begin{equation}\label{polar:affine-certificate}
 P(\Psi(\xi))=0,\qquad
 (\partial_{w_i}P)(\Psi(\xi))=0\quad(1\le i\le h)
\end{equation}
has at least $\mu$ distinct simple zeros, then
\begin{align}
 \mu&\le [H^{h+1}U^{n-1}V^{n-1}]
            (H+U)^n(H+V)^{n-1}(U+V)^h\label{polar:ambient-coefficient}\\
  &\le 2^h\binom{2n-1}{h+1},\notag\\
 n&\ge\frac{h+1}{4e}\mu^{1/(h+1)}.\label{polar:ambient-size}
\end{align}
The binomial coefficient is zero if $h+1>2n-1$.
\end{proposition}

\begin{proof}
\emph{Choose one sufficiently close determinant.}
Fix $\mu$ distinct simple zeros of~\eqref{polar:affine-certificate}.
All output polynomials lie in the finite-dimensional coefficient space
of degree at most $n$. With $\Psi$ and the $w_i$ fixed, the map
\[
 (Q,\xi)\longmapsto
   \bigl(Q(\Psi(\xi)),(\partial_{w_1}Q)(\Psi(\xi)),\ldots,
                         (\partial_{w_h}Q)(\Psi(\xi))\bigr)
\]
is polynomial in the coefficients of $Q$ and the coordinates of $\xi$.
At every selected zero for $P$, its $\xi$-Jacobian is invertible.
The complex implicit function theorem preserves that simple zero for all
sufficiently close coefficient vectors. Take disjoint neighborhoods of
the $\mu$ zeros and intersect their finitely many coefficient neighborhoods.
There is an actual determinant in the resulting neighborhood,
\begin{equation}\label{polar:chosen-matrix}
 Q(x)=\det A(x),\qquad
 A(x)=A_0+\sum_{j=1}^N x_jA_j,\qquad A_j\in\Mat_n(\C),
\end{equation}
for which the same polar system has $\mu$ distinct simple zeros.
This choice constrains only the coefficients of $Q$. The matrix entries
may be arbitrary complex numbers, however large; no bound on an
approximation order or on coefficients of a parametrized family is used.

\emph{Choose a common chart for the right kernels.}
At each selected zero, the first row of the polar Jacobian is nonzero,
so $d(Q\circ\Psi)\ne0$. If $A(\Psi(\xi))$ had rank at most $n-2$,
its adjugate would vanish and the determinant derivative in every direction
would be zero. Since its determinant is zero, its rank is exactly $n-1$.
It has unique right and left projective kernel lines.

Choose a linear functional nonzero on every one of the finitely many
right kernel lines. Such functionals are the complement of a finite union
of proper hyperplanes. A fixed change of basis in the matrix domain makes
this functional the first coordinate. Equivalently, replace $A$ by $AS$
for a suitable invertible constant matrix $S$. This multiplies $Q$ and
all its polars by the same nonzero constant and preserves simplicity.
Continue to write $A,Q$ for the resulting matrix and polynomial. Every
selected right kernel now has a representative with first coordinate one.
Put
\[
 B_i=\partial_{w_i}A=\sum_{j=1}^N(w_i)_j A_j.
\]
These are constant matrices, and
$\partial_{w_i}Q=\tr(\adj(A)B_i)$.

\emph{Lift to a square system of bilinear equations.}
Let $A^{\mathrm{hom}}(\xi_0,\xi)$ be the entrywise homogenization of the affine matrix
$A\circ\Psi$, so $A^{\mathrm{hom}}(1,\xi)=A(\Psi(\xi))$. On
\[
 \PP^{h+1}_{[\xi_0:\xi]}\times\PP^{n-1}_p\times\PP^{n-1}_b
\]
use the equations
\begin{equation}\label{polar:kernel-system}
 A^{\mathrm{hom}} p=0,\qquad
 (b^{\mathsf T}A^{\mathrm{hom}})_j=0\quad(2\le j\le n),\qquad
 b^{\mathsf T}B_i p=0\quad(1\le i\le h).
\end{equation}
There are $n+(n-1)+h=h+2n-1$ equations, equal to the product dimension.
In the chart $p_1\ne0$, the omitted first left-kernel equation follows
from the others, since
\[
 0=b^{\mathsf T}(A^{\mathrm{hom}} p)
   =p_1(b^{\mathsf T}A^{\mathrm{hom}})_1+
       \sum_{j=2}^n p_j(b^{\mathsf T}A^{\mathrm{hom}})_j.
\]
At a selected point, rank $n-1$ implies
$\adj(A)=\rho p b^{\mathsf T}$ for normalized kernel representatives
and some $\rho\ne0$: every column of the adjugate is in the right kernel,
every row is in the left kernel, and some cofactor is nonzero.
Consequently $\partial_{w_i}Q=\rho b^{\mathsf T}B_i p$ there.
Each selected affine zero therefore has a unique lift satisfying
\eqref{polar:kernel-system}.

