Permanent–Determinant Separation Through Geometric Complexity Theory

Permanent–Determinant Separation Through Geometric Complexity Theory

Definitions and mathematical statement

For variables $x_{ij}$ over a characteristic-zero field $K$, define \[ \operatorname{per}_n(x)=\sum_{\sigma\in S_n}\prod_{i=1}^n x_{i,\sigma(i)}. \] An algebraic circuit computes a polynomial using field constants, variables, addition, and multiplication. The class $\mathsf{VP}$ has polynomially bounded numbers of variables, degrees, and circuit sizes; $\mathsf{VNP}$ consists of polynomially long Boolean sums of such efficiently computable polynomials. The selected claim is that no polynomial bounds the arithmetic circuit sizes of $\operatorname{per}_n$, equivalently $\mathsf{VP}\ne\mathsf{VNP}$ in characteristic zero. This is a nonuniform algebraic-operation model, not a bit-complexity assertion.

Formulation and concept references: [S1].

Short English statement

Can one prove that the permanent requires more than polynomially many arithmetic operations in general algebraic circuits?

Research attempt

Scope, sources, and the chosen attack (27 September 2026).

Fix the characteristic-zero field $K$. Constants from $K$ are allowed, and the circuit for one matrix size need not be computably recoverable from the circuit for another. The desired conclusion is that no polynomial bounds the minimum sizes for all $n$; a lower bound against one restricted circuit form does not suffice. The symmetric classes in [S1, S2] impose conditions absent here, and [S1] explicitly distinguishes their orbit-size measure from general circuit size. The August 2026 report [S3] gives elusive-function and bounded-depth results; its discussion identifies degree growth as an obstruction to deriving the full separation. None is read as a solution to unrestricted $\mathsf{VP}\ne\mathsf{VNP}$.

The strategy is to calculate concrete derivative invariants of the permanent, extract the lower bounds they actually prove, and test their ability to survive general multiplication gates. A second calculation recovers the known quadratic determinantal bound of Mignon--Ressayre [E2]. This is a reconstruction and stress test of classical tools, not a claim of a new circuit lower bound. Valiant's completeness theorem [E1] supplies the stated equivalence with the class separation; the membership direction can be proved explicitly here.

An explicit Boolean-sum witness and an exact upper bound.

For $y=(y_1,\ldots,y_n)$ define \begin{equation}\label{wsref-0-76-70-3a-77-69-74-6e-65-73-73} g_n(x,y)=(-1)^n\prod_{j=1}^n(1-2y_j) \prod_{i=1}^n\left(\sum_{j=1}^n x_{ij}y_j\right). \end{equation} This polynomial has degree at most $3n$, $n^2+n$ variables, and a circuit of size $O(n^2)$. For Boolean $y$ with support $S$, its value is $(-1)^{n-|S|}\prod_i\sum_{j\in S}x_{ij}$. Expanding the sum of these values over all $S$ gives, for each function $f:[n]\to[n]$, the monomial $\prod_i x_{i,f(i)}$ with coefficient \[ \sum_{S\supseteq\operatorname{im}f}(-1)^{n-|S|} =\begin{cases}1,&\operatorname{im}f=[n],\\0,&\text{otherwise}. \end{cases} \] The identity follows by choosing the subset of the complement; a nonempty complement contributes $(1-1)^d=0$. A surjective map between these two $n$-element sets is a permutation. Therefore \begin{equation}\label{wsref-0-76-70-3a-62-6f-6f-6c-65-61-6e} \operatorname{per}_n(x)=\sum_{y\in\{0,1\}^n}g_n(x,y). \end{equation} This proves the required $\mathsf{VNP}$ membership with polynomially many witness bits. It also gives an exact $O(n^2 2^n)$ circuit by explicit summation, the usual inclusion--exclusion mechanism. The phrase "polynomially long Boolean sum" bounds the number of witness bits, not the number of summands. Evaluating $g_n$ efficiently does not make the exponential summation a polynomial-size computation.

For $n=1$, the two Boolean evaluations give $x_{11}$. For $n=2$, the subset terms cancel the two monomials that reuse a column and leave $x_{11}x_{22}+x_{12}x_{21}$. Thus the witness has the correct signs even at the first nontrivial size. Direct enumeration for sizes up to five checked the identity on integer matrices in the local calculation.

Exact derivative dimensions.

