Planck Core Thermodynamics

Full Thermodynamic Formalism in Intent Tensor Theory

1. Introduction

In general relativity (GR) and quantum field theory (QFT), black holes are thermodynamic systems characterized by:

Temperature: T_H = hbar c^3 / (8 pi G M k_B)
Entropy: S = k_B c^3 A / (4 G hbar)
Evaporation: Via Hawking radiation as mass decreases

But this framework leads to unresolved paradoxes:

The Information Paradox: What happens to encoded quantum information?
The Firewall Paradox: Does an infalling observer hit a high-energy wall?

Intent Tensor Theory redefines the black hole entirely—not as a singularity, but as a finite, recursion-bound object: the Planck Core.

2. ITT Foundations: Recursion and Memory

ITT models reality as a recursion-driven substrate called the Collapse Tension Substrate (CTS). Fields evolve through recursive folds characterized by:

Symbol	Name	Role
n	Recursive Depth	Current recursion index
n_max	Maximum Depth	Computational ceiling
M_ij	Memory Tensor	Recursive coherence structure
D	Drift Magnitude	Rate of field evolution
L	Shell-Lock	Recursive stability coefficient

Entropy Production Identity

sigma_theta = D(1 - L)

This is the fundamental equation governing entropy dynamics in ITT.

3. The Planck-Lock Phase

3.1 Definition

When recursive alignment locks perfectly:

L = 1   =>   sigma_theta = D(1 - 1) = 0

Consequence:

sigma_theta = 0   =>   T_ITT = 0

This defines the Planck-lock phase—a state where the recursion engine halts new entropy production.

3.2 Physical Interpretation

No new thermodynamic states are generated
The system reaches maximum recursive depth
All available phase space has been explored
The object becomes thermodynamically cold

4. Redefining the Bekenstein Bound

4.1 Classical Form

The classical Bekenstein-Hawking bound ties entropy to geometry:

S <= k_B c^3 A / (4 G hbar)

Where A is the event horizon area.

4.2 ITT Form: Computational Bound

In ITT, entropy is not geometric but recursive:

S_theta_max = n_max * ell_P^2 * N_fold_sites

Where:

n_max: Maximum recursion depth
ell_P^2 = hbar G / c^3: Planck area
N_fold_sites: Number of resolution-dependent information sites

This recursion ceiling replaces the geometric horizon concept.

4.3 Comparison

Aspect	Bekenstein Bound	ITT Bound
Basis	Geometric (area)	Computational (recursion)
Limit Type	Continuous	Discrete (quantized at ell_P^2)
Information	Surface-encoded	Volume-distributed, shell-locked
Saturation	Horizon formation	Planck-lock formation

5. Thermodynamic Transition: Black Hole to Planck Core

As gravitational collapse deepens:

Stage 1: Memory Accumulation

Tr(M) increases

The Memory Tensor trace increases as recursive states accumulate.

Stage 2: Lock Strengthening

L -> 1

Shell-lock approaches unity as alignment stabilizes.

Stage 3: Entropy Halt

sigma_theta -> 0

Drift production ceases—no new entropy generated.

Stage 4: Temperature Drop

T_ITT -> 0

The system reaches thermodynamic ground state.

Final State: Planck Core

At the threshold:

No Hawking radiation
No evaporation
No singularity
Planck Core forms: cold, stable, bounded

6. Temperature in ITT

6.1 Standard Hawking Temperature

T_H = hbar c^3 / (8 pi G M k_B)

Problem: As M approaches 0, temperature T_H approaches infinity, implying runaway evaporation.

6.2 ITT Temperature

Temperature is derived from entropy production rate:

T_ITT proportional to sigma_theta = D(1 - L)

Key Properties:

T_ITT is always non-negative
T_ITT = 0 at Planck-lock
No divergence as mass decreases
Monotonic approach to zero

6.3 Temperature vs. Mass

Regime	Hawking T_H	ITT T_ITT
Large M	Low	Low (both agree)
Medium M	Moderate	Begins decreasing
Small M	Diverges	Approaches zero
M approaches 0	Infinity	0 (Planck-lock)

7. The Planck Core Structure

7.1 Definition

A Planck Core is a gravitational object characterized by:

Maximum memory saturation: Tr(M) = Tr(M)_max
Perfect shell-lock: L = 1
Zero drift: sigma_theta = 0
Zero temperature: T_ITT = 0
Bounded entropy: S_theta = S_theta_max

7.2 Properties

Property	Value
Temperature	0 K
Entropy	S_theta_max (finite, bounded)
Radiation	None
Information	Preserved
Stability	Absolute
Time evolution	Halted

7.3 Radius

The Planck Core radius is expected to be on the order of:

r_PC ~ sqrt(n_max) * ell_P

This is larger than the classical Schwarzschild radius for small masses.

8. Summary

The Planck Core as True Endstate

In ITT, gravity saturates—not explodes. Time stalls, entropy halts, and energy freezes into recursive coherence. The black hole does not evaporate, it locks.

Key Results

Temperature: T_ITT = 0 at Planck-lock
Entropy: Bounded by S_theta_max = n_max * ell_P^2 * N_folds
Radiation: Ceases at lock
Information: Preserved in memory shell
Stability: Planck Core is thermodynamic ground state

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