epoch settlement — measured 2026-10-17

Receipt for tru feat/epoch-settlement (spec: specs/epoch.md; chain contract: soft3 specs/chain-economy.md §4–§5, rewards.md §16). Machine: Apple M4 Max (12 P + 4 E cores, 48 GB), macOS 26.4, rustc 1.95 release. Data: the burial snapshots' cyberlinks_indexed.csv.gz converted by eval/links_bin.py — pussy 29,112 links / 16,271 particles, bostrom 2,949,732 links / 3,143,650 particles. Synthetic stake per link (no stake exists on chain yet): uniform 10⁶, or Pareto(α = 1.16, scale 10⁶, cap 10¹⁵). Particle ids are hashed (random order, as real hemera ids), except where noted.

Harnesses: rs/examples/epoch_bench.rs (modes legacy, epoch, settle, phi-only, gradient-only), rs/examples/epoch_shapley.rs, and the ignored tests bostrom_phi_within_budget / bostrom_settlement_within_budget in rs/tests/epoch_settlement.rs. Peak memory is /usr/bin/time -l maximum RSS (whole process, harness inputs included) and, for the API calls alone, a counting allocator's heap peak above the live baseline.

1. before — the legacy engine (FocusingGraph::build + compute_focusing)

Every link at its stake, Context::none(), default FocusingParams.

Findings:

• the combinatorial heat kernel's cost is O(τ·λ_max) and λ_max is the largest weighted degree: at uniform stake one bostrom solve is a day and a half of one core. Its cost depends on the stake distribution, so it cannot carry a consensus deadline.
• the λ₂ estimate (120 power iterations) is wrong on these graphs: bostrom uniform reports λ₂ = 37,253 > λ_max = 36,026, and pussy reports 1.96 for a disconnected graph whose λ₂ is 0. κ and T(ε) inherit the error. T is also order-dependent (nodes indexed by first appearance).
• 2⁻³² resolution: bostrom's φ sums to 0.999966 (n rounding errors), and a mean entry of 2⁻²¹·⁵ keeps about three significant digits.
• the soft3 spec's own baseline (chain-economy §7.2, genesis weights): one bostrom solve 48 s, 2.8 GB.
• deterministic across two runs (bit-identical) in every case.

2. after — the epoch engine (tru::focusing::epoch)

Positions: one per link, valence +1, every 13th passive (0), every 31st against (−1, subtracting); net_axons → 2,848,062 axons over 3,061,768 particles on bostrom (25,878 / 15,965 on pussy). PHI_STEPS = 48, cold from uniform.

2.1 φ*

Every run: bit-identical across two runs, across 1/4/12 threads, and with the input list reversed (and, in the tests, rotated and with an axon split in two). The blessed conformance vectors computed in a release build pass in the debug build. No zero entry; bostrom's smallest φ is 1.7 × 10⁹ units of 2⁻⁶⁰ (1.5 × 10⁻⁹).

Convergence from uniform, bostrom power-law, ℓ1 to the 192-step reference:

Observed rate ≈ 0.75 per step against the a-priori κ = 0.775; pussy and uniform stake match within a factor 1.3. PHI_STEPS = 48 is pinned on this: below 10⁻⁷ in ℓ1 on both graphs and stakes, two orders under the ε = 10⁻⁶ the heat degree is set to. The rounding floor (reference last-step drift) is 4 × 10⁻¹¹.

Memory (bostrom, power-law): phi(axons) heap peak 645 MB above its input (the axon list itself is 217 MB; the returned φ + ids 116 MB); gradient 960 MB. Whole-process max RSS of the phi-only harness 1.49 GB, gradient-only 1.89 GB, both including the harness's 2.9 M positions and their net-weight pass. The graph's arrays are ≈ 406 MB.

Locality (bostrom, per step, one thread): particles in hashed-id order 450 ms; in first-appearance (chain-height) order 228 ms; breadth-first relabelling (shipped) 131 ms. The relabelling changes no bit (each output entry is an exact integer function of its row).

2.2 settlement (chain-economy §5.1: solve 1 on A_cur, solves 2 + 3 on A_mid)

Previous boundary: all but the last 10,000 positions; current: all, every 97th earlier position doubled — 40,307 changed positions.

The ignored test bostrom_settlement_within_budget measured 26.1 s for one thread. Against SETTLE_BUDGET = 900 s that is a 26–35× margin on one core and ≈ 130× on twelve; pussy settles in 0.11 s. (The single-thread wall times carry ±25 % noise: a legacy benchmark held one core throughout.)

2.3 before / after

3. attribution against exact Shapley

epoch_shapley: random graphs of 8–15 particles and 16–60 axons (weights 10⁶–10⁷); five players — axons whose weight changes between the snapshots; v(S) = J∪(φ*(A_prev + Δ_S)) − J∪(φ*(A_prev)) over the union particle set, φ* at 256 steps, exact Shapley over all 2⁵ subsets. Error = Σ|est − Shapley| / Σ|Shapley| per case.

