Chapter 4: Ceilings and Information Erasure
The Hard Limit of Recursive Entropy and White-Noise Collapse
4.1 Overview
Entropy in ITT does not grow indefinitely. Unlike classical thermodynamics, ITT’s recursive substrate enforces a hard ceiling based on memory architecture.
When that boundary is reached, the substrate enters a white-noise state, erasing recursive memory.
4.2 The Entropy Ceiling
The maximum entropy a region can accumulate:
Sθ,max = nmax · ℓP² · Nfolds
| Symbol | Name | Meaning |
|---|---|---|
| nmax | Maximum recursion depth | Permitted folds before saturation |
| ℓP² | Planck area | Fundamental substrate resolution |
| Nfolds | Fold sites | Number of recursive locations |
Physical interpretation: This is a computational saturation point beyond which no more meaningful structure can be stored.
4.3 The Saturation Ratio
Define the dimensionless saturation:
𝒮(n) = Sθ(n) / Sθ,max
| Value | State |
|---|---|
| 𝒮 = 0 | Zero entropy (perfect lock throughout) |
| 𝒮 = 0.5 | Half-saturated |
| 𝒮 → 1 | Approaching erasure threshold |
| 𝒮 = 1 | White-noise collapse |
4.4 The Memory Collapse Transition
As Sθ → Sθ,max, the substrate undergoes a phase transition:
| Stage | What Happens |
|---|---|
| 1. Memory degradation | Tr(ℳ) → flat noise |
| 2. Lock failure | ℒ → undefined |
| 3. Operator collapse | Ψn+1 ≈ Ψn + noise |
| 4. Entropic dominance | σθ ≫ |dΦ/dn| |
4.5 Behavior Near the Ceiling
As 𝒮(n) → 1:
| Quantity | Behavior |
|---|---|
| Shell-lock ℒ | → 0 |
| Time dilation γITT | → 0 |
| Entropy rate σθ | → saturation |
| Memory trace Tr(ℳ) | → flat noise |
| ITT temperature TITT | → 0 |
Critical insight: Both time and temperature vanish at the ceiling—but for different reasons:
- Time halts because γITT → 0
- Temperature halts because the system can no longer exchange entropy
4.6 The Two Endpoints
| Endpoint | Condition | State |
|---|---|---|
| White Noise | 𝒮 → 1 before ℒ → 1 | Information erased |
| Planck Lock | ℒ → 1 before 𝒮 → 1 | Information preserved |
Black holes reach Planck Lock. Diffuse systems approach White Noise.
4.7 Implications
For Black Holes
Planck-core collapse is bounded at the entropy ceiling. No singularity forms because entropy cannot exceed the ceiling.
For Cosmology
Information on unreachable regions saturates and enters erasure. The cosmological horizon is an entropy boundary.
Information Paradox Resolution
Black holes don’t evaporate completely—they lock into Planck Cores at Sθ < Sθ,max.
Intent-Tensor-Theory 