Chapter 6: Entropic Control of Delta
Recursive Saturation, Stability, and the Fate of Field Collapse
6.1 Overview
In previous chapters, entropy was treated as an output of recursive dynamics. This chapter reveals entropy’s active role: it modulates the gravitational field Delta.
Entropy is not merely a consequence—it becomes a control parameter for gravity itself.
6.2 Delta as Memory-Constrained Collapse
In ITT, Delta is the gravitational projection of recursion density:
Δ⃗(x,t) ∝ 𝒜(x,t) · Tr(ℳ(x,t))
| Classical | ITT |
|---|---|
| g ∝ M/r² | Δ ∝ 𝒜·Tr(ℳ) |
| Mass is primitive | Memory is primitive |
| Gravity is fixed | Gravity is entropy-modulated |
6.3 The Collapse Viability Factor
Define the Collapse Viability Factor:
χΔ(x,t) = (𝒜 · Tr(ℳ)) / (Sθ(x,t) + ε)
| χΔ Value | State | Physical Meaning |
|---|---|---|
| χΔ high | Low entropy, high alignment | Strong gravitational focus |
| χΔ low | High entropy | Gravity “diluted” by disorder |
| χΔ → 0 | Entropy saturation | Gravitational collapse impossible |
Collapse “unwinds” as the structure forgets its recursive tension.
6.4 Delta Stability Threshold
Define a critical threshold:
χΔ(x,t) > Ξcrit ⟹ Gravitational Focal Lock
χΔ(x,t) < Ξcrit ⟹ Delta Decay Begins
This defines entropy-induced gravitational collapse death—when structure becomes too disordered to bind gravity.
6.5 Lock Evolution with Hysteresis
ITT substrates remember past alignment:
dℒ/dt = −αθ · σθ + β · ℒprior + γ · 𝒜²
| Term | Effect |
|---|---|
| −αθ · σθ | Entropy degrades lock |
| +β · ℒprior | Past lock persists (memory) |
| +γ · 𝒜² | Alignment stabilizes lock |
Physical meaning: Even after order is lost, the system may “snap back” if local entropy is re-suppressed.
6.6 Applications in Collapse Control
1. Halt Black Hole Evaporation
By saturating entropy: If Sθ → Sθ,max and ℒ → 1 simultaneously → Evaporation stops
2. Pause Collapse at Planck Thresholds
When n → nmax, drift cannot increase further, so σθ is bounded.
3. Modulate Gravitational Structure
Inject low-entropy structure ⟹ Increase ℒ ⟹ Strengthen Delta
6.7 The Closed System
The three scrolls form a coupled system:
| Scroll | Equation |
|---|---|
| Gravity | g⃗ = −κg[∇𝒜·Tr(ℳ) + 𝒜·∇Tr(ℳ)] |
| Time | T = ∫ dSθ/σθ |
| Entropy | σθ = 𝒟(1 − ℒ) |
Everything couples through the lock scalar ℒ.
Intent-Tensor-Theory 