Faster blocks cannot drive constant-product LP losses to zero once the reference price jumps. Bundi proves it as an exact inequality for symmetric jump laws satisfying his aggregation hypothesis, and it is the durable result in the paper. The 8.4-second optimal block time in the same paper is a much softer object, and the paper's own sensitivity table says so.
LVR measures how much a passive LP gives up against a continuously rebalanced portfolio holding the same instantaneous exposure. Under geometric Brownian motion, the frictionless rate for a full-range Uniswap-style pool is sigma-squared times pool value over eight. Add a swap fee and Poisson blocks. The rate then gets multiplied by the stationary probability that an arriving block finds the pool mispriced beyond the fee band. That probability collapses as block time shrinks, which is the standing technical argument for sub-second slots.
Bundi swaps geometric Brownian motion for a Merton jump-diffusion and redoes the calculation. Apply the It\u00f4-L\u00e9vy formula to the pool value function and the zero-fee loss splits in two. One piece is the classical diffusion term. The other is a jump term, proportional to jump intensity times the expected squared concavity gap, and it carries no block time at all. With fees and finite blocks he writes the log mispricing between pool and reference price as a state driven by two point processes. Exogenous jumps push it out of the no-arbitrage band. Block arrivals push it back to the band edge. He then works with what he calls the separated rate, the version where those two channels do not interact. There the diffusion term carries the known multiplier. The jump term is a functional of the jump law alone, with no block time in it. For symmetric Merton jumps that functional collapses to a discount in fee over jump size.
The lever is smaller than the sub-second case assumes
Calibration is Binance ETH/USDT five-minute closes over January 2020 to June 2026, 2,373 days. Sigma comes from the continuous part of realised variance, via bipower variation: 0.8156. Jump intensity and jump standard deviation come from the Lee-Mykland jump-detection test at the 1% level, with the detected jump sizes corrected for their diffusive component. Those two land at 283.3 a year and 0.0192. At the Uniswap 0.05% tier and $1M of pool value, the rate at Ethereum's 12-second slot is 471 bp/yr against a jump floor of 125. 73% of the loss is schedule-addressable.
Then the convergence. Cut block time by a factor of 240, from 12 s to 50 ms, and only 89% of the diffusion term goes away. The rate is still 162 bp/yr, 29% above the floor. The two channels are equal at 750 ms. The diffusion term does not fall to a tenth of the floor until 5.5 ms.
Solana's 400 ms slot sits at 221 bp/yr, with only 43% of it diffusion.
In the Milionis accounting, fees split LVR between arbitrageurs and LPs rather than reducing it. On the diffusion channel the LP share goes to one as blocks shorten. On the jump channel the recovery is fixed at 4.1% at this calibration. The discount is a function of fee over jump size, and no block time enters it. The lever against the floor is fee-tier design.
We could not test any of this on our own data. Block-level mispricing needs pool reserves and swap events at block resolution, and our crypto history is daily and one-minute bars.
Can jumps really be schedule-invariant?
The mechanism requires a jump to open a gap off-chain and then be cleared at essentially full size at the next block, by an arbitrageur who is capital-unconstrained and has zero latency. Bundi flags the idealization himself, and flags it in the right place: it is weakest exactly in the sub-second regime the floor argument concerns. The scales he calibrates sit far apart. Jump standard deviation runs about 38 times the fee band. Jumps arrive on the order of 10^-4 per block at 12 s. A jump that clears the band therefore clears it decisively. Whether one searcher captures the whole gap inside a 400 ms slot is my own doubt rather than his.
The assumption I would push hardest on is symmetry. Proposition 1 and the exact floor theorem both need it, and it is imposed rather than estimated. Bundi sets m to zero. The sample point estimate is -0.0012, with a 99% interval of [-0.0026, +0.0003]. That interval contains zero, though on my reading it sits almost entirely below it. Bundi is clear about which results survive general jump laws (Theorems 1, 2, 4) and which do not (the remainder bound and the floor). The headline result is the floor, and Theorem 3 requires symmetric nu, which for Merton means m = 0. Two other choices are conceded in the paper: Poisson block arrivals rather than a deterministic proof-of-stake schedule, and a full-range pool rather than concentrated liquidity. The first is conservative. The second means the full-range floor is not a lower bound for a concentrated position.
