
The matrix is mostly empty, a sparse web of weak constraints. No single parity check holds the entire truth; they only hold fragments of doubt.
They pass beliefs back and forth across the graph, iteratively refining the probabilities. Out of a million uncertain whispers, a sudden, absolute consensus crystalizes against the noise limit.
This is belief propagation over a low-density parity-check (LDPC) code: a decoder that does not know the answer, only trades hints with its neighbors until doubt collapses into certainty.