XORSystem
plain-language theorem explainer
A system of XOR constraints over n boolean variables is a list in which each entry names a subset of variables together with the required parity of their sum modulo 2. Researchers building backpropagation solvers for hybrid CNF-plus-parity instances cite the type when stating consistency predicates and step relations. The declaration is a one-line abbreviation that expands directly to the list type of the sibling constraint structure.
Claim. An XOR system on $n$ variables is a finite list of constraints, where each constraint is a pair consisting of a list of variable indices from $1$ to $n$ together with a required parity bit.
background
The sibling structure pairs a list of variable indices with a boolean parity target; satisfaction requires that the parity of the selected bits equals the target. The module places these parity clauses beside ordinary CNF clauses to form mixed satisfiability problems. Upstream results supply the J-cost convexity and local eight-tick update rules that embed discrete constraint systems inside the Recognition Science derivation of complexity from ledger factorization and spectral emergence.
proof idea
The declaration is a one-line abbreviation that expands directly to List of the sibling constraint structure, carrying no additional computation or proof obligations.
why it matters
The type is the second parameter of BackpropSucceeds and BPStep, allowing the backpropagation relation to enforce parity while preserving compatibility with CNF models. It supplies the interface needed for the completeness statements under isolation invariants. In the framework the construction encodes parity constraints that arise from the eight-tick octave and ledger factorization steps of the foundational forcing chain.
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