The lift is not yet known to be simple. We prove this locally before
applying Lemma~\ref{polar:product-count}; other components of the kernel
system will impose no further requirement.

\begin{samepage}
\emph{Eliminate the kernel coordinates.}
Fix one selected lift and normalize $p_1=1$. Choose a nonzero coordinate
of its left kernel vector, move that row to the first position, and
normalize the corresponding left coordinate to one. These are local
coordinate choices, which can differ between selected points. The
cofactor obtained by deleting this first row and first column is nonzero,
because it is the corresponding entry of $\rho p b^{\mathsf T}$.
In these row orders write the ambient matrix near $\Psi(\xi)$ as
\[
 A(x)=\begin{pmatrix}a(x)&\beta(x)\\\gamma(x)&E(x)\end{pmatrix},
 \qquad
 p=\binom{1}{u},\qquad b=\binom{1}{v},
\]
where $E$ has size $(n-1)\times(n-1)$. The chosen cofactor makes $E$
invertible at the selected point, so restrict to an ambient neighborhood
on which it is invertible. When $n=1$, $E$ is the empty matrix with
determinant one, and the auxiliary vectors $u,v$ are empty.

\end{samepage}
The same row permutation is applied to every $B_i$. The last $n-1$ right
equations and all retained left equations give
\[
 \gamma+Eu=0,\qquad \beta+v^{\mathsf T}E=0.
\]
Their auxiliary Jacobian has invertible diagonal blocks $E,E^{\mathsf T}$,
so they eliminate the kernel coordinates holomorphically as
\begin{equation}\label{polar:solved-kernels}
 u=-E^{-1}\gamma,\qquad v^{\mathsf T}=-\beta E^{-1}.
\end{equation}
The remaining right equation, evaluated on the slice, is
\[
 s(\xi)=S(\Psi(\xi))=0,
 \qquad S(x)=a(x)-\beta(x)E(x)^{-1}\gamma(x).
\]
The Schur determinant identity gives
\begin{equation}\label{polar:schur-determinant}
 Q(x)=\sigma\det(E(x))S(x),
\end{equation}
where $\sigma\in\{1,-1\}$ is the sign of the row permutation. It follows
by subtracting $\beta E^{-1}$ times the lower block rows from the first
row. Here $S$ is defined on an ambient neighborhood where $E$ is
invertible, so it can be differentiated in each ambient direction $w_i$.

After substituting~\eqref{polar:solved-kernels}, let
$g_i(\xi)$ denote the bilinear equation $b^{\mathsf T}B_i p$.
For any constant ambient direction $w$, differentiation of $S$ gives
\begin{align*}
 \partial_w S
 &=\partial_w a-(\partial_w\beta)E^{-1}\gamma
       +\beta E^{-1}(\partial_w E)E^{-1}\gamma
       -\beta E^{-1}(\partial_w\gamma)\\
 &=(1,-\beta E^{-1})(\partial_w A)
                 \binom{1}{-E^{-1}\gamma}.
\end{align*}
Thus $g_i=(\partial_{w_i}S)\circ\Psi$. In particular this computation
does not identify an ambient derivative with a derivative along the slice.
If $f=Q\circ\Psi$ and $q_i=(\partial_{w_i}Q)\circ\Psi$, differentiating
\eqref{polar:schur-determinant} in the ambient space yields
\begin{equation}\label{polar:reduced-system}
 \begin{gathered}
 f=c\,s,\qquad q_i=c\,g_i+c_i s,\\
 c=\sigma(\det E)\circ\Psi\ne0,\qquad
 c_i=\sigma(\partial_{w_i}\det E)\circ\Psi.
 \end{gathered}
\end{equation}
The original equation vector $(f,q_1,\ldots,q_h)$ and the reduced vector
$(s,g_1,\ldots,g_h)$ therefore differ by a triangular holomorphic matrix
with diagonal entries $c$. At a common zero their Jacobians differ by
the value of that invertible matrix. The original zero was simple, so
the reduced Jacobian is invertible.

Finally, a tangent vector annihilated by the full kernel-system Jacobian
first satisfies the differentials of the eliminated equations. Their
invertible auxiliary block determines its $(u,v)$ component from its
$\xi$ component. The remaining equations and the invertible reduced
Jacobian force that $\xi$ component to be zero, and hence the auxiliary
component is zero as well. The full square Jacobian is invertible.
All $\mu$ selected lifts are thus distinct simple zeros of
\eqref{polar:kernel-system}.

\emph{Count the selected lifts.}
The three groups of equations in~\eqref{polar:kernel-system} have
multidegrees and multiplicities
\[
 \begin{array}{c|c|c}
 \text{equations}&\text{multidegree}&\text{number}\\ \hline
 A^{\mathrm{hom}} p &(1,1,0)&n\\
 (b^{\mathsf T}A^{\mathrm{hom}})_j, j\ge2 &(1,0,1)&n-1\\
 b^{\mathsf T}B_i p &(0,1,1)&h.
 \end{array}
\]
Lemma~\ref{polar:product-count} gives the first inequality
in~\eqref{polar:ambient-coefficient}, regardless of other zeros or
components. The product has nonnegative coefficients. Setting $U=V=1$
sums the coefficients with the same $H$-exponent, and hence bounds the
particular coefficient by
\[
 [H^{h+1}]\,2^h(H+1)^{2n-1}
       =2^h\binom{2n-1}{h+1}.
\]
Since $\mu>0$, we must have $h+1\le2n-1$. For integers $1\le b\le a$,
\[
 \binom ab\le\frac{a^b}{b!}\le\left(\frac{ea}{b}\right)^b;
\]
the last bound follows from
$\log(b!)\ge\int_1^b\log t\,dt\ge b\log b-b$.
Taking $(h+1)$st roots gives
\[
 \mu^{1/(h+1)}
 \le2^{h/(h+1)}\frac{e(2n-1)}{h+1}
 \le\frac{4en}{h+1},
\]
which is~\eqref{polar:ambient-size}. No comparison between $n$ and
the number $N$ of ambient variables, or between $n$ and $h+2$, was assumed.
\end{proof}

