Math overview

The four layered pieces of Inflexion’s math and the one-line mental model: MaxIL (how much capital) x fairRate (how risky) x load (how scarce) multiply into the premium.

Everything Inflexion charges an LP factors into four layered pieces that multiply into a single premium. They are deliberately orthogonal — each one isolates a different kind of information — and that separation is what lets the protocol publish an exact, closed-form price on-chain instead of a fitted curve.

One-line mental model. MaxIL is pure geometry (the how much capital). fairRate is pure stochastics (the how risky). The load stack is pure inventory state (the how scarce). They multiply:

$$\text{premium} \;=\; \underbrace{\text{fairRate}}_{\text{how risky}}\;\cdot\;\underbrace{\text{MaxIL}}_{\text{how much capital}}\;\cdot\;\underbrace{(1 + \text{load})}_{\text{how scarce}}$$

fairRate · MaxIL is the FairValue published on-chain; (1 + load) is the underwriter’s charge over fair value, hard-capped by invariant I10.

The four layers, in stacking order

1 — The IL formula

The Uniswap v3 settle-path math: V_hold (affine) vs V_lp (concave in range). realized_IL = max(0, V_hold − V_lp). This is the quantity settle() actually pays.

2 — MaxIL

The boundary IL — max(IL(Pa), IL(Pb)). Pure geometry, frozen at creation. It is both the load-bearing payout cap and the unit of risk.

3 — FairValue

FairPremium = E_Q[min(IL, MaxIL)], an exact closed-form Φ-sum. fairRate = FairPremium / MaxIL is the dimensionless S-curve in σ²·T.

4 — The load stack

baseLoad + util_skew + dispersion_skew, clamped to maxLoad by invariant I10. The underwriter’s charge over fair value, driven by inventory state.

Plus two companion topics that sit alongside the stack: the no-bad-debt math (why the cap makes FULL mode structurally solvent) and the theory anchors (the LVR / convexity results that make the IL claim priceable in the first place).

How the layers stack

Each layer carries exactly one kind of dependence — and only that kind. That is the whole design: it lets the protocol publish three prices for one position by moving a single dial.

Layer	Formula	Depends on	Carries
IL	$\max(0,\; V_{\text{hold}} - V_{\text{lp}})$	$P_a, P_b, L, P_0, P_T$	the realized settle quantity
MaxIL	$\max\big(IL(P_a),\, IL(P_b)\big)$	geometry only ($P_a, P_b, L, P_0$)	how much capital
fairRate	$E_Q[\min(IL, \text{MaxIL})]\,/\,\text{MaxIL}$	$a{=}P_a/P_0,\, b{=}P_b/P_0,\, \sigma_{\text{ref}},\, T$	how risky
FairPremium	$\text{fairRate}\cdot \text{MaxIL}$	the above	the published fair value
load	$\min(\text{baseLoad}+\text{util}+\text{disp},\; \text{maxLoad})$	inventory state	how scarce
premium	$\text{FairPremium}\cdot(1+\text{load})$	all of the above	the LP’s actual charge

Read top to bottom and the orthogonality is the point:

• MaxIL carries no vol and no time. It is pure geometry, frozen at creation, and identical across the three durations (7/30/90d) for a given position. It is also L-independent in the fair-rate sense — fairRate depends only on the ratios $a = P_a/P_0$, $b = P_b/P_0$, plus $\sigma$ and $T$, never on L. That is exactly what makes positions fungible to an underwriter: an MM quote is per-market (a load + an optional MaxIL-ratio band + capacity), never per-NFT.
• fairRate carries all the vol/time dependence. It is the S-curve in $\sigma^2 T$ — the fraction of MaxIL the claim is worth under the risk-neutral measure. So one position has one MaxIL but three FairPremiums (7/30/90d); only fairRate moves between them.
• The load carries only inventory state — vol regime, utilization, concentration. It never touches settle(), the MaxIL formula, or the no-bad-debt invariants. It is bounded by code (I10).

The factorization FairPremium = fairRate · MaxIL is the single most important structural choice in the protocol. Because MaxIL carries none of the vol/time dependence, the cvAMM can publish three durations for the same position by computing the geometry once and evaluating only the cheap S-curve three times — and a market maker can quote a whole market with a single load number, because every position in it is normalized by its own MaxIL.

A worked walk-through of the multiply

One position priced end-to-end (from the load-stack worked example; full numbers on the worked-example page): a 50,000 USDC position, ±10% geometric range, $\sigma_{\text{ref}} = 60\%$, baseLoad = +15%, skews = 0.

