Foundations

AIMM is assembled from published work: inventory-based market making (Avellaneda-Stoikov, Bergault et al.), the LVR result of Milionis et al., asset-liability accounting from early stableswap design, oracle-anchored quoting from earlier on-chain market makers, and concentrated-liquidity research. This page traces each borrowed mechanism to its source, states what AIMM changed, and records the designs evaluated and not adopted. The vision the design serves is in the Manifesto.

1. What AIMM is built on

Eight statements. Each one is enforced by the contracts, and each links to the page that specifies it.

1. Price comes from outside the pool. Reserves set depth, never direction. There is no on-chain price EMA, no TWAP and no reserve mid. Anchor Path Pricing
2. Inventory moves the mid, never the spread. Coverage deviation shifts the reservation price; the spread stays symmetric around it. Inventory Management
3. The spread is a sum of risk terms, not a rate card. A volatility band, the feed’s confidence interval and a mark-staleness surcharge, composed over every leg of the path. Spread & Fees
4. Depth is data, not code. A clamped quartic I-spline sets liquidity at every price; changing one is a timelocked write, not a contract upgrade. Liquidity Shaping
5. One pool holds N assets and prices any pair along the unique anchor-tree path: up to the lowest common ancestor, then down. Two stables never consult the base mark. Anchor Path Pricing
6. Solvency is pooled. Coverage is tracked per leg, but every mint and exit settles at one pool-level rate, so no exit can price ahead of another. Inventory Management
7. A drained leg is priced, not written down. The convex coverage wall makes the next drain expensive, and the toll it keeps raises the pool rate. Liabilities are never rewritten. Inventory Management
8. Adverse selection is priced, not excluded. No solver set, no last look, no whitelist by default: informed one-way flow pays through the skew, the benign majority pays the fee floor. Toxic Flow Mitigation

Sections 2 to 15 derive those statements from their sources. Sections 18 and 19 measure the result against the same prior art. Orbital and RFQ/intents were evaluated and not built. Papers are listed in §16, venue whitepapers in §19.6.

2. Inventory-Based Market Making: The Avellaneda-Stoikov Framework

2.1. The reservation price

AIMM’s pricing traces to Marco Avellaneda and Sasha Stoikov’s 2008 paper “High-Frequency Trading in a Limit Order Book”, which formalized how a rational market maker quotes asymmetrically against inventory risk. Rather than quoting symmetrically around mid, the optimal strategy adjusts quotes to encourage trades that reduce inventory:

where $r$ = reservation price, $s$ = mid-market price, $q$ = inventory quantity (positive = long), $\gamma$ = risk aversion, $\sigma$ = volatility, $\tau =T-t$ = time remaining. When $q>0$ the reservation price shifts below market: the maker sells lower to reduce exposure. The converse applies for short positions.

2.2. Optimal spread

$k$ = order arrival intensity. Higher volatility or risk aversion widens the spread; more frequent order flow tightens it.

Framework assumptions, applying to both formulas. AS-2008 assumes a finite horizon and exponential order-arrival intensity $\lambda =A·\exp (-k\delta )$. AIMM, as an infinite-horizon AMM, uses the stationary-limit reformulation (Guéant-Lehalle-Fernandez-Tapia 2012): the $\gamma {\sigma }^2\tau$ inventory penalty is replaced by its stationary analog.

2.3. AIMM’s adaptation

The volatility band $S_v$ is the symmetric spread component; the skew shifts the mid only, so coverage never enters the spread (Spread & Fees).

3. Asset-Liability Management: Platypus & Wombat

3.1. The two-ledger innovation

Platypus Finance (2021) and later Wombat Exchange were the earliest on-chain asset-liability AMMs. Uniswap-style AMMs track only reserves; Platypus tracked assets (tokens the pool holds) against liabilities (tokens it owes LPs). Two ledgers buy single-sided deposits, an explicit per-token solvency figure, and per-token liability growth.

3.2. Coverage ratio

Coverage = Assets / Liabilities.

Both deployed it for stable pairs only, and both used coverage as a slippage modifier on a modified invariant: coverage moves where a trade lands on a fixed curve. No volatility term, no depth shaping, no inventory theory behind the slip function.

3.3. AIMM’s extension

AIMM keeps the two ledgers and the ratio and replaces the pricing:

• Price is decoupled from reserves. Price derives from the mark and spline profiles; reserves determine depth, not direction.
• Coverage is the inventory input to an Avellaneda-Stoikov reservation price (§2). A mid shift, not fee asymmetry.
• Coverage restores itself through pricing rather than through a write-down of LP claims. The convex coverage wall (kappaCovBps) tolls any swap draining a walled leg further, superlinearly as $c$ falls, and the skew moves the mid toward flow that refills it. LP solvency is pooled, in the sense of §1 statement 6, so a drained leg against a filled one is inventory location rather than an LP loss. Both act on the trade path. The one write-down of LP claims is a yield-venue loss, charged to that leg’s LPs (Hooks).

Mechanics: Inventory Management.

