Metric Emergence and Invariance Breaking
This section examines how metric structure can emerge within DEF regimes
and how invariance breaking arises when constraint balance is lost.
Metrics are treated here as regime-level representations,
not as fundamental primitives.
Metric structure as a representational layer
Section titled “Metric structure as a representational layer”In DEF, a metric is not assumed a priori.
A metric becomes admissible only when:
- relational structure can be compared consistently,
- ordering relations are stable,
- and interaction coupling admits symmetry.
Metric structure is therefore a secondary representation of a closure-stable kernel.
Conditions for metric emergence
Section titled “Conditions for metric emergence”Metric emergence requires that:
- the Existence-side coupling (Structure · Space),
- and the Happening-side coupling (Exchange · Dynamics),
remain mutually compatible under transformation.
Formally, this corresponds to satisfaction of the core constraint:
(S · R) ≡ (X̂ · D)
When this equivalence holds uniformly across admissible transformations, relations may be coordinatized by distances and intervals.
Relation to Lorentz-invariant regimes
Section titled “Relation to Lorentz-invariant regimes”In regimes where:
- constraint tension is minimized,
- phase ordering is reversible,
- and coupling symmetry is preserved,
the emergent metric admits Lorentz-invariant form.
The Minkowski metric is thus interpreted as a balanced kernel representation,
encoding symmetric coupling between configuration and interaction.
Metric asymmetry and invariance breaking
Section titled “Metric asymmetry and invariance breaking”Metric invariance breaks when constraint symmetry is lost.
This occurs when:
- Existence-side relations dominate over Happening-side relations, or vice versa,
- exchange–dynamics coupling becomes anisotropic,
- or phase resolution fails to restore balance.
In such regimes:
- metric interpretation may persist locally but fail globally,
- invariant intervals may become direction- or scale-dependent,
- preferred frames or directions may emerge.
Invariance breaking as regime transition
Section titled “Invariance breaking as regime transition”In DEF, invariance breaking does not imply inconsistency.
Instead, it signals:
- transition between regimes,
- partial loss of coordinatizability,
- or movement toward divergent closure.
Lorentz violation is therefore interpreted as a structural indicator,
not a fundamental breakdown of physical law.
Metric loss and non-metric regimes
Section titled “Metric loss and non-metric regimes”Some regimes may not admit metric representation at all.
In such cases:
- relational ordering cannot be embedded into distance measures,
- time cannot be globally defined,
- or interaction lacks stable coordinatization.
DEF explicitly allows for non-metric regimes,
even when closure remains intact.
Observational consequences
Section titled “Observational consequences”Observers embedded in metric regimes will:
- infer distances and durations,
- formulate invariant laws,
- and describe interactions geometrically.
Observers near invariance-breaking boundaries may observe:
- anisotropies,
- scale-dependent propagation,
- or breakdown of spacetime description.
These observations reflect regime properties, not ontological failure.
This section does not propose specific metric forms
or predict concrete symmetry-breaking signatures.
It clarifies the structural conditions under which metric descriptions are possible, stable, or lost within DEF.
Metric emergence and invariance breaking are thus understood as
regime-dependent representational phenomena, grounded in DEF constraint structure.