qector-math-foundationslisted

# QECTOR Math Foundations - Strict Ground Truth Rules Source of authority: QECTOR Decoder v3 reference manual v1.0.0 (DOI `10.5281/zenodo.21941046`), chapters 2, 3, 4-13, 15, 19, 20, 22. The source document is not redistributed here. 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 }`. In the stabilizer / CSS setting, where the relevant checks are self-orthogonal (`H H^T = 0`), `im(H^T)` is a subspace of `ker(H)`; only then does `ker(H) \ im(H^T)` describe non-trivial logical operators. Arbitrary matrices require code-provided stabilizer and logical matrices. - 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)**. For a valid stabilizer / CSS sector with `rank(H) = r`, self-orthogonal checks, and