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+# LMSR-based Multi-Asset AMM
+
+Abstract
+We propose a multi-asset automated market maker (AMM) whose pricing kernel is the Logarithmic Market Scoring Rule (LMSR) (Hanson, 2002.) The AMM maintains a convex potential $C(\mathbf{q}) = b(\mathbf{q}) \log\!\big(\sum_i e^{q_i / b(\mathbf{q})}\big)$ over normalized inventories $\mathbf{q}$ and sets the effective liquidity parameter proportional to pool size, $b(\mathbf{q}) = \kappa \, S(\mathbf{q})$ with $S(\mathbf{q}) = \sum_i q_i$ and fixed $\kappa>0$. This choice preserves scale-invariant responsiveness while retaining LMSR’s softmax pricing structure under a quasi-static-$b$ view. We derive closed-form expressions for asset-to-asset swaps in the induced two-asset subspace, including exact-in, exact-out, limit-hitting (swap-to-limit), and capped-output variants. We discuss numerical stability techniques (e.g., log-sum-exp reformulations, guarded domains) and a balanced two-asset specialization that enables polynomial approximations with provable error bounds. The protocol is parameter-fixed at deployment (no governance over $\kappa$ or fees), yielding reproducible behavior and transparent depth calibration. Analytical and numerical evidence suggest that this LMSR AMM combines desirable theoretical properties (convexity, path-independent cost differences at constant $b$, multi-asset support) with practical robustness (monotonicity preservation and conservative fallbacks). We outline liquidity operations (proportional and single-asset joins/exits), fee accrual to LPs via state appreciation, and risk considerations.
+
+Keywords
+AMM; LMSR; cost-function market maker; multi-asset liquidity; bounded loss; convex optimization; fixed-point arithmetic; numerical stability.
+
+2) Executive Summary
+
+Motivation and problem
+- Multi-asset liquidity often faces a trade-off between simplicity (CFMM invariants) and expressivity (risk sharing across many assets). We pursue an AMM that natively supports many assets while delivering predictable depth and strong theoretical guarantees.
+
+Mechanism overview
+- The AMM’s kernel is the LMSR cost function
+  $$C(\mathbf{q}) = b(\mathbf{q}) \log\!\left(\sum_i e^{q_i / b(\mathbf{q})}\right),\qquad b(\mathbf{q})=\kappa\,S(\mathbf{q}),\;\; S(\mathbf{q})=\sum_i q_i.$$
+- Instantaneous marginal price ratios follow the softmax structure under quasi-static $b$:
+  $$P(\text{base}\to\text{quote}\mid \mathbf{q})=\exp\!\left(\frac{q_{\text{quote}}-q_{\text{base}}}{b(\mathbf{q})}\right).$$
+
+Key properties
+- Convex potential and softmax gradient (at constant $b$), multi-asset support via a single potential.
+- Path independence of cost differences under constant $b$; pairwise price ratios preserved under $b=\kappa S$ via a common additive gradient term.
+- Bounded-loss intuition from LMSR extends proportionally with $b$; scale invariance via $b=\kappa S$.
+- Closed forms for two-asset reductions enable exact-in/exact-out and limit-hitting swaps with monotonicity.
+
+Main contributions
+- Fixed-parameter LMSR AMM: $b=\kappa S$ with immutable $\kappa$ and fees post-deployment.
+- Stability techniques: log-sum-exp evaluation, domain guards, ratio shortcuts, and conservative fallbacks.
+- Balanced two-asset optimization: polynomial approximations with error bounds and dispatcher preconditions.
+- Evaluation plan: analytical depth comparisons, numerical accuracy, monotonicity/no-negative-arbitrage checks, and gas microbenchmarks.
+
+3) Background and Related Work
+
+AMM landscape
+- CFMMs define implicit invariants over reserves. Examples include:
+  - Constant product (e.g., Uniswap): $x y = k$, offering simple two-asset liquidity and predictable slippage profiles.
+  - Constant mean (e.g., Balancer): $\prod_i q_i^{w_i} = \text{const}$, generalizing to many assets with weights.
+  - Stableswap (e.g., Curve): combines constant sum and constant product to target low-slippage near parity.
+- LMSR differs by specifying a convex cost function $C(\mathbf{q})$; prices are given by its gradient (or ratios thereof), rather than from a multiplicative invariant.
+
+LMSR primer and adaptation to AMMs
+- The classical LMSR potential with constant $b$ is
+  $$C(\mathbf{q}) = b \log\!\left(\sum_i e^{q_i/b}\right),\qquad \frac{\partial C}{\partial q_i} = \frac{e^{q_i/b}}{\sum_k e^{q_k/b}}.$$
+- In our AMM setting, we parameterize $b(\mathbf{q})=\kappa S(\mathbf{q})$ for scale-invariant responsiveness, and use a quasi-static-$b$ view to compute instantaneous price ratios:
+  $$P(\text{base}\to\text{quote})=\frac{e^{q_{\text{quote}}/b}}{e^{q_{\text{base}}/b}}=\exp\!\left(\frac{q_{\text{quote}}-q_{\text{base}}}{b}\right).$$
+- Two-asset reduction yields closed forms for exact-in and exact-out trades (see Sections 5–6 and Appendix A).
+
+Related LMSR applications and prior DeFi adaptations
+- LMSR originates from scoring-rule-based market making in information markets. Subsequent adaptations explored cost-function AMMs and variants that tailor $b$ to achieve targeted responsiveness. We adopt a proportional $b$ tied to total pool size for transparency and predictable scaling across liquidity regimes.
+
+When LMSR is preferable and limitations
+- Preferable when:
+  - Multi-asset exposure and cross-asset risk sharing are primary goals.
+  - A convex potential with softmax-driven pricing is desired for analytical tractability and monotonic behavior.
+- Limitations and design choices:
+  - With $b=\kappa S$, gradient components include a common additive term; we rely on pairwise ratios for pricing.
+  - Extreme imbalances can induce steep price moves; capacity caps and domain guards mitigate numerical and economic edge cases.
+
+4) System Model and Notation
+
+4.1 State, assets, and units
+- Assets and indices: We consider $n \ge 2$ assets indexed by $i \in \{0,\dots,n-1\}$.
+- State vector: $\mathbf{q} = (q_0,\dots,q_{n-1}) \in \mathbb{R}_{\ge 0}^{\,n}$ denotes normalized internal quantities held by the pool. Each $q_i$ is in common “internal units” so cross-asset operations are comparable. In practice, token amounts are scaled to a common unit (e.g., $10^{-\text{decimals}}$ normalization) and represented with fixed-point arithmetic; equations here are stated over reals for clarity.
+- Size metric: $S(\mathbf{q}) := \sum_i q_i$. This is the aggregate pool size used both to summarize liquidity and to set the effective liquidity parameter.
+- Liquidity parameterization: The effective LMSR liquidity parameter is $b(\mathbf{q}) := \kappa \cdot S(\mathbf{q})$, where $\kappa > 0$ is a fixed, deployment-time constant. Intuitively, $b$ scales linearly with the pool’s total size, preserving responsiveness under proportional rescaling of $\mathbf{q}$.
+- Fees: Let $f_{\text{swap}} \in [0,1)$ denote the swap fee applied at the token layer (separate from the fee-free pricing kernel). Optionally a protocol fee $f_{\text{proto}}$ can be taken as a fraction of the swap fee. All derivations of kernel prices below are fee-free; fees are applied as multiplicative factors to input/output amounts outside the kernel.
