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coefficient 0 is the intercept when present.","Values are exact strings: integer, decimal (\"3.5\") or fraction (\"7/2\"), never JSON numbers or exponents; comparisons are exact and outputs canonical reduced fractions. difference = claimed - actual.","Ragged rows, len(y) != rows or empty x are invalid_input; a claimed vector of the wrong length is valid:false dimension_mismatch, checked first.","Rank is exact; dependent_column is the first column of X that is a combination of earlier columns. rank < k (including fewer samples than columns) is singular_design: the least-squares coefficients are not unique.","Check order: dimensions, then rank (singular_design; no coefficient is compared because the solution is ambiguous), then coefficients in index order (coefficient_mismatch). coefficients is the exact least-squares solution when rank = k, else null.","Limits: 256 samples, 8 features, 30 digits per sample value, 1000 digits per claimed value, 9 claimed coefficients, 256 claimed residuals, and common denominators of at most 100 digits for the x values and for the y values and 1000 digits for the claimed coefficients; over a limit is limit_exceeded."],"limits":{"max_samples":256,"max_features":8,"max_sample_digits":30,"max_claim_digits":1000,"max_data_denominator_digits":100,"max_claim_denominator_digits":1000},"pricing":{"status":"unpriced","charge_usd":null},"input_schema":{"type":"object","additionalProperties":false,"required":["x","y","intercept","coefficients"],"properties":{"x":{"type":"array","minItems":1,"maxItems":256,"items":{"type":"array","minItems":1,"maxItems":8,"items":{"type":"string","pattern":"^-?(0|[1-9][0-9]*)(\\.[0-9]+|/[1-9][0-9]*)?$"}}},"y":{"type":"array","minItems":1,"maxItems":256,"items":{"type":"string","pattern":"^-?(0|[1-9][0-9]*)(\\.[0-9]+|/[1-9][0-9]*)?$"}},"intercept":{"type":"boolean"},"coefficients":{"type":"array","maxItems":9,"items":{"type":"string","pattern":"^-?(0|[1-9][0-9]*)(\\.[0-9]+|/[1-9][0-9]*)?$"}}}},"output_schema":{"type":"object","additionalProperties":false,"required":["valid","reason","rank","coefficients","first_discrepancy","reference_version"],"properties":{"valid":{"type":"boolean"},"reason":{"type":["string","null"],"enum":[null,"dimension_mismatch","singular_design","coefficient_mismatch"]},"rank":{"type":"integer","minimum":0},"coefficients":{"type":["array","null"],"items":{"type":"string","pattern":"^-?(0|[1-9][0-9]*)(/[1-9][0-9]*)?$"}},"first_discrepancy":{"type":["object","null"],"additionalProperties":false,"required":["check","index","claimed","actual","difference","expected_length","claimed_length","rank","dependent_column"],"properties":{"check":{"type":"string","enum":["dimensions","rank","coefficients"]},"index":{"type":["integer","null"],"minimum":0},"claimed":{"type":["string","null"]},"actual":{"type":["string","null"]},"difference":{"type":["string","null"]},"expected_length":{"type":["integer","null"],"minimum":0},"claimed_length":{"type":["integer","null"],"minimum":0},"rank":{"type":["integer","null"],"minimum":0},"dependent_column":{"type":["integer","null"],"minimum":0}}},"reference_version":{"type":"string","const":"NIST/SEMATECH e-Handbook of Statistical Methods sec. 4.1.4.1 and 4.4.3.1 (ordinary least squares, normal equations); 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