soft3/specs/chain-economy — the focus economy of the root chain.md

Owner order, 2026-10-17: "the goal is foculus, stakes with valence, and mining — not just relaunching a network on the new stack". This page makes the root chain of chain.md pay for knowledge from its first block: a cyberlink carries locked stake and a valence, the chain cuts its history into epochs of a fixed number of blocks, every full node computes the focus φ* of the staked graph at each epoch boundary with tru's tri-kernel, attributes the epoch's change of syntropy to the stake that caused it by one gradient solve, and pays the attributed stake from a focus pool fed by a share of the block subsidy. Proof-of-work miners keep the rest of the subsidy and all fees.

Binding decisions (coordinator, 2026-10-17), the contract of this page:

1. block ordering stays Nakamoto proof of work (chain.md §1, §5);
2. an epoch is a fixed number of blocks; the epoch beacon is the id of the epoch's boundary block; the boundary is a consensus object (closes cyber/launch row 41; foculus's sequential-squaring VDF is retired from the root chain);
3. one φ*: tru's fixed-point tri-kernel; foculus::phi is retired;
4. epoch settlement is verified by recomputation on every full node, no proof on the critical path; attribution of Δφ⁺ to links is the gradient (integrated-gradient) form of Shapley of tru/specs/rewards.md §4, computed by one adjoint solve, never by per-contributor recomputes;
5. a cyberlink carries a locked stake (conviction) and a valence v ∈ {−1, 0, +1}; only staked links earn;
6. everything on this page is active from genesis: no shadow mode, one rule set at launch.

This page amends chain.md: §2.2 (coinbase amount), §2.4 (cyberlinks carry stake), §3.2 (dimensions written), §3.4–§3.5 (transition, subsidy split), §7 (rule set 1 now carries the economy) and §10 item 2 (superseded). Where the two pages differ, this page wins for the economy and chain.md for everything else. It descends from tru/specs/rewards.md and fixes the parameters rewards.md leaves open for the root chain (rewards.md §16 records them on the tru side).

Design rules, as in chain.md: integers and fixed point only, never a float in a consensus path; one valid byte form per value; every derived quantity is a pure function of finalized chain state.

0. parameters

Chain-wide constants, identical on pussy and bostrom. S0 and SUBSIDY are chain.md §0.

Derived, pussy (S0 = 1,099,389,810,279,114,036, SUBSIDY = 10,458,426,657) and bostrom (S0 = 482,287,925,778,234, SUBSIDY = 4,587,974): epochs per day 12; first settlement at height 360 (epoch 1); focus pool per epoch at the cap 240 · floor(SUBSIDY / 2).

1. what is consensus at genesis

Decision 6 is read as: everything this page specifies is in force from block 0, under rule set 1. The pool of chain.md §11 stays rule set 2: it needs verifying keys that the proof gates of privacy-frozen.md have not produced, and activating it with no keys would make every spend invalid.

2. epochs

2.1 numbering and boundary

epoch(h)    = ceil(h / L)            // h ≥ 1; height 0 belongs to epoch 0
boundary(E) = E · L                  // the last block of epoch E, E ≥ 0

Epoch 0 is the genesis block alone. Epoch E ≥ 1 is the blocks (E − 1) · L + 1 … E · L. The boundary block of epoch E is the block at height E · L on the chain being validated; the boundary state G_E is the state after that block (chain.md §3.4), and its root is that block's state_root. Two nodes agree on E, the boundary and the boundary state from headers and bodies alone — no gossip of epoch objects, no state copy (launch row 41's gate, met by construction).

2.2 length

L = 240 blocks, two hours at the target interval. The reasons, in order of force:

• observability: the chain launches on 2026-11-05 and the economy has to be seen working that day — the first payout lands at height 360, about three hours after genesis, and twelve settlements a day give a red team twelve chances a day to break one;
• pipeline: settlement of epoch E is applied at boundary(E) + D (§2.5); L ≥ D + 1 keeps at most one settlement in flight, so a node never computes two epochs at once;
• cost: a settlement is three tri-kernel solves over the whole graph (§5.1); at the measured 3–5 minutes on bostrom's 3 M links (§7.2) twelve a day cost a full node under an hour of one core per day;
• path resolution: the integrated gradient is evaluated once per epoch (§5.2), so a shorter epoch follows the path of the graph more finely; the sum over epochs telescopes to the same total change of syntropy for any L, so L moves only how credit is split along the path, and two hours is coarse enough that ordering by latency inside an epoch earns nothing (all links of one epoch are one coalition).

A day (2,880 blocks) is the alternative and the natural length once the graph outgrows recomputation; changing L is a scheduled fork like any constant.

bbg::state::EPOCH_BLOCKS (100) is bbg's storage pruning cadence and is unrelated to this clock.

2.3 beacon

beacon(E) = block_id(boundary(E))         // chain.md §1.1, 32 bytes

The beacon is the hash of the boundary block, nothing else. At genesis no rule reads it: settlement by recomputation (§5) is deterministic and needs no randomness. It is recorded in the epoch record (§5.6) and is the seed of the ticket rule set (§7.3), where its one weakness matters: the miner of a boundary block can withhold a found block to draw another beacon, paying the block's reward per draw. That bias is bounded by the subsidy and is accepted for the ticket rule set's orderings, which are only sampling seeds; nothing that pays a winner takes its randomness from a single beacon.

