Ethereum uses a constant BLOB_BASE_FEE_UPDATE_FRACTION when updating the blob base fee each block, with the proportional change in the fee computed as:
math.exp((blob_gas_used - TARGET_BLOBS * GAS_PER_BLOB) / BLOB_BASE_FEE_UPDATE_FRACTION)
A new value for this constant has been included in each EIP that changes the target blob gas. It turns out that there is a simple generalization for computing the constant, consistent with both the value set in EIP-4844 and EIP-7691, that would be appropriate for the blob parameter only (BPO) hardforks of EIP-7892. This generalized update fraction calculation, also specified here, is:
BLOB_BASE_FEE_UPDATE_FRACTION = round(MAX_BLOBS * GAS_PER_BLOB / (2 * math.log(1.125)))
Mathematically, we can express the ideal (unrounded) value as
where m is the max blob gas per block. To see why this generalizes both EIP-4844 and EIP-7691, we will now set up the equations used by these EIPs and derive the generalized formula from them.
Exposition
Define t as the target blob gas per block. Under EIP-4844, the target is at half of the max: t=m/2, and the blob base fee is designed to increase or decrease by a maximum of 1.125 and 1/1.125 per block when blob usage is at its maximum or minimum, respectively. At max blob gas consumed, the equation for the proportional change in the fee becomes
Taking the natural logarithm of both sides and rearranging for b, then substituting t=m/2, yields the specified equation:
Under EIP-7691, the BLOB_BASE_FEE_UPDATE_FRACTION is derived as “the mid-point between keeping the responsiveness to full blobs and no blobs constant”. The b that keeps the response to “full blobs” constant is
The b that keeps the response to “no blobs” constant is
which via the logarithm rule: \ln(1/x) = -\ln(x), becomes
The “mid-point” (b_\text{full}+b_\text{empty})/2 simplifies to the generalized equation:
Application
The generalized equation gives satisfactory percentage changes at any ratio. Table 1 outlines a few examples.
| Max:Target | Full % | Empty % | Full factor 1.125^k | Empty factor 1.125^k |
|---|---|---|---|---|
| 5{:}4 | +4.82\% | -17.18\% | k = \tfrac{2}{5} | k = -\tfrac{8}{5} |
| 4{:}3 | +6.07\% | -16.19\% | k = \tfrac{1}{2} | k = -\tfrac{3}{2} |
| 3{:}2 | +8.17\% | -14.53\% | k = \tfrac{2}{3} | k = -\tfrac{4}{3} |
| 2{:}1 | +12.50\% | -11.11\% | k = 1 | k = -1 |
| 3{:}1 | +17.00\% | -7.55\% | k = \tfrac{4}{3} | k = -\tfrac{2}{3} |
| 4{:}1 | +19.32\% | -5.72\% | k = \tfrac{3}{2} | k = -\tfrac{1}{2} |
| 5{:}1 | +20.74\% | -4.60\% | k = \tfrac{8}{5} | k = -\tfrac{2}{5} |
Table 1. Impact for full and empty blocks across various max:target ratios with the generalized update fraction.
Note in columns 4 and 5 that the base fee changes by a factor of 1.125^k, where k is a function of the consumed blob gas g:
There is a complementary-ratio symmetry around 2:1. The magnitude of k is preserved when the target is moved an equal distance above or below the halfway point (m/2). Specifically, k_\text{full}(m{:}t) = -k_\text{empty}(m{:}(m-t)) and k_\text{full}(m{:}(m-t)) = -k_\text{empty}(m{:}t), which for example applies to the complementary ratios 3{:}1 and 3{:}2 as well as 4{:}1 and 4{:}3.
Extension to gas normalization
The generalized equation naturally extends to the gas normalization approach outlined in EIP-7999. This approach decouples the update fraction from any specific resource limit by being more direct: it normalizes the gas delta (g-t) by dividing it by the max gas m (referred to in EIP-7999 as the limit) when updating the running excess_gas. This allows a single, universal update fraction — independent of m — to be used for all resources, regardless of their individual limits.
The two approaches differ only in when the division by m occurs. In the generalized equation, m is placed in the numerator of the update fraction b. Since b itself is used as the denominator in the fee update exponent, (g-t)/b, this effectively places m in the denominator of the overall calculation.
Ultimately, the final fee change factor:
remains the same. However, the normalization makes the architecture more robust, ensuring the price remains stable if a resource’s limit is changed, as that limit is already part of the normalized excess_gas update.
Concluding points
The purpose of this post was the following:
- To expose why the proposed equation generalizes previous EIPs.
- To present the effect of the proposed equation across various max:target ratios, illustrating reasonable percentage changes and the complementary-ratio symmetry.
- To set the stage for further optimization in EIP-7999. The final piece of the puzzle is to take advantage of the generalization properties of the equation, but apply m at an earlier stage, already when updating the
excess_gas. - During the last ACDE, several participants only recognized the general properties after the discussion had concluded (via chat). This post can serve as a venue for a more informed conversation on establishing the proposed equation as the default for the update fraction going forward.