+
+4.2 Cost function and prices
+- Cost function: We use the LMSR cost function
+  $$C(\mathbf{q}) \;=\; b(\mathbf{q}) \,\log\!\left(\sum_{i} e^{\,q_i / b(\mathbf{q})}\right).$$
+  For numerical stability we compute it via a log-sum-exp formulation. Define $y_i := q_i / b(\mathbf{q})$ and $M := \max_i y_i$; then
+  $$C(\mathbf{q}) \;=\; b(\mathbf{q}) \left( M + \log \sum_{i} e^{\,y_i - M} \right).$$
+- Price shares (softmax): Define the unnormalized weights $w_i := e^{q_i / b(\mathbf{q})}$ and $W := \sum_i w_i$. The price share of asset $i$ is
+  $$\pi_i(\mathbf{q}) := \frac{w_i}{W} \;=\; \frac{e^{q_i / b(\mathbf{q})}}{\sum_j e^{q_j / b(\mathbf{q})}}.$$
+  Note: With $b$ constant, the gradient satisfies $\partial C/\partial q_i = \pi_i$. With $b = \kappa \, S(\mathbf{q})$, $\partial C/\partial q_i$ includes an additive term common across $i$; differences in marginal prices still track $\pi_i$ up to an additive constant, and the pairwise ratios below remain unchanged.
+- Pairwise marginal price ratio: The instantaneous marginal price of “base” in units of “quote” is
+  $$P(\text{base}\to\text{quote}\mid \mathbf{q}) \;=\; \exp\!\left(\frac{q_{\text{quote}} - q_{\text{base}}}{b(\mathbf{q})}\right).$$
+  This equals $w_{\text{quote}}/w_{\text{base}}$ and is invariant to the common softmax denominator, and remains valid under the quasi-static-$b$ swap model (holding $b$ constant over an infinitesimal trade or a single pricing step; see Appendix A).
+
+4.3 Swap quantities and conventions
+- Exact-in: Given an input amount $a \ge 0$ of asset $i$, the fee-free kernel determines the output amount $y \ge 0$ of asset $j$ by integrating the marginal price along the $i\to j$ path with $b$ held quasi-static at its pre-trade value (see Appendix A).
+- Exact-out: Given a desired output $y \ge 0$ of asset $j$, the fee-free kernel solves for the required input $a \ge 0$ of asset $i$ (inverse of exact-in).
+- Price limits (swap-to-limit): A user can provide a maximum acceptable marginal price ratio $\Lambda > 0$ for $p_i/p_j$. If the marginal price trajectory would exceed $\Lambda$ before consuming the full $a$, the swap truncates at the unique $a_{\text{lim}}$ that reaches $\Lambda$ (see Appendix A).
+- Capacity caps: Outputs cannot exceed the available balance of the out-asset; if a formula would produce $y > q_j$, we cap to $q_j$ and solve inversely for the corresponding input.
+
+4.4 Units, scaling, and normalization
+- Token decimals: For each token $i$ with decimals $d_i$, on-chain amounts are normalized to a common internal unit so that arithmetic over $\mathbf{q}$ is coherent. Let $s_i$ be the scale factor implied by $d_i$; normalized internal balances are proportional to on-chain token balances via $s_i$.
+- Scale invariance: If $\mathbf{q}$ is scaled by $\lambda > 0$ (all assets multiplied by the same $\lambda$), then $S$ scales by $\lambda$ and $b = \kappa S$ scales by $\lambda$; prices as pairwise ratios, $P(\text{base}\to\text{quote})$, are invariant to this rescaling.
+
+4.5 Assumptions
+- Tokens are standard fungible assets with deterministic, non-rebasing balances; no transfer fees are embedded at the kernel level.
+- Swaps and liquidity operations are atomic; the mechanism is permissionless (subject to access rules at the wrapper layer).
+- No external price oracles are required by the kernel; price discovery is endogenous via the cost function.
+- Numerical computations are performed in fixed-point with explicit domain guards (see Appendix B).
+
+5) AMM Design and LMSR Formulation
+
+5.1 Convex potential and invariant view
+- We model the pool by the LMSR potential
+  $$C(\mathbf{q}) \;=\; b(\mathbf{q}) \,\log\!\left(\sum_{i=0}^{n-1} e^{\,q_i / b(\mathbf{q})}\right), \qquad b(\mathbf{q}) = \kappa\,S(\mathbf{q}),\;\; S(\mathbf{q})=\sum_i q_i,$$
+  which induces a softmax over coordinates of $\mathbf{q}$. For pricing, we operate under a quasi-static-$b$ view for an infinitesimal step, recovering the usual LMSR gradient structure (see Section 6 and Appendix A).
+- Intuition: $C$ is a convex potential in $\mathbf{q}$ when $b$ is treated as a constant parameter; gradient components are softmax probabilities. This convexity underpins path independence of cost differences and no-arbitrage properties in the constant-$b$ setting. With $b=\kappa S(\mathbf{q})$, the gradient picks up a common additive term across coordinates; pairwise price ratios, which drive swaps, remain governed by softmax differences (Appendix A).
+
+5.2 Kappa parameterization: $b=\kappa \cdot S$
+- Rationale: Choosing $b$ proportional to pool size preserves responsiveness under proportional rescalings of inventory. If all $q_i$ are multiplied by $\lambda>0$, then $S$ and $b$ scale by $\lambda$, while price ratios
+  $$P(\text{base}\to\text{quote}\mid \mathbf{q}) \;=\; \exp\!\left(\frac{q_{\text{quote}}-q_{\text{base}}}{b(\mathbf{q})}\right)$$
+  remain invariant.
+- Responsiveness: Smaller $\kappa$ yields smaller $b$ for a given $S$, producing steeper price impact (more responsive market); larger $\kappa$ produces deeper liquidity and gentler impact.
+- Conformance: The induced prices coincide with the classic LMSR softmax ratios under quasi-static-$b$, ensuring consistency with LMSR’s scoring-rule interpretation for marginal moves.
+
+5.3 Choice of $S$ and alternatives
+- Default choice: $S(\mathbf{q})=\sum_i q_i$ (the $\ell_1$ size metric) is transparent, easy to compute, and scale-consistent across assets in normalized units.
+- Alternatives:
+  - Weighted sum: $S_w(\mathbf{q})=\sum_i w_i q_i$ for exogenous weights $w_i>0$; may bias depth across assets.
+  - Quadratic norm: $S_2(\mathbf{q})=(\sum_i q_i^2)^{1/2}$; increases $b$ more when inventory is concentrated, potentially dampening extreme price moves.
+- Trade-offs: Simplicity, composability, and transparency argue for $S=\sum_i q_i$. Alternatives can tailor depth but complicate interpretation and comparability.
+
+5.4 Fixed-parameter policy
+- The proportionality constant $\kappa$ and fee parameters are set at deployment and remain immutable. This yields reproducible behavior, eliminates governance risk, and allows users to evaluate depth and price impact ex-ante for a given $S$.
+
+5.5 Bounded-loss and capital efficiency implications
+- Classical LMSR with constant $b$ admits a bounded-loss guarantee of $b\ln n$ in appropriate units. With $b=\kappa S(\mathbf{q})$, scale invariance implies an instantaneous bound per unit of $S$ that is proportional to $\kappa \ln n$; as liquidity scales, so does absolute depth and the notional bound.
+- Capital efficiency: For fixed $\kappa$, larger $S$ linearly increases price depth while preserving price ratios; for fixed $S$, tuning $\kappa$ linearly trades off depth versus responsiveness. The two-asset closed forms in Section 6 quantify this via $y(a)$ and its slope at the origin $y'(0)=\frac{r_0}{1+r_0}$.
