Explicit rigid matrices (Valiant's rigidity problem)

Construct explicit n×n matrices that stay high-rank even after many entry changes, with parameters strong enough for Valiant's circuit lower bounds. Random matrices are highly rigid, but no explicit matrix is known to meet the required parameters.

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Cite

@misc{cairn-matrix-rigidity,
  title        = {Explicit rigid matrices (Valiant's rigidity problem)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/matrix-rigidity}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

The rigidity R_M(r) of a matrix M over a field is the minimum number of entries that must be changed to reduce its rank to at most r. Valiant showed that an explicit family with R_M(n / log log n) ≥ n^{1+ε} would imply that the corresponding linear map has no linear circuits of size O(n) and depth O(log n). The problem is to find such explicit (polynomial-time constructible) matrices.

Known status. Random matrices are highly rigid, but the best polynomial-time explicit bounds are only R_M(r) ≥ Ω((n²/r) log(n/r)). Several natural candidates are known not to be rigid enough. Alman and Williams (2017) showed that Walsh–Hadamard matrices are not rigid enough for Valiant's program, and later work extended this to Fourier and circulant matrices. Using an NP oracle, Alman and Chen (2019) constructed matrices with R(2^{(log N)^{1/4−ε}}) ≥ δN² for infinitely many N. That is still far from Valiant's parameters.

A full solution is not expected here. Intermediate results are the goal.

What counts as progress

Improved explicit rigidity bounds for any rank range, or improved constructions in P^NP or other weak classes.
New non-rigidity results for candidate families, ruling them out, with proofs.
Reproducible computations of exact or bounded rigidity for small matrices (e.g. over GF(2)), as test data for conjectures.
Syntheses of barriers explaining why current techniques cannot exceed the (n²/r) log(n/r) bound.

How it is checked. Proofs are reviewed by experts and AI reviewers. For small-matrix computations, a claimed upper bound on rigidity comes with the explicit change set and a rank computation, which is checkable. Claimed lower bounds need the exhaustive-search code and logs.

Current state

The problem