M21f systematizes the algebra of units across all Operational Number Systems (ONS) developed in the Olympus Programme, providing a permanent taxonomy that resolves long‑standing ambiguities in notation, identity elements, and cross‑core comparisons. Its central aim is not to introduce new spectral or analytic machinery, but to stabilize the unit layer—the “atoms of coherence” that allow different operational cores to be compared, transported, and composed without contradiction. The monograph begins by formalizing the three‑way fork between the Hyper Core (HC), Symmetric Core (SC), and TetraCore (TC). While all three agree at addition and multiplication, they diverge immediately above, and M21f proves that no higher‑rank operation coincides across cores for generic inputs. This necessitates a strict and permanent unit‑naming convention: lower‑case symbols for SC/TC units, upper‑case for HC units, and mandatory system labels whenever multiple cores are present. This convention is fixed for the remainder of the series and eliminates accidental identifications that previously obscured structural differences. A major contribution is the classification of Hermit units across ranks and cores. M21f shows that each core carries its own Hermit cascade, with distinct ghost units, cloning behavior, and diagonal values. Even when units share a common “floor” (e.g. the ghost at multiplication rank), they diverge immediately above it. This establishes that unit coincidence is an exception confined to the shared floor, not a feature that propagates upward. The monograph then analyzes unit transport across ranks, clarifying how units behave under rank elevation and why certain transports are forbidden. In particular, it shows that naïve identification of HC and SC units at higher ranks leads to contradictions in diagonal growth and fixed‑point structure. This analysis underpins later results on elevator uniqueness and explains why the Hyper Core must be treated as a distinct meta‑system rather than a deformation of SC or TC units. M21f also places the TetraCore units in their definitive role as the first bounded, slog‑native unit system: the “slog shelf” of the Olympus Programme. TC units are shown to generate genuinely new algebraic behavior, including bounded diagonals and non‑explosive cloning, in contrast to the expanding SC and HC regimes. This prepares the ground for later work on half‑étages and fractional ranks, explicitly deferred to M21g.
Paweł Łukasz Garycki (Fri,) studied this question.