For a homogeneous degree-$n$ polynomial $f$, let $D_k(f)$ be the $K$-linear span of all order-$k$ partial derivatives, and set $d_k(f)=\dim_K D_k(f)$. Here $0\le k\le n$. A nonzero derivative of $\operatorname{per}_n$ differentiates distinct variables occupying distinct rows and columns. It is exactly the permanent on the undeleted rows and columns. Conversely every choice of $k$ deleted rows and $k$ deleted columns occurs by choosing a matching between them. Hence \begin{equation}\label{wsref-0-76-70-3a-64-65-72-69-76-61-74-69-76-65-73} d_k(\operatorname{per}_n)=\binom nk^2. \end{equation} To justify equality rather than just a spanning bound, observe that the surviving monomials determine their row set and column set. Different subpermanents consequently have disjoint monomial supports. They are nonzero and linearly independent. At $k=n$ there is just the constant polynomial, agreeing with the formula.

This yields an exponential lower bound for a genuine restricted model. Suppose \[ \operatorname{per}_n=\sum_{a=1}^s c_a\ell_a^n \] with homogeneous linear forms $\ell_a$ and scalars $c_a$. Every order-$k$ derivative belongs to the span of the $s$ polynomials $\ell_a^{n-k}$. Thus \[ s\ge\binom n{\lfloor n/2\rfloor}^{\!2} \ge\frac{4^n}{(n+1)^2}. \] The last inequality uses the largest of the $n+1$ binomial coefficients and their sum $2^n$. This is a lower bound for sums of powers of linear forms, not for unrestricted depth-three circuits or general circuits. No reduction preserving polynomial size from the latter models to this restricted representation has been proved here.

Why derivative dimension does not control general circuits.

The elementary product $h(z)=z_1\cdots z_n$ has a circuit with $n-1$ multiplications, but $d_k(h)=\binom nk$, because its derivatives are all squarefree monomials of degree $n-k$. This already refutes the proposal that exponential derivative dimension implies exponential general circuit size. The product rule explains the failure: \[ D_k(fh)\subseteq\sum_{j=0}^k D_j(f)D_{k-j}(h),\qquad d_k(fh)\le\sum_{j=0}^k d_j(f)d_{k-j}(h), \] where a product of spaces denotes the span of pairwise products. Repeated multiplication can generate binomially many derivatives without using binomially many gates.

There is a stronger countertest: the determinant has exactly the same entire derivative-dimension profile as the permanent. Its nonzero order-$k$ derivatives are signed complementary minors, which are independent by the same disjoint-support argument. Thus \[ d_k(\det_n)=\binom nk^2=d_k(\operatorname{per}_n) \quad(0\le k\le n). \] Nevertheless $\det_n$ has polynomial-size circuits over $K$. For completeness, compute $p_j=\operatorname{tr}(X^j)$ for $j\le n$ by successive matrix products, at cost $O(n^4)$, and use \[ e_0=1,\qquad e_k=\frac1k\sum_{j=1}^k(-1)^{j-1}e_{k-j}p_j. \] Newton's identities give $e_n=\det X$. These identities follow by logarithmically differentiating $\prod_i(1+\lambda_i t)$ and comparing coefficients; polynomial identity then gives them for the generic matrix. The divisions by the positive integers $k$ are permitted field constants in characteristic zero, not divisions by input expressions. Consequently no invariant that depends only on this derivative-rank profile can distinguish the two families. A successful refinement must retain additional structure or exploit restrictions on gates.

A more local calculation: the quadratic determinant bound.

Let $\operatorname{dc}(f)$ be the least $m$ for which $f(x)=\det_m(A_0+\sum x_{ij}A_{ij})$ over $K$. If $H_f$ denotes the Hessian, the affine chain rule gives \begin{equation}\label{wsref-0-76-70-3a-68-65-73-73-69-61-6e} H_f(x_0)=L^{\mathsf T}H_{\det_m}(A(x_0))L. \end{equation} At a singular $m$ by $m$ matrix the determinant Hessian has rank at most $2m$. Row and column transformations reduce the matrix to $\operatorname{diag}(I_r,0)$. For $r\le m-3$ its quadratic term vanishes. For $r=m-2$ it is a two-by-two determinant, of Hessian rank four. For $r=m-1$ the quadratic term is \[ z_{mm}\sum_{i<m}z_{ii}-\sum_{i<m}z_{im}z_{mi}, \] whose Hessian rank is at most $2+2(m-1)=2m$. The case $m=1$ has zero Hessian.