+
+6) Pricing and Swap Mechanics
+
+6.1 Instantaneous pricing from the gradient
+- Define unnormalized weights $w_i = e^{q_i/b}$ and $W=\sum_k w_k$. Under quasi-static $b$,
+  $$\pi_i(\mathbf{q})=\frac{\partial C}{\partial q_i}=\frac{w_i}{W},\qquad
+  \frac{\pi_{\text{quote}}}{\pi_{\text{base}}}=\frac{w_{\text{quote}}}{w_{\text{base}}}=\exp\!\left(\frac{q_{\text{quote}}-q_{\text{base}}}{b}\right).$$
+  We interpret $P(\text{base}\to\text{quote}) := \pi_{\text{quote}}/\pi_{\text{base}}$ as the instantaneous marginal price ratio.
+
+6.2 Cost differences and asset-to-asset swaps
+- Conceptually, an asset-to-asset trade from $i$ to $j$ of sizes $(+a,-y)$ satisfies
+  $$\Delta C \;=\; C(\mathbf{q} + a\,\mathbf{e}_i - y\,\mathbf{e}_j) - C(\mathbf{q}),$$
+  and the two-asset reduction with quasi-static $b$ yields a closed-form relation between $a$ and $y$ (Appendix A):
+  $$y(a) \;=\; b \,\ln\!\Big( 1 + r_0 \,\big(1 - e^{-a/b}\big) \Big),\qquad r_0 := \exp\!\left(\frac{q_i - q_j}{b}\right).$$
+- The inverse exact-out mapping for a target $y$ is
+  $$a(y) \;=\; b \,\ln\!\left(\frac{r_0}{\,r_0 + 1 - e^{\,y/b}\,}\right).$$
+
+6.3 Limit-hitting swaps and swap-to-limit
+- For a user-specified price limit $\Lambda > 0$ on $p_i/p_j$, the marginal price trajectory $r(t)=r_0 e^{t/b}$ hits the limit at
+  $$a_{\text{lim}} \;=\; b \,\ln\!\left(\frac{\Lambda}{r_0}\right),\qquad
+    y_{\text{lim}} \;=\; b \,\ln\!\Big( 1 + r_0 \,\big(1 - r_0/\Lambda\big) \Big).$$
+- Capacity cap: If $y_{\text{lim}} > q_j$, cap to $y=q_j$ and use the inverse mapping to compute the implied input
+  $$a_{\text{cap}} \;=\; b \,\ln\!\left(\frac{r_0}{\,r_0 + 1 - e^{\,q_j/b}\,}\right).$$
+
+6.4 Properties: monotonicity, uniqueness, and stability
+- Monotonicity and uniqueness: For feasible states, $y(a)$ is strictly increasing and concave in $a$; the inverse $a(y)$ is strictly increasing and convex in $y$. The limit-hitting $a_{\text{lim}}$ is unique when $\Lambda>r_0$.
+- Numerical stability: Evaluations use log-sum-exp style reformulations, direct ratio formation for $r_0$, and argument guards for $\exp$ and $\ln$. Domain checks ensure denominators (e.g., $r_0 + 1 - e^{y/b}$) remain positive. See Appendix B for details.
+
+7) Liquidity Operations
+
+7.1 Pool initialization and bootstrap
+- Let the seed inventory be $\mathbf{q}^{(0)} \in \mathbb{R}_{\ge 0}^{\,n}$ with $S^{(0)} := \sum_i q_i^{(0)} > 0$. The effective liquidity is $b^{(0)} = \kappa\,S^{(0)}$.
+- LP supply: We take LP supply proportional to the size metric, $L := \eta\,S(\mathbf{q})$, with a fixed conversion $\eta>0$. Without loss of generality we set $\eta=1$ so that
+  $$L \;=\; S(\mathbf{q}) \;=\; \sum_i q_i.$$
+  At bootstrap, the seeder mints $L^{(0)} = S^{(0)}$ LP shares against $\mathbf{q}^{(0)}$.
+- LP price in units of asset $k$: Define the marginal value of one unit of $S$ in asset $k$ as
+  $$P_L^{(k)}(\mathbf{q}) \;=\; \frac{1}{S(\mathbf{q})}\,\sum_{j=0}^{n-1} q_j \,\exp\!\left(\frac{q_j - q_k}{b(\mathbf{q})}\right),$$
+  which aggregates the marginal exchange rates from each asset into $k$ (cf. Section 6).
+
+7.2 Proportional deposit (mint)
+- A proportional deposit scales all coordinates by $(1+\alpha)$ for some $\alpha \ge 0$:
+  $$\mathbf{q}' \;=\; (1+\alpha)\,\mathbf{q},\qquad \Delta q_i \;=\; \alpha\,q_i.$$
+- Minted LP shares are linear in the size-metric increase:
+  $$\Delta L \;=\; L' - L \;=\; S(\mathbf{q}') - S(\mathbf{q}) \;=\; \alpha\,S(\mathbf{q}).$$
+- The post-deposit liquidity is $b'=\kappa\,S(\mathbf{q}')=(1+\alpha)\,b$.
+
+7.3 Single-asset deposit (exact-in)
+- A contributor provides amount $a$ of asset $i$ and receives a proportional growth $\alpha \ge 0$ such that the system state can be rebalanced to $(1+\alpha)\,\mathbf{q}$ by swapping from $i$ into $j\ne i$ along the fee-free kernel. For each $j \ne i$, target out-amount $y_j := \alpha\,q_j$ requires input
+  $$x_j(\alpha) \;=\; b \,\ln\!\left(\frac{r_{0,j}}{\,r_{0,j} + 1 - e^{\,y_j/b}\,}\right),\qquad
+    r_{0,j} \;:=\; \exp\!\left(\frac{q_i - q_j}{b}\right).$$
+- The total input required to realize proportional growth $\alpha$ is
+  $$a_{\text{req}}(\alpha) \;=\; \alpha\,q_i \;+\; \sum_{j\ne i} x_j(\alpha).$$
+- The minted shares are
+  $$\Delta L \;=\; \alpha\,S(\mathbf{q}),$$
+  where $\alpha$ is the unique solution to $a_{\text{req}}(\alpha)=a$ on its feasible domain (see Appendix A.5).
+
+7.4 Multi-asset deposit (arbitrary vector)
+- Given a deposit vector $\mathbf{a} \in \mathbb{R}_{\ge 0}^{\,n}$, decompose it into:
+  - a proportional component $\bar{\alpha} := \min_i \{ a_i / q_i \}$ (with convention $a_i/q_i=+\infty$ if $q_i=0$ and $a_i>0$), which mints $\bar{\alpha}\,S(\mathbf{q})$ shares and updates $\mathbf{q}$ proportionally, and
+  - residuals $\tilde{\mathbf{a}} := \mathbf{a} - \bar{\alpha}\,\mathbf{q}$ that can be contributed via single-asset deposit(s) using 7.3 (sequence or batching).
+- The total minted shares are additive in the realized proportional growths:
+  $$\Delta L \;=\; \left(\bar{\alpha} + \sum_{m} \alpha_m\right)\,S(\mathbf{q})$$
+  where each $\alpha_m$ solves $a_{\text{req}}(\alpha_m)$ for a residual leg.
+
+7.5 Proportional withdrawal (burn)
+- Burning $\Delta L$ LP shares effects a proportional redemption with factor
+  $$\alpha \;=\; \frac{\Delta L}{S(\mathbf{q})} \in (0,1],\qquad \mathbf{q}' \;=\; (1-\alpha)\,\mathbf{q}.$$
+- The holder receives $\alpha\,q_i$ units of each asset $i$; equivalently, $L' = L - \Delta L = S(\mathbf{q}')$ and $b'=(1-\alpha)\,b$.