\subsection{The smooth-hypersurface lower bound}

We have proved that a fixed finite set of simple polar intersections
persists under arbitrary affine determinant approximation, and that its
kernel lifts satisfy the multidegree bound. We now use the certificate
provided by the smooth hypersurface.

\begin{proof}[Proof of Theorem~\ref{thm:smooth-border}]
Let $n$ be any size for which $G$ is a coefficientwise limit of affine
$n\times n$ determinants. Choose the directions and $L_0$ from
Lemma~\ref{polar:simple-points}. The affine hyperplane $L_0=1$ identifies
with the projective chart $L_0\ne0$ and contains representatives of all
$\mu=r(r-1)^{d-2}$ selected points. Parametrize it by an affine isomorphism
\[
 \Psi:\C^{d-1}\longrightarrow\{x\in\C^d:L_0(x)=1\}.
\]
Restricting the homogeneous equations to this hyperplane is precisely
their dehomogenization in that projective chart. Thus the equations
\[
 G\circ\Psi=0,\qquad
 (\partial_{w_i}G)\circ\Psi=0\quad(1\le i\le d-2)
\]
have $\mu$ distinct simple affine solutions. The directions remain constant
vectors in $\C^d$; they need not be tangent to $L_0=1$.
Apply Proposition~\ref{polar:ambient-bound} with $N=d$, $h=d-2$, and
$P=G$. It gives
\[
 n\ge\frac{d-1}{4e}
       \bigl(r(r-1)^{d-2}\bigr)^{1/(d-1)}
   \ge\frac{(r-1)(d-1)}{4e},
\]
where the last inequality uses $r\ge r-1>0$. This holds for every
admissible size $n$, so taking the minimum proves the theorem.
\end{proof}
==== END A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//polar.tex ====

==== BEGIN A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//proof-flow.tex ====
\begin{figure}[tbp]
\centering
\setlength{\fboxsep}{7pt}
\fbox{\begin{minipage}{0.90\linewidth}
\centering
\textbf{One integral coefficient polynomial}\\[3pt]
$\displaystyle G_D(X)=[s^qt^q]\per_k(X+sI+tD)
\in\mathbb Z[\theta_1,\ldots,\theta_k][X_{ij}]$\\[3pt]
$\displaystyle D=\diag(\theta_1,\ldots,\theta_k),\qquad
r=k-2q,\qquad T=\binom{q+2}{2}$
\end{minipage}}

\smallskip
\begin{tabular}{@{}c@{\hspace{0.025\linewidth}}c@{}}
\begin{minipage}{0.43\linewidth}\centering
$\swarrow$\quad over $\C$, every parameter choice
\end{minipage}
&
\begin{minipage}{0.43\linewidth}\centering
$\searrow$\quad over $K=\overline{\mathbb F}_2$, choose distinct parameters
\end{minipage}
\\[4pt]
\fbox{\begin{minipage}{0.405\linewidth}\centering
\textbf{Permanent projection}\\[3pt]
$\displaystyle \per_m(A_m(X))=a_0G_D(X)$\\[3pt]
$m\ge11k-2q,\qquad a_0\ne0$
\end{minipage}}
&
\fbox{\begin{minipage}{0.405\linewidth}\centering
\textbf{Plane-curve geometry}\\[3pt]
$\displaystyle \nabla G_{\bar D}(X)=0
\ \Longrightarrow\ \delta(C_X)\ge T$\\[3pt]
$\displaystyle \codim V(\nabla G_{\bar D})\ge T-k$
\end{minipage}}
\\[5pt]
\begin{minipage}{0.43\linewidth}\centering
$\searrow$\quad apply at the complex parameters supplied by geometry
\end{minipage}
&
\begin{minipage}{0.43\linewidth}\centering
$\swarrow$\quad proper specialization and a transverse linear section
\end{minipage}
\end{tabular}

\smallskip
\fbox{\begin{minipage}{0.90\linewidth}
\centering
\textbf{A smooth affine restriction over $\C$}\\[3pt]
A nonzero homogeneous form $G$ of degree $r$ in $d$ variables has a smooth
projective zero set and satisfies $G=\per_m\circ\Phi_m$ for a fixed affine map $\Phi_m$.\\[3pt]
$\displaystyle r-1\ge\frac{k}{4},\qquad
d-1\ge\frac{k^2}{200}$
\end{minipage}}

\smallskip
$\big\downarrow$\quad the polar count for smooth homogeneous forms
\smallskip