Geometry first — one MaxIL for all three durations:

$$\text{MaxIL} = 1{,}280\ \text{USDC} \quad (2.56\%\ \text{of }V_0)$$

Then the S-curve does all the work — fairRate is the only thing that changes:

duration	fairRate	FairPremium = fairRate·MaxIL	premium @ +15% load
7d	44.9%	574 USDC	661 USDC (1.32% of V₀)
30d	70.8%	906 USDC	1,042 USDC (2.08% of V₀)
90d	82.9%	1,061 USDC	1,221 USDC (2.44% of V₀)

Over 90d the ±10% band is touched ~75% of the time versus ~24% over 7d — the S-curve prices exactly that. The collateral (MaxIL) is constant; only the risk dial moves.

What’s exact, and the one residual assumption

Two of these layers are exact closed forms, not approximations:

• MaxIL is exact geometry. IL(P) = V_hold − V_lp is convex on $[P_a, P_b]$ (affine minus a strictly concave function), so its in-range max is always at an endpoint — proven for any entry $P_0$, centered or not. See the MaxIL derivation.
• fairRate is an exact finite Φ-sum (≈6–10 Black–Scholes-class terms), evaluated live per quote. It has no calibrated coefficients, no Monte Carlo, no lookup table — the capped v3 payoff is piecewise-linear/convex in the terminal price, and each arm integrates to a standard lognormal interval moment in the normal CDF $\Phi$. See the Phi-sum page.

The single modeling assumption is risk-neutral GBM with $\mu = 0,\ r = 0$ (forward = $P_0$). The Φ-sum is exact given that model. The residual is covered two ways: a deliberately conservative $\sigma_{\text{ref}}$ (an EWMA vol oracle with a hard floor that sits above calm ETH vol), and the forward-vol premium left as MM alpha — forward-looking MMs price off implied vol and correct the pool’s structural backward-looking bias (Pillar 3).

The closed form is verified ≡ the reference Python implementation (itself verified ≡ dense quadrature ≡ Monte Carlo) to max_abs_err ≈ 5.1e-11, and the on-chain Stylus Φ-sum matches 50-digit mpmath fixtures to ≤ 1e-12 on fairRate. The Stylus erf is integer fixed-point and machine-precise precisely because Stylus bans WASM floating point at activation, which is why the Stylus oracle is production and the Solidity FairValueOracle is only a CI cross-check.

The two separate guarantees the math gives you

The math underwrites two distinct claims — and they must never be merged.

The no-bad-debt structural core is three lines, and it is orthogonal to the premium and oracle-independent — it holds for any price path and any (even wrong) $\sigma$:

$$\begin{aligned} \text{collateral}_{\text{FULL}} &= \text{MaxIL} && \text{(locked at creation)} \\ \text{covered\_payoff} &= \min(\text{realized\_IL},\, \text{MaxIL}) \le \text{MaxIL} && \text{(by the cap)} \\ \therefore\quad \text{payout} &\le \text{collateral} = \text{MaxIL} && \text{(no bad debt: I1, I2)} \end{aligned}$$

The heavy Monte Carlo run confirmed max_payout_frac = 1.000000 on every simulated path, over a ±99% (and a widened −6…+6 log-move) grid, with zero exceptions. Full treatment on the no-bad-debt math page.

The second claim is a different animal entirely — it is about the underwriter, not the LP:

A wrong or too-low $\sigma_{\text{ref}}$ can make the pool under-charge load, so premium income stops covering payouts and NAV compresses — junior absorbs it first, senior in the tail. That can cost depositors money but can never create LP bad debt in FULL: collateral = MaxIL ≥ payout regardless of what $\sigma$ was. The vol oracle is solvency-load-bearing for the I10 cap and depositor solvency only, not for the FULL no-bad-debt invariant. Two separate guarantees, never merged.

Where to go next

The IL formula

V_hold vs V_lp, the three regimes, the guarded subtraction, and the Q64.96 fixed-point implementation.

MaxIL derivation

The convexity proof, why the cap is load-bearing, the unit-of-risk argument, and reference MaxIL/V₀ magnitudes.

The Phi-sum

The exact closed form: piecewise payoff, lognormal interval moments, cap-crossing prices, and verification to machine precision.

σ_ref

The EWMA vol oracle: max(σ_short, σ_long, floor), why the max is mandatory, and its precise load-bearing scope.

The load stack

baseLoad (the CVaR-gap envelope), util_skew, dispersion_skew, and the I10 clamp that makes overcharge impossible by code.

No-bad-debt math

The three-line structural core, the I1–I10 invariants, and the dual-tranche waterfall.

Worked example

One position, three prices, with the verified [80, 100, 125] fixture numbers to the wei.

Theory anchors

Lipton–Lucic–Sepp (2025) static replication and Milionis–Moallemi–Roughgarden (2022) LVR — why the claim is priceable.

Want the protocol-level picture instead of the formulas? Start at the three pillars (published FairValue, the cvAMM pool, MM competition), or see how the math plugs into a trade on the pricing stack and the lifecycle.