+
+7.6 Single-asset withdrawal (exact-out)
+- A holder burns $\Delta L$ shares (i.e., $\alpha=\Delta L/S(\mathbf{q})$) and requests payout exclusively in asset $i$. Starting from $\mathbf{q}_\text{local}=(1-\alpha)\,\mathbf{q}$, for each $j\ne i$:
+  - withdraw $\alpha\,q_j$ units of $j$, and
+  - swap $j \to i$ along the fee-free kernel using the two-asset closed form. The candidate out-amount is
+    $$y_{j\to i} \;=\; b \,\ln\!\Big( 1 + r_{0,j}\,\big(1 - e^{-a_j/b}\big) \Big),\quad
+      a_j := \alpha\,q_j,\quad r_{0,j} := \exp\!\left(\frac{q^{\text{local}}_j - q^{\text{local}}_i}{b}\right),$$
+    with $b=\kappa\,S(\mathbf{q})$ evaluated at pre-burn or quasi-static local state.
+- If the computed $y_{j\to i}$ would exceed $q^{\text{local}}_i$, cap to capacity and invert to solve the implied input (Appendix A.6). The total single-asset payout is
+  $$Y_i \;=\; \alpha\,q_i \;+\; \sum_{j\ne i} y_{j\to i}.$$
+
+7.7 Share issuance, pricing, and dilution
+- With $L=S$, share issuance is linear in the size-metric; proportional joins/exits preserve relative ownership. The instantaneous LP price in units of asset $k$ is $P_L^{(k)}(\mathbf{q})$ from 7.1. Under joins, $P_L^{(k)}$ remains unchanged for proportional deposits; under single-asset joins, it adjusts according to the realized rebalancing path.
+
+7.8 Fee accrual to LPs and value capture
+- Swap fees are taken at the token layer and retained in the pool balances, increasing $S(\mathbf{q})$ relative to $L$ and thereby raising $P_L^{(k)}$ for all $k$. This constitutes implicit fee accrual to LPs via state appreciation rather than explicit distributions.
+
+7.9 Edge cases and operational notes
+- Tiny liquidity: When $S$ is small, $b=\kappa S$ is small and price impact is steep; deployments SHOULD enforce a minimum bootstrap $S^{(0)}$ and/or minimum minted $L^{(0)}$.
+- Extreme imbalances: As some $q_j \to 0$, price ratios $\exp((q_{\text{quote}}-q_{\text{base}})/b)$ can become large; capacity caps ($y \le q_j$) and positivity checks on logarithm arguments ensure safe evaluation.
+- Asset additions/removals: Changing the asset set alters $n$ and the pricing manifold. Deployments typically fix the asset universe; adding/removing assets is best handled via new pool instances with fresh initialization.
+
+8) Fees and Incentives (Static Parameters)
+
+8.1 Fee model and placement
+- Let the swap fee rate be $f_{\text{swap}} \in [0,1)$, and let the protocol capture a fraction $\phi \in [0,1]$ of that fee (so LPs receive the remaining $1-\phi$ share via state appreciation).
+- We apply fees outside the fee-free pricing kernel. For an exact-in trade with submitted input $a$ on asset $i$, the effective kernel input is
+  $$a_{\text{eff}} \;=\; (1 - f_{\text{swap}})\,a.$$
+  The fee amount is $a - a_{\text{eff}}$, of which $\phi\,(a - a_{\text{eff}})$ accrues to the protocol and $(1-\phi)\,(a - a_{\text{eff}})$ to LPs (retained in the pool state).
+- The fee-free kernel computes the out-amount $y_{\text{ker}}$ using $a_{\text{eff}}$; the user receives $y_{\text{user}} = y_{\text{ker}}$ (or a fee-adjusted variant if fees are taken from output instead). The invariant and closed forms remain unaffected because pricing is computed on $a_{\text{eff}}$.
+
+8.2 Economic impact
+- Effective price and slippage: With input-side fees, the user’s effective marginal price scales by $(1 - f_{\text{swap}})^{-1}$ for small trades; total slippage decomposes into kernel slippage (from the LMSR curve) plus a constant offset due to fees.
+- LP returns: Fees retained in the pool increase $S(\mathbf{q})$ relative to outstanding $L$ and thus raise LP share value. Protocol revenue scales with $\phi$ and trade flow; LP revenue scales with $(1-\phi)$.
+
+8.3 Static-parameter policy (immutability)
+- Parameters $\kappa$, $f_{\text{swap}}$, and $\phi$ are set at deployment and are immutable thereafter. Benefits include:
+  - Predictability: depth and fee impact are ex-ante auditable for a given $S$.
+  - Governance minimization: no discretionary levers to be toggled post-deployment.
+  - Composability: integrators can rely on stable behavior across time.
+
+9) Risk Analysis and Theoretical Properties
+
+9.1 Convexity and path independence
+- With constant $b$, $C(\mathbf{q}) = b\log\!\big(\sum_i e^{q_i/b}\big)$ is convex and cost differences are path independent:
+  $$\Delta C \;=\; C(\mathbf{q}+\Delta\mathbf{q}) - C(\mathbf{q}) \quad \text{depends only on the endpoints}.$$
+- With $b=\kappa S(\mathbf{q})$, $\partial C/\partial q_i$ includes a common additive term (Section 4); pairwise ratios
+  $$P(\text{base}\to\text{quote}\mid \mathbf{q}) \;=\; \exp\!\left(\frac{q_{\text{quote}}-q_{\text{base}}}{b(\mathbf{q})}\right)$$
+  remain valid under a quasi-static-$b$ view, which is the pricing lens used for infinitesimal (or discretized) steps.
+
+9.2 Bounded loss (intuition) and capital efficiency
+- For constant $b$, the classic LMSR worst-case loss is $b\ln n$ in the payout numéraire. Under $b=\kappa S$, this scales proportionally with $S$, giving an instantaneous per-unit-$S$ bound proportional to $\kappa \ln n$.
+- Capital efficiency follows from linear scaling: increasing $S$ (or $\kappa$) linearly deepens liquidity. The two-asset exact-in form
+  $$y(a) \;=\; b\,\ln\!\Big(1 + r_0(1 - e^{-a/b})\Big)$$
+  exhibits $y'(0)=\frac{r_0}{1+r_0}$ and curvature $\frac{\mathrm{d}^2 y}{\mathrm{d}a^2}<0$, quantifying the marginal depth and diminishing returns for larger $a$.
+
+9.3 Sensitivity to $b$ and reserve scales
+- Scale invariance: If $\mathbf{q}\mapsto \lambda \mathbf{q}$, then $S\mapsto \lambda S$, $b\mapsto \lambda b$, and $P(\text{base}\to\text{quote})$ is unchanged. Thus depth in notional terms scales linearly with $\lambda$.
+- As $b$ increases (via larger $S$ or $\kappa$), the function $y(a)$ becomes less curved (greater depth), reducing slippage for a given input size $a$.
+
+9.4 Failure modes and mitigations
+- Thin liquidity: Small $S$ implies small $b$, steep impact, and sensitivity to large orders. Mitigations: enforce minimum bootstrap $S^{(0)}$, external routing safeguards, and user-specified price limits $\Lambda$.
+- Extreme concentration: As some $q_j \to 0$, prices can become very large; capacity caps ($y\le q_j$) and limit-hitting logic prevent pathological outputs.
