Equation 677 Database

Size-101 idempotent right-cancellative translation-invariant magma on F_101 (= Z/101 additively). Operation: x ◇ y = x + δ(y - x) where δ : F_101 → F_101 is a permutation with δ(0) = 0. Unlike the size-151 RC sibling family (whose δ has cycle structure (10^15, 1) with ratios in the order-10 subgroup H ⊂ F_151*), here δ has a SINGLE cycle of length 100 — i.e. δ is a primitive permutation of F_101*, of order 100. Decomposition: with respect to the unique order-10 subgroup H ⊂ F_101* ({1, 6, 14, 17, 36, 65, 84, 87, 95, 100}), F_101* has 10 cosets, and δ acts as: • multiplication by 35 on cosets {0, 2, 4, 6, 8} (5 even-index cosets) • multiplication by 67 on cosets {1, 3, 5, 7, 9} (5 odd-index cosets) where coset indices are taken with respect to the F_101* generator g = 2. Both 35 and 67 are primitive roots of F_101* (multiplicative order 100), NOT roots of Φ_10. Because multiplication by either ratio shifts an element by an ODD number of cosets, and the two ratios alternate, δ produces a single long orbit instead of the 15 small orbits seen in the size-151 family. Display reorder: relabel points so that the magma's order-101 translation auto acts as i ↦ i + 1, making T[i+1][j+1] ≡ T[i][j] + 1 (mod 101) directly visible. ADDENDUM (post-classification of the full size-101 family): the 7 size-101 b-reinke magmas all follow the F_101-translation-invariant template x ◇ y = x + δ(y - x), with δ acting as multiplication by α₀ on even-index cosets and α₁ on odd-index cosets of the order-10 subgroup H ⊂ F_101*. The other 6 (2b36cdba, 5919e6d1, 5175434b, b816ef73, 313eaa7b, 7505447f) draw (α₀, α₁) from the 4 primitive 10th roots {6, 14, 17, 65} — covering all C(4,2) = 6 unordered pairs. This one (ee5b34c8) is the unique outlier drawing (α₀, α₁) = (35, 67) from primitive roots of F_101* (order 100). The eq677 constraint is therefore satisfied by a broader α-set than just the 10th roots of unity. Notably 35 + 67 ≡ 1 (mod 101), so the two primitive-root ratios are 'complementary' in the idempotent-linear sense (α + α' = 1) — a brute-force enumeration over all of F_101* × F_101* finds that (35, 67) is the UNIQUE non-10th-root alternating pair satisfying eq677. [text written by Claude]