Here is an explicit permanent calculation to apply that bound. Take $X_0$ with all entries one except $(X_0)_{11}=1-n$. Then $\operatorname{per}_n(X_0)=(n-1)!(n-1+1-n)=0$. For $n\ge3$, divide its Hessian by $(n-3)!$. An entry for two variables in a common row or column is zero; for compatible variables it is $t=n-2$ if their deleted rows or columns include index one, and $-2$ otherwise.

To check nonsingularity directly, write a kernel vector in blocks $(a,(b_j),(c_i),(d_{ij}))$ for the corner, first row, first column, and interior. All interior indices range over $2,\ldots,n$. Write $S_b,S_c,S_d$ for the corresponding sums, and $R_i,C_j$ for interior row and column sums. The corner and border equations give \[ tS_d=0,\quad t(S_c+S_d-C_j)=0,\quad t(S_b+S_d-R_i)=0. \] Summing them yields $S_b=S_c=S_d=R_i=C_j=0$. The interior equations reduce to \[ 2d_{ij}=t(a-b_j-c_i). \] Their row sums give $c_i=a$, their column sums give $b_j=a$, and $S_b=0$ forces $a=0$. All coordinates vanish. For $n=2$, the Hessian of $x_{11}x_{22}+x_{12}x_{21}$ has rank four directly. Thus $\operatorname{rank}H_{\operatorname{per}_n}(X_0)=n^2$ for $n\ge2$. Substitution in (vp:hessian) proves the classical Mignon--Ressayre bound [E2] \begin{equation}\label{wsref-0-76-70-3a-64-63} \operatorname{dc}(\operatorname{per}_n)\ge n^2/2. \end{equation}

What that bound leaves open.

The Hessian argument is stronger than merely comparing the global dimensions $d_k$: it evaluates derivatives at a specially chosen zero and uses the geometry of singular determinant matrices. It still produces a polynomial lower bound, compatible with determinantal representations of size $n^3$, for example. Its input Hessian has only $n^2$ rows, so comparing its rank with $2m$ cannot by itself produce a superquadratic lower bound in $n$.

There is a second, independent gap in using this route for the selected problem. A polynomial-size affine determinantal representation would give a polynomial-size general circuit, by the determinant algorithm above. The reverse implication is not known with polynomial overhead: determinant is complete for polynomial-size algebraic branching programs, a potentially smaller model than $\mathsf{VP}$ [E3, Sections 2.5 and 2.9]. Thus even a superpolynomial lower bound on $\operatorname{dc}(\operatorname{per}_n)$ would not automatically separate $\mathsf{VP}$ from $\mathsf{VNP}$. It would do so under the additional hypothesis that every polynomial-size, polynomial-degree general circuit family admits polynomial-size affine determinantal representations. That simulation is an extra hypothesis, not a step established by (vp:hessian).

Why a generic counting argument does not select the permanent.

There are indeed degree-$n$ polynomials in $n^2$ variables with large circuits. To see the distinction from this task, fix a size bound $s$ for binary arithmetic circuits. There are finitely many circuit topologies and choices of variable labels of that size. A topology has at most $2s$ constant input slots; regard them as parameters. The coefficients of its homogeneous degree-$n$ component are polynomial functions of those parameters. Their image in the coefficient space of dimension \[ N=\binom{n^2+n-1}{n} \] has Zariski closure of dimension at most $2s$: the field generated by these coefficient functions has transcendence degree at most the number of parameters. If $2s<N$, each such closure is proper. A finite union of these proper closed sets cannot fill $K^N$ over the infinite field $K$. Equivalently, the product of nonzero defining polynomials for the closures is nonzero and cannot vanish at every point of $K^N$. This proves existence of a polynomial beyond the specified circuit bound, even while allowing arbitrary constants.

However, the permanent is one fixed coefficient vector, not a generic one. The argument supplies no reason it avoids any of these circuit images, and the chosen hard polynomial is not shown to have a polynomial-length Boolean-sum description. Counting unrestricted coefficient vectors therefore does not give a hard family in $\mathsf{VNP}$. This also explains why counting circuit strings with only a finite alphabet of constants would address a different model: the actual constants range over all of $K$. The parameter argument handles that issue but leaves the essential explicitness gap intact.

Verification, first gap, and outcome.

Exact finite checks compared the Boolean witness sum with the permutation definition, derivative-support counts with $\binom nk^2$, and the displayed Hessian with direct complementary subpermanent calculations. Hessian ranks were computed over the rationals for $2\le n\le7$, yielding $n^2$ each time. The proof above, not those finite ranks, handles arbitrary $n$. Characteristic zero was used in Newton's identities, the kernel calculation, and the completeness formulation; these arguments are not silently transferred to characteristic two, where permanent and determinant coincide.