3.4. Oracle-anchored proactive market making: DODO PMM

DODO’s Proactive Market Maker (2020) is the first on-chain form of the §2 reservation price, and the first on-chain AMM to abandon reserve-derived price discovery for an imported mark with a curve shaped around it. PMM prices around an oracle mid i with a slippage coefficient k; from the 2020 whitepaper, §3:

with $B$ base reserves and $B_0$ the oracle-set target. k → 0 collapses toward constant-sum (flat, stableswap-like); k → 1 recovers constant-product impact. Between the two branches the quote is inventory-aware: as one asset depletes, PMM makes it dearer to buy and cheaper to sell, pulling rebalancing flow back to target.

AIMM inherits the imported mark (the keeper mark plays i), the inventory-driven mid shift (computeInventorySkew) and a tunable flatness knob (k is the predecessor of vol-scaled dispersion plus the quartic I-spline). It changes all three, and §4.2 and §12.1 record how: a piecewise-linear surrogate over coverage $c$ rather than a quadratic over $B/B_0$, a per-asset depth curve adapting every swap rather than one static global k, and a multi-asset anchor tree on a self-owned keeper mark rather than a Chainlink-anchored pair.

3.5. Shared-pool single-sided deposit: GMX GLP and GM

GMX is the closest precedent for the deposit shape: both generations let a provider deposit one asset into a pool holding several. GLP (V1) is one global basket, so deposit BTC and you hold the index; composition is steered by target weights and a dynamic mint/burn fee, cheaper for an underweight token and dearer for an overweight one, a fee-side version of the pressure BTR applies on the mid. GM (V2) is a set of isolated two-token markets whose receipt prices as (pool value + net pending PnL) / supply, so a one-sided depositor still holds both sides plus the traders’ open PnL; its deposit and withdrawal price impact, paying for improving the pool’s balance and charging for worsening it, is the production mechanism closest to coverage-driven inventory skew.

Three differences from BTR are structural. What the provider holds: GLP gives an index share and GM a two-sided market share, while BTR keeps the claim on the leg deposited into, with per-leg liabilities, a per-leg receipt token and a withdrawal in that same asset, so the position needs no rebalancing. What they are counterparty to: GMX is a perpetuals venue, so a holder is counterparty to leveraged traders and trader profit is paid out of pool value, whereas a BTR provider faces only adverse selection on swap flow, priced by the spread and the skew. What “coverage” means: GMX bounds exposure with a reserve factor, available liquidity = pool tokens × reserve factor − reserved tokens, enforced by blocking, so at zero available liquidity a provider waits for positions to close and there is no haircut, because there is no per-asset solvency figure to be short against. BTR’s coverage is that figure (§3.2), enforced by pricing, and LPs settle at one pool-level rate, so a drained leg does not queue its providers.

Single-sided deposit into a multi-asset pool is an asset-liability idea that GMX productionised at scale first; doing it for a perpetuals book is what pushed GMX toward a pooled index claim rather than per-leg liabilities.

4. Oracle-Guided Pricing: Swaap’s Matrix Market Maker

4.1. MMM and the stochastic spread

Swaap Finance introduced the Matrix Market Maker (MMM), “an oracle-guided, multi-asset AMM. Its goal is to function like a passive Index ETF, aiming for near-zero impermanent loss by decoupling price discovery (from oracles) from the act of market-making” (introduction). Instead of volume-based fees it runs a “stochastic spread mechanism” adjusting fees on real-time volatility, charging higher premiums for riskier trades, which directly influenced AIMM’s volatility band $S_v$. Multi-asset weighting uses a constant geometric mean product $\prod _i{x}_{i^{w_i}}=k$ for quantity $x_i$ at weight $w_i$, so the pool rebalances to maintain allocations. Swaap v2 then moved to “a non-custodial RfQ market-making infrastructure” with on-chain drawdown circuit breakers and last look, prioritizing professional market makers over passive LPs.

4.2. AIMM’s divergence

Tail risk is gated by per-asset depeg bands checked against an independent reference feed (Mark Relayer, Flow Guards).

4.3. The optimal-pricing theory: Bergault, Bertucci, Bouba and Guéant

Swaap’s MMM and AIMM are two implementations of one theory, the most direct academic parent of AIMM’s design: Automated Market Makers Designs beyond Constant Functions (2022) and Price-Aware Automated Market Makers: Models Beyond Brownian Prices and Static Liquidity (2024), which extend the §2 objective from a single quoting agent to an AMM referencing an imported mark.

Their result: for a maker quoting bid/ask markups δ(t,z) around an imported mark S_t, maximizing excess-PnL-vs-HODL under a quadratic-variation risk penalty, the mean-variance-optimal quotes are markups linear in the inventory-value deviation and in trade size, with slope driven by the effective instantaneous variance. That holds under Heston-Bates, Stein-Stein-with-jumps and Hawkes dynamics, beyond the Brownian, static-liquidity assumptions of earlier AMM analyses.

Validation against the papers concluded AIMM is a faithful on-chain implementation of this optimal-quote structure. §2 is the inventory-theory root, §4.1 a peer implementation, and this line the optimal-AMM theory AIMM instantiates.

