Relational physics is the operational layer that the derivation produces at step 17 (Profiles and Physics), when Geometry acts on the contents of Profiles. Four structures emerge: Observable, State, Evolution, and Measurement. Together they constitute the physics of the relational field — not an application of the derivation to physics but a structure that the derivation forces.
In standard mathematics, the four structures correspond to elements of observable algebras, states on those algebras, dynamical semigroups (evolution), and positive operator-valued measures (measurement). The relationship between evolution and measurement — the fact that how things change and how things are observed constrain each other — appears in quantum mechanics as the Heisenberg vs. Schrodinger picture. In the relational derivation, this relationship is Residuation: the same adjunction pattern that governs Implication in the Heyting algebra and Flow-Nucleus interaction in the dynamics.
The derivation produces these structures as follows:
- A Filter carves out a Profile — a complete relational universe with its own Terms, Judgements, Order, Flow, Nucleus, and Geometry.
- Within a Profile, Observation is localized: an Observable is what can be seen from a particular position in a particular Profile.
- A State is what remains fixed under a Profile’s Flow — the structures that directed transformation does not currently change.
- Evolution is Flow applied to a State: how the state changes. Measurement is Nucleus applied to a State: how the state is observed.
- Evolution and Measurement interact through Residuation — the third and final level where this pattern recurs (after logic and dynamics).
The physics is internal to each Profile. Different Profiles may yield different Observables, States, Evolution, and Measurement from the same underlying field, because each Profile reconstructs the derivation with its own perspective. This is the relational analog of frame-dependence in physics: what you observe depends on where you stand, but the structure governing observation is the same everywhere.
The old formulation of relationality explored connections to conservation laws, transport equations, detailed balance (forward-reverse symmetry), entropy (monotonicity under flow-measure evolution), and spectral decomposition. These remain areas for further development within the canonical framework.