Scale, Quantization, and Exchange
This section examines how scale structure and quantization can arise within DEF regimes, with particular emphasis on the operative mode Exchange (X̂).
Quantization is treated here as a regime-level outcome of closure constraints, not as a primitive axiom.
Scale as a regime property
Section titled “Scale as a regime property”In DEF, scale is not assumed a priori.
A regime exhibits scale structure when:
- compositions remain closed but non-uniform across magnitudes,
- constraint tension changes with coupling intensity,
- or phase traversal behaves differently at different interaction resolutions.
Scale is therefore interpreted as a structural property of admissible composition,
not as a fundamental coordinate.
Exchange as the carrier of discreteness
Section titled “Exchange as the carrier of discreteness”Among the operative modes:
- Structure (S) supports persistence,
- Space (R) supports relational separation,
- Dynamics (D) supports ordered change,
- Exchange (X̂) supports interaction and transfer.
DEF treats Exchange as the most natural locus for granularity, because:
- interaction often occurs in bounded events,
- transfer admits minimal units in many realizations,
- and closure constraints naturally enforce boundedness.
Thus, quantization is interpreted as discrete admissibility in exchange pathways.
Quantization as constrained admissibility
Section titled “Quantization as constrained admissibility”Within DEF, quantization can be expressed structurally as:
- not all exchange magnitudes are admissible,
- admissible exchange occurs in discrete steps,
- closure is preserved only for specific exchange increments.
This does not imply a particular unit.
It implies a constraint-induced discrete spectrum of admissible exchange operations.
Relation to finite closure
Section titled “Relation to finite closure”Finite closure requires that:
- recursive compositions do not generate unbounded operator hierarchies,
- self-referential liftings remain bounded,
- perturbations can be resolved within the kernel.
Quantization can support finite closure by:
- preventing arbitrarily small destabilizing perturbations,
- limiting runaway recursion through minimal increments,
- and stabilizing phase transitions under repeated interaction.
Thus, quantization is interpreted as one possible mechanism by which regimes achieve bounded closure.
Scale dependence and invariance
Section titled “Scale dependence and invariance”In metric and Lorentz-compatible regimes, invariances suggest smooth transform structure.
However, if quantization is present:
- invariance may be approximate,
- scale-dependent deviations may appear,
- and symmetry may hold only within specific bands of interaction resolution.
In DEF terms, this corresponds to:
- regime realizations where constraint balance holds only within a finite scale window,
- and invariance breaks outside that window.
Exchange-driven regime transitions
Section titled “Exchange-driven regime transitions”Exchange plays a special role in regime transitions:
- increasing exchange intensity can increase constraint tension,
- increasing exchange granularity can change phase traversal properties,
- and asymmetric exchange coupling can induce invariance breaking.
Thus, quantization and scale effects are not merely descriptive;
they are potential structural drivers of transition boundaries.
This section does not claim:
- a specific quantization scale,
- a specific discrete spectrum,
- or a specific physical realization (energy, information, charge, etc.).
It states only that:
- quantization is structurally admissible in DEF,
- and may be required in some regimes to preserve finite closure.
Scale structure and quantization are therefore treated as regime-level consequences of
Exchange under closure constraints, providing a bridge between structural kernel theory
and discrete physical descriptions.