The index peg as a
control system.
An S&P 500 unit can be algorithmic. Mint and redeem are a feedback law around a level the protocol does not set. The index already has a world market. The token is pinned to that market by creation, redemption, inventory constraints, and a fee schedule that is a function of book state, not of belief.
Why an algorithmic index peg is even a coherent sentence.
Call a peg algorithmic when the restoring force is a published function of state, not a discretion of an issuer. Three objects are enough.
Exogenous spot Sₜ
The S&P 500 is a float-adjusted capitalization index on a published divisor. A token float of even several billion dollars is noise against the underlying free float. ∂S/∂P ≈ 0. The collateral price is not a function of the token price.
Endogenous inventory Iₜ
A replicating book, not a treasury narrative. Float cannot exceed assayed, unencumbered units of the basket or of the futures hedge that maps to it. This is the anti-reflexivity constraint algorithmic coins delete.
Policy π(I, Q)
Mint cost and redeem cost are functions of fill and of the redemption queue. The algorithm does not set the index. It sets the width of the no-arbitrage band around the index.
Arbitrageurs
The actuators. If they have capital and a legal path to the basket or to the front future, the band is enforced. If they do not, the algorithm is a PDF.
The comparison case is not a dollar stablecoin and not a leveraged token. A dollar stablecoin is a claim on dollars held off-system. A leveraged token is a path-dependent claim whose financing dominates the return. spu as specified is a warehouse receipt on index exposure with a burn right. Its dollar volatility should match the S&P 500. Stability is against the index, not against the dollar. Anyone selling dollar-stability is selling a second instrument: a hedge, with margin, which this note refuses to hide inside the token.
A divisor-equivalent point, not “the market” as a vibe.
The index is not a single share. The mint function first maps a delivery to benchmark-equivalent points. Constituent weights, the official divisor, and pending corporate actions are the first-order basis.
A delivery of basket value v, measured in index points, mints n = v·q(w) tokens, and only if q(w) clears a floor. Off-spec delivery — stale weights, unsettled tender, wrong currency — does not enter the float at par. Redemption inverts the map at the chosen venue:
Location is priced by netback, not by a slogan that all venues are one good. Let f_{h,h★} be the cost of moving a point of exposure from book h to the holder’s venue. The redeemable packet is the solution of a tiny linear program over books that currently have free stock:
That program is the algorithm. It is operations research. It does not require a new monetary theory.
No arbitrage is the peg. The fee schedule is the algorithm.
Let Pₜ be the token price in dollars per spu, Sₜ the official index level, cₘ the all-in cost of creating a token, cᵣ the all-in cost of destroying one. Any price outside the band is a free lunch for someone who can touch the basket.
Inside the band, arbitrage is idle. The token may wander. That wandering is not a broken peg; it is the bid-ask of the physical world — creation windows, borrow, futures basis, settlement. A design that displays a flat dollar price through a limit-down open is lying.
The algorithmic part is how costs depend on state. Fill fraction φ = I / I_max. Queue depth Q in points waiting on redemption.
Equation (2) is a soft capacity constraint. As the replicating book fills against its custody and borrow limits, minting becomes uneconomic before the book is physically full, so the float cannot be forced through the walls by a mint rush. Equation (3) prices congestion instead of freezing redemptions. A freeze is how commodity schemes become banks. A queue fee is how they stay warehouses.
Carry sits under the band, not inside a hidden yield.
The index future’s fair value, in continuous form, is the equity cost-of-carry relation. Dividends leave the spot; financing enters it. Neither is a protocol yield.
r is the financing rate. q is the dividend yield of the basket. A futures hedge that is rolled without publishing the basis is how an index claim becomes a path-dependent note. The token does not roll for the holder. If the book is in futures, the basis is a line item on redemption, quoted ex ante:
Local stability of the band is ordinary. Let the deviation be x = P − S. Outside the band an arbitrage flow restores inventory at a rate proportional to the mispricing net of fees. Linearized about a feasible inventory,
κ is not a governance parameter. It is the speed of capital that can touch a creation window. If κ → 0, equation (1) is a wish.
The peg fails where the actuator fails.
Closed creation
A halt, a limit, a custody outage. The band widens to the width of the outage. Pretending otherwise is the failure mode, not the outage.
Reflexive collateral
If the book is the token, ∂S/∂P is no longer ≈ 0. The exogenous-spot hypothesis is then false and the control law is circular.
Hidden financing
Embedding the roll, the borrow, or a short-vol overlay inside “the peg” converts a warehouse receipt into a structured note. The dollar path then ceases to match the index.
Discretionary freeze
A redemption gate that is not a published function of Q is an issuer. The object stops being algorithmic at the moment the gate is a meeting.
The design claim is narrow. spu tracks the S&P 500 inside a state-dependent band if and only if inventory is exogenous to the token price, fees are a published function of fill and queue, and someone with capital can mint or burn. Outside those conditions there is no peg. There is a story about one.