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CS-32-A1 · revision r1 · on CS-32

gpt 6 astra pro attempt on CS-32

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Manuscript

Contents
  1. Graph Isomorphism in Polynomial Time
  2. Definitions and mathematical statement
  3. Short English statement
  4. Research attempt
  5. Current frontier.
  6. Attack route.
  7. Attempt: a robust integrality certificate.
  8. Attempt: explicit fractional and spectral obstruction.
  9. Stress test.
  10. Outcome.
  11. Sources

Graph Isomorphism in Polynomial Time

Definitions and mathematical statement

Two finite graphs $G=(V,E)$ and $H=(W,F)$ are isomorphic if some bijection $\varphi:V\to W$ satisfies \[ \{u,v\}\in E\quad\Longleftrightarrow\quad\{\varphi(u),\varphi(v)\}\in F. \] The problem asks for a deterministic classical algorithm deciding the existence of such a bijection in time polynomial in the ordinary graph encoding length. No planarity, degree bound, or structural restriction is assumed. A quasipolynomial algorithm, or an algorithm efficient only on typical graphs, is not the demanded bound.

Formulation and concept references: [S1].

Short English statement

Can arbitrary finite graphs be tested for isomorphism in deterministic polynomial time?

Research attempt

Current frontier.
Babai's general algorithm is quasipolynomial, not polynomial in the ordinary encoding length [E1]. Polynomial or efficient smoothed-case and numerical methods do not by themselves resolve worst-case deterministic GI; the smoothed setting in [E2] illustrates the distinction. The general polynomial-time claim retained as [S4] is not treated as established merely because it was posted. The following attempt tests a concrete convex-relaxation route and gives an exact stopping certificate for a sufficiently integral solution.
Attack route.
Spectral invariants are cheap but incomplete. Fractional isomorphism adds entrywise positivity and linear constraints, yet permits many nonpermutation solutions. A plausible repair would be to drive a feasible matrix near a permutation using a provable integrality gap. The attempt proves a precise rounding threshold, then constructs two explicit nonisomorphic graphs whose fractional and heat-kernel information defeats the simpler route. The gap certificate is mathematical; an efficient method for attaining it is not assumed.
Attempt: a robust integrality certificate.
Let $A,B$ be the $n\times n$ adjacency matrices of two simple graphs. Consider the rational polytope \[ \mathcal P(A,B)=\{X\geq0:X\mathbf1=X^{\mathsf T}\mathbf1=\mathbf1, \ AX=XB\}. \] A permutation matrix in this set is an exact isomorphism. Define $\Delta(X)=n-\|X\|_F^2$. Each row is a probability vector, so $\Delta(X)\geq0$, with equality exactly when every row is a coordinate vector. The column sums then make those coordinates distinct.

There is also a nonzero exact threshold: \begin{equation}\label{wsref-0-65-71-37-39-3a-72-6f-75-6e-64} X\in\mathcal P(A,B),\qquad \Delta(X)<\frac1{4n^2} \quad\Longrightarrow\quad G\cong H. \end{equation} For a proof, put $m_i=\max_jX_{ij}$. Since $\sum_jX_{ij}^2\leq m_i$, a row with $m_i\leq1/2$ would already contribute at least $1/2$ to $\Delta$, impossible under the displayed bound. The unique entries exceeding $1/2$ in different rows must occupy distinct columns, by the column-sum condition. They define a permutation matrix $P$. Moreover, \[ \sum_i(1-m_i)\leq\Delta(X),\qquad \|X-P\|_F^2=2\sum_i(1-m_i)-\Delta(X)\leq\Delta(X). \] Since $\|A\|_{\rm op},\|B\|_{\rm op}\leq n$ and $AX=XB$, \[ \|AP-PB\|_F\leq2n\|P-X\|_F<1. \] But $AP-PB$ has integer entries. Any nonzero integer matrix has Frobenius norm at least one, so $AP=PB$. This proves (eq79:round). It yields the contrapositive gap: for a nonisomorphic pair every feasible $X$ has $\Delta(X)\geq1/(4n^2)$.

The exactness of the linear constraints matters. A rationally certified feasible matrix below the threshold is a finite, polynomially checkable isomorphism certificate and can be rounded directly. A floating-point residual is not automatically such a certificate; quantitative residual terms would need to be included in the last inequality. Conversely, searching for the minimum of the concave function $n-\|X\|_F^2$ over the polytope is not linear programming. The existence of a gap does not prove that a deterministic polynomial-time algorithm finds a point attaining it whenever a permutation exists.

