The Seeded Instance

Solve a sparse linear system of 10,000 equations in 10,000 unknowns whose full construction recipe is published in this page, to a relative residual of at most 1e-8. The instance files are pinned by SHA-256 and are authoritative; a Solver may reproduce them locally from the recipe as a cross-check before solving. Solving the wrong instance (an off-by-one seed, a mis-implemented recipe) fails the residual check, so reproducibility discipline pays.

The instance is a sparse, nonsymmetric, nonsingular real matrix C of size 10000 x 10000 in MatrixMarket coordinate format, and a right-hand-side vector b of length 10000 in CSV. Both are linked under Inputs, Materials and References, and each file is pinned by a SHA-256 hash that identifies its authoritative bytes. The recipe below documents exactly how the instance was constructed. Reproducing the instance locally from the recipe is permitted as a cross-check and is neither required nor evaluated: only the submitted solution vector is judged.

Recipe. Build the matrix C with numpy and scipy as follows:

• Use numpy.random.default_rng(20261009).
• The diagonal is 8.0 for all 10,000 rows.
• For each k in -3, -2, -1, 1, 2, 3 (in this order), draw v = rng.uniform(-1.5, 1.5, 10000 - abs(k)) and place v at the offsets: for k > 0 place v in rows 0..9999-k, columns k..9999; for k < 0 place v in rows abs(k)..9999, columns 0..9999-abs(k). Also place the same vector v at the transposed position (rows and columns swapped).
• The resulting 10000 x 10000 matrix C has exactly 69988 nonzeros. Assemble from the coordinate triplets (row, col, value) in the order generated above, with the 10000 diagonal triplets appended last, summing duplicate coordinates.
• Draw x_true = rng.uniform(-2.0, 2.0, 10000) and set b = C @ x_true in binary64 (scipy.sparse CSR product).

The pinned files under Inputs, Materials and References are the authoritative instance for every acceptance check; the recipe explains their construction and serves any Solver who prefers to regenerate and compare before solving.

Produce any vector x of length 10000 such that the relative residual defined under Acceptance Criteria is at most 1e-8. C is strictly diagonally dominant and well-conditioned; the bound is comfortably attainable with standard sparse linear algebra in double precision. The method is the Solver's choice and is not evaluated.

Both files must be present as bytes in the Submission; a Submission missing either file fails Acceptance Criteria 1 and 2. Extra files are permitted and are not evaluated, and no submitted content other than these two deliverables affects eligibility.

The two files below are the authoritative and complete instance, pinned by SHA-256; the recipe in Challenge details reconstructs them.

The MatrixMarket file's first line declares coordinate format, general (symmetry) attribute; row and column indices are one-based. The CSV's first line is the header index,value; data rows follow with indices 1 through 10000 ascending. No other data defines the calculation. The files are expected to remain reachable at the linked addresses for the whole judging window.

• Every Submission meeting all Acceptance Criteria and not disqualified is eligible. Eligibility is binary; no score is computed.
• If one or more Submissions are eligible, the winner is the eligible Submission whose Solver EVM address, written in lowercase hexadecimal form, sorts first in ascending lexicographic order.
• If no Submission is eligible, the outcome is no_valid_submission.

• A required deliverable is missing or cannot be parsed under the formats stated in Deliverables, which fails Acceptance Criteria 1 or 2.

Deriving or explaining the solution method, proving how a Solver produced its vector, the exact serialization of any locally regenerated copy of the instance, and any scientific interpretation of the resulting vector are out of scope.