5. Dynamic Pegging: Curve v2 Cryptoswap

Curve v2 Cryptoswap introduced dynamic repegging: the concentrated-liquidity center moves automatically with an internal price oracle, an exponential moving average of trade prices applied in N-dimensional price space. Per Curve’s documentation, “a repegging algorithm which is run after each swap” keeps swaps near the equilibrium point. That internal oracle determines where liquidity concentrates, not only how trades price.

Cryptoswap tracks xcp_profit and xcp_profit_real so repegging cannot destroy LP value: it “undoes p adjustment if it causes xcp_profit_real-1 to fall lower than half of xcp_profit-1”, constraining curve updates to cases where accumulated fee revenue exceeds incurred losses. The residual is a loss type Uniswap does not have:

“Crypto Pool is vulnerable to permanent loss (which cannot be recovered even if the price does) due to changes in the curve itself when repegging the equilibrium price.”

AIMM performs no dynamic invariant update: shape changes go through timelock, never through trade-path repegging, and the pool quotes the raw keeper mark with no on-chain EMA (Mark Relayer).

6. Concentrated Liquidity: Gyroscope E-CLP

Gyroscope’s E-CLP trades along the curve of an ellipse, formed by transforming a circle through a stretch ($\lambda$), a rotation ($\varphi$) and a displacement ($\alpha$, $\beta$) that sets the price bounds. Gyroscope reports E-CLPs “can improve capital efficiency by upwards of 75% over StableSwap pools”.

E-CLP parameters are set by the pool deployer, not by LPs: “instead of users setting price bounds themselves, the pool deployer takes on the responsibility of calibrating and establishing the trading parameters upon launch.” AIMM takes the same stance on who shapes the curve but places the authority differently: an AIMM pool’s parameters sit with the poolAdmin seat named at creation, behind the Admin timelock (Admin).

The ellipse is single-peaked and smooth, so it approximates most desired curves but cannot express double-peaked distributions, flat regions with sharp cutoffs, or arbitrary asymmetry. Splines lift those limits:

Spline mechanics: Liquidity Shaping.

7. Volatility-Based Fees: LFJ v2 Liquidity Book

LFJ’s Liquidity Book (formerly Trader Joe v2) introduced surge pricing: a base fee plus a variable fee that rises with volatility. Volatility is measured by an on-chain Volatility Accumulator, which “is able to calculate instantaneous volatility for each Liquidity Pool, without relying on any outside oracles by tracking transactions across bins”: more bin crossings per trade means more price movement. The rationale is that IL “is highest during the most volatile times when the market tries to correctly price assets”, so surge fees compensate LPs exactly then.

AIMM’s volatility band implements the same idea against a pushed σ instead of bin crossings:

$S_v=\sum _i{f}_{\min ,i}+\frac{{\sigma }_p·{\nu }_p}{100·\mathrm{B}\mathrm{P}\mathrm{S}}$

in PBPS (1 unit = 0.0001%), where $f_{\min ,i}$ are the per-leg minFeePbps floors (there is no hardcoded base; the base is the configured floor), ${\sigma }_p$ is path volatility by quadrature over legs, ${\nu }_p$ the endpoint-maximum vegaBps, and BPS = 10,000.

8. CLMM/DLMM Trade-offs and Their Mitigations

Mainstream concentrated-liquidity DEXs have addressed several of these via V4 hooks, custom curves and active management.

8.1. The measured costs

Uniswap v3’s concentrated liquidity buys capital efficiency at three costs. IL is amplified once price leaves the range (Kaiko Research), management is a standing burden that smaller LPs cannot carry, so “their positions tend to drift out of range, suffering from IL and failing to capture any trading fees”, and the aggregate is negative: “Between V3’s launch and September 20th, analyzed pools saw over $100B in trading volume, earning LPs approximately $200M in fees. However, LPs lost more than $260M to impermanent loss, resulting in a net loss of over $60M.” (Rekt News)

A fourth cost is unique to concentrated liquidity. JIT liquidity adds depth before a large trade, captures the fees and removes it in the same block, which “reduces price impact for the trader by increasing the pool size, but dilutes the trading fees distributed amongst existing LPs”. Research identified 36,671 JIT attacks over 20 months generating 7,498 ETH profit, and it is a whales’ game: “JIT transactions are significantly larger than regular transactions, generally with a minimum size of $10 million.”

8.2. What the venue does not supply

Those quotes describe the symptom. The cause is that a concentrated-liquidity DEX supplies pool infrastructure, not market making, leaving three cost centres to a participant it presents as passive: range selection, venue selection across fee tiers, and execution of the rebalancing trade itself. Each is a quantitative problem with an infrastructure cost and no venue primitive behind it. First-person account, and the reason BTR exists: Manifesto §1.2.

AIMM moves all three inside the pool: the mark comes from outside so there is no range to select, one pool spans the asset set so there is no tier to choose, and inventory is priced rather than traded, so there is no rebalance to execute. The last has a precondition the first two do not. Pricing substitutes for rebalancing only where flow comes back the other way, so a pair pool cannot net across legs at all while a multi-asset core nets internally provided its legs are correlated enough that flow draining one tends to arrive against another. That is why cores are grouped by asset class, not by listing availability (Pool Composition).

