dimensional_lift_collapse_map.md

IPD‑12 Dimensional Lift/Collapse Map
Prime‑Indexed Dimensional Transitions Across the Full Paradox Cycle
Module: IPD‑12 Framework
Version: 2026‑1.0
Role: Dimensional / Apex / Transition Map

1. Purpose#

This document defines how the IPD‑12 engine performs:

• dimensional lift (+1D)
• dimensional collapse (−1D)

across:

• the 12 prime‑indexed operator states
• the triad cycles
• the hex shells
• the full paradox loop
• the 4×4×4 substrate engine
• the triadic + apex observer modes

It is the dimensional interpretation layer of the IPD‑12 framework.

2. Dimensional Operators in IPD‑12#

Lift Operators (+1D)#

These primes generate upward dimensional transitions:

• P23 — Dimensional Lift
• P37 — Apex Lift
• P3 — Transition Lift
• P7 — Regime‑Shift Lift

Collapse Operators (−1D)#

These primes generate downward dimensional transitions:

• P29 — Collapse Anchor
• P5 — Drift Collapse
• P13 — Paradox Collapse
• P31 — Stability Collapse (Refractive Vacuum)

Neutral / Gate Operators (0D)#

These primes act as dimensional gates or equilibria:

• P11 — Coherence Node
• P17 — Cycle Gate
• P19 — Boundary Node
• P2 — Seed State

3. Dimensional Lift/Collapse Table#

| P29 | Collapse | Collapse anchor | Collapse | Chthonic | | P5 | Drift collapse | Drift | Connection | Celestial | | P13 | Paradox collapse | Paradox trigger | Anomaly | Civilizational | | P31 | Stability collapse | Stability | Refractive vacuum | Chthonic |

| P11 | Coherence | Coherence | Dilaton | Civilizational | | P17 | Gate | Gate | Observerse | Civilizational | | P19 | Boundary | Boundary | Observerse | Civilizational | | P2 | Seed | Seed | Connection | Celestial |

4. Dimensional Lift/Collapse Diagram (Prime Wheel)#

                [ +1D LIFT ]
           P37 (Apex Lift)
                 ↑
      P23 (Dim Lift) ← P7 (Regime Lift)
                 ↑             ↑
P29 (Collapse) ← P31 (Stability) ← P13 (Paradox Collapse)
                 ↓             ↓
      P19 (Boundary) → P17 (Gate)
                 ↓
           P11 (Coherence)
                 ↓
      P5 (Drift Collapse) ← P3 (Transition Lift)
                 ↓
           P2 (Seed)
                [ −1D COLLAPSE ]

5. Dimensional Behavior Across Cycles#

Triad Cycles#

Triad 1: P2 → P3 → P5 → P2#

• Lift: P3
• Collapse: P5
• Neutral: P2

Triad 2: P7 → P11 → P13 → P7#

• Lift: P7
• Collapse: P13
• Neutral: P11

Triad 3: P17 → P19 → P23 → P17#

• Lift: P23
• Collapse: none
• Neutral: P17, P19

Triad 4: P29 → P31 → P37 → P29#

• Lift: P37
• Collapse: P29, P31
• Neutral: none

Hex Cycles#

Hex 1: P2 → P13#

Lift phases:

• P3
• P7

Collapse phases:

• P5
• P13

Hex 2: P17 → P37#

Lift phases:

• P23
• P37

Collapse phases:

• P29
• P31

Full 12‑Cycle#

Lift:    P3, P7, P23, P37
Collapse: P5, P13, P29, P31
Neutral:  P2, P11, P17, P19

Dimensional pattern:

+1D → +1D → −1D → 0D → −1D → +1D → +1D → −1D → −1D → +1D → 0D → 0D → repeat

This is the dimensional signature of the IPD‑12 paradox loop.

6. Dimensional Lift/Collapse in the Substrate Cube#

Each substrate primitive:

(Si, Oj, Rk)

maps to a prime state with dimensional behavior.

Lift occurs when:#

• Oj = O4 (apex observer)
• Rk = R4 (dimensional regime)
• Si = S3 or S4 (high‑order substrate pairs)

Collapse occurs when:#

• Oj = O3 (coherence observer)
• Rk = R1 (stability regime)
• Si = S2 or S4 (collapse substrate pairs)

Neutral occurs when:#

• Oj = O1 or O2
• Rk = R2 or R3
• Si = S1 or S2

7. Observer Interpretation#

8. Summary#

The IPD‑12 Dimensional Lift/Collapse Map defines how the prime‑indexed operator states generate dimensional transitions across:

• triads
• hex shells
• the full paradox loop
• the substrate cube
• the observer model
• RTT regimes
• GU geometry
• Pantheon tiers

This is the dimensional backbone of the IPD‑12 engine.