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 K₄ (a, b) = sqrt (a b slogB (a) slogB (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 fₜ (x) = x^ (1-t) * B^ (xᵗ), while Ctet is the HC-symmetric tetrational mean. The Hypertriad consists of ten objects, including Ctet, HCt, LC, RC, CLC, CRC, Atown, TAddB, TMultB, TCHolB, divided into: SC (ln, base-free): Atown TC (slog, base-dependent): TAddB (a, b) = tetB (slogB (a) + slogB (b) ) TMultB (a, b) = tetB (slogB (a) * slogB (b) ) chiral/a-ONS sector: LC, RC, HCt The Central Conjecture proposes recovery of the HC core via symmetrisation: Symₚrot (HCt) = Ctet. The rank‑4 Hermit unit is defined through the shifted quadratic z² - z = i*pi/2, z = slogB (nu), nu = tetB ( (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 (Atown), TC core (Ctet), and the chiral sector, with two axes: HC-elevator and chirality. M28a defines this structure and its conjectural closure.