Defenses in detail: Toxic Flow Mitigation.

9. Multi-Asset Routing: Prior Art

N tokens require N(N-1)/2 pairs for direct trading. Prior solutions: all-pairs liquidity (Balancer, every pair with explicit depth under a geometric mean invariant, capital-inefficient for large N); base currency routing (FX, all quotes relative to USD so $A\to B$ routes $A\to \mathrm{U}\mathrm{S}\mathrm{D}\to B$); graph-based routing (DEX aggregators, external path search, not pool-native).

9.1. AIMM’s anchor tree

A multi-anchor tree rooted at the base token. Every asset has exactly one anchor (parent), chosen for correlation rather than for being the numeraire, and a swap prices along the unique path via the lowest common ancestor.

$\mathrm{U}\mathrm{S}\mathrm{D}\mathrm{T}\to \mathrm{D}\mathrm{A}\mathrm{I}$ turns at USDC and never reads the WBTC mark; $\mathrm{s}\mathrm{t}\mathrm{E}\mathrm{T}\mathrm{H}\to \mathrm{W}\mathrm{E}\mathrm{T}\mathrm{H}$ is a single edge off one stETH/ETH feed. Why a tree and not a flat star: a correlated pair should be one edge with its own feed, its own $\sigma$ and its own fee floor, and forcing it through an uncorrelated numeraire adds two uncorrelated marks to a pair that has none. The structure mirrors FX cross-rates, equity makers quoting against cash, and BTC or USDT base pairs on crypto exchanges.

What the tree does not provide is a parent-depeg breaker: an anchored child inherits its parent’s depeg risk with no automatic breaker (Anchor Path Pricing §7.1). Full topology, cycle-safety conditions, leg accounting and fee composition: Anchor Path Pricing.

10. Circular/Orbital Market Makers

10.1. Polar coordinate AMMs

The mathematics below traces to Paradigm’s Orbital paper and to academic work on Concentrated Circular Market Makers (CCMM) (Tolstikov et al.). Orbswap is one implementing DEX; its lite-paper presents the formulas as images and is not the canonical source.

Sphere invariant (n stablecoins, reserves $x_i$, radius $r$), with the equal-price point at $q=r(1-1/\sqrt{n})$ along the diagonal $\overrightarrow{v}=\frac{1}{\sqrt{n}}(1,\dots ,1)$:

Polar ticks are hyperplanes $\overrightarrow{x}·\overrightarrow{v}=k$. On a tick boundary the pool degenerates to a lower-dimensional sphere of radius $s=\sqrt{r^2-(k-r\sqrt{n}{)}^2}$; combining an interior spherical tick with that boundary sphere yields a torus, and Orbswap extends the construction to a superellipse invariant for tunable concentration.

Reported capital efficiency against Curve for n=5 stables (Paradigm Orbital paper): ~15× at a 0.90 depeg threshold, ~150× at 0.99, pegged-only in v1, with LST extensions theoretical. Risk isolation is structural, because the sphere geometry penalizes any single coordinate diverging far from the diagonal: “if one of the stablecoins depegs, the others can all trade at efficient prices, while the depegged one will become worthless much faster than a traditional AMM curve.”

10.2. Why AIMM did not pursue this

Polar transforms and torus invariants add computational overhead and more complex Newton iterations, the construction is optimized for symmetric pools of correlated assets and cannot express asymmetric profiles, and a polar “tick” is harder for an LP to read than a price range. Cartesian splines are simpler, express arbitrary profiles, and keep a direct price/depth correspondence.

For a pure stables or pure LST pool, Orbital is likely the better primitive. For mixed-volatility deployments (USDC + WBTC + WETH + LSTs together), only AIMM applies: the spherical invariant cannot price volatile-vs-pegged. The two compete on BTR’s stables-pool deployments; there Orbswap has an edge on intrinsic depeg isolation, and AIMM compensates with oracle sync, inventory feedback and regime-adaptive fees.

11. Spline-Based Liquidity Profiles

AIMM’s spline-based liquidity curves appear to be novel in DeFi AMM design: a literature search found no prior implementations of interpolating or shape-constrained splines for AMM pricing functions.

AIMM uses a clamped quartic I-spline: the depth curve is the monotone integral of a C2 density, so marginal liquidity never jumps, nondecreasing control weights make the curve monotone at any degree (a non-monotone segment would imply negative marginal liquidity), moving one control weight affects only nearby segments, and stored prefix integrals make range integration exact and O(1). Fritsch-Carlson monotone cubic Hermite interpolation was considered and rejected: C1 only, with a value-discontinuity risk in the density at knots under its asymmetric tangent clamp.

The standing objection to splines in an AMM is cost. Here the curve is stored packed, evaluation is a directory lookup plus a Horner evaluation, and a range integral is a difference of stored prefix integrals, so traversal cost is flat in trade size. That makes an arbitrary shape competitive with a closed-form invariant. Packing, slot layout, exact gas figures and the curve mathematics: Liquidity Shaping.