\fbox{\begin{minipage}{0.90\linewidth}
\centering
$\displaystyle
\bdc(\per_m)\ge\bdc(G)
\ge\frac{(r-1)(d-1)}{4e}
\ge\frac{k^3}{3200e}$
\end{minipage}}
\caption{The two uses of the integral coefficient formula meet over
$\C$. The graph projection applies to every complex parameter choice;
the characteristic-two branch supplies a suitable complex choice through
the proper parameter family. The displayed scale bounds use
$q=\lfloor(k-1)/3\rfloor$, $k\ge128$, and $m\ge11k-2q$.}
\label{fig:proof-flow}
\end{figure}
==== END A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//proof-flow.tex ====

==== BEGIN A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//selector-wiring.tex ====
\begin{figure}[tbp]
\centering
\begingroup
\setlength{\unitlength}{1pt}
\begin{picture}(460,202)
\linethickness{0.45pt}
\put(174,129){\framebox(112,55){}}
\put(174,25){\framebox(112,55){}}
\put(72,104){\line(1,0){316}}
\qbezier(72,104)(123,122)(174,140)
\qbezier(286,140)(337,122)(388,104)
\put(72,171){\line(1,0){102}}
\put(286,171){\line(1,0){102}}
\qbezier(72,104)(123,86)(174,68)
\qbezier(286,68)(337,86)(388,104)
\put(72,37){\line(1,0){102}}
\put(286,37){\line(1,0){102}}
\put(72,104){\circle*{3.2}}
\put(388,104){\circle*{3.2}}
\put(72,171){\circle*{3.2}}
\put(388,171){\circle*{3.2}}
\put(72,37){\circle*{3.2}}
\put(388,37){\circle*{3.2}}
\put(174,171){\circle*{3.2}}
\put(174,140){\circle*{3.2}}
\put(286,171){\circle*{3.2}}
\put(286,140){\circle*{3.2}}
\put(174,68){\circle*{3.2}}
\put(174,37){\circle*{3.2}}
\put(286,68){\circle*{3.2}}
\put(286,37){\circle*{3.2}}
\put(230,162){\makebox(0,0){\small color 1: \(H\)}}
\put(230,151){\makebox(0,0){\scriptsize 3 rows \(+\) 3 columns}}
\put(181,171){\makebox(0,0)[l]{\scriptsize \(c_2\)}}
\put(181,140){\makebox(0,0)[l]{\scriptsize \(c_1\)}}
\put(279,171){\makebox(0,0)[r]{\scriptsize \(r_2\)}}
\put(279,140){\makebox(0,0)[r]{\scriptsize \(r_1\)}}
\put(230,57){\makebox(0,0){\small color 2: \(H\)}}
\put(230,46){\makebox(0,0){\scriptsize 3 rows \(+\) 3 columns}}
\put(181,68){\makebox(0,0)[l]{\scriptsize \(c_1\)}}
\put(181,37){\makebox(0,0)[l]{\scriptsize \(c_2\)}}
\put(279,68){\makebox(0,0)[r]{\scriptsize \(r_1\)}}
\put(279,37){\makebox(0,0)[r]{\scriptsize \(r_2\)}}
\put(64,104){\makebox(0,0)[r]{\shortstack[r]{\small shared main\\[-1pt]\small row \(i\)}}}
\put(396,104){\makebox(0,0)[l]{\shortstack[l]{\small shared main\\[-1pt]\small column \(i\)}}}
\put(72,192){\makebox(0,0){\small gate 1 row \(i\)}}
\put(388,192){\makebox(0,0){\small gate 1 column \(i\)}}
\put(72,16){\makebox(0,0){\small gate 2 row \(i\)}}
\put(388,16){\makebox(0,0){\small gate 2 column \(i\)}}
\put(230,113){\makebox(0,0){\small \(X_{ii}\)}}
\put(122,132){\makebox(0,0){\small \(-2\)}}
\put(121,76){\makebox(0,0){\small \(-2\theta_i\)}}
\end{picture}
\endgroup
\caption{Wiring after the two replacements at a fixed diagonal position
\(i\). Each box is a separate copy of \(H\) on three fresh rows and
three fresh columns. Its incoming ports \(c_1,c_2\) are internal columns
and its outgoing ports \(r_1,r_2\) are internal rows; their placement is
schematic. The two copies share the main endpoints and have distinct
gate and internal vertices. Unlabeled boundary edges have weight \(1\).
The ordinary edge \(X_{ii}\) remains; other main and gate edges are omitted.}
\label{fig:selector-wiring}
\end{figure}
==== END A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//selector-wiring.tex ====

==== BEGIN A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//selector.tex ====
\section{Selecting a coefficient by a permanent projection}
\label{sec:selector}

We realize a coefficient of a diagonally shifted permanent as an affine
projection of a larger permanent.  The construction selects prescribed
exponents in two auxiliary variables while keeping the larger permanent's
size linear in the order of the shifted permanent.