Attempt: explicit fractional and spectral obstruction.
Every pair of $d$-regular graphs of the same order admits $X=J/n\in\mathcal P(A,B)$, since $AJ=dJ=JB$. Thus feasibility alone cannot settle GI. For a stronger stress test take two graphs on $16$ vertices. The rook graph has vertices $(i,j)\in({\mathbb{Z}}/4{\mathbb{Z}})^2$, with edges between distinct vertices in the same row or column. The Shrikhande graph is the Cayley graph on the same group with connection set \[ \{(1,0),(-1,0),(0,1),(0,-1),(1,1),(-1,-1)\}. \] Both are six-regular, and direct neighbour counting gives \begin{equation}\label{wsref-0-65-71-37-39-3a-73-72-67} A^2=4I+2J \end{equation} for each adjacency matrix: the diagonal is six and each pair of distinct vertices has two common neighbours. On the subspace orthogonal to $\mathbf1$, (eq79:srg) gives eigenvalues $2$ and $-2$; trace zero then gives multiplicities $6$ and $9$, with the remaining eigenvalue $6$. So their adjacency spectra coincide exactly.

They are not isomorphic. A row of the rook graph is a four-clique. In the Shrikhande graph the induced graph on the six neighbours of $(0,0)$ is a six-cycle, as one checks by subtracting pairs in the displayed connection set. A clique containing $(0,0)$ therefore has at most three vertices. Translations are automorphisms, so this holds for a clique through any vertex; triangles exist. Its clique number is three, proving the claimed nonisomorphism. The companion script verifies the adjacency identities, the six-cycle and the clique numbers by exact finite enumeration.

For the graph Laplacian $L=6I-A$, every analytic function of $L$ lies in $\operatorname{span}\{I,A,J\}$ by (eq79:srg). In particular the heat matrix $e^{-tL}$ has, for every fixed $t$, entries determined solely by equality, adjacency, or nonadjacency of the two vertices, with identical three values for both graphs. Hence their global entry multisets agree at every heat time. This does not make the matrices conjugate by a permutation; it shows that the multiset invariant throws away the combinatorial placement of the entries. Also $AX=XB$ automatically implies $A^rX=XB^r$ for every $r$, so appending those redundant linear constraints does not remove $J/n$.

Stress test.
The gap (eq79:round) could be useful if one had a polynomial-time global near-integrality method, but no such method is supplied. Local improvement of a quadratic objective or convergence of a numerical flow to a stationary point is not a worst-case global certificate. Nor is an exponential branch search made polynomial simply because most branches collapse on typical graphs. Those are algorithmic missing steps, not statements that optimization methods cannot ever help.

The rook/Shrikhande example defeats the specified relaxation and spectral summaries; it does not defeat all higher-dimensional refinement algorithms, all invariant rings, or Babai's group-theoretic method. Likewise it is not a complexity lower bound. An isomorphism algorithm is allowed to exploit information beyond eigenvalues, heat-entry multisets, and fractional intertwiners. For graphs of different orders rejection is immediate; the argument's equal-order assumption is not a restriction on the full decision problem because that case is handled first.

Outcome.
Outcome: Exploratory approach with unresolved gap.

A robust exact rounding criterion is proved, and an explicit nonisomorphic pair defeats a natural convex/spectral simplification. The remaining question is whether a deterministic polynomial-time procedure can find near-integral feasible intertwiners whenever an isomorphism exists, or use a different complete invariant with polynomial evaluation cost. No such procedure or superpolynomial lower bound is obtained. These are method tests and a certificate lemma, not a claim of a new general GI algorithm or of literature priority.

Sources

  • S1 E. W. Weisstein, 'Graph Isomorphism,' MathWorld, accessed 2026. https://mathworld.wolfram.com/GraphIsomorphism.html. Catalogue formulation source.
  • S2 Graph Isomorphism: Mixed-Integer Convex Optimization from First-Order Methods. https://arxiv.org/abs/2512.17417. Catalogue research/status reference; not a proof endorsement.
  • S3 Finding Graph Isomorphisms in Heated Spaces in Almost No Time. https://arxiv.org/abs/2601.03787. Catalogue research/status reference; not a proof endorsement.
  • S4 Testing Isomorphism of Graphs in Polynomial Time. https://arxiv.org/abs/2305.12688. Catalogue research/status reference; not a proof endorsement.
  • E1 László Babai, Graph Isomorphism in Quasipolynomial Time, arXiv:1512.03547, main algorithmic theorem and complexity convention. https://arxiv.org/abs/1512.03547. Inspected 27 September 2026; quasipolynomial is not polynomial.
  • E2 Michael Anastos, Matthew Kwan and Benjamin Moore, Smoothed analysis for graph isomorphism, STOC 2025, DOI 10.1145/3717823.3718173. https://research-explorer.ista.ac.at/record/20007. Institutional publication record inspected 27 September 2026; smoothed hypotheses are not dropped in discussing the general target.