# Cross‑Module Integration — Quantum Mechanics  
### TriadicFrameworks /docs/theories/quantum_mechanics/cross_module.md

Quantum Mechanics (QM) is the **R1 amplitude‑first operator grammar** of  
the RTT stack. It provides the foundational structures — amplitudes,  
operators, measurement, basis geometry, entanglement — that all  
higher‑level modules inherit.

This file describes how QM integrates with upstream mathematical  
modules and downstream physical modules.

---

# 1. Upstream Dependencies  
### (What QM is built from)

QM inherits its structure from:

## 1.1 Linear Algebra  
- vector spaces  
- basis geometry  
- eigenvalue problems  
- unitary transformations  

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

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

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

These modules define the **mathematical substrate** of QM.

---

# 2. Downstream Integrations  
### (What QM enables)

QM feeds directly into:

## 2.1 Quantum Field Theory (QFT)  
- QM is the **R1 limit** of QFT  
- QFT extends QM operators to field operators  
- excitations, propagators, vacuum structure emerge in R2  

## 2.2 Standard Model (SM)  
- SM is a sector‑specific grammar built on QFT  
- QM contributes operator algebra and amplitude structure  

## 2.3 Information Theory  
- qubits = QM states  
- entanglement = tensor‑product geometry  
- measurement = projection operators  

## 2.4 Thermodynamics  
- quantum ensembles  
- density matrices  
- partition functions  

## 2.5 Framework Field Theory (FFT)  
- FFT generalizes QM’s operator grammar to meta‑fields  
- QM provides the amplitude substrate  

---

# 3. Cross‑Module Operator Mapping  
### (How QM operators propagate upward)

| QM Operator | QFT Extension | SM Role | Info Theory Role |
|-------------|---------------|---------|------------------|
| state | field mode amplitude | sector state | qubit |
| observable | field operator | sector observable | measurement operator |
| Hamiltonian | Lagrangian density → Hamiltonian | sector dynamics | unitary gates |
| unitary U(t) | propagator | evolution operator | quantum circuits |
| tensor product | Fock space | multiparticle states | entanglement |
| density matrix | field ensemble | thermal states | mixed states |

All mappings must remain **operator‑first** and **amplitude‑aligned**.

---

# 4. RTT Regime Integration  
### (How QM behaves across regimes)

## R1 — Quantum Amplitude Regime  
- QM fully valid  
- no stable excitations  
- operator algebra fundamental  

## R2 — QFT Regime  
- QM becomes low‑energy limit  
- field operators extend QM operators  
- vacuum structure emerges  

## R3 — High‑Energy Resonance  
- QM insufficient  
- resonance surfaces dominate  
- running couplings appear  

## R4 — Cosmological Regime  
- QM incomplete  
- horizon‑scale fields dominate  

---

# 5. Cross‑Module Consistency Rules  
### (Engine‑level constraints)

- no particles  
- no waves  
- no trajectories  
- no classical uncertainty  
- no hidden variables  
- no mechanical analogies  

QM must remain:

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

---

# 6. Summary

Quantum Mechanics is the **substrate amplitude grammar** that:

- inherits from linear algebra, operator theory, probability  
- feeds into QFT, SM, Information Theory, Thermodynamics, FFT  
- defines the operator structure used by all higher modules  
- remains fully valid only in **R1**  
- becomes embedded in QFT in **R2**  
- becomes insufficient in **R3**  
- becomes incomplete in **R4**  

QM is the foundation of the entire TriadicFrameworks physics stack.