cycle_diagrams.md

IPD‑12 Cycle Diagrams
Triads • Hexes • Full 12‑Cycle Paradox Loop
Module: IPD‑12 Framework
Version: 2026‑1.0
Role: Cycle / Paradox / Regime Visualization

1. Purpose#

This document visualizes the intransitive cycle structure of the IPD‑12 engine:

• 4 triad cycles
• 2 hex cycles
• 1 full 12‑cycle paradox loop

These cycles define the paradox‑stable behavior of the prime‑indexed operator system.

2. Triad Cycles (4)#

Each triad is a 3‑node intransitive loop:

A → B → C → A

Triad 1 — Transition Cycle#

P2 → P3 → P5 → P2

   P2
    ↓
   P3
    ↓
   P5
    ↓
   P2

Meaning: seed → transition → drift → seed
RTT: drift/transition loop
GU: connection geometry
Pantheon: celestial order cycle

Triad 2 — Paradox Cycle#

P7 → P11 → P13 → P7

   P7
    ↓
   P11
    ↓
   P13
    ↓
   P7

Meaning: regime shift → coherence → paradox → regime shift
RTT: paradox trigger loop
GU: curvature ↔ anomaly
Pantheon: civilizational paradox cycle

Triad 3 — Boundary Cycle#

P17 → P19 → P23 → P17

   P17
     ↓
   P19
     ↓
   P23
     ↓
   P17

Meaning: gate → boundary → dimensional lift → gate
RTT: boundary regime cycle
GU: observerse geometry
Pantheon: civilizational ↔ chthonic boundary

Triad 4 — Apex Cycle#

P29 → P31 → P37 → P29

   P29
     ↓
   P31
     ↓
   P37
     ↓
   P29

Meaning: collapse → stability → apex → collapse
RTT: dimensional regime cycle
GU: collapse ↔ refractive vacuum ↔ anomaly apex
Pantheon: chthonic apex cycle

3. Hex Cycles (2)#

Each hex cycle is a 6‑node shell, combining two triads.

Hex Cycle 1 — Regime‑1 → Regime‑2 Shell#

P2 → P3 → P5 → P7 → P11 → P13 → P2

P2 → P3 → P5 → P7 → P11 → P13 → P2

Meaning: transition → drift → regime shift → coherence → paradox → transition
RTT: Regime‑1/2 shell
GU: connection → curvature → anomaly
Pantheon: celestial → civilizational

Hex Cycle 2 — Regime‑2 → Regime‑3 Shell#

P17 → P19 → P23 → P29 → P31 → P37 → P17

P17 → P19 → P23 → P29 → P31 → P37 → P17

Meaning: gate → boundary → lift → collapse → stability → apex → gate
RTT: Regime‑2/3 shell
GU: observerse → collapse → refractive vacuum → anomaly
Pantheon: civilizational → chthonic

4. Full 12‑Cycle Paradox Loop#

This is the complete IPD‑12 paradox engine, spanning all regimes:

P2 → P3 → P5 → P7 → P11 → P13 →
P17 → P19 → P23 → P29 → P31 → P37 → P2

ASCII diagram:

        P37 → P2 → P3
       ↑             ↓
     P31             P5
       ↑             ↓
     P29             P7
       ↑             ↓
     P23 ← P19 ← P17 ← P13 ← P11

Interpretation#

RTT:

• full paradox loop
• regime traversal (0 → 1 → 2 → 3 → 0)

GU:

• connection → curvature → anomaly → observerse → collapse → refractive vacuum → anomaly apex

Pantheon:

• celestial → civilizational → chthonic → apex

5. Cycle → Regime Mapping#

6. Cycle → Substrate Mapping#

Cycles traverse the 4×4×4 substrate engine:

• Triads: 3‑step substrate traversal
• Hexes: 6‑step shell traversal
• Full cycle: 12‑step dimensional traversal

Each step corresponds to a substrate primitive:

(Si, Oj, Rk)

7. Summary#

This document provides the complete cycle visualization for the IPD‑12 framework:

• 4 triads
• 2 hex shells
• 1 full paradox loop

These cycles define the paradox‑stable, prime‑indexed behavior of the IPD‑12 engine and integrate with RTT, GU, Pantheon, and the substrate cube.