Constant Depth Curve: mathematical proof

Companion to PNK/USD market maker: model & path-independence validation.

Setup

Consider a bonding curve on a CLOB with token holdings $x(P)$ and USD holdings $y(P)$ as functions of price $P$ , operating over a bounded range $[P_a, P_b]$ .

Uniform depth constraint: Define depth over a symmetric log-price band of half-width $h > 0$ . At any price $P$ such that both $Pe^{-h}$ and $Pe^{h}$ lie within $[P_a, P_b]$ , the USD available on each side must equal a constant $D$ :

y(P) - y(Pe^{-h}) = D \quad \text{(bid depth)}

y(Pe^{h}) - y(P) = D \quad \text{(ask depth)}

The effective domain for the constraint is $P \in [P_a e^{h}, P_b e^{-h}]$ . Outside this interior band, the full symmetric log-band is no longer contained in $[P_a, P_b]$ , so the available depth on the truncated side is strictly less than $D$ .

Note: if the band is instead defined arithmetically as $P(1\pm f)$ , the bid and ask sides involve different log-widths ( $\ln(1/(1-f)) \neq \ln(1+f)$ ), making exact two-sided uniformity impossible with a single curve. The log-price formulation avoids this asymmetry.

1. The functional equation

Substituting $u = \ln P$ and $Y(u) = y(e^u)$ , both constraints reduce to:

Y(u) - Y(u - h) = D \quad \forall\, u \in [\ln P_a + h,\; \ln P_b - h]

This is a finite-difference equation: $Y$ has constant increments over equal steps $h$ in log-price space.

2. Solving the functional equation

The general smooth solution is:

Y(u) = \frac{D}{h} \cdot u + \phi(u)

where $\phi$ is any smooth $h$ -periodic function ( $\phi(u) = \phi(u - h)$ ), since:

Y(u) - Y(u-h) = \frac{D}{h} \cdot h + \phi(u) - \phi(u-h) = D

To obtain a homogeneous curve with no oscillatory structure in log-price, we select the canonical non-oscillatory solution $\phi = \text{const}$ . This is the scale-invariant choice: the curve has constant USD depth per unit log-price, with no oscillatory structure or preferred frequencies in log-price space.

Note: monotonicity of $y(P)$ alone is insufficient to force $\phi = \text{const}$ . A small-amplitude periodic $\phi$ (e.g., $\phi(u) = \varepsilon \sin(2\pi u / h)$ with $\varepsilon < D/2\pi$ ) preserves $Y'(u) > 0$ everywhere while introducing oscillatory microstructure. Such solutions are mathematically valid but economically unnatural — they would create a curve where the marginal selling rate oscillates with price, with no economic justification for the preferred frequency. We exclude them by construction.

Thus:

Y(u) = \frac{D}{h} \cdot u + \text{const}

Substituting back and setting $y(P_a) = 0$ (boundary condition: all capital is in tokens at the lower bound):

\boxed{\,y(P) = C \cdot \ln(P / P_a)\,}

where:

C = \frac{D}{h}

3. Deriving x(P)

Along a differentiable bonding curve with marginal price $P$ , trade accounting gives:

dy = -P \cdot dx

(When price rises by $dP$ , the curve sells $dx$ tokens at price $P$ , receiving $P \cdot dx$ in USD.)

From $y(P) = C \ln(P/P_a)$ :

\frac{dy}{dP} = \frac{C}{P}

Therefore:

\frac{dx}{dP} = -\frac{1}{P}\frac{dy}{dP} = -\frac{C}{P^2}

Integrating with boundary condition $x(P_b) = 0$ (all capital is in USD at the upper bound):

\boxed{\,x(P) = C\left(\frac{1}{P} - \frac{1}{P_b}\right)\,}

4. The invariant

From $x(P)$ and $y(P)$ :

x + \frac{C}{P_b} = \frac{C}{P} \quad \Rightarrow \quad P = \frac{C}{x + C/P_b}

And:

e^{y/C} = \frac{P}{P_a}

Combining:

\boxed{\,\left(x + \frac{C}{P_b}\right) \cdot e^{y/C} = \frac{C}{P_a}\,}

This defines the state manifold of the curve. Infinitesimal trades respecting $dy = -P \cdot dx$ preserve this invariant.

5. Verification of uniform depth

Bid depth at price $P$ (USD spent as price falls from $P$ to $Pe^{-h}$ ):

D_{bid}(P) = y(P) - y(Pe^{-h}) = C\ln\frac{P}{P_a} - C\ln\frac{Pe^{-h}}{P_a} = C \cdot h = D \;\checkmark

Ask depth at price $P$ (USD received as price rises from $P$ to $Pe^{h}$ ):

D_{ask}(P) = y(Pe^{h}) - y(P) = C\ln\frac{Pe^{h}}{P_a} - C\ln\frac{P}{P_a} = C \cdot h = D \;\checkmark

Both sides are exactly $D$ , for all $P$ in the effective domain $[P_a e^{h}, P_b e^{-h}]$ .

6. Derived properties

Portfolio value:

V(P) = x(P) \cdot P + y(P) = C\left(1 - \frac{P}{P_b} + \ln\frac{P}{P_a}\right)

Range boundaries from capital. Given $X$ tokens and $Y$ USD at current price $P_0$ , with $C$ held fixed (i.e., target depth $D$ and band half-width $h$ fixed):

USD sets the downside: $P_a = P_0 \cdot \exp(-Y/C)$ , which is always positive.
Tokens set the upside: $P_b = C \cdot P_0 / (C - X \cdot P_0)$ . This requires $X \cdot P_0 < C$ for a finite positive $P_b$ . If $X \cdot P_0 = C$ , then $P_b = \infty$ . If $X \cdot P_0 > C$ , no finite or infinite positive upper bound exists within this curve family for that $C$ .

These are independent — adding USD extends the downside without affecting the upside, and vice versa.

CLOB grid order size. On a log-uniform grid with spacing $\delta$ , each order is approximately $C \cdot \delta$ USD, uniform at every price level.

7. Relationship to arithmetic percentage bands

If the depth band is defined arithmetically as $[P(1-f), P(1+f)]$ instead of $[Pe^{-h}, Pe^{h}]$ , the bid and ask log-widths differ:

Bid: $h_{bid} = \ln(1/(1-f))$
Ask: $h_{ask} = \ln(1+f)$

No single log-affine curve can produce exact constant depth on both sides simultaneously. The relative excess of bid depth over ask depth is:

\frac{D_{bid}}{D_{ask}} - 1 = \frac{\ln(1/(1-f))}{\ln(1+f)} - 1 = f + O(f^2)

At $f = 0.02$ , bid depth exceeds ask depth by ~2%, which is negligible in practice. But exact two-sided uniformity under arithmetic bands is impossible with this curve family.