2.4 the snapshot

At the boundary of E the following are frozen as G_E, and settlement reads nothing else:

• every position (card, §3.2): its author, axon, valence, stake and payout owner;
• the genesis weights (§4.1), constant for the life of the chain;
• the axon list (axon_edges) giving (p, q) of every axon;
• the total native supply S0 + boundary(E) · SUBSIDY;
• the side high-water marks H⁺, H⁻ of every axon as left by the settlement of epoch E − 1 (§5.3).

The focus pool balance is read at the settlement block (§5.5), not at the boundary.

2.5 settlement height, finality and reorgs

Settlement of epoch E ≥ 1 is a step of the state transition of the block at height

settle(E) = boundary(E) + D          // D = FINALITY_DEPTH = 120

• When settle(E) is applied, boundary(E) is 120 deep; by chain.md §5.2 no node that saw it in time reorganizes below it, so the snapshots G_{E−1} and G_E are final.
• Everything settlement reads is a function of the two final snapshots and of the pool balance at settle(E), and the pool balance is itself a function of height once the boundary is final (§6.2). A reorganization that replaces the block at settle(E) (or any block after the boundary) therefore reproduces the same payouts at the same height; payouts are spendable at once and need no maturity.
• A reorganization that replaces the boundary block itself (fewer than 120 blocks deep) replaces the snapshot; nodes compute settlement for the boundary on their best chain and recompute when it changes. Each candidate boundary costs its miner a valid block; the recompute is bounded by §7.2.
• A node starts computing at the boundary and has D blocks (one hour) to finish. A node that has not finished cannot validate settle(E) and waits; a miner that has not finished cannot build it. This is the liveness bound, and §7.2 is the margin.

3. positions: staked cyberlinks

3.1 identities

card(author, p, q) = hemera(D_CARD  ‖ author ‖ p ‖ q)        // 20 + 96 = 116 bytes in
stake_key(card)    = hemera(D_STAKE ‖ card)                  // 21 + 32 = 53 bytes in
focus_key(τ)       = hemera(D_FOCUS ‖ τ)                     // 21 + 32 = 53 bytes in

A position is one card: one author's stake on one axon (p, q). Its stake is the native balance at stake_key(card); the focus pool of a chain is the native balance at focus_key(τ_native). Both are raw keys of the balances dimension (chain.md §3.2), like the turnstile T(τ) of §11.5: a transparent key hemera(owner ‖ token) hashes 64 bytes and the input length is in the capacity, so no transfer, coinbase, claim or genesis frame can address a stake or the pool. A transfer to a card id credits hemera(card ‖ τ), an ordinary balance that no key controls — a burn by the sender's choice, never stake.

Domain tags, ASCII with a terminating NUL, disjoint from every tag of chain.md:

3.2 the card record

Positions live in bbg's cards dimension, keyed by the card id. The record grows from CardRecord { owner, particle } to

bbg change requested (rung 3): CardRecord gains author, valence and since, field order as listed, DIMENSION_VERSION 4. The stake amount is not a field: it is the balance at stake_key(card), so conservation (§6.3) counts it with no new rule. A card exists exactly while its stake is positive; at zero the record is deleted. (p, q) of an axon is read from axon_edges, which every link already writes.

bbg's own insert moves amount between balances[hemera(p ‖ τ)] and balances[hemera(q ‖ τ)] and writes energy/weight; the chain does not use that path. Every cyberlink, staked or not, calls insert with amount = 0 (topology only, as rule set 1 does today), and stake accounting is the chain's, through the balances and cards dimensions.

NeuronRecord.focus, .karma and .stake stay 0 (§3.7); ParticleRecord is not written by the economy (φ* is published by digest and over HTTP, §5.6, §11 — writing 3 M particle records per epoch would cost about half an hour of root updates at the measured 0.4–0.9 ms each).

3.3 lock: kind 2 with amount > 0

The cyberlink encoding of chain.md §2.4 is unchanged (153-byte body, 257 bytes). Rule set 1 now admits amount > 0:

• amount > 0 requires token = τ_native; any other token with a positive amount is invalid;
• c = card(from, p, q); if the card exists, tx.valence must equal the card's valence (a position changes side only by unlocking to zero and locking again);
• from must hold amount + fee of spendable native coin (coinbase maturity applies, chain.md §2.2);
• effects: balances[hemera(from ‖ τ)] −= amount + fee; balances[stake_key(c)] += amount; the card is created with owner = from, particle = axon(p, q), author = from, valence = tx.valence if absent; since = height in either case; bbg insert with amount = 0; nonce += 1.

amount = 0 keeps its meaning: a bare assertion, topology only, no card, no weight in φ*, no reward.

