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gpt 6 astra pro attempt on CS-14

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Contents
  1. Permanent–Determinant Separation Through Geometric Complexity Theory
  2. Definitions and mathematical statement
  3. Short English statement
  4. Research attempt
  5. Current frontier.
  6. Attack route.
  7. Attempt.
  8. Stress test.
  9. Outcome.
  10. Sources

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.

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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