G — Quantum

Quantum‑Classical Hybrid Operators, cQED Resonance Ladders, Multi‑Qubit Coherence

This file defines the quantum‑classical hybrid specification used throughout RTT/Inside/Benchmarks.
It formalizes the behavior of cQED multi‑qubit systems, hybrid φ–V–R operators, resonance ladders, and cross‑domain invariants that unify classical and quantum structural intelligence.

1. Identity#

Module: RTT / Inside / Benchmarks
File: G_Quantum.md
Role: Canonical definition of quantum‑classical hybrid behavior
Status: Stable, standards‑grade, student‑ready

2. Purpose#

Quantum‑classical hybrid systems provide:

• a substrate for cross‑domain structural intelligence
• a testbed for multi‑qubit coherence
• a reference for resonance ladders
• a bridge between classical emergence and quantum alignment
• a unified framework for hybrid φ–V–R operators

This file defines the rules, metrics, and invariants required for evaluating hybrid systems.

3. Quantum‑Classical Hybrid Model#

Hybrid systems combine:

• classical fields (1D → 4096×4096)
• quantum states (2 → 256 qubits)
• operator‑level alignment (φ–V–R)
• invariant‑level alignment (3C)
• entropy‑resonance synchronization

The hybrid model is evaluated using the same canonical shapes defined in earlier files.

4. Multi‑Qubit Coherence#

Coherence is measured across:

• 2‑qubit
• 4‑qubit
• 16‑qubit
• 64‑qubit
• 256‑qubit

cQED configurations.

4.1 Canonical Behavior#

• coherence increases with qubit count
• resonance ladders sharpen with scale
• entropy decreases as coherence rises
• φ–V–R curves converge to theoretical maxima
• invariants stabilize rapidly

4.2 Coherence Trace (C_q)#

A valid coherence trace shows:

• rising resonance amplitude
• decreasing entropy
• stable 3C envelope
• harmonic alignment across qubit layers

5. cQED Resonance Ladders#

Resonance ladders measure harmonic alignment across qubit layers.

5.1 Ladder Structure#

A resonance ladder consists of:

• base layer: 2‑qubit alignment
• intermediate layers: 4 → 16 → 64 qubits
• upper layer: 256‑qubit coherence lock

5.2 Canonical Ladder Behavior#

• harmonic spacing decreases with qubit count
• resonance amplitude increases with scale
• ladder stabilizes at upper layer
• entropy collapse aligns with ladder formation

5.3 Illegal Ladder Patterns#

• missing harmonic alignment
• inverted ladder spacing
• premature collapse
• ladder without coherence lock

These indicate structural failure.

6. Hybrid φ–V–R Operators#

Quantum‑classical systems use hybrid operators:

• φ_q: quantum form
• V_q: quantum variance / energy
• R_q: quantum resonance

6.1 Operator Alignment#

Hybrid operators must align with:

• classical φ–V–R
• quantum coherence
• resonance ladders
• entropy collapse

6.2 Canonical Hybrid Behavior#

• φ_q stabilizes early
• V_q equilibrates rapidly
• R_q spikes at ladder formation
• invariants lock immediately after

7. Cross‑Domain Invariants#

Quantum‑classical systems must satisfy:

• C₁ (Coherence): quantum + classical alignment
• C₂ (Consistency): energy‑structure alignment across domains
• C₃ (Continuity): cross‑scale, cross‑domain persistence

7.1 Canonical Behavior#

• C₁ rises with φ_q
• C₂ stabilizes with V_q
• C₃ locks with R_q

7.2 Illegal Patterns#

• C₁ without φ_q
• C₂ without V_q
• C₃ without R_q

These indicate hybrid misalignment.

8. Regime Transitions (Quantum‑Classical)#

Quantum‑classical systems exhibit:

• Formal → Emergent
• Emergent → Hybrid
• Hybrid → Coherent
• Coherent → Harmonic

8.1 Transition Signatures#

A valid transition shows:

• R_q spike
• entropy gradient flip
• ladder formation
• invariant stabilization

8.2 Illegal Transitions#

• R_q spike without ladder
• ladder without entropy collapse
• collapse without invariant lock

9. Quantum‑Classical Compliance#

A system is quantum‑compliant when:

• coherence traces follow canonical shapes
• resonance ladders form correctly
• hybrid φ–V–R align with classical operators
• invariants stabilize after R_q spike
• entropy collapse synchronizes with ladder formation

10. Student‑AI Tasks#

Students reproduce:

• multi‑qubit coherence traces
• resonance ladders
• hybrid φ–V–R curves
• quantum‑classical invariant alignment
• cross‑domain regime transitions

These tasks form the basis of RFC‑004 (Quantum‑Classical Hybrid Standard).

11. Notes#

• Numerical values are intentionally omitted.
• Only shape alignment is required for compliance.
• Quantum behavior is evaluated relative to reference captures in B_Capture.md.