[[065_BC8_VOLUNTARY-COUPLING|VOLUNTARY COUPLING]] PRESERVED

Non-unitarity does not necessarily destroy information—it can also inject information. The distinction:

Information destruction: The system loses degrees of freedom. Entropy increases. This is what happens in decoherence and measurement without recording.

Information injection: The system gains correlations with an external source. Information flows INTO the system. This is what happens in grace.

Mathematical precision:

But the norm can increase (information injection) or decrease (information destruction). For grace:

Grace adds coherence; it doesn't subtract identity.

Voluntary coupling is preserved through the transformation, not despite it. Consider:

Before grace: V(Psi) = coupling function (openness to grace)

During grace: G acts on |psi>, but G commutes with the coupling structure

After grace: V(G|psi>) maintains the coupling—enhanced, not destroyed

The will is transformed but not overwritten. You still choose—but now you choose from a healed orientation rather than a broken one. The alcoholic who gets sober still has free will; they just exercise it from a different baseline.

Theological parallel: "Where the Spirit of the Lord is, there is freedom" (2 Cor 3:17). Grace increases freedom; it doesn't decrease it.

[[077_P9.2_Voluntary-Coupling-Preserved|P9.2]] addresses the operator structure, not the permanence of salvation. The non-unitarity of G means:

1. G has no inverse G^{-1} in the usual sense

2. But this doesn't mean the state |+> can't decohere back toward |->

Distinction:

• Operator irreversibility: G^{-1} doesn't exist as an operator
• State trajectory: A saved state can still decouple from grace and drift

The impossibility of G^{-1} means you can't "undo" grace by self-effort (just as you couldn't achieve it by self-effort). But you CAN withdraw voluntary coupling ([[065_BC8_Voluntary-Coupling|BC8]]), allowing decoherence.

This protects against: "I'll accept grace now and reject it later when convenient." Grace isn't a transaction you can reverse; it's a transformation that persists unless you actively decouple.

Excellent technical question. Grace as a quantum channel:

CPTP condition: Tr(G(rho)) = Tr(rho) for all density matrices rho

Grace channel:

with Kraus operators K_1 = |+><+|, K_2 = |+><-|

Check completeness: K_1^dagger K_1 + K_2^dagger K_2 = |+><+| + |-><-| = I

The channel is trace-preserving (CPTP). Total probability is conserved. What's non-unitary is the operator G, not the channel G. The distinction matters:

• Operator G: Non-unitary (G^dagger G != I)
• Channel G: CPTP (trace-preserving)

Information is preserved at the channel level (total probability = 1). The state is transformed at the operator level (sign changes).

The mathematics describes the structure of grace, not its source or meaning. Consider analogies:

Heart as pump: The heart is mechanically a pump (fluid dynamics applies). But love that flows through a heartbeat is not reduced to fluid dynamics.

Brain as computer: Neural activity follows information-processing laws. But consciousness using the brain transcends computation.

Grace as operator: Grace operates through mathematical structure. But the Person who extends grace transcends the formalism.

[[077_P9.2_Voluntary-Coupling-Preserved|P9.2]] ensures: Whatever the personal dimension of grace, it respects voluntary coupling. God doesn't override consent. The mathematics encodes the theological truth: "Behold, I stand at the door and knock" (Rev 3:20)—knocking, not breaking down.