+- Numerical edge cases: Guard $\exp$/$\ln$ domains, ensure denominators like $r_0 + 1 - e^{y/b} > 0$, and prefer ratio-based computations (Appendix B).
+- No-arbitrage hygiene: Maintain monotonicity of $y(a)$ and its inverse $a(y)$; avoid rounding that could create free lunches (see Section 12).
+
+10) Numerical Methods and Implementation Considerations
+
+10.1 Fixed-point arithmetic and precision policy
+- Representation: Quantities are computed in fixed-point; equations are presented over reals for clarity. Let $F$ denote the fractional precision (bits or decimal places).
+- Range limits: For stability of exponentials and logarithms, enforce
+  $$|x| \le X_{\max} \quad \text{when evaluating } \exp(x),\qquad u > 0 \quad \text{when evaluating } \ln(u).$$
+  Practical choices take $X_{\max}$ large enough to cover the operating envelope while preventing overflow.
+- Rounding: Use round-toward-zero or round-to-nearest consistently, prioritizing order-preservation (see 10.6). Avoid mixed rounding modes within a single expression.
+
+10.2 Stable exp/log evaluation (log-sum-exp)
+- Cost and shares are evaluated via a log-sum-exp recentering. Define $y_i := q_i/b$, $M := \max_i y_i$, then
+  $$C(\mathbf{q}) \;=\; b\left(M + \log \sum_i e^{\,y_i - M}\right),\qquad
+    \pi_i \;=\; \frac{e^{\,y_i - M}}{\sum_k e^{\,y_k - M}}.$$
+  Centering at $M$ prevents overflow/underflow when $y_i$ are far apart.
+- Ratio formation: Compute ratios directly to avoid extra $\exp$/$\ln$ where possible, e.g.
+  $$r_0 \;=\; \exp\!\left(\frac{q_i - q_j}{b}\right)$$
+  rather than $e^{q_i/b}/e^{q_j/b}$.
+
+10.3 Reformulations for numerical stability
+- Use $\ln(1+u)$ and $e^x-1$ style identities for small arguments:
+  $$\ln(1+u) \approx u - \frac{u^2}{2} \quad (|u|\ll 1),\qquad e^{x}-1 \approx x + \frac{x^2}{2} \quad (|x|\ll 1),$$
+  switching to series forms when $|u|$ or $|x|$ are below thresholds to reduce cancellation.
+- Inverse mapping stability: For exact-out inversion
+  $$a(y) \;=\; b \,\ln\!\left(\frac{r_0}{\,r_0 + 1 - e^{\,y/b}\,}\right),$$
+  compute $E:=e^{y/b}$ once; if $E\approx 1$, use series for $E-1$ to avoid subtractive cancellation.
+- Precompute reciprocals: Cache $b^{-1}$ to replace divisions by multiplications and reduce dispersion.
+
+10.4 Algorithm selection, termination, and convergence
+- Closed forms: Prefer the exact two-asset formulas for exact-in and exact-out when applicable (Sections 6 and A.2–A.3).
+- Root-finding for proportional joins: Solve $a_{\text{req}}(\alpha)=a$ via bracketing and bisection on a monotone map:
+  $$a_{\text{req}}(\alpha) \;=\; \alpha q_i + \sum_{j\ne i} b \ln\!\left(\frac{r_{0,j}}{\,r_{0,j} + 1 - e^{\,\alpha q_j/b}\,}\right).$$
+  Terminate when the interval width is below $\varepsilon$ or the function gap is within tolerance. Monotonicity guarantees uniqueness.
+- Limit-hitting: Compute $a_{\text{lim}}=b\ln(\Lambda/r_0)$ directly; validate $\Lambda>r_0$ and that intermediate expressions stay in-range.
+
+10.5 Performance vs precision trade-offs
+- Caching: Reuse common subexpressions (e.g., $b$, $b^{-1}$, $r_{0,j}$) across loops and iterative steps.
+- Operation count: Prefer fused operations and single-pass accumulations (e.g., recentered $\sum e^{\cdot}$ with on-the-fly rescaling).
+- Approximation regions: In designated near-balanced regimes (Section 11), switch to polynomial approximations to avoid transcendental calls, respecting global error budgets.
+
+10.6 Error analysis, monotonicity, and arbitrage-safety
+- Error budgeting: Allocate a maximum relative error $\epsilon_{\text{rel}}$ to each primitive ($\exp$, $\ln$, polynomials), ensuring the composed map (e.g., $y(a)$) meets end-to-end bounds.
+- Monotonicity preservation: Ensure numerical implementations of $y(a)$ are strictly increasing and of $a(y)$ are strictly increasing by:
+  - enforcing positive denominators (e.g., $r_0 + 1 - e^{y/b} > 0$),
+  - clamping intermediate “inner” terms to $(0,\infty)$,
+  - preferring formulations without subtractive cancellation near boundaries.
+- Arbitrage hygiene: Use conservative branches (cap-and-invert) when near capacity or domain boundaries to avoid nonphysical outputs (negative or exceeding balances).
+
+11) BalancedPair Optimization (Dedicated Section)
+
+11.1 Applicability conditions
+- Define $\delta := (q_i - q_j)/b$ and $\tau := a/b$. The balanced regime is characterized by
+  $$|\delta| \le \delta_\star,\qquad |\tau| \le \tau_\star,$$
+  with design thresholds $(\delta_\star,\tau_\star)$ chosen so that polynomial approximations meet the global error budget while preserving monotonicity.
+
+11.2 Balanced 2-asset closed form and small-argument expansions
+- The exact two-asset mapping is
+  $$y(a) \;=\; b \,\ln\!\Big(1 + r_0 (1 - e^{-a/b})\Big),\qquad r_0=e^{\delta}.$$
+- For $|\delta|\ll 1$ and $|\tau|\ll 1$, use expansions
+  $$e^{\pm x} \approx 1 \pm x + \frac{x^2}{2},\qquad \ln(1+u) \approx u - \frac{u^2}{2},$$
+  yielding
+  $$y(a) \;\approx\; b \left[ r_0 \tau - \frac{1}{2} r_0 \tau^2 \right] + \mathcal{O}\!\left(\tau^3,\, |\delta|\,\tau^2\right),\quad r_0 \approx 1 + \delta + \frac{\delta^2}{2}.$$
+- Symmetry: At $\delta=0$, the mapping satisfies $y(a)\approx \tfrac{a}{2} - \tfrac{a^2}{4b} + \cdots$, reflecting equal liquidity on both sides.
+
+11.3 Polynomial approximations without $\exp/\ln$
+- Construct minimax polynomials $P_d(x)\approx e^{x}$ on $[-\tau_\star,0]$ and $Q_d(u)\approx \ln(1+u)$ on $[0,u_\star]$, where $u_\star$ is induced by the range of $r_0(1-e^{-\tau})$ in the regime.
+- Compose
+  $$\tilde{y}(a) \;=\; b \, Q_d\!\Big( r_0 \,\big(1 - P_d(-\tau)\big) \Big),\qquad \tau=\frac{a}{b},$$
+  with $r_0$ optionally approximated by a low-degree polynomial in $\delta$ when $|\delta|\le \delta_\star$.
+- Error bounds:
+  $$\big|e^{x}-P_d(x)\big| \le e^{\tau_\star}\,\frac{\tau_\star^{\,d+1}}{(d+1)!},\qquad
+    \big|\ln(1+u)-Q_d(u)\big| \le \frac{u^{\,d+1}}{(d+1)\,(1-u)^{\,d+1}},$$
+  ensuring $\big|y(a)-\tilde{y}(a)\big| \le \epsilon$ for a target $\epsilon$ via appropriate $d$, $\delta_\star$, $\tau_\star$, $u_\star$.