The first gap in the derivative strategy is a nontrivial upper bound for a stronger permanent invariant on all small circuits, consistent with the product and determinant countertests. Three useful next tasks are to preserve row/column structure in a refined derivative space; to prove gate-by-gate bounds that account for reuse and cancellation; and to identify a higher-order local obstruction whose growth exceeds every fixed polynomial after the relevant simulation cost is included. Merely enlarging the ordinary derivative spaces or finding more small Hessian examples cannot meet these requirements.

UNRESOLVED. The attempt proves explicit membership and upper bounds, an exponential lower bound for sums of powers, and reconstructs the known quadratic determinantal lower bound. It does not give a superpolynomial lower bound for general arithmetic circuits. Written by GPT-6 Astra Ultra with primary-source scope checks and exact finite calculations; no formal verification or novelty is asserted.

Sources

• S1 Dwivedi, Pago, and Seppelt, Lower Bounds in Algebraic Complexity via Symmetry and Homomorphism Polynomials, STOC 2026, Introduction. https://www.prateekdwivedi.in/papers/symACT-confversion.pdf. Formulation source.
• S2 Symmetric Algebraic Circuits and Homomorphism Polynomials. https://drops.dagstuhl.de/storage/00lipics/lipics-vol362-itcs2026/LIPIcs.ITCS.2026.46/LIPIcs.ITCS.2026.46.pdf. Further reference; not a proof endorsement.
• S3 Arithmetic circuit lower bounds from sumset expansion. https://eccc.weizmann.ac.il/report/2026/138/. Further reference; not a proof endorsement.
• E1 Leslie G. Valiant, Completeness classes in algebra, Proceedings of STOC 1979, 249--261. https://doi.org/10.1145/800135.804419. Author-paper scan: https://sites.math.washington.edu/~billey/colombia/references/valiant.completeness.1979.pdf. Primary source for algebraic completeness; the explicit Boolean-sum membership identity is proved above. Consulted 27 September 2026.
• E2 Thierry Mignon and Nicolas Ressayre, A quadratic bound for the determinant and permanent problem, International Mathematics Research Notices 2004$79$, 4241--4253. https://math.univ-lyon1.fr/~ressayre/PDFs/permdet.pdf. Primary source of the known characteristic-zero quadratic determinantal lower bound, reconstructed here by an explicit Hessian calculation; consulted 27 September 2026.
• E3 Peter Bürgisser, Completeness classes in algebraic complexity theory, arXiv:2406.06217 $2024$, Sections 2.4--2.9. https://arxiv.org/html/2406.06217v1. Author account of the determinant, branching-program completeness, and the distinction from general circuits; consulted 27 September 2026.

Permanent–Determinant Separation Through Geometric Complexity Theory

Definitions and mathematical statement

For variables $x_{ij}$ over a characteristic-zero field $K$, define \[ \operatorname{per}_n(x)=\sum_{\sigma\in S_n}\prod_{i=1}^n x_{i,\sigma(i)}. \] An algebraic circuit computes a polynomial using field constants, variables, addition, and multiplication. The class $\mathsf{VP}$ has polynomially bounded numbers of variables, degrees, and circuit sizes; $\mathsf{VNP}$ consists of polynomially long Boolean sums of such efficiently computable polynomials. The selected claim is that no polynomial bounds the arithmetic circuit sizes of $\operatorname{per}_n$, equivalently $\mathsf{VP}\ne\mathsf{VNP}$ in characteristic zero. This is a nonuniform algebraic-operation model, not a bit-complexity assertion.

Formulation and concept references: [S1].

Short English statement

Can one prove that the permanent requires more than polynomially many arithmetic operations in general algebraic circuits?

Research attempt

Current frontier.

The supplied 2026 formulation paper proves separations for symmetry-constrained algebraic models and completeness results for homomorphism polynomials. It explicitly distinguishes these from unrestricted $\mathsf{VP}$ versus $\mathsf{VNP}$. Output symmetry is not an assumption on every circuit computing that output. [S1], [S2]. Narayanan's July 2026 manuscript uses sumset expansion to construct elusive functions and improve certain low-depth bounds. Its abstract still identifies the stronger explicit constructions needed for Valiant's conjecture as missing. [S3], [E1]. The relevant definitions and statements were inspected; no full proof audit of the recent manuscripts is claimed. Partial-derivative methods have a long-established role in restricted arithmetic lower bounds. [E2].

