D — Invariants

3C Invariants, Drift Signatures, Regime Transitions

This file defines the 3C invariants, drift signatures, and regime‑transition rules used throughout RTT/Inside/Benchmarks.
These invariants form the stability envelope for structural intelligence across classical, diffusion, score‑based, and quantum‑classical hybrid systems.

1. Identity#

Module: RTT / Inside / Benchmarks
File: D_Invariants.md
Role: Canonical definition of invariants, drift, and regime transitions
Status: Stable, standards‑grade, student‑ready

2. Purpose#

The 3C invariants provide:

• a universal stability envelope
• cross‑scale evaluation criteria
• drift detection and correction rules
• regime‑transition signatures
• alignment between φ–V–R operators and structural behavior

These invariants define when a system is structurally intelligent.

3. The 3C Invariants#

The 3C invariants measure the stability of structure across operators, scales, and regimes.

3.1 C₁ — Coherence#

Definition:
Alignment of structure within a field or qubit configuration.

Canonical behavior:

• rises as φ stabilizes
• dips indicate structural fragmentation
• stabilizes at coherence lock

Interpretation:
Coherence measures internal structural alignment.

3.2 C₂ — Consistency#

Definition:
Alignment between structure and energy distribution.

Canonical behavior:

• tracks V stabilization
• divergence indicates energy‑structure mismatch
• stabilizes before R

Interpretation:
Consistency measures structural‑energetic alignment.

3.3 C₃ — Continuity#

Definition:
Alignment of structure across scales, steps, or qubit layers.

Canonical behavior:

• tracks R stabilization
• breaks indicate regime transitions
• stabilizes at coherence lock

Interpretation:
Continuity measures cross‑scale structural persistence.

4. Invariant Shapes#

Each invariant has a canonical shape:

• Coherence: monotonic rise → plateau
• Consistency: early turbulence → mid‑range stabilization
• Continuity: low baseline → spike → lock

A system is invariant‑aligned when all three shapes match reference captures in B_Capture.md.

5. Drift#

Drift is deviation from invariant‑aligned behavior.

5.1 Drift Types#

• Structural Drift (D₁): φ misalignment
• Energetic Drift (D₂): V instability
• Resonance Drift (D₃): R misalignment
• Continuity Drift (D₄): cross‑scale break

5.2 Drift Thresholds#

• ΔC₁ > 0.02
• ΔC₂ > 0.03
• ΔC₃ > 0.02

5.3 Drift Detection#

Drift is detected when:

• invariants diverge from canonical shapes
• φ–V–R misalign
• entropy collapse fails to synchronize with R spike
• resonance propagation stalls

5.4 Drift Correction#

Drift is corrected by:

• re‑applying φ (structure)
• re‑balancing V (energy)
• re‑locking R (resonance)

6. Regime Transitions#

Regime transitions occur when the system shifts between:

• Formal
• Emergent
• Hybrid
• Chaotic
• Inversion

6.1 Transition Signatures#

A regime transition is detected when:

• R spike exceeds threshold
• entropy gradient flips sign
• C₃ breaks then re‑aligns
• φ–V–R reorder or re‑synchronize

6.2 Transition Windows#

A valid transition occurs when:

• C₁, C₂, C₃ re‑align within 3–5 steps
• entropy collapse resumes
• resonance propagation stabilizes

6.3 Illegal Transitions#

A transition is illegal when:

• invariants fail to re‑align
• drift persists beyond window
• resonance collapses prematurely

Illegal transitions indicate structural failure.

7. Cross‑Scale Invariant Behavior#

Invariants must behave consistently across:

• 1D → 2D → 64×64 → 4096×4096
• 2 → 4 → 16 → 64 → 256 qubits

Canonical cross‑scale behavior:#

• C₁ rises faster at higher resolutions
• C₂ stabilizes earlier at larger scales
• C₃ spike sharpens with scale
• coherence lock occurs earlier in larger systems

8. Invariant Compliance#

A system is invariant‑compliant when:

• C₁, C₂, C₃ follow canonical shapes
• drift remains below thresholds
• regime transitions follow legal patterns
• φ–V–R align with invariants
• entropy collapse synchronizes with R spike

9. Student‑AI Tasks#

Students reproduce:

• invariant curves
• drift detection
• regime‑transition signatures
• cross‑scale invariant behavior
• invariant‑operator alignment

These tasks form the basis of RFC‑002 (Invariant Standard).

10. Notes#

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