3.4 unlock: kind 6

unlock (kind 6): from: neuron[32] ‖ nonce: u64 ‖ p: particle[32] ‖ q: particle[32] ‖ amount: u64 ‖ fee: u64

Body 120 bytes, NSIG1 signature, 224 bytes in total; envelope, txid and statement of chain.md §2.1. Valid iff the signature verifies for from, the nonce is current, c = card(from, p, q) exists, 0 < amount ≤ balances[stake_key(c)], and height ≥ card.since + MIN_LOCK. Effects: balances[stake_key(c)] −= amount; balances[hemera(from ‖ τ_native)] += amount; then the fee is debited from that balance (an unlock may pay its fee from what it returns); the card is deleted if its stake reaches 0; nonce += 1. Partial unlocks are allowed; any increase restarts the lock for the whole position (since). Rewards already paid are not touched: they were credited to the payout owner at settlement and are ordinary balances. What an unlock gives up is future credit, through the high-water mark of §5.3.

MIN_LOCK = 20,160 blocks (seven days). It must be at least L + D so a position increased in epoch E is still locked at settle(E) (§5.4); seven days makes the lock a costly signal — conviction — rather than a formality, and short enough that canary users are not trapped. The owner may change it (§12).

3.5 payout owner: kind 7

payout (kind 7): from: neuron[32] ‖ nonce: u64 ‖ p: particle[32] ‖ q: particle[32] ‖ payout: [u8;32] ‖ fee: u64

Body 144 bytes, NSIG1, 248 bytes in total. Valid iff card(from, p, q) exists and the nonce and fee rules hold. Effect: card.owner = payout. The payout owner is any 32-byte account id; rewards of a position go to balances[hemera(owner ‖ τ_native)]. This is privacy-frozen.md §7 ("rewards attach to links, not neurons; the payout key in the link receives what the link earns") in its transparent form: settlement pays positions, never neurons, and the payout account is the position's choice.

Kinds 4 and 5 remain the pool's (chain.md §11.7). Kinds 8 (settlement ticket), 9 (fold step) and 10 (report) are reserved for the later rule sets of §7.3–§7.4; under rule set 1 they are invalid.

3.6 fees

Kinds 6 and 7 pay fee ≥ len · MIN_FEE_PER_BYTE like every signed kind; fees go to the block's miner through the coinbase (chain.md §2.6). There is no fee on settlement: it is a state transition of the settlement block, not a transaction, so no one can censor a payout or be paid to include it.

3.7 karma, will and price at launch

A_eff of rewards.md §3 multiplies stake by karma κ and by a market factor f(price); tru's Context also carries will. At genesis:

• κ = 1 for every neuron, and nothing persists karma. Karma is the accumulated BTS score (truth-scoring.md); there are no reports on chain at launch (§7.4), so karma has no input, and a field of constants is not worth a record. There is a second reason to wait: privacy-frozen.md §7 makes identity-weighted schemes opt-in disclosures once the pool is live, so karma must be designed to attach to positions or payout keys, and that design belongs to the reports rule set, not to a launch rush. NeuronRecord.karma stays 0 and means "neutral".
• f(price) = 1: the ICBS market (launch row 25) is not on chain.
• will = 0: positions already are the conviction primitive; will (a neuron-wide budget auto-shared across its links) needs its own lock kind and is deferred with the reports rule set.

4. the focus graph

4.1 genesis weights

The burial graph arrives with no locked stake (chain.md §4.1 applies every genesis link with amount = 0). To give φ* a body on day one, every genesis link frame carries a genesis weight, computed by every node from the genesis file alone:

B(n)  = native balance of n in the genesis file (its b frame for τ_native, 0 if none)
L(n)  = number of l frames authored by n
g(f)  = max(1, floor(B(author(f)) / L(author(f))))       // per l frame f
G_pq  = Σ g(f) over the l frames f with (p, q)

Genesis weights are passive weight on the rank axis (rewards.md §9): they enter φ* like a v = 0 position, never change, cannot be unlocked, earn nothing (their change is always zero, §5.2), do not count in θ (§6.1) and are not balances, so conservation does not see them. They are a pure function of the genesis file, which the chain id commits to; nothing new is stored. The reasons for this rule and its trade-offs are §8.

4.2 axon weight and valence

For each axon (p, q) at a boundary, with s(c) the stake of card c:

S⁺_pq = Σ s(c) over cards on (p, q) with v ∈ {0, +1}
S⁻_pq = Σ s(c) over cards on (p, q) with v = −1
A_pq  = max(0, G_pq + S⁺_pq − S⁻_pq)                       // u128, then ≤ u64 range in practice

Valence is the direction of a position's push on the edge (rewards.md §9, two axes):

• v = +1: active stake for the edge; adds weight; earns when its addition raised syntropy (§5);
• v = 0: passive stake; adds weight exactly like +1 (it moves rank) and never earns — "a purchase, not an investment" (rewards.md §9);
• v = −1: active stake against the edge; subtracts weight down to zero; earns when its subtraction raised syntropy.

The net is clipped at zero per axon, which is arithmetic.md's "negative stake is clipped to zero before matrix construction" read per axon: the adjacency stays in the nonnegative cone, and stake beyond what zeroes the edge does nothing and earns nothing (its gradient is zero, §5.2). An axon with A_pq = 0 is absent from the focus graph; a particle that is an endpoint of no present axon has no focus.