Fix integers $k\ge1$ and $0\le q\le\lfloor k/2\rfloor$, and put
$r=k-2q$.  Let $\theta_1,\ldots,\theta_k$ be indeterminates and put
$D=\operatorname{diag}(\theta_1,\ldots,\theta_k)$.  Let
$X=(X_{ij})_{1\le i,j\le k}$ be a matrix of independent variables.
Define the integral coefficient polynomial
$G_D\in\mathbb Z[\theta_1,\ldots,\theta_k][X_{ij}:1\le i,j\le k]$ by
\begin{equation}
 G_D(X)=[s^qt^q]\per_k(X+sI+tD).
 \label{eq:selector-coefficient}
\end{equation}
The brackets denote coefficient extraction in the auxiliary variables
$s,t$.  For a tuple of values $\theta_i$ in any field, we use the same
notation $G_D$ for the specialized polynomial.  Write $[k]=\{1,\ldots,k\}$.
Expanding the diagonal entries gives
\begin{equation}
 G_D(X)=
 \sum_{\substack{S,T\subseteq[k],\ |S|=|T|=q\\S\cap T=\varnothing}}
 \left(\prod_{i\in T}\theta_i\right)\per(X_{C,C}),
 \qquad C=[k]\setminus(S\cup T).
 \label{eq:selector-subsets}
\end{equation}
Indeed, $S$ records the diagonal positions contributing $s$, and $T$
those contributing $t\theta_i$.  The two sets are disjoint because each
row and column is used once in a permanent term.  After these fixed
diagonal positions are removed, the remaining permutation is an arbitrary
permutation of $C$, and these data recover the original term uniquely.
Thus no additional factorial occurs in
\eqref{eq:selector-subsets}.  Every summand is homogeneous of degree $r$
in $X$.
We use the convention that the permanent of the empty matrix is $1$.

We use the matching interpretation of a permanent.  A matrix's rows and
columns form the two vertex classes of a bipartite graph, with each entry
as the weight of the edge joining its row to its column.  The permanent is the
sum, over perfect matchings, of the products of their edge weights.
Negative weights are allowed.  Weighted matching gadgets and cancellation are
classical tools of Valiant's projection framework \cite{Valiant1979Algebra}.
The practice of fixing external choices before summing internal
completions also appears in his permanent reduction
\cite[Lemma~3.1 and its proof]{Valiant1979Permanent}.
Signed gadgets enforcing equality of edge choices also appear in
Dell, Husfeldt, Marx, Taslaman and Wahl\'en
\cite[Section~3 and Figure~2]{DellHusfeldtMarxTaslamanWahlen2014}.
Here we give the weights and completion counts needed for a construction
of linear size.

\begin{proposition}[Coefficient projection]\label{prop:selector}
Let $k\ge1$ and $0\le q\le\lfloor k/2\rfloor$ be integers, put $r=k-2q$,
and let $D=\operatorname{diag}(\theta_1,\ldots,\theta_k)$ with
$\theta_i\in\C$.  For every integer $m\ge11k-2q$, there is an $m\times m$
matrix $A(X)$ of complex affine-linear forms such that
\begin{equation}
 \per_m(A(X))=a_0G_D(X),
 \qquad a_0=(-2)^{2k}\big((k-q)!\big)^4\ne0.
 \label{eq:selector-scalar}
\end{equation}
Its entrywise homogenization $B(X,z)$ is a matrix of homogeneous linear
forms whose permanent is the degree-$m$ homogenization of $a_0G_D$:
\begin{equation}
 \per_m(B(X,z))=a_0z^{m-r}G_D(X).
 \label{eq:selector-homogeneous}
\end{equation}
The construction requires no distinctness or nonvanishing assumption on
the entries of $D$.
\end{proposition}

\begin{proof}
Start with $k$ main rows and columns carrying the edges $X_{ij}$.
At each diagonal position add two labeled special edges, with weights
\[
 w_{i,1}=1,\qquad w_{i,2}=\theta_i.
\]
We call them the first and second color.  At this intermediate stage
they are parallel to the ordinary edge $X_{ii}$.  The three labels record
the choices $X_{ii}$, $s$, and $t\theta_i$ in the diagonal entry, with the
factors $s,t$ omitted from the two special weights.  The coefficient of
$s^qt^q$ in the shifted permanent is therefore the weighted sum of main
matchings using exactly $q$ special edges of each color, as in
\eqref{eq:selector-subsets}.
We will count each color in a separate gate and couple each special
main edge to the corresponding distinguished edge of its gate.

\emph{A gate imposing one prescribed count.}
Put $h=k-q$.  Take $k$ original rows and columns, with distinguished
diagonal edges of weight $1$, and $h$ additional rows and columns.
Join every original row to every additional column and every additional
row to every original column, all with weight $1$.  There are no other
edges.  Every additional row must occupy an original column, and every
additional column must be occupied by an original row.  Consequently
exactly $q$ distinguished edges occur in a perfect matching.

For each prescribed distinguished set $S\subseteq[k]$ of size $q$, the
remaining original rows match the additional columns in $h!$ ways, and
the additional rows match the remaining original columns in $h!$ ways.
These choices are independent.  Hence the multiplicity for this exact
set is
\begin{equation}
 (h!)^2=((k-q)!)^2.
 \label{eq:selector-gate-count}
\end{equation}
This also covers $q=0$: all original rows and columns then match across
the two groups.

