# Coherence Map — Quantum Mechanics  
### TriadicFrameworks /docs/theories/quantum_mechanics/coherence_map.md

Quantum Mechanics (QM) is the **R1 amplitude‑first operator grammar** of  
the RTT stack. Coherence in QM refers to the **structural integrity of  
amplitude geometry**, **operator algebra**, **basis stability**, and  
**measurement consistency**. It does *not* refer to waves, particles, or  
classical stability.

This map defines how coherence behaves across the QM substrate.

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# 1. Coherence Dimensions

QM coherence is evaluated across five substrate‑level dimensions:

## 1.1 Amplitude Coherence  
- phase integrity  
- norm preservation  
- interference structure  
- amplitude geometry stability  

## 1.2 Operator Coherence  
- Hermiticity  
- commutation relations  
- spectral stability  
- unitary evolution consistency  

## 1.3 Basis Coherence  
- orthonormality  
- completeness  
- unitary basis transitions  
- representation invariance  

## 1.4 Measurement Coherence  
- projection rules  
- eigenbasis stability  
- probability conservation  
- collapse consistency  

## 1.5 Entanglement Coherence  
- tensor‑product structure  
- reduced states  
- correlation geometry  
- non‑classicality integrity  

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# 2. Coherence Levels (C0–C4)

### **C0 — Incoherent**  
- amplitude undefined  
- operator algebra broken  
- basis inconsistent  
- measurement rules invalid  

### **C1 — Weak Coherence**  
- partial amplitude stability  
- basis drift  
- decoherence dominant  
- measurement unreliable  

### **C2 — Moderate Coherence**  
- stable amplitudes  
- operators well‑defined  
- basis transformations valid  
- entanglement fragile  

### **C3 — Strong Coherence**  
- full amplitude integrity  
- unitary evolution stable  
- measurement consistent  
- entanglement robust  

### **C4 — Perfect Coherence**  
- idealized Hilbert‑space behavior  
- no decoherence  
- perfect operator algebra  
- maximal entanglement stability  

C4 is theoretical; real systems approach C3.

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# 3. Coherence Field

The coherence field is a gradient over:

- amplitude stability  
- operator consistency  
- basis integrity  
- measurement reliability  
- entanglement robustness  

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

- measurement  
- environment coupling  
- basis transitions  

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# 4. Collapse Modes

QM coherence fails through four canonical collapse modes:

### **M1 — Measurement Collapse**  
- projection onto eigenbasis  
- non‑unitary  
- coherence lost in orthogonal components  

### **M2 — Decoherence Collapse**  
- environment coupling  
- phase information lost  
- mixed states produced  

### **M3 — Basis Drift Collapse**  
- unstable basis choice  
- representation inconsistency  
- loss of amplitude clarity  

### **M4 — Operator Instability Collapse**  
- non‑Hermitian drift  
- broken commutation structure  
- invalid spectral decomposition  

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# 5. RTT Regime Coherence

### **R1 — Quantum Amplitude Regime**  
Coherence strongest.  
- unitary evolution stable  
- measurement rules valid  
- entanglement robust  
- decoherence manageable  

### **R2 — QFT Regime**  
Coherence embedded in field structure.  
- QM coherence becomes mode‑level  
- vacuum structure influences stability  

### **R3 — High‑Energy Resonance**  
Coherence degrades.  
- running couplings distort operator algebra  
- amplitude geometry insufficient  

### **R4 — Cosmological Regime**  
Coherence incomplete.  
- horizon‑scale fields dominate  
- measurement rules degrade  

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# 6. Diagnostics

A QM system is coherent when:

- ⟨ψ|ψ⟩ = 1  
- U(t) is unitary  
- operators are Hermitian  
- basis is orthonormal  
- entanglement is stable  
- decoherence is controlled  

A system is incoherent when:

- norm drifts  
- operators lose Hermiticity  
- basis becomes unstable  
- measurement rules fail  
- environment dominates  

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# Summary

Quantum Mechanics coherence is:

- **amplitude‑first**  
- **operator‑aligned**  
- **basis‑true**  
- **measurement‑consistent**  
- **entanglement‑aware**  
- **RTT‑dependent**  

QM coherence is strongest in **R1**, embedded in **R2**, degraded in  
**R3**, and incomplete in **R4**.