4.3 canonical input

The input to tru at a boundary is the list of present axons (p, q, A_pq), sorted ascending by p ‖ q bytewise, one entry per axon. Particles are indexed in ascending order of their 32-byte id. Nothing else enters: no author, no karma, no price (§3.7). This removes the order dependence of tru's current FocusingGraph::build (nodes indexed by first appearance, which moves λ_max and the step count) — the chain hands tru one canonical order, and tru's epoch entry point must index particles by ascending id.

4.4 φ*, syntropy and determinism

φ*(A)  = the tru tri-kernel fixed-point iterate of the canonical graph A
J(φ)   = syntropy, tru/specs/focusing.md (D_KL(φ ‖ u) over the graph's particles)

• φ* is tru's: the composite operator, its parameters (λ_d, λ_s, λ_h, teleport, τ, heat terms), the fixed-point scale and the normalization are normative in tru/specs and in tru's epoch entry point at the revision pinned in release/phase1.toml. A change of any bit of φ* for a given input is a consensus change and moves the conformance vectors (§10).
• iteration count: every solve starts from the uniform vector and runs exactly PHI_STEPS steps — a constant, never derived from the data. tru's T(ε) = ⌈log(1/ε)/log(1/κ)⌉ sets the constant once: the measured κ̂ ≈ 0.74 on bostrom (tru eval/spectral-gap-bostrom.md) gives T(10⁻⁶) = 46, so 48 is the recommended value; tru's EpochParams fixes it. No step count that depends on a power iteration, a convergence test or a warm start enters consensus: a cold start makes settlement of epoch E a function of G_{E−1} and G_E alone, so a node can check any epoch from its two snapshots.
• continuity (amended 2026-10-17): every particle's out-row carries a ground weight w0_p = ⌊in_p / 4⌋ that teleports to the stake prior, so the row normalises by out_p + w0_p (tru/specs/epoch.md §2). Without it a particle with focus and no out-axon sends all of it to the prior and its first out-axon, at any positive weight, redirects all of it — φ* jumps, and no gradient attributes a jump (measured: median error 34–45 % against exact Shapley on such epochs; with the ground and four nodes 0.65–3.0 %, tru audit/epoch-settlement-2026-10.md §6). A particle without in-weight is unchanged, one without out-weight is pure teleport as before.
• arithmetic: integer and fixed point only (tru/specs/arithmetic.md); the result must not depend on thread count; division and rounding rules are tru's and are pinned by the vectors. At 3 M particles a mean focus entry is 2⁻²¹·⁵, so the fixed point must carry at least 2⁻⁶⁰ resolution (tru's epoch engine scales by 2⁶⁰; the legacy 2⁻³² Fx keeps three significant digits there and is not adequate for gradients).

5. settlement

5.1 the solves

Settlement of epoch E reads G_{E−1} and G_E and computes (amended 2026-10-17: four path nodes instead of the one midpoint):

A_prev = A(G_{E−1}),  A_cur = A(G_E)                         // §4.2
A_i    = A_prev + floor(t_i · (A_cur − A_prev) / 2^32)       // per axon present in either, i = 1..ATTR_NODES
φ*_E   = φ*(A_cur)                                           // solve 1, published (§5.6)
g_i    = ∂J(φ*(A))/∂A at A = A_i                             // per node: one forward, one adjoint solve
ḡ      = Σ_i floor(w_i · g_i / 2^32)                          // tru `path_gradient`

(t_i, w_i) are the Gauss–Legendre nodes and weights on [0, 1] at 2^32 (tru/specs/epoch.md §4): t = 298207500, 1417379916, 2877587380, 3996759796; w = 747012592, 1400471056, 1400471056, 747012592 (the weights sum to exactly 2^32). A settlement is 1 + 2 · 4 = 9 solves.

g_pq is the derivative of syntropy at the fixed point with respect to the weight of axon (p, q), propagated through the fixed-point map by its adjoint (rewards.md §2: the envelope theorem makes the value of a new link the reshaping of the landscape, ⟨∇_A J, ΔA⟩, never the induced shift of φ). The adjoint iterates λ ← ∇_φ J + (∂F/∂φ)ᵀ λ from λ = 0 for PHI_STEPS steps and then reads g = λᵀ ∂F/∂A; tru/specs owns the transposed operators and the scale of g (a signed fixed-point integer per present axon of A_i; zero for absent axons).

ḡ is the four-node Gauss–Legendre quadrature of the integrated gradient along the straight path from A_prev to A_cur, i.e. the Aumann–Shapley value of rewards.md §4. The node count was chosen by measurement against exact Shapley (tru audit/epoch-settlement-2026-10.md §6): one node (the midpoint) stays near 6 % median error whenever a particle appears; four nodes hold every class of change under 5 % and weight changes ≤ 30 % at 0.04 %. One path gradient serves every position, which is decision 4.