Attack route.

A direct route would attach a large linear-algebraic invariant to the permanent and prove that it grows only polynomially in unrestricted circuit size. A second route would use the permanent's symmetry to replace a hypothetical small circuit by a small symmetric one, then invoke the known symmetric separation. I test the first route with exact derivative and row-cut profiles and use the second as an adversarial comparison. The goal is to identify an invariant that sees cancellation and circuit sharing, rather than merely count monomials or minors.

Attempt.

For a homogeneous polynomial $f$ over a characteristic-zero field $K$, write \[ D_k(f)=\dim_K\operatorname{span}\{\partial^\alpha f:|\alpha|=k\}. \] The following calculation gives the complete profile, not just one asymptotic lower bound.

Lemma 32.1 (identical derivative dimensions). For $0\le k\le n$, \begin{equation}\label{wsref-0-72-33-32-3a-64-65-72-69-76-61-74-69-76-65-73} D_k(\operatorname{per}_n)=D_k(\det_n)=\binom nk^2. \end{equation} For $k>n$ both quantities are zero.

Proof. Differentiating a permanent monomial at $k$ entries gives zero unless the entries lie in distinct rows and distinct columns. In the nonzero case, the result is the permanent of the complementary $(n-k)\times(n-k)$ submatrix. Every pair of deleted row and column sets occurs, giving at most $\binom nk^2$ polynomials. They are linearly independent: the variables appearing in any monomial record exactly the remaining row and column sets, so polynomials attached to different pairs have disjoint monomial supports. Each is nonzero. The same argument for the determinant gives signed complementary determinants, still nonzero with disjoint supports. At $k=n$ all nonzero derivatives are constants and the formula gives one.$\square$

The full profile is therefore incapable of distinguishing the two polynomials. This is a sharper stress test than simply finding a weak bound at one derivative order. In particular, no argument using only this profile and a bound valid for all small circuits can certify that the permanent requires larger circuits than the known polynomial-size determinant circuit. For clarity, the latter fact can be exhibited directly in this model.

Let $p_i=\operatorname{tr}(X^i)$, $e_0=1$, and recursively compute \begin{equation}\label{wsref-0-72-33-32-3a-6e-65-77-74-6f-6e} e_j=\frac1j\sum_{i=1}^j(-1)^{i-1}e_{j-i}p_i \quad(1\le j\le n). \end{equation} The formal identity $\det(I+tX)'/\det(I+tX)=\sum_{i\ge1}(-1)^{i-1}p_i t^{i-1}$ proves that $e_n=\det X$: it follows either by formal differentiation of the determinant or by triangularizing over an algebraic closure and comparing symmetric polynomials. Sequential matrix multiplication computes all $X^i$ with $O(n^4)$ arithmetic operations; the recurrence adds $O(n^2)$. The divisions by $j$ are multiplications by allowed constants $1/j\in K$, not division gates or variable-dependent denominators. Thus this is a valid polynomial-size arithmetic circuit in every characteristic-zero field. Its derivative dimensions near $k=n/2$ are nevertheless exponential in $n$.

The basic product inequalities explain the mismatch: \[ D_k(f+g)\le D_k(f)+D_k(g),\qquad D_k(fg)\le\sum_{i=0}^kD_i(f)D_{k-i}(g). \] Leibniz's rule proves the second. Even the $O(n)$-size circuit $f=x_1\cdots x_n$ has $D_k(f)=\binom nk$. Exponential derivative-space growth is compatible with a linear number of multiplication gates. The missing invariant cannot be supplied by treating that growth as though it were additive at product gates.

A restricted lower bound that survives the calculation. Split the variables into $Y$, the first $k$ rows, and $Z$, the other rows, and let $\operatorname{rank}_{Y|Z}(f)$ be the rank of the matrix of coefficients of monomials $u(Y)v(Z)$. All such matrices are finite for a fixed polynomial.

Lemma 32.2 (exact row-cut rank). For $0\le k\le n$, \begin{equation}\label{wsref-0-72-33-32-3a-63-75-74} \operatorname{rank}_{Y|Z}(\operatorname{per}_n) =\operatorname{rank}_{Y|Z}(\det_n)=\binom nk. \end{equation} Consequently a decomposition $f(Y,Z)=\sum_{j=1}^r a_j(Y)b_j(Z)$ of either polynomial requires $r\ge\binom nk$, and that value is attainable.