+
+11.4 Dispatcher logic and safe fallback
+- Preconditions:
+  - check $|\delta|\le \delta_\star$, $|a|/b \le \tau_\star$, and positivity of intermediate “inner” terms,
+  - ensure capacity is respected ($\tilde{y}(a)\le q_j$) or switch to cap-and-invert branch.
+- If any precondition fails, fall back to the general closed-form path with full transcendental evaluations.
+- Price-limit compatibility: When a price limit $\Lambda$ is set, verify that the approximated trajectory respects $r(t)\le \Lambda$; otherwise, revert to exact limit-hitting computation $a_{\text{lim}}=b\ln(\Lambda/r_0)$.
+
+11.5 Invariant preservation and monotonicity guarantees
+- Enforce $\tilde{y}'(a) > 0$ on the approximation domain by design (choose $P_d$, $Q_d$ that are monotonically increasing on their intervals and validate numerically).
+- Guard inner arguments to keep them in $(0,\infty)$, preventing nonphysical outputs or $\ln$ domain violations.
+- Capacity and nonnegativity: Clamp to $[0, q_j]$ and use inverse mapping to reconcile inputs in cap branches.
+
+11.6 Gas and performance analysis
+- Eliminating transcendental calls in the balanced regime reduces cost to a small fixed number of polynomial evaluations and multiplications.
+- Dispatcher overhead is minimal (few comparisons and a couple of scaled differences). Overall, the optimization provides substantial speedups in near-parity trades while maintaining accuracy guarantees.
+- The fallback ensures worst-case performance is bounded by the general path.
+
+12) Protocol Safety: Numerical and Invariant Guarantees
+
+12.1 Invariant checks and fail-fast conditions
+- Domain guards:
+  - Size metric: $S(\mathbf{q}) = \sum_i q_i > 0$, hence $b=\kappa S > 0$.
+  - Valid indices and nonnegative inputs/outputs for user-facing operations.
+  - Exponential and logarithm arguments within bounded, valid domains; prefer log-sum-exp recentering.
+- Capacity and limit checks:
+  - Out-amounts are capped by available balances ($y \le q_j$).
+  - Price-limit trades enforce $\Lambda > r_0$ and truncate at $a_{\text{lim}}=b\ln(\Lambda/r_0)$.
+- Consistency guards:
+  - Denominators (e.g., $r_0 + 1 - e^{y/b}$) must be positive.
+  - Reciprocal quantities (e.g., $1/b$) are computed once and reused to avoid drift.
+
+12.2 Precision-induced error handling and rounding
+- Monotonicity-first policy: Prefer formulations that preserve order (e.g., log-sum-exp, ratio formation for $r_0$).
+- Conservative rounding: When a decision boundary is approached (e.g., inner argument of $\ln$ near zero), choose the conservative branch (cap-and-invert) rather than extrapolating.
+- Bounded evaluations: Enforce $|x| \le X_{\max}$ for $\exp(x)$ to prevent overflow; clamp inputs that would violate this bound and surface clear errors to callers.
+
+12.3 Verification targets
+- Structural properties:
+  - Convexity under constant $b$; gradient softmax identities.
+  - Pairwise price ratios consistent with $P(\text{base}\to\text{quote})$ under quasi-static $b$.
+- Numerical properties:
+  - Monotonicity of $y(a)$ and $a(y)$, uniqueness of $a_{\text{lim}}$ when $\Lambda>r_0$.
+  - Path-independence of $\Delta C$ in constant-$b$ tests; bounded relative error within predefined budgets.
+
+12.4 Testing approach
+- Property-based tests across randomized states $\mathbf{q}$, input sizes, and asset pairs, including adversarial edge cases (tiny $S$, extreme $r_0$, near-capacity).
+- Boundary tests for domain guards (e.g., $S\downarrow 0$, $\Lambda \downarrow r_0$, inner argument of $\ln$ near zero).
+- Differential tests against high-precision reference implementations for $y(a)$, $a(y)$, and price ratios.
+- No-negative-arbitrage checks under rounding: ensure discrete effects cannot be exploited for profit with zero risk.
+
+13) Deployment Model and Parameter Fixity
+
+13.1 Immutable parameters and non-upgradability
+- The pool is deployed with a fixed asset set and immutable parameters $(\kappa, f_{\text{swap}}, \phi)$, where $\kappa>0$ determines $b(\mathbf{q})=\kappa S(\mathbf{q})$, $f_{\text{swap}}$ is the swap fee rate, and $\phi$ is the protocol share of fees.
+- Contracts are deployed in a non-upgradable, ownerless configuration. No governance can modify $\kappa$, fees, or the asset universe after deployment.
+
+13.2 Deployment inputs and initialization
+- Deployment specifies: the asset list, normalization conventions, and parameter tuple $(\kappa, f_{\text{swap}}, \phi)$. Initialization requires a seed inventory $\mathbf{q}^{(0)}$ with $S^{(0)}=\sum_i q_i^{(0)}>0$, yielding initial liquidity
+  $$b^{(0)} \;=\; \kappa \, S^{(0)}.$$
+- A minimum bootstrap size $S^{(0)}$ SHOULD be enforced to avoid thin-liquidity regimes at genesis.
+
+13.3 Reproducibility and transparency
+- Given $(\kappa, f_{\text{swap}}, \phi)$ and the initial state $\mathbf{q}^{(0)}$, the AMM’s pricing map is fully determined for all subsequent states via
+  $$C(\mathbf{q}) \;=\; b(\mathbf{q}) \,\log\!\left(\sum_i e^{q_i / b(\mathbf{q})}\right),\qquad b(\mathbf{q})=\kappa\,S(\mathbf{q}).$$
+- Observability: Public views expose instantaneous price ratios $P(\text{base}\to\text{quote})$, price shares $\pi_i$, LP price $P_L^{(k)}$, and size metric $S(\mathbf{q})$.
+
+13.4 Operational implications
+- Asset changes: To add or remove assets, deploy a new pool instance and migrate liquidity; existing pools remain immutable.
+- Ecosystem integration: The fixed-parameter model simplifies routing and valuation, enabling integrators to precompute depth profiles for anticipated $S$ regimes.
+- Risk discipline: The absence of admin levers eliminates governance risk but requires careful parameter selection at deployment.
+
+15) Limitations and Future Work
+
+15.1 Limitations
+- Quasi-static $b$ assumption: Pricing steps treat $b$ as locally constant. While pairwise ratios are exact for infinitesimal moves, finite trades are evaluated with closed forms derived from the two-asset reduction; model fidelity remains high but is not a full global-gradient integration under state-dependent $b$.
+- Thin-liquidity vulnerability: For small $S$, $b=\kappa S$ is small and price impact is steep; users should rely on price limits $\Lambda$ and integrators should enforce minimum bootstrap sizes.
+- Extreme concentration and domain edges: As some $q_j \to 0$, ratios $\exp((q_{\text{quote}}-q_{\text{base}})/b)$ become large; capacity caps and domain guards prevent nonphysical outcomes but can truncate trades.
+- Expressivity: The mechanism is not tuned for pegged pairs like stableswap, nor does it encode cross-asset correlations by default.
+- Numerical constraints: Fixed-point range/precision, exponential/log bounds, and approximation regimes impose domain restrictions for safe operation.
+- Static parameters: Immutability removes governance agility; mis-specified $\kappa$ or fees require new deployments.
