C — φ–V–R Operator Standard
Operator Grammar, Invariants, Drift Boundaries, Composability
This file defines the φ–V–R operator standard used throughout RTT/Inside/Benchmarks.
It specifies the operator grammar, invariant behavior, drift boundaries, and composability rules required for structural intelligence evaluation across classical, diffusion, score‑based, and quantum‑classical hybrid systems.
1. Identity#
Module: RTT / Inside / Benchmarks
File: C_Operators.md
Role: Canonical definition of φ–V–R operators
Status: Stable, standards‑grade, student‑ready
2. Purpose#
The φ–V–R operator standard provides:
- a unified operator grammar
- cross‑scale operator behavior
- invariant‑aligned operator expectations
- drift boundaries
- composability rules
- reference shapes for φ(t), V(t), and R(t)
These operators form the core structural engine for RTT/Inside/Benchmarks.
3. Operator Definitions#
3.1 φ — Form Operator#
Definition:
φ measures the emergence, stability, and propagation of structure.
Canonical behavior:
- increases monotonically during emergence
- stabilizes at structural equilibrium
- aligns with Coherence (C₁)
Interpretation:
φ tracks what is forming.
3.2 V — Variance / Energy Operator#
Definition:
V measures the distribution, flow, and stabilization of energy or variance across the system.
Canonical behavior:
- early turbulence
- mid‑range stabilization
- alignment with Consistency (C₂)
Interpretation:
V tracks how structure is energized and distributed.
3.3 R — Resonance Operator#
Definition:
R measures cross‑scale alignment, emergence, and coherence lock.
Canonical behavior:
- low baseline
- resonance spike at regime transition
- stabilization at coherence lock
- alignment with Continuity (C₃)
Interpretation:
R tracks why structure coheres across scales.
4. Operator Grammar#
The φ–V–R grammar defines how operators are expressed, composed, and evaluated.
4.1 Syntax#
φ[x], V[x], R[x] φ(t), V(t), R(t) φ∘V∘R
Where:
xis a field, state, or qubit configurationtis a timestep or operator step∘denotes operator composition
4.2 Composition Rules#
Rule 1 — Order Matters#
φ ∘ V ∘ R ≠ R ∘ V ∘ φ
Rule 2 — Canonical Composition#
The canonical operator chain is:
φ → V → R
Rule 3 — Stability Requirement#
A composition is valid when:
- φ stabilizes before V
- V stabilizes before R
- R spike precedes 3C stabilization
5. Operator Invariants#
Operators must respect the 3C invariants:
- C₁ — Coherence aligns with φ
- C₂ — Consistency aligns with V
- C₃ — Continuity aligns with R
A system is invariant‑aligned when:
- φ, V, R follow canonical shapes
- 3C invariants stabilize within expected windows
- drift remains below threshold
6. Drift Boundaries#
Drift is deviation from invariant‑aligned operator behavior.
6.1 Drift Thresholds#
- φ drift: Δφ > 0.03
- V drift: ΔV > 0.05
- R drift: ΔR > 0.02
6.2 Drift Detection#
Drift is detected when:
- φ fails to stabilize
- V oscillates after mid‑range
- R spike misaligns with entropy collapse
- 3C invariants diverge
6.3 Drift Correction#
Drift is corrected by:
- re‑applying φ
- re‑balancing V
- re‑locking R
7. Cross‑Scale Operator Behavior#
Operators must behave consistently across:
- 1D → 2D → 64×64 → 4096×4096
- 2 → 4 → 16 → 64 → 256 qubits
Canonical cross‑scale behavior:#
- φ increases faster at higher resolutions
- V stabilizes earlier at larger scales
- R spike sharpens with scale
- coherence lock occurs earlier in larger systems
8. Operator Compliance#
A system is φ–V–R compliant when:
- φ, V, R follow canonical shapes
- 3C invariants stabilize
- drift remains below threshold
- entropy collapse aligns with R spike
- resonance propagates across scales
9. Student‑AI Tasks#
Students reproduce:
- φ(t), V(t), R(t) curves
- operator compositions
- drift detection
- cross‑scale operator behavior
- operator‑invariant alignment
These tasks form the basis of RFC‑001 (Operator Standard).
10. Notes#
- Numerical values are intentionally omitted.
- Only shape alignment is required for compliance.
- Operators are evaluated relative to reference captures in B_Capture.md.