5.2 attribution per position

For a card c on axon (p, q) with side σ(c) = +1 for v ∈ {0, +1} and −1 for v = −1:

Δs(c) = s_E(c) − s_{E−1}(c)        // stake on c's side; a card absent at a boundary has s = 0 there
a(c)  = Δs(c) · σ(c) · ḡ_pq                                  // signed, 256-bit integer (ḡ is 0 where the axon is absent at every node)

A card that changed side between the two boundaries (unlocked to zero and locked again with another valence) is a decrease on its old side and an increase on its new one. Attribution is linear in stake within a side, so splitting a stake across many cards or many neurons changes no total (Sybil-neutral, rewards.md §4), and positions of one side of one axon share its credit pro rata to what each added. Genesis weights never change, so their attribution is always zero.

The epoch's total change of syntropy J(φ*_E) − J(φ*_{E−1}) is recorded (§5.6) and gates nothing: the budget (§5.5) is fixed in tokens, the attribution only divides it, and an epoch with no positively attributed active stake pays nothing.

5.3 the new-high rule

A position earns only for stake that takes its axon side above its high-water mark. Per axon and side, with S^σ from §4.2 and H^σ the mark left by the previous settlement (0 for an axon never staked):

N^σ = max(0, S^σ_E − H^σ_{E−1})                     // new height reached this epoch
R^σ = Σ Δs(c) over cards on that side with Δs(c) > 0   // raw increase
x(c) = floor(max(0, a(c)) · N^σ / R^σ)              // R^σ > 0 whenever Δs(c) > 0
M^σ = max(H^σ_{E−1}, S^σ_E)
H^σ_E = S^σ_E + floor((M^σ − S^σ_E) · 65,410 / 65,536)

Above the mark the side earns in full; a side that fell and climbs back earns nothing for the climb until the mark has decayed (half-life 360 epochs, 30 days). Without the rule a holder could unlock, relock and be paid again for restoring the same edge, from a second key if needed; with it, recycling stake on one edge earns at most the decay. The price is borne by an honest staker who restores an edge someone else abandoned within the month; it is named in §9. Marks are kept per side only for axons that ever held a card; an axon with S⁺ = S⁻ = 0 and marks below 1 drops them.

5.4 gates

A card c earns in epoch E iff all hold:

• active valence: v ≠ 0 (passive stake moves rank and earns nothing, rewards.md §9);
• increase: Δs(c) > 0, the position added stake during the epoch;
• conviction: guaranteed by MIN_LOCK ≥ L + D (§3.4): stake added in epoch E cannot leave before settle(E); a position that added stake is paid on its stake at G_E;
• surprise: the weight is ρ(c) · x(c) with ρ(c) = RHO_LAUNCH = 1 for every active position. There are no reports on chain at genesis: BTS (rewards.md §5) needs each contributor's meta-prediction of the crowd, and the valence alone is a side, not a prediction. ρ = 1 is safe at launch because copying is already priced by the gradient: a copy on the same axon in the same epoch splits that axon's credit pro rata, and a copy in a later epoch meets a smaller gradient on a saturating edge. The reports rule set (§7.4) replaces it.

There is no per-neuron cap. A cap per identity is free to evade by splitting (one key per position costs nothing) and contradicts privacy- frozen.md §7, where rewards attach to links; the cap is per position and proportional to the stake it adds (§5.5), which splitting cannot move.

5.5 budget, caps and rounding

At settle(E), with F the focus pool balance after this block's coinbase (§5.7):

B      = floor(F / POOL_RELEASE)                             // the epoch's budget
X      = Σ x(c) over earning cards                           // u256
cap(c) = floor(Δs(c) · R_MAX_PPM / 1,000,000)
pay(c) = min(cap(c), floor(B · x(c) / X))                    // u256 intermediate, result u64
P      = Σ pay(c)                                            // P ≤ B ≤ F
F'     = F − P

• If X = 0 nothing is paid and the whole pool carries to the next epoch.
• Rounding is floor for every recipient; the dust B − Σ floor(B·x/X) and everything a cap withholds stay in the pool. No recipient is favoured by position, id or tie, and conservation is exact by construction.
• Releasing a quarter of the pool smooths payouts: a quiet epoch carries its accrual forward, and one epoch can never take more than a quarter of what accumulated. In a steady state the pool holds four epochs of accrual and pays one per epoch.
• The cap bounds what one epoch can pay one position to 1 % of the stake it added: a single position that is the only earner of a quiet epoch takes its 1 %, not the pool. In the expected regime the cap does not bind (pussy at θ = 1 %: about 2.5 × 10¹¹ PUSSY per epoch over an expected 10¹⁴ of new active stake, 0.25 %).

5.6 payout and the epoch record

Each pay(c) > 0 credits balances[hemera(owner(c) ‖ τ_native)], with owner(c) the card's payout owner in G_E, and debits the pool. Payouts are liquid at once (§2.5).

