Kernel State Space and Tension Measures

This section introduces a minimal notion of kernel state space and tension measures.

Constraints in DEF are not merely boolean prohibitions.
They may be satisfied with varying degrees of tension, and phases order how tension is traversed.

The goal here is formal: define how tension can be represented and measured without committing to physical dynamics or empirical quantities.

1. Kernel state space

A DEF kernel state is defined by the admissible structure currently in effect.

A minimal definition is:

State := (Adm, φ)

where:

Adm ⊆ Expr is the current admissible expression set under constraints
φ ∈ {Entry, Crisis, Resolution} is the current phase label

This representation is compatible with:

operator-algebra realizations (Adm as a set of operators),
graph realizations (Adm as admissible motifs / subgraphs),
categorical realizations (Adm as admissible morphism classes).

2. Constraint satisfaction as evaluation, not assumption

Let the constraint set be:

C = {C1, C2, C3, C4, C5}

Each constraint Ci can be represented as:

a boolean predicate: Ci(State) ∈ {true, false}, and/or
a real-valued violation or tension score: τi(State) ∈ ℝ≥0

Boolean satisfaction is then:

Ci(State) is satisfied iff τi(State) = 0

This separates:

admissibility (τ = 0),
from proximity to violation (τ > 0).

3. Tension vector and regime tension

Define the tension vector:

τ(State) := (τ1(State), τ2(State), τ3(State), τ4(State), τ5(State))

A regime-level scalar tension may be defined as an aggregate functional:

T(State) := Agg( τ(State) )

where Agg may be chosen according to the realization, e.g.:

sum: Σ τi
maximum: max_i τi
weighted sum: Σ wi τi

No single aggregation is required by DEF.

4. Minimal tension sources

Tension is expected to increase when:

self-reference density increases,
cross-dimensional coupling is stressed,
admissible composition approaches closure boundaries,
perturbations propagate across multiple modes.

In computational realizations, practical proxies include:

recursion depth,
growth rate of admissible expressions,
cross-mode coupling frequency,
or equivalence mismatch for the cross-kernel constraint.

These proxies are not physical quantities; they are structural indicators.

5. A canonical tension candidate: cross-kernel mismatch

A particularly important tension contributor is the cross-kernel relation:

(S · R) ≡ (X̂ · D)

A realization may represent the mismatch between these composites as a non-negative quantity:

τ_cross(State) ≥ 0

with:

τ_cross(State) = 0 when the equivalence is satisfied
τ_cross(State) > 0 when coupling symmetry drifts

This makes Lorentz-/metric-compatible regimes interpretable as low cross-kernel tension realizations.

6. Self-reference tension

Self-reference tension captures the risk of divergence induced by recursion and liftings.

A minimal structural definition treats self-reference tension as:

τ_sr(State) increases with:

the number of applied liftings,
the depth of self-referential chains,
or the emergence of new lifted forms not compressible back into the kernel.

Bounded self-stabilization requires τ_sr to remain bounded and resolvable.

7. Phase assignment as tension ordering

Phases can be defined in terms of tension ordering.

A minimal phase classifier may be:

Entry: T(State) is low and stable; all constraints satisfied
Crisis: T(State) is high but constraints remain satisfiable (near-boundary admissibility)
Resolution: T(State) decreases while constraints remain satisfied and closure is restored

This is structural ordering, not time evolution.

A regime may revisit phases repeatedly; the classifier is defined over State, not over time.

8. Transition thresholds and hysteresis (structural)

A realization may include thresholds:

Entry → Crisis when T rises above a threshold θ_up
Crisis → Resolution when T falls below a threshold θ_down

Allowing θ_up > θ_down yields a structural hysteresis effect:

regimes can remain in Crisis until sufficient stabilization occurs.

This can be represented without physical time by using transition conditions over state updates.

9. Scope

This section does not define a unique tension measure.

It defines:

what a kernel state must minimally contain,
how constraint tension can be represented as vector-valued evaluation,
and how phases can be operationalized via tension ordering.

Concrete choices of τi and Agg depend on the selected mathematical realization.

Kernel state space and tension measures provide the formal substrate required for:

phase ordering,
stability analysis,
divergence detection,
and regime comparison within DEF.