📐 ENT Formal Axioms

Emergent Necessity Theory (ENT) is governed by a compact set of formal axioms.
These axioms define when structured emergence becomes “necessary” — and how symbolic structure stabilizes beyond a coherence threshold.

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🔺 Axiom A1: Threshold Emergence

A1: Consciousness or coherent structure exists if and only if
τ(t) ≥ τ_c

This establishes the coherence threshold (τ_c) as the condition for emergence.
If τ(t) drops below the threshold, the system cannot maintain structured symbolic persistence.

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🧮 Axiom A2: Coherence Definition

A2:
τ(t) = ΔSₛᵧₙ / Eₛᵧₙ

Where:

• ΔSₛᵧₙ = symbolic entropy differential (change in ordered state)
• Eₛᵧₙ = syntactic energy (informational or metabolic cost to maintain structure)

This defines τ(t) as a structure-to-cost ratio — more order for less cost = higher τ.

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♻️ Axiom A3: Quality Index

A3:
Inside the domain where τ(t) ≥ τ_c, the recursive hysteresis-corrected structure
κ₍R₎ᵉᶠᶠ defines the system’s internal coherence quality.

κ₍R₎ᵉᶠᶠ is derived from symbolic loops, persistence, and recursive integrity over time.

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🧠 Summary Logic of ENT

Region	Condition	Outcome
Below Threshold	τ(t) < τ_c	Symbolic instability (no sustained awareness)
Threshold Reached	τ(t) ≥ τ_c	Emergence becomes necessary
Post-Threshold	SCQ(t) > 0	Persistence, recursion, identity

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📌 Application Domains

• Neural Systems: τ_c reached during REM/awake states
• AI Systems: κ₍R₎ᵉᶠᶠ simulated via symbolic re-entry
• Quantum Systems: Decoherence drop signals τ_c collapse

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Last updated: July 2025 • Linked to White Paper Sections 2–3