The least cardinality is \[ \boxed{\Delta[\mathbb Z/127\mathbb Z]=13}. \] One attaining set is \[ A=\{0,1,5,11,19,38,61,78,80,81,93,102,109\}. \] The companion verifier forms all 169 ordered differences and obtains every residue modulo 127. Each nonzero residue occurs at least once.

Elementary counting gives only \(|A|\geq12\): an \(m\)-element set has at most \(m(m-1)\) nonzero ordered differences. The exclusion of cardinality 12 comes from exhaustive computation. Wiedemann computed minimum cyclic difference covers through modulus 133. Haanpää later searched all Abelian groups through order 127 with an orderly backtrack algorithm and reported the same minimum cardinalities as Wiedemann for every cyclic group in that range. Their independent results give 13 at modulus 127. Together with the displayed basis, this settles the value.

Let \(A\) have \(m\) elements. The identity is represented by \(a-a\). The ordered pairs \((a,b)\) with \(a\ne b\) number \(m(m-1)\), so \[ |A-A|\leq1+m(m-1). \] Covering a group of order 127 therefore requires \[ m(m-1)\geq126. \] This gives \(m\geq12\). At cardinality 12 there are 132 ordered nonzero differences available for 126 residues, leaving an excess of only six. Equivalently, the 66 unordered pairs must cover all 63 inverse classes with at most three repetitions. That small collision budget is the pruning rule used in exhaustive searches for this case.

Wiedemann's 1992 computation extended the table of lexicographically first minimum cyclic difference covers through modulus 133. Its modulus-127 result has cardinality 13, which excludes a 12-element cover.

Haanpää independently computed minimum difference covers for every finite Abelian group of order at most 127. His search recursively extends subsets, rejects affine-equivalent copies, and prunes a partial set once its repeated differences exceed the final collision allowance. For a cyclic group, the equivalence mappings used by the canonicity test are exactly affine maps \(x\mapsto ux+c\) with \(u\) a unit. The orderly-search theorem proves that every canonical subset is visited. Section 5 reports agreement with Wiedemann on the minimum cardinality of every cyclic group through order 127.

For this modulus, affine normalization may send any ordered pair of distinct elements to \((0,1)\). A hypothetical 12-set would then have to cover the 63 nonzero inverse classes using 66 unordered pairs, so only three repeated inverse classes are allowed. This is the exact monotone collision prune described by the published method. The two implementations supply independent exhaustive evidence for the lower endpoint 13.

The program computes every residue \((a-b)\bmod127\) for the displayed set. It verifies 127 covered residues and checks all 126 nonzero multiplicities. Among the nonzero residues, 100 occur once, 24 occur twice, and two occur four times. These multiplicities sum to the expected \(13\cdot12=156\) ordered nonzero differences.

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