12. Synthesis: How AIMM Combines Prior Art

12.1. Component attribution

Quoting off a fresh per-asset mark removes the classical stale-curve LVR channel. It does not remove push-latency LVR or OEV, and nothing in the protocol rebates either.

13. RFQ & Intent-Based Protocols

Intent-based protocols outsource execution to specialized solvers or market makers: users express desired outcomes and solvers compete to fulfill them.

They buy better execution for sophisticated users, MEV protection through batch auctions and cross-chain composability. The cost is a permissioned quote layer: a limited solver set, off-chain quote servers and signature aggregation, latency requirements only well-capitalized operators meet, and identifiable entities providing quotes.

AIMM did not take that route. Anyone can trade without solver intermediation: the pool prices the swap itself, so there is no quote intermediary and no whitelisted counterparty. Listing is curated, and a leg can restrict deposits to an allowlist (§15). The only off-chain component is the mark push (Mark Relayer). The tradeoff is slightly worse execution in extreme conditions.

14. Oracle-Based & Hybrid AMMs

A parallel line of AMMs prices proactively off oracles rather than reactively off reserves, reducing LVR by quoting closer to true market value. The UAMM paper (“UAMM: Price-oracle based Automated Market Maker”) formalizes the design space, showing oracle-based pricing can theoretically eliminate LVR at the cost of oracle dependency.

Every oracle-based design shares one vulnerability: if the oracle fails, the AMM fails, through manipulation, latency, downtime or compromise. Hybrid designs (Swaap v2, Valantis) add solver or quoter dependency on top of it.

AIMM’s answer is to shorten the lag rather than add a second trust layer. It prices off a fresh per-asset mark pushed on a deviation band $\theta$ or a heartbeat, not off reserves and not off an internal lagging average (Mark Relayer). Two mechanisms price the risk between pushes: coverage-driven inventory skew moves the mid on flow (Inventory Management), and the volatility, confidence and staleness terms price the push-latency window (Spread & Fees).

15. Permissionless Design Philosophy

Any centralization in a DeFi primitive creates a regulatory chokepoint, a single point of failure, a gatekeeper able to extract rent and a censorship surface. AIMM’s position layer by layer:

Against the alternatives in §13 and §14, which need a solver whitelist, an off-chain quoter or a curated oracle provider, the signed-mark dependency is one option rather than a requirement: a cash-collateral 1:1 leg priced by the par helper needs no push for its mid, though the helper fits nothing else. Per-asset depeg bands run against the lane’s reference tier in the same mark store, so no third-party oracle network sits on the quote path.

Resilience is bounded rather than absolute. Privileged keys are scoped and revocable and parameter changes queue through Admin.requestOp timelocks; a stalled mark is priced through the staleness premium, and once the feed TTL expires, legs on that feed fail closed rather than quoting stale.

16. References

The papers statements on this page rest on. Venue whitepapers are footnoted where they are compared, in §19.6.

17. Where each mechanism is specified

18. Capital efficiency

A BTR pool holds 5-15 blue-chip assets in one contract, so one reserve backs every market its asset participates in instead of being split across pair pools. This section prices that sharing: a volume-per-capital metric, depth per dollar for a pair-pinned concentrated-liquidity venue and for a BTR pool, then the two composed with contention, cross-pair routing and mark-sync effects. The output is a range with its assumptions stated, not a headline multiplier.

18.1. Volume capital ratio (VCR)

For small $S$ under constant-product, $f(D,S)\approx DS$. Shared inventory changes $D_{ij}$.

18.2. UniV3 baseline (pair-pinned)

With $N$ assets, UniV3 needs $N(N-1)/2$ pair pools. On an even TVL split, $T_p=2T/\left(N\right(N-1\left)\right)$ and per-side depth is $T/\left(N\right(N-1\left)\right)$. The active-range factor $\alpha \in [1,\mathrm{\infty })$, often $5$-$50\times$ on blue chips, multiplies one pair only:

18.3. BTR pool (shared inventory)

One pool, $N$ assets, even split: $R_i=T/N$. Anchor-path pricing quotes every directed pair off a common tree (Pricing), so depth on $(h,j)$ is the full $R_h$, independent of $j$ and subject to §18.4:

holding ${\alpha }^{\prime }\approx \alpha$ under base-token flow. At $N=5$ and $T=\mathrm{\$}1\mathrm{B}$ that is 10 UniV3 pools holding $\mathrm{\$}50\mathrm{M}$ of USDC against each pair, against one pool whose $\mathrm{\$}200\mathrm{M}$ USDC reserve backs all four: identical per-asset reserves, $4\times$ the depth per pair. The multiplier removes an artificial partition; it does not invent capital. Per-$N$ figures after the contention discount are tabulated in §18.8.

18.4. Contention

The $(N-1)\times$ multiplier assumes non-concurrent trades. Aggregate outflow of the base token in one block is still capped at $R_h=T/N$.

If concurrent sizes $S_1,S_2$ hit two base-token pairs, the second trade sees $R_h-S_1$ and quotes with skew. Expected-case multiplier stays near $(N-1)$; worst-case concurrent drain on every base-token pair goes toward $1\times$. Indicative hold: $\sim 70$-$90\mathrm{\%}$ of nominal under realistic flow.

