{ Math

Formal model

Intents, CoW, auction, and signer welfare as equations.

This page is the protocol as written math. Same objects as the rest of the dome — intents, CoW, surplus, bond — without the prose around them.

Intent

I ∈ 𝒜 × 𝒜 × ℝ₊ × ℝ₊ × ℕ × Sig

(9.1)

Limit price p_lim(I) = q_min / q_in. Feasibility is (2.2). Partial fills scale q_min linearly.

Coincidence of wants

CoW(I, J) ⇔ a_out(I) = a_in(J) ∧ p_lim(I) · p_lim(J) ≤ 1

(9.2)

The feasible uniform rate sits in the closed interval

r* ∈ [ p_lim(I) , p_lim(J)^-1 ]

(9.3)

//R.05 Uniform clearing rate r* inside the CoW interval

Picking r*

Archerdome picks the rate that maximises total surplus inside the interval. Ties break toward the older intent.

r* = arg max_r S_I(r) + S_J(r)

(9.4)

Auction

Each solver v submits a bundle F_v of feasible fills. The winner maximises

max_v Σ_{i ∈ F_v} π_v(i) − c_gas(F_v)

(9.5)

subject to the bond (7.1) and no overlapping claims on the same intent.

Signer welfare

u(I) = π_u = ½ max(0, q_out − q_min)

(9.6)

The signer is strictly better off than the limit they signed whenever a fill improves, and never worse. That is the only welfare claim the protocol makes.