Exact floor, bounded remainder
The zero-fee decomposition is clean and general, depending on the jump law only through one moment. The fee-and-block-time formula is a definition plus a bound. Bundi says the equation defines the remainder and carries no content until that remainder is bounded. He bounds it to [-0.653, +0.849] bp/yr at 12 s and [-0.355, +0.464] at 50 ms, never more than 0.29% of the rate. Quadrature puts the realised value at +0.165 and +0.052. Those are small numbers and the bound is honest.
The bound rests on a mixing condition that is asserted and checked numerically rather than proved for the relevant kernel. The reported ratio peaks at 1.27 near eta = 0.8, against the 2 assumed. It settles near 0.38 in the fast-block corner, thinnest where the band is narrow relative to a block's diffusion.
The floor itself needs none of this, which is the paper's best move. Theorem 3 gets there by a chain of three inequalities, never touching the stationary law, which has no closed form once intensity is positive.
Eight seconds, conditional on a cost nobody measures
Netting the separated rate against per-block consensus cost gives a cubic with a unique root, solvable by Cardano. The invariances are the striking part. Pool value drops out of the first-order condition exactly. Under the separated rate, jumps shift the level of LP loss and drop out of the first-order condition entirely. Under the exact rate the shift is about a second.
The 8.4 seconds is another matter. Consensus cost is proxied by Ethereum gross issuance, about $1.7B a year against $208B of ETH. Bundi scales that by 0.83, because current staking sits above Drake's security-optimal quarter of supply. He then attributes it pro-rata by value secured, at 4.8 x 10^-6. He says plainly that issuance is a compensation policy rather than a measured resource cost. It overstates the cost where issuance subsidizes growth, and understates it where MEV and fees fund validators. Run the cost from 500 down to 2 (x 10^-5 USD per block) and the optimum runs 15.4, 8.4, 3.4, 1.4, 0.20 seconds. Nearly two orders of magnitude. Volatility does comparable work: 0.30, 0.815, 1.50 gives 62.3, 8.4, 2.5 s.
The gain is thin as well. Moving Ethereum from 12 s to 8.4 s lowers the objective from 539 to 533 bp/yr. Six basis points, under 2%. The objective sits within 1 bp/yr of its minimum across [7.4, 9.7] s. Accounting for the remainder widens the plausible minimizer to [7.3, 9.9] s, which Bundi concedes is not resolvable against the flatness.
In fairness, he does not rest on the point estimate. His conclusion rests on an ordering. Ethereum's 12 s is somewhat too long under an LP-LVR-only objective, while the sub-second targets pursued by some chains overshoot what LVR alone would justify. That ordering is the claim worth attacking, and his own cost sensitivity attacks it. At c = 2 x 10^-5 USD per block the optimum is 0.20 seconds. Sub-second slots then land on the right side of the tradeoff, which flips the second half of the verdict. The ordering holds only if consensus really costs something close to issuance.
I would read the paper for the floor and for the ratio of fee to jump size, and treat the eight seconds as a placeholder. The regime grid makes the point better than the point estimate does. At 12 s the rate spans 55 bp/yr in the calm, jump-light corner to 3,597 in the stressed, jump-heavy one, a factor of 65, with the diffusion share running from 1% to 98%. Any claim about whether faster blocks help this pool is a claim about which corner it is in today.
One thing would change my mind on the floor. The entire level, 125 bp/yr, is set by the jump intensity and the jump size, and nothing else enters it. Both come from a jump-detection test run at the 1% level on five-minute closes from a single exchange. That estimator cannot separate a true price discontinuity from microstructure noise or an exchange-specific gap. Re-estimate them on tick data across venues and the 125 moves. The structural result does not.