whitepaper rendering fix

doc/whitepaper.md

@@ -8,78 +8,106 @@ Multi-asset liquidity typically trades off simplicity and expressivity. Classica
 
 ## System Model and Pricing Kernel
 We consider $n\ge 2$ normalized assets with state vector $\mathbf{q}=(q_0,\dots,q_{n-1})\in\mathbb{R}_{\ge 0}^{\,n}$ and size metric $S(\mathbf{q})=\sum_i q_i$. The kernel is the LMSR cost function
+
 $$
 C(\mathbf{q}) = b(\mathbf{q}) \log\!\left(\sum_{i=0}^{n-1} e^{q_i / b(\mathbf{q})}\right), \qquad b(\mathbf{q})=\kappa\,S(\mathbf{q}),\quad \kappa>0.
 $$
+
 For numerical stability we evaluate $C$ with a log-sum-exp recentering. Let $y_i := q_i/b(\mathbf{q})$ and $M:=\max_i y_i$. Then
+
 $$
 C(\mathbf{q}) \;=\; b(\mathbf{q}) \left( M + \log \sum_{i=0}^{n-1} e^{\,y_i - M} \right),
 $$
+
 which prevents overflow/underflow when the $y_i$ are dispersed. Quantities are represented in fixed-point with explicit range and domain guards; equations are presented over the reals for clarity.
 
 ## Gradient, Price Shares, and Pairwise Prices
 With $b$ treated as a constant parameter, the LMSR gradient recovers softmax shares
+
 $$
 \frac{\partial C}{\partial q_i} \;=\; \frac{e^{q_i/b}}{\sum_k e^{q_k/b}} \;=:\; \pi_i(\mathbf{q}),
 $$
+
 so that the ratio of marginal prices is $\pi_j/\pi_i = \exp\!\big((q_j-q_i)/b\big)$. When $b(\mathbf{q})=\kappa S(\mathbf{q})$ depends on state, $\frac{\partial C}{\partial q_i}$ acquires a common additive term across $i$ from $\partial b/\partial q_i$, but pairwise ratios remain governed by softmax differences. We therefore use a quasi-static-$b$ view for pricing steps, holding $b$ fixed at the pre-trade state for the infinitesimal move, and define the instantaneous pairwise marginal price ratio for exchanging $i$ into $j$ as
+
 $$
 P(i\to j \mid \mathbf{q}) \;=\; \exp\!\left(\frac{q_j - q_i}{b(\mathbf{q})}\right).
 $$
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 This ratio drives swap computations and is invariant to proportional rescaling $\mathbf{q}\mapsto \lambda\mathbf{q}$ because $b$ scales by the same factor.
 
 ## Two-Asset Reduction and Exact Swap Mappings
 Swaps are computed in the two-asset subspace spanned by the in-asset $i$ and out-asset $j$, with all other coordinates held fixed under a quasi-static-$b$ step. Let
+
 $$
 r_0 \;:=\; \exp\!\left(\frac{q_i - q_j}{b}\right), \qquad b \equiv b(\mathbf{q})\;\text{ held quasi-static}.
 $$
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 Along the $i\!\to\! j$ path, the instantaneous ratio evolves multiplicatively as $r(t)=r_0\,e^{t/b}$ where $t$ denotes cumulative input of asset $i$. In the two-asset reduction the infinitesimal output satisfies
+
 $$
 \mathrm{d}y \;=\; \frac{r(t)}{1+r(t)}\,\mathrm{d}t.
 $$
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 Integrating from $t=0$ to $t=a$ yields the exact-in closed form
+
 $$
 y(a) \;=\; b \,\ln\!\Big( 1 + r_0 \,\big(1 - e^{-a/b}\big) \Big).
 $$
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 This mapping has $y(0)=0$, is strictly increasing and concave in $a$, and satisfies $y'(0)=\frac{r_0}{1+r_0}$ with asymptote $\lim_{a\to\infty} y = b\,\ln(1+r_0)$. The inverse exact-out mapping follows by solving for $a$ in terms of target $y$. Writing $E:=e^{y/b}$, we obtain
+
 $$
 a(y) \;=\; b \,\ln\!\left(\frac{r_0}{\,r_0 + 1 - E\,}\right),
 $$
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 which is strictly increasing and convex for $y\in\big[0,\, b\ln(1+r_0)\big]$. These two expressions are the workhorses for exact-in and exact-out swaps in our kernel.
 
