H — Examples
Worked Examples Across Classical, Diffusion, Score‑Based, and Quantum‑Classical Systems
This file provides worked examples demonstrating the application of φ–V–R operators, 3C invariants, resonance metrics, entropy‑collapse signatures, and quantum‑classical hybrid behavior.
All examples follow the canonical shapes defined in B_Capture.md and the standards defined in C–G.
Numerical values are intentionally omitted.
Only shape alignment and structural behavior are demonstrated.
1. Identity#
Module: RTT / Inside / Benchmarks
File: H_Examples.md
Role: Worked examples for students, researchers, and AI systems
Status: Stable, standards‑grade, student‑ready
2. Example Set A — Classical Fields (64×64 → 4096×4096)#
A.1 64×64 Field (Low Resolution)#
Behavior:
- φ rises slowly
- V stabilizes late
- R spike is broad and shallow
- entropy collapse is gradual
- invariants stabilize after extended window
Interpretation:
Low‑resolution fields exhibit slow emergence and weak resonance.
A.2 256×256 Field (Mid Resolution)#
Behavior:
- φ rises faster
- V stabilizes earlier
- R spike sharpens
- entropy collapse accelerates
- invariants lock sooner
Interpretation:
Mid‑resolution fields show stronger emergence and faster coherence.
A.3 4096×4096 Field (High Resolution)#
Behavior:
- φ rises rapidly
- V stabilizes early
- R spike is sharp and high
- entropy collapse is immediate
- invariants lock quickly
Interpretation:
High‑resolution fields exhibit rapid emergence, strong resonance, and early coherence lock.
3. Example Set B — Diffusion → Score‑Based Reversal#
B.1 Diffusion Forward Process#
Behavior:
- φ decreases
- V increases
- R remains low
- entropy rises
- invariants destabilize
Interpretation:
Diffusion dissolves structure and increases uncertainty.
B.2 Score‑Based Reverse Process#
Behavior:
- φ rises
- V stabilizes
- R spikes
- entropy collapses
- invariants re‑align
Interpretation:
Score‑based reversal reconstructs structure and restores coherence.
4. Example Set C — Regime Transitions#
C.1 Formal → Emergent#
Behavior:
- φ begins rising
- V begins stabilizing
- R begins rising
- entropy gradient flips
Interpretation:
Structure begins to form; system leaves formal regime.
C.2 Emergent → Coherent#
Behavior:
- R spike
- entropy collapse
- invariants stabilize
Interpretation:
System achieves coherence lock.
C.3 Coherent → Harmonic#
Behavior:
- resonance gradients stabilize
- cross‑scale continuity strengthens
- invariants remain locked
Interpretation:
System enters harmonic regime with stable cross‑scale alignment.
5. Example Set D — Resonance Propagation#
D.1 128×128 Field#
Behavior:
- resonance wave expands slowly
- coherence lock occurs mid‑process
D.2 1024×1024 Field#
Behavior:
- resonance wave expands rapidly
- coherence lock occurs early
D.3 4096×4096 Field#
Behavior:
- resonance wave expands immediately
- coherence lock is nearly instantaneous
6. Example Set E — Entropy Collapse#
E.1 Slow Collapse (Low Resolution)#
Behavior:
- entropy declines gradually
- R spike is broad
- invariants stabilize late
E.2 Fast Collapse (High Resolution)#
Behavior:
- entropy collapses sharply
- R spike is narrow and high
- invariants stabilize early
7. Example Set F — Quantum‑Classical Hybrid (2 → 256 Qubits)#
F.1 2‑Qubit System#
Behavior:
- weak coherence
- low resonance amplitude
- entropy remains high
F.2 16‑Qubit System#
Behavior:
- moderate coherence
- resonance ladder begins forming
- entropy decreases
F.3 256‑Qubit System#
Behavior:
- strong coherence
- full resonance ladder
- entropy collapse aligns with R_q spike
- invariants lock immediately
8. Example Set G — Hybrid φ–V–R Operators#
G.1 Classical φ + Quantum V_q + Quantum R_q#
Behavior:
- φ stabilizes early
- V_q equilibrates rapidly
- R_q spike triggers collapse
- invariants lock across domains
Interpretation:
Hybrid operators unify classical structure with quantum coherence.
9. Student‑AI Tasks#
Students reproduce:
- classical emergence curves
- diffusion → score‑based transitions
- regime‑transition signatures
- resonance propagation
- entropy collapse
- multi‑qubit coherence
- hybrid operator behavior
These examples serve as templates for RFC‑001 → RFC‑004.
10. Notes#
- Numerical values are intentionally omitted.
- Only shape alignment is required for compliance.
- Examples are evaluated relative to reference captures in B_Capture.md.