The settlement block writes one value into the time dimension (empty in rule set 1 until now): time[settle(E)] = hemera(D_EPOCH ‖ record), with

record = epoch: u64 ‖ boundary: u64 ‖ beacon: [u8;32] ‖ boundary_root: [u8;32]
       ‖ particles: u32 ‖ axons: u32 ‖ phi: [u8;32] ‖ syntropy: i64 ‖ delta_syntropy: i64
       ‖ focus_bps: u16 ‖ budget: u64 ‖ paid: u64 ‖ pool_after: u64 ‖ rewarded: u32 ‖ payouts: [u8;32]
                                                                                        // 198 bytes
phi     = hemera(D_PHI ‖ n: u32 ‖ φ*_E[0]: u64 ‖ … ‖ φ*_E[n−1]: u64)   // particles ascending, tru's raw fixed point
payouts = Merkle root over leaves hemera(D_PAYOUT_LEAF ‖ card ‖ owner ‖ amount: u64),
          cards ascending, amount > 0 only, nodes hemera(D_PAYOUT_NODE ‖ L ‖ R),
          odd node promoted unchanged (chain.md §1.7); 32 zero bytes when none

syntropy and delta_syntropy are J(φ*_E) and J(φ*E) − J(φ*{E−1}) in tru's raw fixed point, signed; focus_bps is the split in force for the blocks of epoch E + 1 (§6.1). The record, the payout list and φ*_E are kept by every node and served over HTTP (§11). Through state_root every header from settle(E) on commits to φ*_E: two nodes that disagree on one focus bit disagree on the root.

5.7 order inside a block

For the block at height h, after the checks of chain.md §5.5 steps 1–6:

1. coinbase: credit the miner SUBSIDY − cut(h) + fees; credit the focus pool cut(h) (§6.2); coins[τ_native].total_supply += SUBSIDY;
2. if h = settle(E) for some E ≥ 1: the settlement of epoch E — payouts, pool debit, high-water marks, time[h];
3. the transactions in body order (chain.md §3.4);
4. conservation (§6.3); state_root.

A transaction in the settlement block sees the payouts (a payout owner may spend a reward in the block that pays it).

6. emission

6.1 the split

The subsidy (chain.md §3.5) stays SUBSIDY = floor(S0 / 105,120,000) per block, 1 % of genesis supply a year, constant. It is base emission in the sense of rewards.md §10 — paid only "to the two providers that do work: PoW compute and active (v ≠ 0) epistemic risk" — split between them by the allocation curve θ^α of rewards.md §10 (tru allocation::split) with the neutral α = 1/2, evaluated in integers:

active(E)    = Σ s(c) over cards with v ≠ 0 in G_E
supply(E)    = S0 + boundary(E) · SUBSIDY
focus_bps(E) = min(FOCUS_MAX_BPS, isqrt(floor(active(E) · 10^8 / supply(E))))   // u128 inside
cut(h)       = floor(SUBSIDY · focus_bps(epoch(h) − 1) / 10,000)                  // h ≥ 1

isqrt is the floor integer square root; 10^8 makes isqrt return θ^½ in basis points. Blocks of epoch 1 use focus_bps(0) = 0 (no stake at genesis). Examples: θ = 1 % → 999 bps; θ = 25 % → 5,000 bps, the cap.

Why the curve and why the cap:

• the curve is the one rewards.md derives for splitting a security budget between work and risk, already implemented and simulated (launch row 27); it pays knowledge in proportion to how much capital actually took risk on the graph, so an unused economy does not drain the miners;
• per unit of active stake the yield θ^½ · SUBSIDY / (θ · supply) falls as θ grows: the first stakers are paid most, which is the bootstrap the economy needs;
• the cap is the miner security floor: at most half the subsidy leaves proof of work, so the cost of a rewrite never falls below half of what a pure PoW chain of this subsidy buys. rewards.md derives the floor from attack economics (floor ≥ c_sec · TVL/M · r_atk); none of those inputs exists before launch, so the floor is a constant at genesis and its PID loop (launch row 27) is a later rule set.

6.2 coinbase and pool

The coinbase amount rule of chain.md §2.2 becomes

coinbase.amount == SUBSIDY − cut(h) + Σ fees of the block's other txs

and the focus pool receives cut(h) as part of the coinbase step, no transaction. cut(h) depends only on G_{epoch(h)−1}, so once a boundary is final the pool's inflow over the next epoch is a function of height, and the pool balance at settle(E) is the same on every branch above a final boundary (§2.5).

6.3 conservation

The invariant of chain.md §3.4 keeps its form:

Σ balances[·, τ_native] == S0 + h · SUBSIDY

where the sum runs over every key holding native coin: owner balances, dormant legacy owners, stakes stake_key(c), the focus pool focus_key(τ_native), and (rule set 2) the turnstile T(τ_native). The subsidy is minted once per block and split between the miner and the pool; stake moves between an owner and its stake key; settlement moves from the pool to payout owners; fees move from senders to the miner. Nothing burns, supply is a function of height alone. Added per-settlement checks, exposed in /status: P ≤ B ≤ F, the pool never negative, and Σ stake keys equal to the stake total of the cards.

6.4 fees

Fees are unchanged: every fee goes to the miner of the block that includes the transaction, nothing is burned, no fee is routed to the pool. The fee role of rewards.md §8 (burn β, servicer γ, security budget) needs a fee market and paid inference, both cut from phase 1.