18.5. Cross-pair routes

UniV3 often routes $X\to \mathrm{U}\mathrm{S}\mathrm{D}\mathrm{C}\to Y$ across two pools (2 fees, 2 slips, extra gas). A BTR pool does $X\to Y$ along one anchor-tree path in one call (1 fee, 1 slip event, interior legs at mid without impact). On a $100\mathrm{b}\mathrm{p}$ slip budget, fee drag alone is typically lower on the single-pool path.

18.6. Oracle sync

A mark push re-prices every spoke in the same block. UniV3 pools are arbed individually after an external move (LVR to LPs). Classical CFMM LVR scales like ${\sigma }^2\alpha /8$ per unit time (Milionis et al.); quoting at the mark cuts the stale mid channel. Treat fee-margin uplift as ${\beta }_o\in [1.0,1.3]$ in §18.7 (conservative).

18.7. Composition

Example ($w_h=0.7$, $N=5$, ${\gamma }_c=0.8$, ${\mu }_h=1.2$, ${\beta }_o=1.2$):

18.8. Scenario table

The §18.7 ratio at $w_h=0.7$. The first row uses ${\gamma }_c=0.8$, ${\mu }_h=1.2$, ${\beta }_o=1.2$. The second spans ${\gamma }_c\in [0.7,0.9]$, ${\mu }_h\in [0.8,1.3]$, ${\beta }_o\in [1.0,1.3]$.

The contention discount holds these below the raw $N-1$ of §18.3. TVL-invariant: absolute depth scales with $T$, and concurrent multi-pair drains compress toward $1\times$ (§18.4).

18.9. Other venues

Curve and Balancer share inventory under different invariants; BTR combines basket sharing with mark-anchored depth. Composed across §18.5 to §18.7, indicative VCR uplift for $N=5$-$15$ is $3$-$10\times$ at the §18.8 point values and $2$-$12\times$ over its ranges, deployment-dependent, and the contention bound of §18.4 holds throughout. Topology and permissioning for the same venues: §19.1.1.

19. AMM landscape

This section places BTR against the AMMs it borrows from and competes with, on dimensions readable off code or public on-chain activity: topology, capital efficiency, LVR exposure, oracle dependency, execution and operations. It then covers the proprietary-AMM cohort, which shares BTR’s design intent but not its cost environment. The case for the design, as opposed to the comparison, is in the Manifesto.

19.1. Comparison matrix

Spot venues only. Perpetuals venues are compared separately in §3.5, because a pool whose providers are counterparty to leveraged trader PnL has no meaningful reading on capital efficiency at peg or on LVR. Who may list and who may deposit is §15, not a column here.

19.1.1. Topology and price source

19.1.2. What it costs the LP

Capital efficiency is quoted in-distribution, at peg where the venue has one.

19.2. What the tables do not carry

Four peers carry history or structure a cell cannot hold.

Uniswap V4 - the singleton PoolManager holds all pool state, so multi-hop and flash loans get cheaper and hooks can rewrite fee logic and even curves. The pool math is still V3 and per-pool logic remains pair-isolated: no shared inventory across pairs, and hooks fragment liquidity by configuration. This is the distinction §18.9 turns on.

Curve V1 - the hybrid invariant is joint, so one asset’s depeg drains the pool across all assets to varying degrees. UST in May 2022 cratered the pool outright; USDC after SVB in March 2023 drained 3pool but recovered. The LP cost is that a leg you did not choose can empty the leg you did.

Platypus - the original asset-liability AMM (§3), and the cautionary one. A ~$8.5M flash-loan exploit in February 2023 turned on emergencyWithdraw in MasterPlatypus not checking outstanding USP debt, letting the attacker withdraw collateral while leaving USP minted against it. Wombat forked it and ringfenced per-asset coverage. BTR does not: every LP settles at one pool rate, so a loss on one leg is shared by the whole pool (§3.3).

Chainlink-primary designs (DODO PMM, Swaap, am-AMM) - quote every asset off a deviation-and-heartbeat feed, so the mark is stale between updates and arbitrageurs take the lag window; the leak is deployment-dependent and not quantified here. BTR’s mark is a keeper push (per-asset deviation band $\theta$ plus heartbeat) with any lag past the grace priced by the staleness surcharge, and no Chainlink feed sits in the quote path.

Maverick V1/V2 - the one peer whose pool moves LP bins on each swap, through modes that trail directional moves. The closest production answer to the rebalancing problem of §8.2 that stays inside the pair-isolated tick model.

BTR - a curated multi-asset singleton (5-15 blue-chip target: stables, LSTs, ETH, BTC, majors) running the eight statements of §1. Two properties sit behind the rows above. The keeper mark is primary for quoting, with a per-asset agreement band against a second attestation of the same pair and a base-token parity halt (Mark Relayer). And one base-token unit counterparties $N-1$ pair markets at once (§18) instead of fragmenting across $N(N-1)/2$ pairs.

19.3. Where BTR is behind

The wins are the table rows above. The losses are not, so they are stated here.

