Coherence Map — General Relativity

TriadicFrameworks /docs/theories/general_relativity/coherence_map.md#

General Relativity (GR) is a geometric coherence theory of gravity. Coherence in GR is the stability of:

• the metric
• curvature
• geodesics
• causal structure
• regime transitions

Gravity = coherent curvature.
Geodesics = coherence trajectories.
Spacetime = a geometric operator field.

This file defines the coherence dimensions, coherence levels, collapse modes, and regime behavior for GR.

1. Coherence Dimensions#

GR uses five geometric coherence dimensions:

1.1 Metric Coherence#

Stability of the metric as a geometric structure.

A metric is coherent when:

• it is non‑degenerate
• causal cones remain valid
• signature remains stable

1.2 Curvature Coherence#

Stability of curvature as a geometric operator field.

Curvature is coherent when:

• curvature invariants remain stable
• curvature does not diverge
• curvature responds consistently to stress‑energy

1.3 Geodesic Coherence#

Stability of geodesics as coherence trajectories.

Geodesics are coherent when:

• they preserve identity
• they respond consistently to curvature
• they maintain causal compatibility

1.4 Causal Coherence#

Stability of causal structure.

Causal structure is coherent when:

• light cones remain valid
• no causal inversion occurs
• adjacency remains consistent

1.5 Regime Coherence#

Stability across R1 → R3 transitions.

Regime coherence holds when:

• transitions preserve geometric identity
• curvature operators remain valid
• dimensional profiles remain consistent

2. Coherence Levels (C0 → C4)#

Coherence is evaluated on a five‑level geometric scale:

C0 — Incoherent#

• metric invalid
• curvature undefined
• no geodesic structure

System cannot support GR.

C1 — Weak Coherence#

• metric barely stable
• curvature inconsistent
• geodesics unreliable

System supports only primitive geometry.

C2 — Moderate Coherence#

• metric stable
• curvature mostly consistent
• geodesics valid

System supports basic GR structure.

C3 — Strong Coherence#

• metric stable under deformation
• curvature consistent
• geodesics coherent
• causal structure intact
• regime transitions stable

System supports full GR behavior.

C4 — Perfect Coherence (Ideal)#

• metric perfectly stable
• curvature fully consistent
• geodesics perfectly coherent
• causal structure fully preserved
• regime transitions lossless

C4 is theoretical; real systems approach C3.

3. Collapse Modes (Geometric)#

Collapse occurs when geometry fails structurally.

G1 — Metric Degeneracy#

Metric becomes singular or invalid.

G2 — Curvature Divergence#

Curvature becomes unbounded or undefined.

G3 — Geodesic Incoherence#

Geodesics lose identity or causal compatibility.

G4 — Causal Structure Failure#

Light cones collapse or invert.

Collapse is geometric, not probabilistic.

4. Regime Behavior (R0 → R3)#

Coherence behaves differently across RTT regimes:

R0 — Pre‑Geometric#

• no metric
• no curvature
• no geodesics

Coherence undefined.

R1 — Metric Stability#

• metric stable
• causal structure emerges
• minimal curvature

Coherence dominated by metric stability.

R2 — Curvature Operators#

• curvature tensor active
• stress‑energy deforms geometry
• geodesics respond coherently

Coherence dominated by curvature stability.

R3 — Dimensional Curvature#

• curvature becomes dimensional
• geodesics become multi‑layer
• causal structure becomes layered

Coherence dominated by dimensional consistency.

5. Coherence Evaluation Procedure#

To evaluate coherence:

1. Validate metric stability
2. Validate curvature consistency
3. Validate geodesic coherence
4. Validate causal structure
5. Validate regime compatibility

If any step fails → classify collapse mode.

6. Summary#

GR coherence is:

• geometric
• operator‑driven
• curvature‑first
• regime‑aware
• zero drift

Gravity = coherent curvature.
Geodesics = coherence trajectories.
Spacetime = a geometric operator field.