7. mining, honestly

7.1 at launch

"Mining" on the root chain at genesis means two things and only two:

• proof-of-work block production: hemera header grinding under ASERT (chain.md §1), paid SUBSIDY − cut + fees. The work is synthetic, the bootstrap puzzle of rewards.md §8 that hands out the initial stake;
• settlement by recomputation: every full node computes every epoch's solves (§5.1) and checks the settlement block bit for bit. Nobody is paid for this computation; it is the cost of validating, as checking signatures is. The miner of the settlement block chooses nothing in it.

Settlement mining — the lottery in which each hash attempt is a Shapley sample and the winning ticket earns the subsidy (rewards.md §7) — is not active at genesis. What is active is its product: the fair division of Δφ⁺ among staked links, computed exactly (to the integrated-gradient approximation) instead of estimated by sampling.

7.2 cost bound

Measured 2026-10-17 on an M4 Max with tru's epoch engine (tru/specs/epoch.md; receipt tru audit/epoch-settlement-2026-10.md §2, §6; amended 2026-10-17 — the figures of the legacy 2⁻³² engine this section first carried, 48 s and 2.8 GB for one bostrom solve, are superseded). Bostrom: 2.85 M axons over 3.06 M particles (synthetic stake on the burial graph).

Results are bit-identical across thread counts. Against the one-hour window of D blocks a single core has a 48× margin, and SETTLE_BUDGET = 900 s (a quarter of the window) holds up to about twelve times bostrom's graph on one core at linear scaling.

Worst case by construction: at most MAX_BLOCK_TXS · L = 245,760 positions change per epoch, so the graph grows by at most that many axons per epoch (≈ 3 M per day if every block were full of fee-paying staked links — a spam bill of 257 units per link). Settlement cost is linear in present axons and independent of how many changed. The threshold that matters is SETTLE_BUDGET, not one block interval: settlement is pipelined over D blocks, and a single solve already exceeds one 30-second interval at bostrom's size today.

7.3 the ticket rule set (the scaling rule)

When the reference settlement exceeds SETTLE_BUDGET, or is projected to within 90 days at the observed growth, settlement moves to the lottery of rewards.md §7 by scheduled fork (chain.md §7.1, two weeks' notice). That rule set is specified here as far as the root chain fixes it:

• the beacon is beacon(E) of §2.3; the cluster set is the canonical partition of the epoch's changed positions into connected components of overlapping ε-supports (launch row 35);
• a ticket (kind 8) carries (E, cluster, n, miner, m(n), proof); the ordering is π(n) = hemera(D_ORDER ‖ beacon(E) ‖ cluster ‖ n), the win test hemera(D_TICKET ‖ beacon(E) ‖ cluster ‖ n ‖ miner ‖ commit(m(n))) < target(cluster) with the target banded by cluster size (row 4); the tags D_ORDER and D_TICKET are fixed by that rule set;
• tickets are accepted in blocks boundary(E) + 1 … settle(E) − 1; the settlement at settle(E) is the mean of accepted samples per cluster under the same budget, gates, new-high rule, caps and rounding of §5; clusters with fewer than k_min samples defer to the next epoch;
• the PoW share of the subsidy is re-targeted from the header hash to the ticket (rewards.md §8: "the subsidy and the settlement lottery are one act"); the header keeps a hash target for ordering;
• every ticket carries a sound per-ticket proof (foculus#61) and tickets fold per cluster into one accumulator (launch rows 2, 5); full nodes stop recomputing.

Preconditions, each a launch-registry row: ticket proof cost below ticket reward (3), progress-freedom across clusters (4), canonical clusters (35), the hardware profile (36), an O(1) fold (5). None is required at genesis.

7.4 the reports rule set

BTS needs, for every active position, a meta-prediction: the share of active stake the author expects on its side of the axon. A later rule set adds kind 10 report: from ‖ nonce ‖ p ‖ q ‖ prediction: u16 (bps) ‖ fee, computes ρ(c) per epoch by tru's hardened scorer (crowd reference over staked, distinct predictors, bts_scores_hardened), replaces RHO_LAUNCH, starts persisting karma per position or payout key (§3.7), and adds the zero-sum redistribution that is the slashing of rewards.md §5. The serum's properties (launch row 24) and the market (row 25) are its gates.

8. genesis stake

The graph comes home with 2,949,732 bostrom links (2,997,569 genesis frames with passports) and 29,112 pussy links, and with no stake: the old chain's weights were its resource tokens, and chain.md §10 item 3 made every delegated balance liquid at genesis. φ* over a weightless graph is empty, so the burial needs weight from somewhere. Options:

Recommendation: A. It is the only option that is a pure function of the genesis file (no oracle, no new frame, no new state), keeps balances liquid, and puts the burial graph in φ* on day one in the same unit live stake uses. Its fairness cost — rank follows capital at halt — is the rank axis working as rewards.md §9 defines it, and it is bounded: genesis weight earns nothing (only change earns, and genesis weight never changes), so A hands out rank, never income; a holder who sold at genesis keeps rank but no reward stream; and live stake displaces it unit for unit, including v = −1 stake that zeroes a genesis edge (§4.2). Attack surface: none at launch — nobody can add genesis weight after block 0, and the weights are public in the genesis report. The 2026-09-18 decision that "the migrated graph would pay nothing" without the resource tokens is met in this form: the migrated graph ranks from block 0 and pays its authors as soon as they stake on it (their links exist; a lock on an existing link is a new high on an empty side).

