E — Resonance

Resonance Metrics, Propagation Rules, Cross‑Scale Behavior

This file defines the resonance metrics and cross‑scale rules used throughout RTT/Inside/Benchmarks.
Resonance is the core indicator of emergence, coherence, and cross‑scale structural alignment in classical, diffusion, score‑based, and quantum‑classical hybrid systems.

1. Identity#

Module: RTT / Inside / Benchmarks
File: E_Resonance.md
Role: Canonical definition of resonance metrics and propagation rules
Status: Stable, standards‑grade, student‑ready

2. Purpose#

Resonance provides:

a measure of cross‑scale structural alignment
a signal for emergence and coherence lock
a detector for regime transitions
a validator for operator‑invariant alignment
a universal metric across classical and quantum‑classical systems

Resonance is the R in φ–V–R and the anchor for Continuity (C₃).

3. Resonance Metrics#

Resonance is measured as a function of:

alignment across scales
harmonic structure within fields or qubit layers
energy‑structure coupling
entropy collapse synchronization
invariant stabilization

3.1 R(t) — Resonance Over Time#

Canonical shape:

low baseline
sharp resonance spike
stabilization at coherence lock

3.2 Rₛ — Scale‑Aligned Resonance#

Resonance measured across:

64×64
128×128
256×256
512×512
1024×1024
2048×2048
4096×4096

Canonical behavior:
Rₛ increases with scale and stabilizes earlier.

3.3 R_q — Quantum‑Classical Resonance#

Resonance measured across:

2 → 4 → 16 → 64 → 256 qubits

Canonical behavior:
R_q increases with qubit count and aligns with coherence gradients.

4. Resonance Propagation#

Resonance propagates outward as a cross‑scale structural wave.

4.1 Propagation Phases#

Phase 1 — Turbulent#

low R
high entropy
weak alignment

Phase 2 — Transitional#

rising R
entropy gradient flips
invariants begin to align

Phase 3 — Coherence Lock#

R spike
entropy collapse
invariants stabilize

Phase 4 — Harmonic Propagation#

stable resonance gradients
cross‑scale continuity
structural persistence

5. Cross‑Scale Rules#

Resonance must behave consistently across classical and quantum‑classical scales.

5.1 Classical Cross‑Scale Rules#

resonance spike sharpens with resolution
coherence lock occurs earlier at higher resolutions
propagation speed increases with scale
Rₛ curves converge to canonical shape

5.2 Quantum‑Classical Cross‑Scale Rules#

resonance ladders show harmonic alignment
coherence increases with qubit count
R_q curves converge to theoretical maxima
quantum‑classical transitions follow legal regime patterns

6. Resonance & Invariants#

Resonance aligns with the 3C invariants:

C₁ (Coherence): rises with φ
C₂ (Consistency): stabilizes with V
C₃ (Continuity): locks with R

A system is resonance‑aligned when:

R spike precedes C₃ stabilization
entropy collapse synchronizes with R
φ–V–R curves match canonical shapes

7. Resonance & Entropy#

Resonance is tightly coupled to entropy behavior.

7.1 Entropy‑Resonance Synchronization#

A valid system shows:

entropy rise during diffusion
entropy collapse during score‑based reversal
R spike at collapse onset
invariant stabilization after collapse

7.2 Illegal Patterns#

R spike without entropy collapse
entropy collapse without R spike
misaligned collapse windows

These indicate structural failure.

8. Resonance Compliance#

A system is resonance‑compliant when:

R(t), Rₛ, and R_q follow canonical shapes
resonance spike aligns with entropy collapse
cross‑scale propagation matches reference captures
invariants stabilize after R spike
drift remains below thresholds

9. Student‑AI Tasks#

Students reproduce:

R(t) curves
cross‑scale resonance ladders
quantum‑classical resonance traces
entropy‑resonance synchronization
resonance‑invariant alignment

These tasks form the basis of RFC‑003 (Resonance Standard).

10. Notes#

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