 ## Price Limits, Swap-to-Limit, and Capacity Caps
 Users may provide a maximum acceptable marginal price ratio $\Lambda>0$ for $p_i/p_j$. The marginal ratio trajectory $r(t)=r_0 e^{t/b}$ first reaches the limit at the unique
+
 $$
 a_{\text{lim}} \;=\; b \,\ln\!\left(\frac{\Lambda}{r_0}\right),
 $$
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 and the output realized at that truncation is
+
 $$
 y_{\text{lim}} \;=\; b \,\ln\!\Big( 1 + r_0 \,\big(1 - r_0/\Lambda\big) \Big).
 $$
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 Outputs are further bounded by available inventory; if a computed $y$ would exceed $q_j$, we cap at $y=q_j$ and compute the implied input by inverting the exact-out formula,
+
 $$
 a_{\text{cap}} \;=\; b \,\ln\!\left(\frac{r_0}{\,r_0 + 1 - e^{\,q_j/b}\,}\right).
 $$
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 These limit and capacity branches ensure monotone, conservative behavior near domain edges.
 
 ## Liquidity Operations from the Same Potential
 Liquidity is accounted via pool shares $L$ taken proportional to the size metric, and we set $L=S(\mathbf{q})$ without loss of generality. At initialization with seed balances $\mathbf{q}^{(0)}$ the pool sets $L^{(0)}=S^{(0)}$ and $b^{(0)}=\kappa S^{(0)}$. A proportional deposit that scales balances to $\mathbf{q}'=(1+\alpha)\mathbf{q}$ mints $\Delta L = \alpha S(\mathbf{q})$ shares and scales liquidity to $b'=(1+\alpha)b$. Single-asset deposits target a proportional growth while rebalancing through kernel swaps: providing amount $a$ of asset $i$ induces a growth factor $\alpha\ge 0$ satisfying the monotone equation
+
 $$
 a \;=\; a_{\text{req}}(\alpha) \;=\; \alpha q_i \;+\; \sum_{j\ne i} b \,\ln\!\left(\frac{r_{0,j}}{\,r_{0,j} + 1 - e^{\,\alpha q_j/b}\,}\right), \quad r_{0,j}:=\exp\!\left(\frac{q_i-q_j}{b}\right),
 $$
+
 and mints $\Delta L=\alpha S(\mathbf{q})$ upon the unique solution. Proportional withdrawals burn $\Delta L$ and return $\alpha=\Delta L/S(\mathbf{q})$ of each asset, updating $b$ to $(1-\alpha)b$. Single-asset withdrawals redeem $\alpha q_i$ directly and swap each redeemed $\alpha q_j$ for $j\ne i$ into $i$ using the exact-in mapping evaluated on the local post-burn state; any capacity overrun is handled by a cap-and-invert branch as above. Because all operations reduce to the same two-asset closed forms, they inherit monotonicity and uniqueness.
 
 ### Single-Asset Mint and Redeem
 
 #### Single-Asset Mint
 Given a deposit of amount $a>0$ of asset $i$, the pool targets a proportional growth factor $\alpha \ge 0$ so that the post-mint state can be rebalanced to $(1+\alpha)\,\mathbf{q}$ using fee-free kernel swaps from $i$ into each $j\ne i$. For each $j\ne i$, let $y_j := \alpha\,q_j$ and define $r_{0,j} := \exp\!\big((q_i - q_j)/b\big)$. The input required to realize $y_j$ via the exact-out inverse is
+
 $$
 x_j(\alpha) \;=\; b \,\ln\!\left(\frac{r_{0,j}}{\,r_{0,j} + 1 - e^{\,y_j/b}\,}\right),
 $$
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 so the total required input for growth $\alpha$ is
+
 $$
 a_{\text{req}}(\alpha) \;=\; \alpha\,q_i \;+\; \sum_{j\ne i} x_j(\alpha).
 $$
+
 Properties and solver:
 - Monotonicity: $a_{\text{req}}(\alpha)$ is strictly increasing on its feasible domain, guaranteeing a unique solution.
 - Solver: bracket $\alpha$ (e.g., start from $\alpha\sim a/S$ and double until $a_{\text{req}}(\alpha)\ge a$ or a safety cap), then bisection to a small tolerance $\varepsilon$ (e.g., $\sim10^{-6}$ in fixed-point units).
@@ -91,28 +119,36 @@ Burning a proportional share $\alpha \in (0,1]$ returns a single asset $i$ by re
 1) Form the local state after burn, $\mathbf{q}_{\text{local}}=(1-\alpha)\,\mathbf{q}$.
 2) Start with the direct redemption $\alpha\,q_i$ in asset $i$.
 3) For each $j\ne i$, withdraw $a_j := \alpha\,q_j$ and swap $j\to i$ using the exact-in form evaluated at $\mathbf{q}_{\text{local}}$:
+
 $$
 r_{0,j} \;=\; \exp\!\left(\frac{q^{\text{local}}_j - q^{\text{local}}_i}{b}\right),\qquad
 y_{j\to i} \;=\; b \,\ln\!\Big(1 + r_{0,j}\,\big(1 - e^{-a_j/b}\big)\Big).
 $$
+
 4) Capacity cap and inverse: if $y_{j\to i} > q^{\text{local}}_i$, cap to $y=q^{\text{local}}_i$ and solve the implied input via
+
 $$
 a_{j,\text{used}} \;=\; b \,\ln\!\left(\frac{r_{0,j}}{\,r_{0,j} + 1 - e^{\,q^{\text{local}}_i/b}\,}\right),
 $$
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 then update $\mathbf{q}_{\text{local}}$ accordingly.
 5) The single-asset payout is
+
 $$
 Y_i \;=\; \alpha\,q_i \;+\; \sum_{j\ne i} y_{j\to i}, \qquad \text{with LP burned } L_{\text{in}} = \alpha \, S(\mathbf{q}).
 $$
+
 Guards and behavior:
 - Enforce $b>0$, positivity of inner terms (e.g., $r_{0,j} + 1 - e^{y/b} > 0$), and safe exponent ranges; treat any per-asset numerical failure as zero contribution rather than aborting the whole redeem.
 - The mapping is monotone in $\alpha$; the cap-and-invert branch preserves safety near capacity.
 
