# Lineage — Quantum Mechanics  
### TriadicFrameworks /docs/theories/quantum_mechanics/lineage.md

Quantum Mechanics (QM) is the **R1 amplitude grammar** of the RTT stack.  
It provides the operator algebra, amplitude structure, and measurement  
rules that all higher‑level theories inherit. QM is not a particle  
theory and not a wave theory — it is a **non‑classical amplitude  
geometry**.

This lineage traces QM’s development across:

- historical foundations  
- conceptual transitions  
- mathematical structures  
- RTT regime placement  
- cross‑module ancestry  

---

# 1. Historical Lineage

### **1900 — Planck’s Quantization**  
- energy quantized  
- classical continuum breaks  

### **1905 — Einstein (Photoelectric Effect)**  
- amplitude‑based interpretation begins  
- classical wave picture insufficient  

### **1925 — Heisenberg (Matrix Mechanics)**  
- operator algebra introduced  
- observables become matrices  

### **1926 — Schrödinger (Wave Mechanics)**  
- amplitude functions introduced  
- basis representation emerges  

### **1927 — Born (Probability Interpretation)**  
- |ψ|² interpreted as probability density  
- measurement becomes projection  

### **1927 — Dirac (Unified Formalism)**  
- bra‑ket notation  
- operator‑first grammar  
- basis transformations formalized  

### **1930s–1950s — Foundations & Measurement**  
- von Neumann measurement theory  
- decoherence precursors  

### **1960s–Present — Quantum Information**  
- entanglement formalized  
- tensor‑product structure central  
- QM becomes substrate for computation  

---

# 2. Conceptual Lineage

QM emerges from four conceptual transitions:

### **1. From classical states → amplitude states**  
States become vectors in Hilbert space.

### **2. From classical variables → operators**  
Observables become Hermitian operators.

### **3. From trajectories → unitary evolution**  
Motion replaced by phase evolution.

### **4. From determinism → amplitude geometry**  
Probabilities arise from amplitude structure, not ignorance.

---

# 3. Mathematical Lineage

QM inherits its structure from:

### **Linear Algebra**  
- vector spaces  
- basis transformations  
- eigenvalue problems  

### **Operator Theory**  
- Hermitian operators  
- commutators  
- spectral decomposition  

### **Functional Analysis**  
- Hilbert spaces  
- continuous spectra  
- completeness  

### **Fourier Analysis**  
- basis duality (x ↔ p)  
- unitary transforms  

### **Probability Theory**  
- amplitude‑squared interpretation  
- expectation values  

---

# 4. RTT Lineage

QM occupies a specific place in the RTT hierarchy:

### **R1 — Quantum Amplitude Regime**  
QM fully valid.  
No stable excitations.  
Operator algebra fundamental.

### **R2 — QFT Regime**  
QM becomes the low‑energy limit of QFT.  
Excitations become stable.  
Field operators extend QM operators.

### **R3 — High‑Energy Resonance**  
QM insufficient.  
Running couplings and resonance surfaces dominate.

### **R4 — Cosmological Regime**  
QM incomplete.  
Horizon‑scale fields dominate.

---

# 5. Cross‑Module Lineage

QM is the substrate ancestor of:

- **Quantum Field Theory** (field operators, excitations)  
- **Standard Model** (sector grammar built on QFT)  
- **Information Theory** (state classification, entanglement)  
- **Thermodynamics** (quantum ensembles)  
- **Foundations** (measurement, decoherence)  

QM inherits from:

- **Linear Algebra**  
- **Operator Theory**  
- **Probability Theory**  

QM feeds into:

- **QFT** (R2 extension)  
- **FFT** (meta‑field generalization)  
- **Triadic Echo Lattice** (resonance‑time geometry)  

---

# 6. Substrate Lineage Summary

Quantum Mechanics is the convergence point of:

- amplitude geometry  
- operator algebra  
- basis structure  
- measurement rules  
- unitary evolution  
- entanglement structure  

QM is the **R1 amplitude grammar** from which QFT emerges and to which  
QFT collapses when excitations lose stability.