+
+15.2 Future work
+- Adaptive proportionality: Explore principled adaptive $\kappa$ while preserving LMSR properties (e.g., bounded-loss analogues) and avoiding governance risk (e.g., rule-based or oracle-free triggers).
+- Correlated or basket-targeted variants: Incorporate weighted size metrics $S_w(\mathbf{q})$ or correlation-aware formulations, with rigorous analysis of price and risk implications.
+- Enhanced approximations: Extend polynomial or rational approximations with verified remainder bounds, larger safe domains, and automatic dispatchers with certified monotonicity.
+- Off-chain assists and proofs: Use off-chain computation for heavy numerical routines with on-chain verification (e.g., succinct proofs) while maintaining transparency.
+- MEV-aware design: Integrate user-settled price limits, batch auctions, or commit-reveal to mitigate adverse selection and sandwich risk.
+- Layer-2 deployment: Leverage lower fees and faster settlement to widen the safe domain for numerical precision and to support more assets.
+
+16) Conclusion
+
+- We presented a multi-asset AMM whose pricing kernel is the LMSR cost function with an effective liquidity parameter proportional to pool size, $b(\mathbf{q})=\kappa S(\mathbf{q})$. This delivers scale-invariant responsiveness, preserves softmax-derived pairwise price ratios, and supports any-to-any swaps via a single convex potential.
+- Closed-form two-asset reductions provide exact-in, exact-out, and limit-hitting formulas with strong monotonicity and uniqueness properties, while capacity caps and conservative inverses ensure safety at domain boundaries. Liquidity operations (proportional and single-asset joins/exits) follow directly from the same potential framework.
+- A fixed-parameter policy eliminates governance risk and makes depth calibration transparent. Numerical stability is achieved through log-sum-exp reformulations, guarded transcendental domains, and optional balanced-regime polynomial approximations with error bounds.
+- Outlook: This LMSR AMM complements CFMMs by offering multi-asset price discovery under a convex potential with predictable scaling. Future work includes adaptive yet governance-minimized responsiveness, correlation-aware variants, and verifiable off-chain assists—aimed at retaining theoretical guarantees while broadening applicability.
+
+
+
+
+
+
+
+
+
+17) Appendices
+
+A. Full derivations and proofs
+
+A.1 Gradient, price shares, and pairwise prices
+- With $b$ treated as a constant parameter, the LMSR cost $C(\mathbf{q}) = b \log\!\big(\sum_k e^{q_k/b}\big)$ yields
+  $$\frac{\partial C}{\partial q_i} \;=\; \frac{e^{q_i/b}}{\sum_k e^{q_k/b}} \;=\; \pi_i(\mathbf{q}).$$
+- With $b = \kappa \, S(\mathbf{q})$, the total derivative becomes
+  $$\frac{\partial C}{\partial q_i} \;=\; A(\mathbf{q}) \;+\; \frac{e^{q_i/b}}{\sum_k e^{q_k/b}},$$
+  where $A(\mathbf{q})$ is an additive term common to all $i$ that arises from $\partial b/\partial q_i$. Consequently, differences between marginal prices are preserved, and the pairwise marginal price ratio reduces to
+  $$P(\text{base}\to\text{quote}\mid \mathbf{q}) \;=\; \exp\!\left(\frac{q_{\text{quote}} - q_{\text{base}}}{b(\mathbf{q})}\right).$$
+  This is the exchange rate used by the kernel, under the quasi-static-$b$ assumption.
+
+A.2 Two-asset closed form: exact-in
+- Consider a swap from $i$ (in) to $j$ (out) with quasi-static $b$. Let $r_0 := \exp\!\big((q_i - q_j)/b\big)$. Along the trade path, the instantaneous marginal price ratio evolves as $r(t) = r_0 \, e^{t/b}$, where $t$ is the cumulative input of asset $i$.
+- The infinitesimal output satisfies
+  $$\mathrm{d}y \;=\; \frac{r(t)}{1 + r(t)} \,\mathrm{d}t,$$
+  in the two-asset reduction induced by the LMSR gradient. Integrating from $0$ to $a$ yields the closed form
+  $$y(a) \;=\; b \,\ln\!\Big( 1 + r_0 \,\big(1 - e^{-a/b}\big) \Big).$$
+- Properties: $y(0) = 0$, $y'(0) = \frac{r_0}{1 + r_0}$, $y$ is increasing and concave in $a$, and $\lim_{a\to\infty} y = b \,\ln(1 + r_0)$.
+
+A.3 Two-asset closed form: exact-out (inverse)
+- Given $y \ge 0$, invert the relation in A.2. Let $E := e^{y/b}$. Then
+  $$1 + r_0 \,\big(1 - e^{-a/b}\big) \;=\; E
+  \;\Rightarrow\; e^{-a/b} \;=\; 1 - \frac{E - 1}{r_0} \;=\; \frac{r_0 + 1 - E}{r_0}$$
+  $$\Rightarrow\quad a(y) \;=\; -\,b \,\ln\!\left(\frac{r_0 + 1 - E}{r_0}\right) \;=\; b \,\ln\!\left(\frac{r_0}{\,r_0 + 1 - E\,}\right).$$
+- This inverse exists and is unique for $y \in \big[0,\, b \ln(1 + r_0)\big]$.
+
+A.4 Limit-hitting and swap-to-limit
+- Let $\Lambda > 0$ be a maximum acceptable marginal price ratio for $p_i/p_j$. With $r(t) = r_0 \, e^{t/b}$, the limit is reached when $r(t) = \Lambda$, giving the unique truncated input
+  $$a_{\text{lim}} \;=\; b \,\ln\!\left(\frac{\Lambda}{r_0}\right).$$
+- The corresponding output at the limit is
+  $$y_{\text{lim}} \;=\; b \,\ln\!\Big( 1 + r_0 \,\big(1 - e^{-a_{\text{lim}}/b}\big) \Big)
+  \;=\; b \,\ln\!\Big( 1 + r_0 \,\big(1 - r_0/\Lambda\big) \Big).$$
+- If $y_{\text{lim}}$ exceeds the available out-asset balance $q_j$, cap output to $q_j$ and solve for the input required to realize $y = q_j$ using the inverse formula of A.3:
+  $$a_{\text{cap}} \;=\; b \,\ln\!\left(\frac{r_0}{\,r_0 + 1 - e^{\,q_j / b}\,}\right).$$
+
+A.5 Single-asset mint via proportional growth (exact-in to many)
+- Suppose a user contributes amount $a$ of asset $i$ and wishes to increase the pool proportionally by factor $1 + \alpha$ (with $\alpha \ge 0$). For each $j \ne i$, let $y_j := \alpha \, q_j$ be the target out-amount in $j$ when swapping from $i$. Define $r_{0,j} := \exp\!\big((q_i - q_j)/b\big)$. From A.3, the input required to realize $y_j$ is
+  $$x_j(\alpha) \;=\; b \,\ln\!\left(\frac{r_{0,j}}{\,r_{0,j} + 1 - e^{\,y_j / b}\,}\right).$$
+- The self-asset contribution is $\alpha \, q_i$. The total input required is
+  $$a_{\text{req}}(\alpha) \;=\; \alpha \, q_i \;+\; \sum_{j \ne i} x_j(\alpha).$$
+- Solve for $\alpha$ via a monotone root-find on $a_{\text{req}}(\alpha) = a$. The function is increasing in $\alpha$ on its domain, with unique solution when feasible.
