{ "schema": "gretna-junction.original-instruction-proof.v1", "creator": "Gretna Junction / CrownThrive, LLC", "license": "CC BY 4.0", "licenseUrl": "https://creativecommons.org/licenses/by/4.0/", "createdAt": "2026-10-19T08:31:09.557Z", "sourceKind": "Publisher-created primary instruction and finite enumeration; no external factual source.", "method": "Complete deterministic enumeration, not random simulation or performance benchmarking.", "version": "2026.10.19-dawn.pattern.v1", "canonicalUrl": "https://gretnajunction.com/learning/production-care-20261019-dawn/pattern-proof.json", "topicIdentity": "creators-eight-position-pattern-rotation-reflection-workshop", "model": { "positions": 8, "alphabet": [ "0", "1" ], "positionOrder": "clockwise from an initially labeled seam", "rotationRule": "Strings are equivalent if one cyclic rotation equals the other.", "reflectionRule": "Strings are equivalent if a rotation of either the string or its reversal equals the other.", "notAllowed": "Complementing 0 and1, changing spacing, introducing a third mark, or judging artistic quality." }, "proof": { "partition": "Rotation by zero fixes each string; reversing a rotation gives an inverse; composing rotations gives another rotation. Hence sharing a rotation class is reflexive, symmetric and transitive. Including reversal and rotations likewise closes under composition and inverse, so it also partitions the space.", "groupSizes": "Let p be the smallest positive shift restoring a string. Shift8 restores it. Divide8=q*p+r with0<=r