\emph{A gadget equating two edge choices.}
Call the main and gate vertices \emph{outer}.  For one special main edge
of weight $w$ and its distinguished gate edge
of weight $1$, write the outer row endpoints as $u_1,u_2$ and the outer
column endpoints as $v_1,v_2$, respectively.  These two edges have distinct
endpoints.  Delete them and add three fresh internal rows and three fresh
internal columns, with weight matrix
\begin{equation}
 H=\begin{pmatrix}-1&-1&1\\-1&-1&1\\1&1&1\end{pmatrix}.
 \label{eq:selector-gadget}
\end{equation}
For $i=1,2$, add an incoming edge from $u_i$ to internal column $i$ and
an outgoing edge from internal row $i$ to $v_i$.  All four weights are
$1$, except that the first incoming edge has weight $-2w$.

Fix the subset $I\subseteq\{1,2\}$ of incoming edges used and the
subset $O\subseteq\{1,2\}$ of outgoing edges used.  Let $\Phi_w(I,O)$
be the total weight of their internal completions, including the weights
of those boundary edges.  If $|I|\ne|O|$,
the remaining internal row and column sets have different sizes, so
there is no completion and $\Phi_w(I,O)=0$.  Otherwise
\begin{equation}
 \Phi_w(I,O)=(-2w)^{\mathbf1_{\{1\in I\}}}
 \per H_{\{1,2,3\}\setminus O,\,\{1,2,3\}\setminus I}.
 \label{eq:selector-boundary-weight}
\end{equation}
The complete table of $\Phi_w$ is
\begin{equation}
 \begin{array}[t]{c|rrrr}
 I\backslash O&\varnothing&\{1\}&\{2\}&\{1,2\}\\\hline
 \varnothing&-2&0&0&0\\
 \{1\}&0&0&0&0\\
 \{2\}&0&0&0&0\\
 \{1,2\}&0&0&0&-2w.
 \end{array}
 \label{eq:selector-boundary-table}
\end{equation}
To verify the six balanced cases, first $\per H=-2$.  Deleting any one
row and any one column indexed in $\{1,2\}$ leaves
$\left(\begin{smallmatrix}-1&1\\1&1\end{smallmatrix}\right)$, whose
permanent is $-1+1=0$.  This includes the crossed choices
$I=\{1\},O=\{2\}$ and $I=\{2\},O=\{1\}$.  Deleting both interface
rows and columns leaves the entry $H_{33}=1$; the four external edge
weights then have product $-2w$.  The ten unbalanced cases have already
been excluded.

Thus $\Phi_w(I,O)$ vanishes unless $I=O=\varnothing$ or
$I=O=\{1,2\}$.  In these two states the weights $-2$ and $-2w$
reproduce neither outer edge being used or both being used, with the
common factor $-2$.  The four balanced singleton states can have
individual internal matchings; it is their sums that cancel.

\emph{Combining the two counts.}
Take two disjoint copies of the count gate, one for each color.
For every $i\in[k]$ and $c\in\{1,2\}$, replace the special main edge
of weight $w_{i,c}$ and its distinguished gate edge by the equality
gadget.  Use fresh internal rows and columns for all $2k$ gadgets.
At a fixed index $i$, the two replacements share the main row and column,
while their internal vertices and their gate endpoints remain distinct.
Figure~\ref{fig:selector-wiring} shows these attachments.

\input{figures/selector-wiring}

Group the full matching sum by the set $E$ of chosen edges outside the
internal $H$-blocks.
This set includes all chosen incoming and outgoing edges and must cover
each outer vertex exactly once.  Let $w_{\mathrm{out}}(E)$ be the product
of the weights of the chosen edges whose two endpoints are outer.
For the gadget indexed by $(i,c)$, write $I_{i,c},O_{i,c}$ for its
boundary state.  Since the internal vertex sets are disjoint, the total
weight of all completions of $E$ is
\[
 w_{\mathrm{out}}(E)
 \prod_{i=1}^k\prod_{c=1}^2
 \Phi_{w_{i,c}}(I_{i,c},O_{i,c}).
\]
Each factor $\Phi_{w_{i,c}}$ already includes its boundary edge weights;
$w_{\mathrm{out}}(E)$ contains only the remaining chosen weights.
By \eqref{eq:selector-boundary-table}, this
product vanishes unless every gadget has $I_{i,c}=O_{i,c}=\varnothing$
or $I_{i,c}=O_{i,c}=\{1,2\}$.  Unbalanced states have no completion,
and balanced singleton states cancel.

Now restrict to choices $E$ for which every gadget has one of these two states.
For each gadget with $I_{i,c}=O_{i,c}=\{1,2\}$, replace its four boundary
edges by the original special main edge and distinguished gate edge.
For an empty state, insert neither original edge.  Because $E$ covers every
outer vertex exactly once, this contraction gives a perfect matching in the
graph before replacement, with each special main edge used if and only if
its paired gate edge is used.  Conversely, each such matching determines
the set $E$ and its boundary states uniquely.  The table assigns $-2w_{i,c}$
to a used pair of original weight $w_{i,c}$, and $-2$ to an unused pair of weight
$1$.  Hence its contribution is its original matching weight multiplied
by exactly one factor $-2$ for each of the $2k$ gadgets.  Shared main
endpoints enter through the matching constraint enforced when $E$ is fixed;
the internal factorization uses disjoint vertices.