9. attacks

10. conformance vectors to add

To chain-conformance.md, new section 14 ("the focus economy"). Values below were computed on 2026-10-17 with cyber-hemera 0.4 at 7237a23 (the phase1.toml pin) through hemera::hash; integer vectors with Python integers. Others are blessed.

Identities:

card(0x66 × 32, token("boot"), token("pussy")) = 74f0a211d177f290e9c95a6e7edb65cd8705ea96d9713a841d5a086f6476b5b4
stake_key(that card)                             = f7e0cbf8e5cfc501b7e3098b238bce8451b3b0a7fa2e42b69724801115c6b08a
focus_key(token("pussy"))                        = 0fae8671376d879515b2736c2d6d51eafdd9ff40055aa84aaff6e17a0beed4b9
payout_leaf(that card, 0x77 × 32, 1,000)         = 63c3867a76af023aa6417e5a945612289906e8352ea69fe4aea259ce5b152115
payout_node(leaf, leaf)                          = 240a852c1bd8f641f65d6b3746a72a08c0930e5d1bd1feafe4b9cdab914a6ec2
phi digest, n = 2, φ = [2^59, 2^59]              = 3351bfa40d6eaa24c8e2b8755d900dbcd3c3015730a5efcfdf9b6e14c7019a8a
phi digest, tru's conformance graph (below)      = e0ff79997aa697bced38e2cdedb823b189892982bae411933ec623007de18fc8

tru's conformance graph (amended 2026-10-17, tru rs/tests/epoch_settlement.rs vector_graph): particles [b; 32] for b = 1..5, axons (1,2) 1,000,000, (2,3) 2,500,000, (3,1) 700,000, (1,4) 4,000,000, (4,2) 1,200,000, (5,3) 3,000,000, (2,5) 900,000; EpochParams::default() (PHI_STEPS 48, GROUND 1/4): φ* = [219310425963150397, 244918065555344491, 314071252939911789, 227905484039409407, 146716275947897466], J = 31829771616584911. The per-axon gradient and the 4-node path gradient of a fixed change of it are pinned in the same test file.

Integers (pussy S0, SUBSIDY):

Transactions (blessed, signed with the known-answer key of §6): kind 2 with amount 7 and native token creates the card and moves 7 to its stake key (the old TxError::Box(7) vector is retired); kind 2 with amount 7 and a non-native token rejected; kind 2 with the opposite valence on a live card rejected; kind 6 at since + MIN_LOCK − 1 rejected, at since + MIN_LOCK accepted, partial then full (card deleted); kind 6 by a non-author rejected; kind 7 re-points the owner; encodings of kinds 6 and 7 at 224 and 248 bytes, trailing and missing bytes rejected, kinds 8–10 rejected.

Epochs (blessed, private test genesis, trivial target, L and D overridable only under cfg(test) like the PoW limit):

• the genesis weights of the conformance genesis and of pussy's real genesis file: Σ G, the count at the floor, the digest of (p, q, G_pq) sorted;
• φ*_0 of pussy's genesis: phi digest and syntropy, single-threaded and with 8 threads (equal);
• a 2-epoch chain: one +1 lock, one 0 lock, one −1 lock on three axons in epoch 1; the epoch record of epoch 1 at settle(1), the payouts (the passive position earns 0), the pool after, and the state root;
• new-high: lock 100, unlock after MIN_LOCK, relock 100 one epoch later: the relock's x is the decayed remainder only;
• reorg: a 3-block reorg ending at settle(1) reproduces the identical epoch record and root; a reorg replacing boundary(1) changes the record;
• conservation at every height of a 2,000-block simulated chain with random locks, unlocks and payout changes, including the settlement blocks.

Determinism gate: two nodes on two machines (deimos, jupiter) agree on every epoch record of pussy-rc for 24 h; cyberproxy (8 GB) completes every bostrom settlement within SETTLE_BUDGET.

11. node surface

HTTP (chain-wire §7), new routes:

/status gains epoch, next_boundary, last_settled, focus_bps, pool, active_stake, total_stake, and the per-settlement checks of §6.3. The node keeps φ* of the last two boundaries and the epoch records in $home/<chain>/epochs/.

12. open questions for the owner

Only choices that are policy rather than engineering; each has the default this page implements.

Decided 2026-10-17 (coordinator, owner-delegated): the dangling switch is fixed in φ* — the teleporting ground weight GROUND (§4.4) — rather than accepted as an under-payment of first out-links; the attribution runs ATTR_NODES = 4 Gauss–Legendre nodes (§5.1). Ground and node count were chosen by measurement (tru audit/epoch-settlement-2026-10.md §6).

13. implementation map

See chain.md for the chain, chain-wire for the network, chain-conformance for the vectors, tru/specs/rewards.md for the reward law this page instantiates, and cyber/launch for the tracker.