+
+A.6 Single-asset burn via proportional redemption
+- Burning a proportional share $\alpha \in (0, 1]$ reduces the pool balances to $(1 - \alpha)\,\mathbf{q}$. A single-asset payout in asset $i$ aggregates (i) the direct $\alpha \, q_i$ redemption and (ii) the swaps from each asset $j \ne i$ of their redeemed $\alpha \, q_j$ portions into $i$ using A.2 with the local (post-burn) state. If a computed out-amount would exceed the local $q_i$, cap to capacity and solve the inverse for the input used.
+
+A.7 Balanced 2-asset special case and polynomial approximations
+- Near balance, define $\delta := (q_i - q_j)/b$ with $|\delta| \ll 1$ and let $a/b$ be small. Using second-order expansions:
+  $$e^{\pm x} \approx 1 \pm x + \frac{x^2}{2},\qquad \ln(1 + u) \approx u - \frac{u^2}{2},$$
+  we obtain for small $a/b$ and $\delta$:
+  $$r_0 = e^{\delta} \approx 1 + \delta + \frac{\delta^2}{2},$$
+  $$y(a) = b \,\ln\!\big(1 + r_0(1 - e^{-a/b})\big)
+       \approx b \left[ r_0 \left(\frac{a}{b}\right) - \frac{1}{2} r_0 \left(\frac{a}{b}\right)^2 \right]
+       + \mathcal{O}\!\left(\left(\frac{a}{b}\right)^3,\, \delta \left(\frac{a}{b}\right)^2\right).$$
+- In particular, when $\delta \approx 0$,
+  $$y(a) \approx \frac{a}{2} - \frac{a^2}{4b} + \cdots,$$
+  which admits efficient evaluation via fixed low-degree polynomials. This motivates a “balanced pair” dispatcher that, under explicit near-balance preconditions ($|\delta| \le \delta_\star$ and $a/b \le \tau_\star$), uses minimax Chebyshev polynomials for $\exp$ and $\ln$ on compact intervals to meet a specified error budget, and otherwise falls back to the general path.
+
+B. Error bounds and approximation details
+
+B.1 Fixed-point arithmetic and stability policy
+- Representation: All quantities are computed in fixed-point with a wide fractional field; equations are written over reals for exposition. Overflow/underflow and domain errors are prevented with explicit guards.
+- Log-sum-exp: The cost is evaluated as $C = b \left(M + \log \sum_i e^{\,y_i - M}\right)$ with $y_i := q_i/b$ and $M := \max_i y_i$. This ensures stable accumulation even when some $y_i$ are far apart.
+- Exponential guard: The arguments to $\exp(\cdot)$ are restricted to a bounded interval to ensure finite, monotone outputs. A practical bound is $|x| \le 32$ (in internal fixed-point units), which comfortably covers operational regimes while preventing overflow.
+- Ratio shortcuts: Where possible, we form ratios such as $r_0 = \exp\!\big((q_i - q_j)/b\big)$ directly, avoiding separate exponentials and a division, improving both precision and cost.
+
+B.2 Monotonicity and error budgets
+- Kernel monotonicity: The closed forms in A.2–A.4 are strictly increasing in input and satisfy $y'(a) \in (0,1)$ for feasible states. Numerical implementations preserve monotonicity by:
+  - using log-sum-exp,
+  - guarding denominators (e.g., $r_0 + 1 - e^{\,y/b} > 0$),
+  - clamping to capacity when necessary and solving inverses in the capped branch.
+- Error targets: Prices, shares, and swap amounts are computed to within small relative error (e.g., $\le 10^{-9}$ for typical ranges). Guards reject or cap inputs that would violate error or domain constraints.
+
+B.3 Polynomial approximations in balanced 2-asset mode
+- Domains: For $|\delta| \le \delta_\star$ and $\big|a/b\big| \le \tau_\star$, $\exp$ and $\ln$ are approximated by minimax polynomials on compact intervals $[-\tau_\star, \tau_\star]$ and $[\,1 - u_\star,\, 1 + u_\star\,]$, respectively, with $u_\star$ induced by the $\exp$ range.
+- Construction: Coefficients are obtained offline (e.g., via Remez) to minimize the maximum relative error over the domain.
+- Remainder bounds: Standard analytic bounds apply:
+  $$\big|e^{x} - P_d(x)\big| \;\le\; e^{\tau_\star}\,\frac{\tau_\star^{\,d+1}}{(d+1)!},$$
+  $$\big|\ln(1 + u) - Q_d(u)\big| \;\le\; \frac{|u|^{\,d+1}}{(d+1)\,(1 - |u|)^{\,d+1}},\qquad |u| < 1.$$
+  Degree $d$ and domain parameters $(\delta_\star, \tau_\star, u_\star)$ are chosen to meet the global error budget while maintaining monotonicity of the composed swap mapping.
+
+B.4 Implementation notes for stability
+- Prefer single-pass accumulations with on-the-fly recentering for $\sum \exp(\cdot)$.
+- Maintain consistent reciprocals (e.g., precompute $1/b$) to reduce rounding dispersion.
+- Use explicit positivity checks (e.g., size metric $S > 0$, $\exp$ arguments within bounds, denominators $> 0$).
+- When a computed “inner” argument to $\ln(\cdot)$ is $\le 0$ due to rounding, switch to a conservative branch (cap-and-invert) rather than continuing.
+
+C. Additional figures and tables (to be included)
+- Price impact curves vs. constant-product and constant-mean baselines across $\kappa$.
+- Parameter sweeps for $\kappa$ and $S$ showing depth and slippage profiles.
+- Numerical accuracy: worst-case relative error heatmaps for prices and swap amounts; monotonicity checks.
+- Gas microbenchmarks: general path vs. balanced 2-asset approximations; cache effects.
+
+D. Glossary and notation
+- $n$: number of assets.
+- $i, j$: asset indices in $\{0,\dots,n-1\}$.
+- $\mathbf{q}\in \mathbb{R}_{\ge 0}^{\,n}$: vector of normalized internal quantities; $q_i$ is the $i$-th component.
+- $S(\mathbf{q}) = \sum_i q_i$: size metric (aggregate pool size).
+- $\kappa > 0$: fixed liquidity proportionality constant.
+- $b(\mathbf{q}) = \kappa \cdot S(\mathbf{q})$: effective LMSR liquidity parameter.
+- $w_i = e^{q_i / b}$; $W = \sum_i w_i$.
+- $\pi_i = w_i / W$: price share (softmax probability).
+- $P(\text{base}\to\text{quote}) = \exp\!\big((q_{\text{quote}} - q_{\text{base}})/b\big)$: pairwise marginal price ratio.
+- $a$: exact-in input amount for asset $i$ (fee-free kernel).
+- $y$: exact-out output amount for asset $j$ (fee-free kernel).
+- $r_0 = \exp\!\big((q_i - q_j)/b\big)$: initial ratio for an $i\to j$ swap.
+- $\Lambda$: user-specified price limit (maximum acceptable $p_i/p_j$).
+- $\alpha$: proportional growth/redeem factor for liquidity operations.
+
+E. References
+- Hanson, R. (2002). [Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation.](https://mason.gmu.edu/~rhanson/mktscore.pdf)
+- Abernethy, J., Chen, Y., & Waggoner, B. (2013). Low-Regret Learning in Prediction Markets.
+- Uniswap (Hayden Adams et al.). Constant product market maker design docs and whitepapers.
+- Balancer. Constant mean market makers and multi-asset pool design notes.
+- Curve Finance. StableSwap invariant design notes.
+- Fixed-point arithmetic references and standard libraries for 64.64 computations.
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