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.

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

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.

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

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

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

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

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.