# Ergodicity Breaking as GF(3) Conservation Violation
## The Conservation Law
In a balanced system, trits sum to zero:
```
MINUS(-1) + ERGODIC(0) + PLUS(+1) = 0
```
This is **GF(3) conservation**: the triadic balance that enables ergodic exploration of state space.
## What Ergodicity Means
**Ergodic**: Time averages equal ensemble averages. Any trajectory eventually visits all accessible states with the right frequency.
```
lim(T→∞) (1/T) ∫₀ᵀ f(x(t)) dt = ∫ f(x) dμ(x)
```
For skills/events: Over time, you encounter MINUS, ERGODIC, and PLUS events in balanced proportion.
## Breaking Directions
When conservation fails, the system becomes **non-ergodic** - trapped in basins:
### 1. PLUS Accumulation (+1 excess)
```
Direction: Σ trits > 0 (mod 3)
+1 +1 +1 +1
───►───►───►───► Runaway generation
State: Only creation, no validation
Ideas multiply without critique
Unbounded growth → collapse
Example: Repo where every patch is applied,
no discussions, no review
```
**Symptom**: `patch_applied` events dominate. Color palette skews toward PLUS hues.
**Real-world**: Feature creep, tech debt accumulation, "move fast break things" without the fix.
### 2. MINUS Accumulation (-1 excess)
```
Direction: Σ trits < 0 (mod 3)
-1 -1 -1 -1
◄───◄───◄───◄─── Runaway validation
State: Only critique, no creation
Everything rejected
Paralysis → stagnation
Example: Repo with endless discussions,
no patches ever applied
```
**Symptom**: `new_discussion` events dominate. Nothing ships.
**Real-world**: Bikeshedding, analysis paralysis, committee death.
### 3. ERGODIC Collapse (0 dominance)
```
Direction: Only trit=0 events
0 0 0 0
─○─○─○─○─○─○─○─ Pure coordination
State: Shuffling without substance
Movement without change
Bureaucracy
Example: Patches added but never applied,
discussions opened but never resolved
```
**Symptom**: `patch_added` without `patch_applied`. Limbo state.
**Real-world**: Endless CI runs, PRs in "changes requested" forever.
### 4. ERGODIC Absence (0 missing)
```
Direction: Only ±1, no 0
+1 -1 +1 -1 +1 -1
───◄───►───◄───►───◄─── Oscillation
State: Creation/destruction cycle
No stable coordination layer
Whiplash
Example: Patches applied then reverted,
applied then reverted...
```
**Symptom**: High churn, no stable state.
**Real-world**: Revert wars, architectural flip-flops.
## Phase Diagram
```
+1 (PLUS)
▲
/|\
/ | \
/ | \
/ A | \ A = Generative excess
/ | \ B = Validation excess
/ | \ C = Coordination excess
/ ERGODIC \ ● = Balanced (ergodic)
/ ● \
/ / | \ \
/ / | \ \
/ C / | \ B \
▼──────▼────┴────▼──────▼
0 (ERGODIC) -1 (MINUS)
```
The center (●) is the ergodic fixed point where all three trits balance.
## Detecting Violations
### Rolling Window Check
```python
def check_gf3_balance(events, window=9):
"""Check GF(3) conservation over sliding window."""
trits = [event.trit for event in events[-window:]]
balance = sum(trits) % 3
if balance == 0:
return "ERGODIC: balanced"
elif balance == 1: # +1 excess
return "BREAKING: PLUS accumulation"
else: # balance == 2 ≡ -1 (mod 3)
return "BREAKING: MINUS accumulation"
```
### Color Drift Detection
```julia
using Gay
function detect_drift(events, seed)
colors = [Gay.color_at(i, seed=seed) for (i, _) in enumerate(events)]
hues = [c.hue for c in colors]
# Check if hue distribution is uniform (ergodic)
# or clustered (non-ergodic)
variance = var(diff(sort(hues)))
return variance < threshold ? :ergodic : :breaking
end
```
## Restoring Ergodicity
When conservation fails, inject compensating events:
| +1 excess | Add -1 events | Open discussions, request reviews |
| -1 excess | Add +1 events | Apply pending patches, ship |
| 0 excess | Add ±1 events | Decide: either accept or reject |
| 0 absent | Add 0 events | Record patches before applying |
### Automatic Rebalancing Hook
```yaml
# In webhook handler
on_event:
- check_gf3_balance(last_9_events)
- if: violation
then:
notify: "GF(3) drift detected: {direction}"
suggest: "Consider {compensating_action}"
```
## Philosophical Frame
**Ergodicity** = the system explores all possibilities fairly over time.
**Breaking** = the system gets stuck, favoring certain modes:
- All creation → bubble
- All critique → freeze
- All coordination → bureaucracy
- No coordination → chaos
GF(3) conservation is the **symmetry** that preserves ergodicity. Violations are **symmetry breaking** - the system chooses a direction and gets trapped.
The color game makes this visible: balanced systems show diverse palettes, broken systems show color clustering.
## Connection to Pijul
Pijul's patch theory already has commutativity (patches can apply in any order when independent). This is a form of ergodicity at the patch level.
GF(3) conservation extends this to the *event* level: discussions, patches, and applications should flow in balanced proportion for healthy repo evolution.
```
HEALTHY REPO:
discussion → patch → applied → discussion → patch → applied
-1 → 0 → +1 → -1 → 0 → +1
Σ = 0 (mod 3) ✓
SICK REPO:
patch → patch → patch → patch → applied → applied → applied
0 → 0 → 0 → 0 → +1 → +1 → +1
Σ = 0 (mod 3) but distribution wrong
(coordination without discourse)
```
The second case sums to 0 but lacks the -1 feedback loop. It's *statistically* balanced but *dynamically* broken - no discussion means no course correction.