Mathematical derivation of O(n²) vs O(n) scaling.
Token counts measured empirically from cogency agent runs with gpt-4o-mini:
- SYSTEM: Measured from typical tool registry (10 tools) + protocol instructions + user profile
- THINK: Average reasoning block across 50 production turns
- CALLS: Typical single tool invocation with file paths
- RESULTS: Average tool output (file contents, search results, shell output)
SYSTEM = 1500 tokens # Tool schemas + protocol + memory
THINK = 200 tokens # Agent reasoning per turn
CALLS = 100 tokens # Tool invocation JSON
RESULTS = 300 tokens # Tool output
Values are conservative estimates. Actual token usage varies by:
- Tool complexity (read vs search vs edit)
- File sizes and content density
- Profile size (0-2000 chars)
Each turn replays growing history:
Turn 1: SYSTEM + THINK + CALLS = 1800
Turn 2: SYSTEM + [Turn 1] + RESULTS + THINK + CALLS = 2400
Turn n: 1800 + 600(n-1)
Total for n turns:
Σ[1800 + 600(k-1)] for k=1 to n
= 1800n + 600 × n(n-1)/2
= 1500n + 300n²
Server maintains state, only incremental content:
Turn 1: SYSTEM + THINK + CALLS = 1800
Turn 2: RESULTS + THINK + CALLS = 600
Turn n: 600
Total for n turns: 1200 + 600n
| Turns | Replay | Resume | Ratio |
|---|---|---|---|
| 1 | 1,800 | 1,800 | 1.0x |
| 8 | 31,200 | 6,000 | 5.2x |
| 16 | 100,800 | 10,800 | 9.3x |
| 32 | 355,200 | 20,400 | 17.4x |
Efficiency ratio: (1500n + 300n²) / (1200 + 600n)
Limit as n → ∞: n/2
The efficiency advantage grows linearly with conversation depth.