The mathematics
Each epoch releases 1/φ of the epoch before it, and that single rule fixes the supply, the cap, the reward and the shape of the whole instrument.
The golden ratio φ is the one positive number that is exactly one more than its own reciprocal, which is the self-referential identity that gives it every property that follows.
A second identity falls straight out and does the real work later, because it is the fraction of the whole that the first epoch will hold.
Time is divided into epochs of equal length. The first epoch releases an amount E₀, and every epoch after it releases 1/φ of the one before, so the release in epoch n is a golden decay.
This is a gentler descent than a Bitcoin halving. Each step keeps 61.8%; a halving keeps 50%, so the early advantage is real yet less severe.
In real-number mathematics the releases are infinite in number and never reach zero, yet their sum is finite, because a geometric series with ratio 1/φ converges. This is the mathematical heart of a perpetuity: an ideal stream without end whose total is bounded.
So the total supply is exactly φ² times the first epoch, which means the schedule determines the cap. Read the other way, the first epoch is exactly 1/φ² of the whole, which is 38.2%, and after each epoch the fraction of the cap still to come is exactly 1/φ, which is 61.8%.
Emission sets the quantity released. Allocation follows weighted presence through the reward-per-unit method used in the current prototype. Writing e(t) for the golden-decay release rate and T(t) for the total held, the reward index is
and a holder of balance b present from tin to tout claims b · (R(tout) − R(tin)). Extending the same position across more positive release intervals increases its claim. Comparing two different arrival windows also depends on T(s), because competing weighted presence can change between them; the earlier schedule is richer, but crowding can reverse the absolute outcome. A holder who leaves stops accruing at that accounting boundary.
The same schedule, drawn in polar coordinates, is a golden logarithmic spiral. Let each epoch be one quarter-turn and let the radius be the release, and the result is the equiangular spiral whose radius contracts by exactly φ every quarter-turn.
The releases are its radii, the cap is the finite sum of those radii, and the spiral winds for ever inward to a single point whose existence is the convergence itself. The perpetuity and the spiral are one fact seen two ways, and the spiral on the face of this site carries the mechanism.
Everything above pays presence that is free to leave at any instant. A holder may go one step further and bind their presence for a stated number of epochs ahead, and the stream weights bound presence more heavily while the promise stands. The release schedule derives the weight: the release k epochs ahead is φ−k of the present one, so the stream already values the next n epochs at their partial sum, and presence bound for n epochs is worth its present plus everything it has promised.
The curve lands on golden ground at every step: w(0) = 1, w(1) = φ by the founding identity itself, w(2) = 2 because 1/φ + 1/φ² = 1, w(3) = √5, and the limit is φ², the constant that maps the first epoch to the cap. Each promised epoch adds exactly φ−n of weight, so the price of patience is the emission schedule read again from the holder's own present, and no promise, however long, outweighs φ² of free presence.
The covenant carries teeth and a law. A term begins settled, because binding claims whatever has already accrued, so what the term earns is exactly what the term holds. Accrual then banks inside the stream at weight w(n), claimable the moment the term matures, and the position is sealed while the term stands. The ether itself may leave at any instant, whole, because breaking the term forfeits only the banked accrual, which is never minted and stays beneath the cap for ever.
A numerical pass confirms every identity to twelve figures: φ² = φ + 1 and 2 − φ = 1/φ² hold, the ideal emission series sums to E₀ · φ², and the first epoch is 1/φ² of the whole. One four-holder scenario pays the earliest-and-longest 48.6% against 45.5%, 3.1% and 2.7% for early-short, late-long and late-short; it illustrates that configuration and does not establish the ranking under every competing-weight path. An integer simulation checks the current covenant mechanics, and the contract suite carries twenty-five green tests. Those tests supply internal evidence. Independent audit remains outstanding.
The mechanism uses the golden ratio and its identities, a convergent geometric series, a reward-per-unit accumulator and the logarithmic spiral, and nothing else. 3-6-9, vortex arithmetic and sacred geometry are arithmetic without consequences, and none of them enters the mechanism at any point.
The equation leaves four release decisions open: supply S, production cadence, activation block and integer-tail behaviour. Solidity also works in discrete token units, so a sufficiently small release truncates to zero even though the ideal real-number series does not. No Perpetuity contract is published on mainnet; production parameters and addresses will appear only with verified source, final bytecode and audit evidence.