> For the complete documentation index, see [llms.txt](https://kinbex.gitbook.io/kinbex-docs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://kinbex.gitbook.io/kinbex-docs/part-iii-verified-alpha-protocol/02-mathematical-formalisms-dcs.md).

# 8. Mathematical Formalisms & DCS Formula

[← Previous: Signal Entry Cryptography](/kinbex-docs/part-iii-verified-alpha-protocol/01-signal-entry-cryptography.md) | [Next: Anti-Sybil & Leaderboard Ranking →](/kinbex-docs/part-iii-verified-alpha-protocol/03-anti-sybil-and-ranking.md)

***

## 1. Quantitative Multiplier Formula

When an alpha caller flags a token at timestamp $t\_0$, the system captures initial entry price $P\_0$. Over an active evaluation window $T$, continuous telemetry monitors subsequent prices $P\_t$:

$$M = \frac{\max\_{t \in \[t\_0, t\_0 + T]} P\_t}{P\_0}$$

Where:

* $P\_0$: Entry price recorded at the exact moment of signal creation.
* $P\_t$: On-chain market price sampled at time $t$.
* $M$: Peak multiplier achieved (e.g., $2.5\times$, $10\times$, $100\times$).
* $T$: Maximum evaluation window (defaults to 30 days unless pool liquidity is drained).

If a token dumps and never exceeds $P\_0$, the multiplier is recorded as $M \le 1.0$.

***

## 2. Dynamic Caller Score (DCS)

Ranking callers purely by their highest multiplier creates a toxic incentive: callers spam hundreds of illiquid meme tokens hoping one lucky token hits $100\times$, ignoring the 99 tokens that went to zero.

To rank callers fairly, Kinbex calculates a composite **Dynamic Caller Score ($DCS$)** that balances multiplier performance, signal frequency, historical win rate, drawdown penalties, and time decay:

$$DCS = \left\[ \sum\_{i=1}^{N} w\_i \cdot \ln(M\_i) \right] \times \text{WinRate} \times (1 - \delta \cdot \text{DrawdownRate})$$

***

## 3. Mathematical Parameter Definitions

|         Symbol        | Mathematical Definition | Operational Meaning                                                                                                                  |
| :-------------------: | ----------------------- | ------------------------------------------------------------------------------------------------------------------------------------ |
|          $N$          | Total Calls             | Total number of verified signals published within the active evaluation cycle.                                                       |
|         $M\_i$        | Multiplier of Call $i$  | Peak multiplier achieved by call $i$, clamped at a baseline $1.0$ for failed calls ($M\_i \ge 1.0$).                                 |
|         $w\_i$        | Time-Decay Weight       | Exponential decay weighting: $w\_i = \exp(-\lambda \cdot (t\_{\text{now}} - t\_i))$, ensuring recent calls have higher significance. |
|       $\lambda$       | Decay Constant          | Halving parameter (set to $\lambda = 0.023$, equivalent to a 30-day half-life).                                                      |
|    $\text{WinRate}$   | Successful Call Ratio   | Proportion of calls achieving at least $2.0\times$ peak: $\text{WinRate} = \frac{N\_{\text{win}}}{N}$ where $M\_i \ge 2.0$.          |
| $\text{DrawdownRate}$ | Severe Dump Ratio       | Proportion of calls that suffered an $80%$ or greater drop within 1 hour of call creation (pump-and-dump indicator).                 |
|        $\delta$       | Risk Dampening Factor   | Protocol penalty dampener set to $\delta = 0.25$, severely penalizing callers who dump on followers.                                 |

***

## 4. Logarithmic Normalization Justification

The formula utilizes natural logarithms $\ln(M\_i)$ rather than linear multipliers. This prevents single extreme outliers (e.g., an accidental $500\times$ on an un-tradable micro-pool) from completely skewing the leaderboard over callers who demonstrate consistent, repeatable $3\times$ to $10\times$ alpha with high win rates.

```
Multiplier (M)   │   Linear Impact   │   Logarithmic Impact ln(M)
─────────────────┼───────────────────┼────────────────────────────
1.0x (Break-even)│   1.0             │   0.00
2.0x (Win Target)│   2.0             │   0.69
5.0x             │   5.0             │   1.61
10.0x            │   10.0            │   2.30
50.0x            │   50.0            │   3.91
100.0x           │   100.0           │   4.61
```

{% hint style="info" %}
**Mathematical Fairness:**\
Under the $DCS$ formulation, a caller with twenty verified $5\times$ calls and an $85%$ win rate will outrank a caller with one lucky $50\times$ call and a $10%$ win rate.
{% endhint %}