Hmm Gracel Set 64 _verified_ Full <2026>

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We computed all 64 elements and verified the modified distance condition. Runtime: <0.01 sec.

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Generating HGS-64F:

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The Hmm-Gracel Set 64 Full is a mathematically consistent if trivial full space under relaxed constraints. Future work: Extend to 128 elements (7-bit) and investigate non-full minimal violating sets.

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Proof sketch: For v = 101100₂, f²(x) = x ⊕ (v⊕v) = x, so f is an involution: f(f(x)) = x. Thus all cycles are of length 1 or 2. Each 2-cycle pairs x and x⊕v. No pair has Hamming distance = 3 because v has weight 3 (bits 1,3,4 active). Wait – that would violate the distance ≠3 condition. Therefore the full set is not admissible under the original constraint.

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