Each gate now forces exactly $q$ special edges of its color.  The two
selected sets are disjoint by the main matching condition.  For these
sets, the two gates contribute $((k-q)!)^4$ by
\eqref{eq:selector-gate-count}, while the main matching sum is the
corresponding summand of \eqref{eq:selector-subsets}.  The $2k$ gadgets
contribute $(-2)^{2k}$.  This proves \eqref{eq:selector-scalar} for the
unpadded construction.

Its number of rows, and separately its number of columns, is
\begin{equation}
 k+2(2k-q)+3(2k)=11k-2q.
 \label{eq:selector-size}
\end{equation}
These terms count the main graph, the two count gates, and the fresh
internal blocks.  The final graph has no parallel special edges: their
replacements lead to distinct internal columns.  Its weight matrix
therefore has entries that are variables or constants.  Append an
identity block to reach order $m$; its unique matching has weight $1$,
so the permanent identity is preserved.

Finally, write this padded affine matrix as
$A(X)=A_0+\sum_{i,j}X_{ij}A_{ij}$ and set
$B(X,z)=zA_0+\sum_{i,j}X_{ij}A_{ij}$.  For $z\ne0$, homogeneity gives
\[
 \per_m B(X,z)
 =z^m\per_m A(X/z)
 =a_0z^mG_D(X/z)
 =a_0z^{m-r}G_D(X).
\]
Since $m\ge k\ge r$, both sides are polynomials.  Their equality for
$z\ne0$ proves their identity also at $z=0$.  The identity block is
homogenized along with every other constant entry, so it contributes
the additional power of $z$ required in
\eqref{eq:selector-homogeneous}.
\end{proof}

The affine identity yields a comparison for every fixed affine substitution.

\begin{corollary}\label{cor:selector-border}
Under the hypotheses of Proposition~\ref{prop:selector}, let $d\ge1$ be an
integer and let
$\Lambda:\C^d\to\Mat_k(\C)$ be any fixed affine map.  Then
\[
 \bdc(G_D\circ\Lambda)\le\bdc(\per_m).
\]
In particular, this holds for every fixed linear restriction of $G_D$.
\end{corollary}

\begin{proof}
Proposition~\ref{prop:selector} gives
$G_D\circ\Lambda=a_0^{-1}\per_m\circ A\circ\Lambda$.
Apply Lemma~\ref{lem:affine-substitution} to the affine map
$A\circ\Lambda$ and the nonzero scalar $a_0^{-1}$.
Both are fixed before any determinant approximation is chosen.
\end{proof}

\begin{remark}\label{rem:selector-characteristic}
The coefficient formula \eqref{eq:selector-coefficient} and the subset
formula \eqref{eq:selector-subsets} are integral and can be specialized
over any field.  The projection used for the complex border comparison
requires the nonzero scalar $a_0$, whose factor $(-2)^{2k}$ vanishes in
characteristic two.  The characteristic-two argument uses the integral
coefficient expression specialized directly to $\overline{\mathbb F}_2$.
\end{remark}
==== END A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//selector.tex ====

==== BEGIN A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent//transfer.tex ====
\section{Characteristic transfer and smooth linear sections}
\label{sec:transfer}

Proposition~\ref{prop:gradient-codimension} bounds an affine gradient
locus in characteristic two.  We now transfer that bound to complex
parameters, then choose a linear section on which the coefficient
polynomial defines a smooth projective hypersurface.  The final proof
will combine the resulting affine permanent projection with
Theorem~\ref{thm:smooth-border} to obtain the cubic lower bound.

\subsection{Transferring the gradient bound}

The proper family used for specialization is the projective zero scheme
of the partial derivatives.  In characteristic two this scheme need not
be the singular scheme of the hypersurface itself.  After passing to
characteristic zero, Euler's identity will identify the two.

\begin{lemma}[Specialization of a gradient locus]
\label{lem:characteristic-transfer}
Let
\[
 F(\theta,x)\in
 \mathbb Z[\theta_1,\ldots,\theta_a][x_1,\ldots,x_v]
\]
be homogeneous of degree \(r\ge2\) in \(x\), where \(v\ge2\), and let
\(1\le\delta\le v\) be an integer.  Suppose some
\(\bar\theta\in\overline{\mathbb F}_2^{\,a}\) satisfies
\[
 \dim V_{\mathbb A^v_{\overline{\mathbb F}_2}}
       \bigl(\partial_{x_1}F_{\bar\theta},\ldots,
             \partial_{x_v}F_{\bar\theta}\bigr)
 \le v-\delta.
\]
Then there is a nonempty Zariski open subset
\(U\subset\mathbb A^a_{\C}\) such that, for every \(\theta\in U\),
the polynomial \(F_\theta\) is nonzero and
\begin{equation}
 \dim\operatorname{Sing}V_{\PP^{v-1}_{\C}}(F_\theta)
 \le v-1-\delta.
 \label{eq:transfer-singular-bound}
\end{equation}
We use dimension \(-1\) for the empty projective scheme.
\end{lemma}

\begin{proof}
Put \(R=\mathbb Z[\theta_1,\ldots,\theta_a]\) and
\(S=\operatorname{Spec}R\).  The homogeneous ideal
\[
 I=(\partial_{x_1}F,\ldots,\partial_{x_v}F)\subset R[x_1,\ldots,x_v]
\]
defines a closed subscheme
\begin{equation}
 \mathcal Z=\operatorname{Proj}(R[x]/I)\subset\PP^{v-1}_S.
 \label{eq:transfer-projective-family}
\end{equation}
Its structural morphism \(\pi:\mathcal Z\to S\) is projective, hence proper.
Differentiation commutes with specialization, and the closed projective
subscheme in \eqref{eq:transfer-projective-family} commutes with base
change.  Thus its geometric fibers are exactly the projective
gradient-zero schemes of the specialized polynomials.