• Curve V1 on stable-stable in-peg. The amplified invariant gives marginally better near-peg execution for pure stable pairs at scale, with 5+ years of audit hardening and mature gauge incentives.
• Uniswap V3/V4 on the long tail. Permissionless pair deployment plus the tick model fits unknown and experimental pairs better than a curated blue-chip singleton.
• OrbSwap on mathematical isolation. Orbital geometry gives provable per-asset depeg isolation; BTR uses an operational depeg-halt circuit breaker.
• The proprietary cohort on quote freshness, which is §19.4.

One capability sits outside the rows because it is a property of the maths rather than a measured capture: the depth density is a free per-asset choice, the integral of a chosen monotone C2 density, not a fixed invariant. Offset resolution is 1 pbps (0.01 bp, ~100× finer than a Uniswap v3 1-bp tick) and tails are hard-cut at $\pm W$, so 100% of capital sits inside the band where constant-product and Curve have unbounded support. A claim about depth, not about realized fee capture (Liquidity Shaping §4).

19.4. PropAMMs (proprietary oracle AMMs)

The peers in §19.1 are public-invariant AMMs. One high-performing class sits outside that matrix: PropAMMs (also called proprietary or “dark” AMMs), on-chain programs that replace a fixed bonding curve with active, model-based quoting driven by an off-chain pricing engine. They are the closest analogues to BTR’s design intent, so the landscape has to place BTR against them and not only against Uniswap and Curve.

Two properties define the class, and both are observable even where the method is not:

• Oracle-referenced pricing, not a fixed invariant. The curve is centered on an imported mark. Lifinity quotes off Pyth ; the proprietary cohort computes quote math off chain.
• Very high quote cadence, delivered two ways. On Solana the quote is written on chain at high frequency; on EVM it is streamed to a block builder and written only when a taker hits it. The quoting algorithm is closed in both cases; the on-chain footprint is not, so update frequency, quoted spread and size-response are measurable from public activity. Figures below are third-party observations of that footprint and carry no claim about the algorithm producing them.

19.4.1. Solana

The Solana cohort came first and is still the cadence leader: BisonFi, HumidiFi, Tessera, SolFi, with Obric, ZeroFi and Lifinity (the first Solana oracle DEX) in the broader set. Reported on-chain metrics put HumidiFi around 17-75 parameter updates per second (peaks near 74×, on the order of ~143 compute units per update), with peers in the ~10-13× range. ~400 ms slots and ~$0.0001 fees are what make that affordable, and three Solana-specific facts carry the cost down further: sBPF programs at 143-740 compute units, a base fee charged per signature rather than per byte written, and Jito’s auction ranking by tip per compute unit. None of the three has an EVM analogue, so the cadence edge is bound to that environment. Measured against our own oracle: The cheapest signed price feed on the EVM.

19.4.2. EVM, and the block builder

On EVM the same class exists with a different enabler. Rather than paying for a high push cadence, an EVM PropAMM has the block builder put the quote in the block. Titan defines the pattern in its own documentation: “A Proprietary AMM (PropAMM [pAMM]) is an onchain pool in which prices are continuously updated by a market maker. Instead of relying on a passive inventory curve, makers stream signed quote updates to the builder ensuring prices are updated in real-time.” (Titan docs)

Three mechanics matter, all from Titan’s maker and taker pages:

1. Ordering, not gas. “When a taker trades against a pAMM, the latest quote update is guaranteed to be ranked before it in the block.” Titan calls this Application-Controlled Execution.
2. Conditional inclusion. “Quotes can land every block, or only when there is taker flow, depending on the maker session configuration.” In the conditional mode a quote nobody trades against never reaches storage, so an idle pair writes nothing. “Quote updates submitted through this endpoint do not pay priority fees”; makers instead pay a per-pair volume fee on fills.
3. A last look at the builder layer. A per-session freshness buffer b means “a taker is only eligible to trade against a quote update if the taker was received by the builder at least b before that quote update was received”, documented with b = 50 ms. Makers are asked to requote every ~50 ms; a quote idle more than 400 ms is evicted.

Titan built 54.25% of Ethereum L1 blocks in the 24 h to 2026-09-22 18:41 UTC (relayscan.io), which is what makes one builder relationship worth having; no other builder is close, though Quasar documents an equivalent maker lane. A community router (LambdaClass) aggregates the venues, and KyberSwap became the first DEX aggregator to route through the Titan lane on 2026-06-01.

Volume is the one number worth tabulating, because it says which of these is a business and which is a contract. DeFiLlama Prop AMM tag, 7-day volume read 2026-09-22:

Where the evidence is thin. Only Kipseli, ElfomoFi and Bebop publish anything of their own, and none of the eight publishes a launch date. No venue in the Titan registry documents who holds the signing key or how a taker verifies the quote it filled against. Lunarbase’s rotating operator-quorum description rests on one secondary blog post and is not corroborated by its repo. TaurusFi is a registry entry and nothing else. The measured on-chain oracle plumbing for Metric and ElfomoFi, the part of this we verified ourselves from mainnet receipts, is in The cheapest signed price feed on the EVM; the trust surface these venues present to a taker is in Opening the dealing desk everyone hides.