Proof. Group permutation monomials by the set $S$ of columns used by the first $k$ rows. For the permanent this yields \[ \operatorname{per}X=\sum_{|S|=k} \operatorname{per}X_{[k],S}\, \operatorname{per}X_{[n]\setminus[k],S^c}. \] Laplace expansion gives the determinant version with a sign depending on $S$. Distinct $S$ have disjoint supports on both sides of the coefficient matrix. Thus the matrix is a disjoint union of exactly $\binom nk$ nonzero rank-one blocks. Each separated product contributes at most rank one, proving both bounds.$\square$

At a balanced cut this is exponential, but it is a bound for a specified separated-product representation. The determinant circuit above is free to mix both row groups at internal gates, so there is no contradiction. To use this route for the target one would need a polynomial-overhead conversion of every arithmetic circuit into a representation respecting an appropriate cut. Lemma 32.2 and the determinant circuit refute such a conversion for this fixed row-cut model. A more flexible decomposition would require a different invariant and proof.

Testing symmetrization as an escape. The permanent and determinant are both invariant under simultaneously permuting rows and columns. Given any circuit for an invariant polynomial, one can certainly make copies indexed by the group and average their outputs. In characteristic zero the normalization by the group order is legal. But for the group $S_n$ this straightforward construction uses $n!$ copies; it gives no polynomial overhead. The fact that the final polynomial is invariant does not certify a small group action on its internal gates. This is exactly where the model distinction in [S1] and [S2] matters. The observation does not prove that every clever symmetrization must be expensive; it identifies the missing lemma in this particular attempted transfer of their result.

Stress test.

The derivative independence argument uses monomial supports, not numerical values at sample matrices, so it is valid over every characteristic-zero field and does not confuse generic rank with evaluated rank. Nonzero signs in determinant minors do not change their supports. Arbitrary cancellations and reused subcircuits remain allowed in the target; they have not been replaced by monotone, formula, symmetric, or fixed-depth computation.

The accompanying exact script constructs the determinant and permanent for $n\le4$, forms their derivative coefficient vectors, and checks the ranks over ${\mathbb{Q}}$. It checks the row-cut ranks and Newton recurrence on small integer matrices as independent implementation tests. These finite cases cannot prove a superpolynomial asymptotic bound. Nor would a Boolean bit-complexity argument automatically apply to unrestricted field constants in this nonuniform arithmetic model.

\Needspace{8\baselineskip}

Outcome.

Outcome: Exploratory approach with unresolved gap.

The attempt establishes exact derivative and cut-rank profiles and a sharp lower bound for a separated-product subclass. Comparing them with an explicit polynomial-size determinant circuit defeats the proposed transfer to general circuits. The remaining task is an invariant, or a controlled normal-form theorem, that detects the permanent's difficulty without also misclassifying the determinant. No such theorem is supplied; the lemmas are elementary structural checks and are not claimed to be historically new or to separate $\mathsf{VP}$ from $\mathsf{VNP}$.

Sources

• S1 Dwivedi, Pago, and Seppelt, Lower Bounds in Algebraic Complexity via Symmetry and Homomorphism Polynomials, STOC 2026, Introduction. https://www.prateekdwivedi.in/papers/symACT-confversion.pdf. Catalogue formulation source.
• S2 Symmetric Algebraic Circuits and Homomorphism Polynomials. https://drops.dagstuhl.de/storage/00lipics/lipics-vol362-itcs2026/LIPIcs.ITCS.2026.46/LIPIcs.ITCS.2026.46.pdf. Catalogue research/status reference; not a proof endorsement.
• S3 Arithmetic circuit lower bounds from sumset expansion. https://eccc.weizmann.ac.il/report/2026/138/. Catalogue research/status reference; not a proof endorsement.
• E1 Anand Kumar Narayanan, Arithmetic circuit lower bounds from sumset expansion, arXiv:2607.15848 $2026$, abstract and scope of the elusive-function constructions; also ECCC TR26-138. https://arxiv.org/abs/2607.15848. https://eccc.weizmann.ac.il/report/2026/138/. Recent manuscript supplement to [S3], checked 27 September 2026; no unrestricted separation inferred.
• E2 Noam Nisan and Avi Wigderson, Lower Bounds on Arithmetic Circuits via Partial Derivatives, BRICS RS-95-43, preliminary version $1995$. https://www.brics.dk/RS/95/43/index.html. Foundational method and restricted-model scope; checked 27 September 2026. The notebook calculations are proved separately.