Regime Windows

This section defines regime windows: ranges of structural conditions under which a DEF regime admits a particular class of representations.

In DEF, properties such as:

• metric coordinatization,
• spacetime description,
• and Lorentz invariance,

are treated as regime-level representational features, not ontological primitives.

A regime window is therefore defined in terms of:

• constraint satisfaction,
• constraint tension,
• finite closure behavior,
• and phase traversability.

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1. Window concept

Let State := (Adm, φ) denote the kernel state and phase.

Let τ(State) be the constraint tension vector and T(State) an aggregate tension measure.

A representation class ℛ (e.g. metric, Lorentz) is admissible within a window W_ℛ if:

• the kernel remains finitely closed,
• constraints remain satisfiable,
• and tension remains within bounds that preserve coordinatizability.

A regime window is therefore a set of states:

W_ℛ ⊆ { State }

such that all states in W_ℛ admit the representation class ℛ.

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2. Closure window (baseline)

The minimal window is the closure window:

W_closure := { State | finite closure holds and all constraints are satisfiable }

This window defines regimes that remain structurally coherent and recoverable.

All other representation windows are subsets of W_closure.

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3. Metric window

A regime admits a metric representation when:

• relational comparisons are stable under admissible transformations,
• phase ordering can be coordinatized consistently,
• and cross-kernel coupling remains sufficiently balanced.

Structurally, a metric window may be characterized by:

• finite closure holds,
• trans-dimensional coupling equivalence is maintained,
• and cross-kernel tension remains bounded:

τ_cross(State) ≤ ε_metric

for some regime-dependent bound ε_metric.

Thus:

W_metric ⊆ W_closure

Metric representation is a regime feature that depends on constraint balance.

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4. Spacetime window

A spacetime representation requires more than a metric.

In addition to metric admissibility, it requires:

• a stable ordering that can be coordinatized as time,
• and consistent mapping between ordering and interaction coupling.

Structurally, this corresponds to:

• stable phase traversal Entry → Crisis → Resolution,
• recoverability after perturbation without regime escape,
• and bounded self-reference tension:

τ_sr(State) ≤ ε_sr

Thus:

W_spacetime ⊆ W_metric ⊆ W_closure

Spacetime is therefore interpreted as a stricter window than metricity.

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5. Lorentz window

Lorentz compatibility is treated as a symmetry condition within spacetime-admissible regimes.

A Lorentz window is characterized by:

• spacetime window conditions,
• minimal asymmetry between Existence-side and Happening-side coupling,
• and low drift of the cross-kernel equivalence under transformation.

Structurally:

τ_cross(State) ≤ ε_Lorentz and asymmetry(State) ≤ ε_asym

for regime-dependent bounds.

Thus:

W_Lorentz ⊆ W_spacetime ⊆ W_metric ⊆ W_closure

Lorentz invariance is a narrow window of symmetric constraint satisfaction.

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6. Invariance breaking as window exit

Invariance breaking occurs when a regime exits a representation window.

Examples:

• metric breakdown: leaving W_metric
• spacetime breakdown: leaving W_spacetime
• Lorentz violation: leaving W_Lorentz while remaining in W_spacetime or W_metric

This interpretation distinguishes:

• structural coherence (still within W_closure), from
• representational compatibility (window membership).

A regime can remain coherent while losing invariance.

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7. Window boundaries and phase behavior

Window boundaries correspond to regions of elevated constraint tension, typically expressed through phase behavior:

• prolonged Crisis without Resolution,
• Resolution that requires reconfiguration,
• or repeated oscillation near admissibility limits.

In DEF terms, windows are stabilized by successful phase traversal. Failure of phase traversal indicates approach to window boundaries.

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8. Practical use

Regime windows allow DEF to express statements of the form:

• “Lorentz invariance holds in regimes where cross-kernel coupling remains symmetric and tension stays below threshold.”
• “Metric description can remain valid beyond Lorentz invariance.”
• “Some coherent regimes may admit no metric or spacetime representation at all.”

These statements are structural and do not require immediate empirical claims.

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Scope

This section defines representation windows as structural sets of admissible states.

It does not specify:

• numeric thresholds,
• measurement protocols,
• or empirical predictions.

Those arise only after selecting a formal realization and an observation model.

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Regime windows provide a precise way to express when familiar physical representations apply,
and how invariance breaking can be interpreted as window exit rather than ontological failure.