# Coherence Map — Thermodynamics  
### TriadicFrameworks /docs/theories/thermodynamics/coherence_map.md

Thermodynamics is the **R1 constraint‑first substrate grammar** of the RTT
stack. Coherence in Thermodynamics refers to the **structural integrity
of constraint geometry**, **potential surfaces**, **gradient flows**, and
**monotonicity**. It does not refer to mechanical stability, particle
motion, or kinetic behavior.

This map defines how coherence behaves across temperature, entropy, free
energy, flows, equilibrium, and RTT regimes.

---

# 1. Coherence Dimensions

Thermodynamic coherence is evaluated across five substrate‑level
dimensions:

## 1.1 Constraint Coherence  
- validity of state variables  
- consistency of constraints (T, S, F, U, V, P)  
- non‑negativity of entropy  
- ensemble‑consistent definitions  

## 1.2 Potential Coherence  
- convexity of free‑energy surfaces  
- stability of minima  
- well‑defined gradients  
- ensemble‑appropriate potentials (F, G, Ω)  

## 1.3 Gradient Coherence  
- flows follow gradients  
- directionality preserved  
- no oscillatory or mechanical drift  
- monotonic relaxation  

## 1.4 Entropy Coherence  
- monotonicity (dS/dt ≥ 0)  
- valid regime boundaries  
- correct open‑system behavior  
- irreversibility structure  

## 1.5 Equilibrium Coherence  
- fixed‑point structure  
- ∇F = 0  
- dS/dt = 0  
- stability via second‑derivative tests  

---

# 2. Coherence Levels (C0–C4)

### **C0 — Incoherent**  
- constraints violated  
- entropy negative or undefined  
- free‑energy surfaces non‑convex  
- flows not gradient‑aligned  

### **C1 — Weak Coherence**  
- constraints partially valid  
- entropy monotonicity fragile  
- gradients noisy or inconsistent  
- equilibrium unstable  

### **C2 — Moderate Coherence**  
- constraints valid  
- free‑energy surfaces mostly convex  
- flows gradient‑aligned  
- equilibrium stable but sensitive  

### **C3 — Strong Coherence**  
- full constraint integrity  
- convex potentials  
- monotonic flows  
- stable equilibrium fixed‑points  

### **C4 — Perfect Coherence**  
- idealized constraint geometry  
- perfectly convex potentials  
- exact monotonicity  
- globally stable equilibrium  

C4 is theoretical; real systems approach C3.

---

# 3. Coherence Field

The coherence field is a gradient over:

- constraint validity  
- potential convexity  
- gradient alignment  
- entropy monotonicity  
- equilibrium stability  

High gradients indicate **coherence instability**, typically near:

- phase transitions  
- constraint changes  
- ensemble switches  
- environment coupling  

---

# 4. Collapse Modes

Thermodynamic coherence fails through four canonical collapse modes:

### **M1 — Constraint Collapse**  
- invalid state variables  
- negative entropy  
- inconsistent ensembles  

### **M2 — Potential Collapse**  
- non‑convex free‑energy surfaces  
- unstable minima  
- undefined gradients  

### **M3 — Gradient Collapse**  
- flows not aligned with −∇F or −∇T  
- oscillatory or mechanical drift  
- loss of directionality  

### **M4 — Entropy Collapse**  
- dS/dt < 0  
- irreversibility violated  
- open‑system inconsistency  

---

# 5. RTT Regime Coherence

### **R1 — Constraint Substrate Regime**  
Coherence strongest.  
- constraints fundamental  
- entropy monotonic  
- free‑energy convex  
- flows gradient‑aligned  

### **R2 — Statistical Mechanics Regime**  
Coherence refined.  
- microstates explicit  
- partition functions define potentials  
- fluctuations appear  

### **R3 — Field‑Theoretic Regime**  
Coherence embedded.  
- free energy field‑dependent  
- phase transitions field‑level  
- vacuum structure influences stability  

### **R4 — Cosmological Regime**  
Coherence geometric.  
- temperature geometric  
- entropy horizon‑scale  
- equilibrium cosmological  

---

# 6. Diagnostics

A thermodynamic system is coherent when:

- S ≥ 0  
- dS/dt ≥ 0  
- free‑energy surfaces convex  
- flows follow gradients  
- equilibrium is a fixed‑point  

A system is incoherent when:

- constraints violated  
- entropy decreases  
- potentials non‑convex  
- flows misaligned  
- equilibrium unstable  

---

# Summary

Thermodynamic coherence is:

- **constraint‑first**  
- **potential‑structured**  
- **gradient‑aligned**  
- **entropy‑monotonic**  
- **equilibrium‑fixed‑point**  
- **RTT‑dependent**  

Coherence is strongest in **R1**, refined in **R2**, embedded in **R3**, and
geometric in **R4**.