{"id":"modular-arithmetic-eliminates-simulation","text":"Solutions for problems with circular, periodic, or wrap-around structure uniformly reduce to O(1) closed-form expressions via modular arithmetic (min-of-diff-and-complement for shortest arc, mod-then-truncate for cyclic shifts, division-and-modulo for periodic bouncing), eliminating iterative simulation entirely.","truth_value":"IN","source":"","source_url":"","source_hash":"","justifications":[{"type":"SL","antecedents":["circular-distance-idiom-min-diff-n-minus-diff","shift-grid-k-mod-optimization","pillow-holder-o1-time"],"outlist":[],"label":"Three independent solutions use modular arithmetic to achieve O(1) on structurally similar circular/periodic problems — the shared technique constitutes a coherent reduction strategy complementary to algebraic closed-form reductions."}],"dependents":["mathematical-reduction-unifies-simulation-elimination"],"metadata":{"last_reviewed":"2026-06-07T22:02:22","review_result":"pass"},"created_at":"","updated_at":"","reviewed_at":"","verified_at":"","retracted_at":"","explanation":{"steps":[{"node":"modular-arithmetic-eliminates-simulation","truth_value":"IN","reason":"SL justification valid","antecedents":["circular-distance-idiom-min-diff-n-minus-diff","shift-grid-k-mod-optimization","pillow-holder-o1-time"],"label":"Three independent solutions use modular arithmetic to achieve O(1) on structurally similar circular/periodic problems — the shared technique constitutes a coherent reduction strategy complementary to algebraic closed-form reductions."},{"node":"circular-distance-idiom-min-diff-n-minus-diff","truth_value":"IN","reason":"premise"},{"node":"shift-grid-k-mod-optimization","truth_value":"IN","reason":"premise"},{"node":"pillow-holder-o1-time","truth_value":"IN","reason":"premise"}]}}