With increasing the target to 4 and keeping the max at 6, we would end up with asymmetric max-scale rates of 6% (positive) and 11% negative. It would be better to achieve symmetry around the target. Thus we can simply scale the deviation form the taget by a scaling factor derived by comparing the max negative deviation with the max positive deviation. The scaling factor can then be applied directly or to the normalized excess gas.


### Deriving the Scaling Factor

#### Understanding the Asymmetry

- **Maximum Negative Deviation ($D_{\text{neg}}$)**:

$$
D_{\text{neg}} = \text{Target} - \text{Minimum~BlobGasUsed} = T - 0 = T
$$

- **Maximum Positive Deviation ($D_{\text{pos}}$)**:

$$
D_{\text{pos}} = \text{Max~BlobGasUsed} - \text{Target} = M - T
$$

- For $T = 4$ and $M = 6$:

$$
D_{\text{neg}} = 4, \quad D_{\text{pos}} = 6 - 4 = 2
$$

#### Equalizing the Deviations

To achieve symmetry, set the maximum positive and negative deviations equal in magnitude by introducing a scaling factor $S$:

$$
S = \frac{D_{\text{neg}}}{D_{\text{pos}}} = \frac{T}{M - T}
$$

Substituting the values:

$$
S = \frac{4}{6 - 4} = \frac{4}{2} = 2
$$

---

### Adjusted Excess Blob Gas Calculation

- **For $\text{BlobGasUsed} \gt T$**:

$$
\text{AdjustedExcessBlobGas} = (\text{BlobGasUsed} - T) \times S = (\text{BlobGasUsed} - T) \times \left( \frac{T}{M - T} \right)
$$

- **For $\text{BlobGasUsed} \leq T$**:

$$
\text{AdjustedExcessBlobGas} = \text{BlobGasUsed} - T
$$

This adjustment ensures the maximum deviations of the adjusted excess blob gas are equal in magnitude.

---

### Calculating the Update Fraction

To achieve a maximum base fee change of ±12.5% (i.e., $R_{\text{max}} = 1.125$), calculate the update fraction:

- **At Maximum Deviation**:

$$
\left| \text{AdjustedExcessBlobGas}_{\text{max}} \right| = T
$$

- **Exponent at Maximum Deviation**:

$$
\left| \text{Exponent}_{\text{max}} \right| = \frac{\left| \text{AdjustedExcessBlobGas}_{\text{max}} \right|}{\text{UpdateFraction}} = \ln(R_{\text{max}})
$$

- **Solving for UpdateFraction**:

$$
\text{UpdateFraction} = \frac{T}{\ln(R_{\text{max}})}
$$

Substituting the values:

$$
\text{UpdateFraction} = \frac{4}{\ln(1.125)} \approx \frac{4}{0.11778} \approx 33.95
$$

---

### Final Base Fee Update Formula

The base fee per blob gas is updated using:

$$
\text{BaseFeePerBlobGas} = \text{PreviousBaseFeePerBlobGas} \times e^{\left( \dfrac{\text{AdjustedExcessBlobGas}}{\text{UpdateFraction}} \right)}
$$

Where:

- **AdjustedExcessBlobGas** is calculated as:

- **For $\text{BlobGasUsed} \geq T$**:

$$
\text{AdjustedExcessBlobGas} = (\text{BlobGasUsed} - T) \times \left( \dfrac{T}{M - T} \right)
$$

- **For $\text{BlobGasUsed} \leq T$**:

$$
\text{AdjustedExcessBlobGas} = \text{BlobGasUsed} - T
$$

- **UpdateFraction**:

$$
\text{UpdateFraction} = \dfrac{T}{\ln(R_{\text{max}})} \approx 33.95
$$

---

### Verification of the Formula

#### At $\text{BlobGasUsed} = 0$ Blobs

- **AdjustedExcessBlobGas**:

$$
\text{AdjustedExcessBlobGas} = 0 - 4 = -4
$$

- **Exponent**:

$$
\text{Exponent} = \dfrac{-4}{33.95} = -0.11778
$$

- **Base Fee Adjustment**:

$$
\text{BaseFeePerBlobGas} = 1 \times e^{-0.11778} \approx 0.8889
$$

- **Percentage Change**:

$$
\text{Decrease} = 1 - 0.8889 = 0.1111 \text{ or } 11.11\%
$$

#### At $\text{BlobGasUsed} = 6$ Blobs

- **AdjustedExcessBlobGas**:

$$
\text{AdjustedExcessBlobGas} = (6 - 4) \times \left( \dfrac{4}{6 - 4} \right) = 2 \times 2 = +4
$$

- **Exponent**:

$$
\text{Exponent} = \dfrac{+4}{33.95} = +0.11778
$$

- **Base Fee Adjustment**:

$$
\text{BaseFeePerBlobGas} = 1 \times e^{+0.11778} \approx 1.125
$$

- **Percentage Change**:

$$
\text{Increase} = 1.125 - 1 = 0.125 \text{ or } 12.5\%
$$

---