Technical notes · Value (V)

V as Evaluative Gradient in Informational Systems

This page provides the rigorous foundation for Value (V) as the necessary evaluative structure arising from informational self-maintenance. It assumes familiarity with the formal definitions of Information (I) and Awareness (A).

1. Preliminaries

Let S be an informationally integrated system with:

• state space Ω
• transition structure T: Ω → Ω
• internal awareness function A: Ω → Σ, mapping system states to internal informational encodings

We define Value not as subjective preference, but as the structural gradient through which a system stabilizes and extends its informational coherence.

2. The Necessity of Value

An aware system must discriminate between states that strengthen or degrade its own informational coherence. This yields an evaluative ordering over Ω.

Definition (Value Functional). V: Ω → ℝ is any functional such that V(x) > V(y) ⇔ state x increases informational coherence relative to state y.

Value is thus structurally necessary for any system with recursive informational awareness.

3. Derivation of Value from Awareness

Proposition 1. If a system S has awareness A of its own informational state, then a non-trivial Value functional V necessarily exists.

Sketch: Awareness provides internal models over Ω. Any internal model that distinguishes change must encode deviations from favorable or unfavorable informational configurations. This discrimination induces an ordering, which is V.

Proposition 2 (Evaluative Gradient). For any trajectory τ = (x₀, x₁, x₂, …) under T, S exhibits evaluative behavior iff ∃V such that V(xₙ₊₁) ≥ V(xₙ) for non-random transitions.

4. Value as System Stability Criterion

Value encodes the tendency of a system to maintain coherent internal informational structure. High-value states correspond to attractors in the system’s informational dynamics.

5. Conclusion

Value is the mathematically necessary evaluative consequence of a system possessing internal awareness. It formalizes the gradient by which systems regulate themselves, and forms the basis of Meaning (M).