M28a The Hypertriad Foundations of the R=4 Operational Architecture

M28a introduces the Hypertriad, the canonical operational structure at rank R = 4. It shows that the transition from ln to slog (the Logarithm Jump) breaks the rank‑3 Triad coincidence and forces a richer geometry with two étages: a lower SC layer (ln-based) and an upper TC layer (slog-based). The rank‑4 conserved quantity is proposed as K_4(a,b) = sqrt(a b slog_B(a) slog_B(b)), replacing the logarithmic invariant of ranks 2–3. A central theorem proves the Non-Coincidence HCt != Ctet, where HCt (half-caterpillar midpoint) arises from the fluent iterator f_t(x) = x^(1-t) * B^(x^t), while Ctet is the HC-symmetric tetrational mean. The Hypertriad consists of ten objects, including Ctet, HCt, LC, RC, CLC, CRC, Atow[n], TAdd_B, TMult_B, TCHol_B, divided into: SC (ln, base-free): Atow[n] TC (slog, base-dependent): TAdd_B(a,b) = tet_B(slog_B(a) + slog_B(b)) TMult_B(a,b) = tet_B(slog_B(a) * slog_B(b)) chiral/a-ONS sector: LC, RC, HCt The Central Conjecture proposes recovery of the HC core via symmetrisation: Sym_prot(HCt) = Ctet. The rank‑4 Hermit unit is defined through the shifted quadratic z^2 - z = i*pi/2, z = slog_B(nu), nu = tet_B((1 ± sqrt(1 + 2 i pi))/2), establishing the TC analogue of the rank‑3 Hermit structure. Geometrically, the Hypertriad forms a rhombus connecting SC shadow (Atow[n]), TC core (Ctet), and the chiral sector, with two axes: HC-elevator and chirality. M28a defines this structure and its conjectural closure.