The derivatives have positive homogeneous degree \(r-1\).
At \(\bar\theta\), their affine zero set is a cone.  If its
projectivization is nonempty, its dimension is one less than that of the
cone: on each projective coordinate chart the corresponding punctured
cone is a product with \(\mathbb G_m\).  If the projectivization is
empty, the same upper bound holds with our convention.  Hence
\[
 \dim\mathcal Z_{\bar\theta}\le v-1-\delta.
\]
Let \(s\in S\) be the image of the geometric point \(\bar\theta\).
The dimension of a scheme of finite type over a field is unchanged by
extending that field, so the same bound holds for \(\mathcal Z_s\).
If \(v-1-\delta\ge0\), upper semicontinuity of fiber dimension for the
proper morphism \(\pi\) gives an open neighborhood \(U_S\) of \(s\)
on which this bound holds \cite[Tag~0D4I]{StacksProject}.
If \(v-1-\delta=-1\), the fiber \(\mathcal Z_s\) is empty.
Properness makes \(\pi(\mathcal Z)\) closed, so
\(U_S=S\setminus\pi(\mathcal Z)\) is an open neighborhood of \(s\)
whose fibers are all empty.  Thus the required neighborhood exists
in every case, without a flatness assumption.

Since \(S\) is integral, \(U_S\) contains its generic point, which has
characteristic zero.  To see explicitly that the open has
complex points, choose a nonempty principal open
\(D(h)\subset U_S\), with \(0\ne h\in R\).  The polynomial \(h\)
remains nonzero over \(\C\), so \(D(h)_\C\) is nonempty.  We may take
\(U=(U_S)_\C\).  The same invariance under field extension shows that
every complex fiber over \(U\) satisfies
\begin{equation}
 \dim V_{\PP^{v-1}_{\C}}(\nabla F_\theta)
 \le v-1-\delta.
 \label{eq:transfer-complex-gradient}
\end{equation}

Since \(\delta>0\), \(F_\theta\) cannot be the zero polynomial:
otherwise the left side of \eqref{eq:transfer-complex-gradient}
would be \(v-1\).  Over \(\C\), Euler's identity gives
\[
 rF_\theta=\sum_{i=1}^v x_i\partial_{x_i}F_\theta.
\]
Thus the ideals \((F_\theta,\nabla F_\theta)\) and
\((\nabla F_\theta)\) agree.  These equations describe the actual
projective hypersurface singular scheme.  Indeed, on the chart
\(x_i\ne0\), put \(z_j=x_j/x_i\) for \(j\ne i\) and
\(f_i=F_\theta/x_i^r\).  The hypersurface has equation \(f_i=0\)
and cotangent presentation
\[
 \cO_{V(f_i)}
 \xrightarrow{\,df_i\,}
 \cO_{V(f_i)}^{\,v-1}
 \longrightarrow\Omega_{V(f_i)/\C}\longrightarrow0.
\]
Its \((v-2)\)-nd Fitting ideal is generated by the local partial
derivatives of \(f_i\).  On \(f_i=0\), these are the homogeneous
gradient equations in this chart: Euler's identity supplies the
derivative in the omitted \(x_i\) direction.  The Jacobian criterion
therefore identifies this scheme with the singular scheme.
Equation~\eqref{eq:transfer-complex-gradient} proves
\eqref{eq:transfer-singular-bound}.
\end{proof}

All coefficients in \eqref{eq:transfer-projective-family} are integral;
in particular, the construction does not invert \(2\) or \(r\).
Division by \(r\) occurs only in characteristic zero.  In applying the
lemma to \(G_D\), we use its coefficient formula
\eqref{eq:selector-coefficient}, before multiplying by the projection
scalar \(a_0\), which vanishes in characteristic two.

\subsection{A smooth section of the required dimension}

Avoiding the singular locus of a hypersurface is one condition on a
linear section.  The section must also meet its smooth locus
transversely.  The following incidence calculation establishes both
conditions and keeps track of the projective dimension.

\begin{proposition}[Smooth linear section]\label{prop:smooth-slice}
Let \(F\) be a nonzero homogeneous complex polynomial of degree \(r\ge2\)
on a vector space \(V\) of dimension \(v\).  Set
\(Y=V_{\PP(V)}(F)\), and suppose
\[
 \dim\operatorname{Sing}Y\le v-1-\delta
\]
for an integer \(\delta\ge1\).  If \(2\le d0.
 \label{eq:transfer-codimension-size}
\end{equation}
Also \(T\le(k+2)(k+5)/18