 ### LP Pricing vs. an Asset Token
 With LP supply set to $L=S(\mathbf{q})$, the instantaneous price of one LP share in units of asset $k$ aggregates marginal exchange rates from each asset into $k$:
+
 $$
 P_L^{(k)}(\mathbf{q}) \;=\; \frac{1}{S(\mathbf{q})}\,\sum_{j=0}^{n-1} q_j \,\exp\!\left(\frac{q_j - q_k}{b(\mathbf{q})}\right).
 $$
+
 Interpretation: proportional deposits leave $P_L^{(k)}$ unchanged; swap fees retained in the pool increase $S$ relative to outstanding $L$, raising $P_L^{(k)}$ (implicit fee accrual). This expression helps LPs and integrators reason about share valuation and dilution across assets.
 
 ## Numerical Methods and Safety Guarantees
@@ -120,13 +156,17 @@ We evaluate log-sum-exp with recentring, compute ratios like $r_0=\exp((q_i-q_j)
 
 ## Balanced Regime Optimization: Approximations, Dispatcher, and Stability
 Since transcendental operations are gas-expensive on EVM chains, we use polynomial approximations in near-balanced regimes (e.g., stable-asset pairs) while preserving monotonicity and domain safety. Parameterize $\delta := (q_i - q_j)/b$ and $\tau := a/b$ for an $i\!\to\! j$ exact-in step. The exact mapping
+
 $$
 y(a) \;=\; b \,\ln\!\Big(1 + e^{\delta}\,\big(1 - e^{-\tau}\big)\Big)
 $$
+
 admits small-argument expansions for $|\delta|\ll 1$ and $|\tau|\ll 1$. Using $e^{\pm x}\approx 1\pm x+\tfrac{x^2}{2}$ and $\ln(1+u)\approx u - \tfrac{u^2}{2}$, we obtain
+
 $$
 y(a) \;\approx\; b \left[ r_0 \tau - \tfrac{1}{2} r_0 \tau^2 \right] + \mathcal{O}\!\left(\tau^3,\, |\delta|\,\tau^2\right), \qquad r_0=e^{\delta}\approx 1+\delta+\tfrac{\delta^2}{2},
 $$
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 and at $\delta=0$ the symmetry reduces to $y(a)\approx \tfrac{a}{2} - \tfrac{a^2}{4b} + \cdots$.
 
 ### Dispatcher preconditions and thresholds (approx path):
@@ -158,9 +198,11 @@ Under constant $b$, classical LMSR admits a worst-case loss bound of $b \ln n$ i
 
 ## Deployment and Parameter Fixity
 The parameter tuple $(\kappa, f_{\text{swap}}, \phi)$ is set at deployment and remains immutable, with $\kappa>0$ defining $b(\mathbf{q})=\kappa S(\mathbf{q})$, $f_{\text{swap}}$ the swap fee rate, and $\phi$ the protocol share of fees. Given the initial state $\mathbf{q}^{(0)}$ with $S^{(0)}>0$, the induced pricing map is fully determined by
+
 $$
 C(\mathbf{q}) = b(\mathbf{q}) \log\!\left(\sum_i e^{q_i / b(\mathbf{q})}\right), \qquad b(\mathbf{q})=\kappa S(\mathbf{q}),
 $$
+
 and the two-asset closed forms above. Fixity eliminates governance risk, makes depth calibration transparent, and simplifies integration for external routers and valuation tools.
 
 ## Conclusion