19.4.3. PropAMM against BTR

The cost model is the real asymmetry and it favours them: a conditional quote no taker hits costs nothing, so an idle pair is free, while BTR pays the heartbeat whether or not a swap follows. That is the price of the verifiability row, and it is what lets the pool quote on a chain where no builder offers the lane. The same venues make the same trade the other way off Ethereum, where Metric, ElfomoFi and Lunarbase fall back to a per-block or per-swap write with no builder in the path.

On freshness BTR is behind, and on volume the cohort is ahead by an order of magnitude. What does not follow is that BTR has to close the cadence gap. Quoting off an imported mark rather than a stale reserve-derived price is what eliminates the CFMM stale-price LVR channel, and the oracle-AMM literature shows optimized oracle AMMs beat CFMMs for LPs even with a lagged oracle, under adverse selection and systematic arbitrage (Milionis et al. ; Bergault et al. ). Higher gas and slower blocks lengthen the lag term; they do not remove the structural reduction. On EVM, Swaap v2 sits close to the same definition with a published whitepaper, which the proprietary cohort does not have.

Two mechanisms close most of the remaining gap without a push. Between pushes the inventory skew (an Avellaneda-Stoikov reservation price around the frozen mark) reprices the pool on flow, so a fill shifts the quote in the same transaction and no keeper had to notice; a PropAMM has to push a new quote to express the same thing, which is part of why its cadence has to be that high. And the depth curve is recalibratable on chain: between timelocked refits, live quotes scale linearly by dispersion / dispRefPbps, so width tracks volatility with no governance action at all (Liquidity Shaping).

Canonical pricing pipeline, not duplicated here: Inventory Management → Liquidity Shaping → Anchor Path Pricing → Spread & Fees.

19.5. Out of scope

Raydium AMM v4 / CPMM (Solana constant-product, historically hybrid with OpenBook) is a classical CFMM peer, closer to Uniswap V2 than to BTR, and its CLMM is the V3 analogue. Both are omitted from the §19.1 matrix to avoid diluting the oracle-AMM comparison; treat them as V2/V3-class for pricing purposes. Where a peer mechanism maps onto a BTR one, the mapping is stated in that mechanism’s own section: DODO’s i/k in §3.4, Curve v2’s EMA repeg in §5, E-CLP shaping in §6, LFJ’s surge fee in §7, coverage slippage in §3.3.

19.6. References

Adams, H. Uniswap V2 Core (2020). https://uniswap.org/whitepaper.pdf

Adams, H. et al. Uniswap V3 Core (2021). https://uniswap.org/whitepaper-v3.pdf

Uniswap Labs. Uniswap V4 Core (2024). https://github.com/Uniswap/v4-core

Egorov, M. StableSwap - efficient mechanism for Stablecoin liquidity (2019). https://docs.curve.finance/assets/pdf/stableswap-paper.pdf

Egorov, M. Automatic market-making with dynamic peg (2021). https://docs.curve.finance/assets/pdf/whitepaper.pdf

Balancer Labs. Balancer V2 Vault / V3 Hooks (2021, 2024). https://docs.balancer.fi/

DODO Team. DODO: A Next-generation On-chain Liquidity Provider Powered by PMM (2020). https://docs.dodoex.io/

Maverick Protocol. Maverick AMM Whitepaper (2022). https://docs.mav.xyz/

Trader Joe / LFJ. Liquidity Book Whitepaper (2022). https://docs.lfj.gg/concepts/concentrated-liquidity

Platypus Finance. Platypus: An Open Liquidity Pool Protocol for Stableswap (2021). https://cdn.platypus.finance/Platypus_Finance_Whitepaper.pdf

Wombat Exchange. Wombat: 4th-Gen Multichain-Native Stableswap (2022). https://docs.wombat.exchange/

Paradigm / OrbSwap. Orbital AMM Lite Paper (2025). https://orbswap.org/lite-paper

Lifinity. Proactive Market Making on Solana (docs). https://docs.lifinity.io/

Milionis, J., Moallemi, C. C., Roughgarden, T., Zhang, A. L. Automated Market Making and Loss-Versus-Rebalancing (2022). https://arxiv.org/abs/2208.06046

Bergault, P., Bertucci, L., Bouba, D., Guéant, O. Automated Market Makers Designs beyond Constant Functions (2022). https://arxiv.org/abs/2212.00336

Bergault, P., Bertucci, L., Bouba, D., Guéant, O. Price-Aware Automated Market Makers: Models Beyond Brownian Prices (2024). https://arxiv.org/abs/2405.03496

Avellaneda, M., Stoikov, S. High-frequency trading in a limit order book. Quantitative Finance 8(3), 217-224 (2008).

Swaap Labs. Swaap v2: a Market Making Protocol (whitepaper, 2023). https://www.swaap.finance/

Note on proprietary-AMM figures (§19.4). Solana update-cadence and compute-unit figures are third-party on-chain observations from public analytics, not vendor-published specifications; those venues are closed-source and publish no methodology. Treat them as indicative orders of magnitude. EVM volume figures are DeFiLlama’s own Prop AMM tag, read 2026-09-22, and carry that source’s methodology and its coverage gaps.