qector-math-foundationslisted

# QECTOR Math Foundations - Strict Ground Truth Rules Source of authority: `QectorDecoder_v3_Reference_Manual_v1.0.0.pdf` (DOI 10.5281/zenodo.21941046), chapters 2, 3, 4-13, 15, 19, 20, 22. These rules are **normative**: every statement the plugin produces must satisfy them. ## Rule M0 - Provenance and resolve - All arithmetic is over the binary field F2 = {0, 1}; addition is XOR (mod 2), multiplication is AND. Vectors are columns by default. - Syndrome faithfulness is written **H c = s (mod 2)**. - ker(H) = { v : H v = 0 }; im(H^T) sits inside ker(H); errors in ker(H) \ im(H^T) are non-trivial logical operators. - A claim only exists if you can point to (a) this manual/theorem, (b) a live execution recorded in this package, or (c) a surviving artifact with the metadata of Rule M5. Otherwise mark it "not verified". ## Rule M1 - The core theorems (chapter 3) - **Theorem 1 (syndrome faithfulness and correction validity).** For H in F2^(m x n), true error e in F2^n, syndrome s = H e: a decoder returning c is faithful iff H c = s; moreover H c = s implies c + e in ker(H). - **Theorem 2 (logical error criterion).** With rank(H) = r and H c = H e = s, decoding is *logically correct* iff c + e in im(H^T); a logical error occurs iff c + e in ker(H) \ im(H^T). Score on the logical coset, never on raw correction equality (degeneracy is respected). - **Theorem 3 (path-flipping faithfulness of MWPM).** For non-empty defect set D and M a minimum-weight perfect mat