Weight changes on existing axons only (120 cases each):

All kinds of change (new axons, raised, lowered, removed; 200 cases each), split by structure:

Reading:

• on smooth changes the gradient form converges to Shapley as the change shrinks (third order for the midpoint); GL-4 is 3–4× tighter than the midpoint but costs four gradient solves instead of one.
• new particles are continuous (a new particle's focus grows from zero) and stay within single-digit percent.
• the dangling switch is a jump: a particle with focus but no out-axon sends its mass to the teleport prior; its first out-axon, at any positive weight, redirects all of it. φ* is discontinuous there, so no gradient form — midpoint or GL-4 — sees the jump, and exact Shapley pays it to whoever made the switch. The budget is divided by attribution, so nothing is minted unattributed; first out-links are under-paid. Closing it needs a continuous dangling rule (for example, every particle keeps a teleport share c_p/(out_p + c_p) with c_p > 0) — a change of φ*, so a spec decision (rewards.md §16, chain-economy §12).

Completeness on the path (integrated_gradient_is_complete): GL-4 sums to J∪(cur) − J∪(prev) within 3 × 10⁻⁸ relative; the midpoint within 0.35 % of the gross attribution Σ|a|.

The gradient against central finite differences (gradient_matches_finite_differences, 4 graphs, ≈ 50 axons): within 0.2 %.

4. what the chain gets

use tru::focusing::epoch::{Axon, EpochParams, Phi, Gradient, phi, gradient, midpoint, net_axons, PHI_STEPS};

let p = EpochParams { threads: 8, ..EpochParams::default() };   // threads never change a bit
let now: Phi = phi(&a_cur, &p);              // solve 1: particles ascending, φ at 2^60, J at 2^60
let digest_input = now.preimage();           // D_PHI ‖ n ‖ φ…, LE — chain hashes with hemera 0.4
let mid = midpoint(&a_prev, &a_cur);         // ⌊(A_prev + A_cur)/2⌋, union, ascending
let g: Gradient = gradient(&mid, &p);        // solves 2 + 3: g.by_axon[k] for mid[k], nats/unit at 2^112
// a(c) = Δs(c)·σ(c)·g.by_axon[k]  (256-bit; integrated::score_of → nats at 2^60)

J is i128 at 2⁶⁰; the epoch record's syntropy: i64 cannot hold it at that scale (J ≤ ln n ≈ 15 nats needs 64 bits plus sign) — the record should carry i128, or J >> 8 (2⁵², ±2,048 nats) as an i64.

5. open

• the dangling switch (§3) — spec decision.
• the legacy uniform-stake solve was not run to completion (projected from the measured sub-application count and per-pass cost).
• cross-architecture determinism is checked by CI (ubuntu x86-64 runs the blessed vectors); the deimos/jupiter 24 h gate is the chain's.

6. amendment 2026-10-17 — the ground weight and the attribution nodes

Coordinator order: fix the dangling switch in φ*, then choose the ground and the quadrature by measurement. Acceptance: median attribution error against exact Shapley ≤ 5 % in every class of change including first/last out-axon, ≤ 1 % for weight changes ≤ 30 %; settlement on bostrom ≤ 300 s on one thread; PHI_STEPS still reaching ℓ1 ≤ 10⁻⁷; the determinism suite unchanged.

Rule: every out-row carries a teleporting ground weight w0_p = ⌊(γ_d·d_p + γ_i·in_p)/2^32⌋, normalised by N_p = out_p + w0_p (spec epoch.md §2, §4; the adjoint and the gradient carry the matching terms, checked against central finite differences within 0.2 % at γ_d = 1/8, γ_i = 1/4). With γ = 0 the engine reproduces §3's numbers exactly.

6.1 the sweep (epoch_shapley, 5 players, 400 cases per scale; error = Σ|est − Shapley| / Σ|Shapley|)

At ≈ 30 % changes, 200 cases, median: ground on degree (γ_d) vs in-strength (γ_i):

γ_i (in-strength) does as well as γ_d at the same γ and moves less focus (a particle without in-weight is untouched; at in = out, γ_i teleports γ/(1+γ) against γ_d's 2γ/(1+2γ)). The midpoint stays at ≈ 6 % on new particles at every ground — a new particle's φ ln φ grows from zero — so one node cannot meet the acceptance.

γ_i at 400 cases, all classes, median (p90) in %:

(γ_i = 15 % and 20 % fall between; full logs reproduce with epoch_shapley <cases> 5 <scale> 0 <γ_i %>, TRU_MODIFY_ONLY=1 for the weights-only rows.)

Choice: γ_i = 1/4, four nodes. It is the widest margin measured in every row (worst median 3.03 %, the first/last out-axon at ≈ 100 % changes), a dyadic constant, and four nodes cost 75 s of one core on bostrom (§6.3), a quarter of the 300 s bar. Two nodes at 1/4 pass too, with less margin (3.7 %); γ_i = 1/10 passes only at four nodes, 0.3 points under the bar.

The test path_gradient_tracks_exact_shapley pins the protocol rule on constructed first/last-out-axon cases: median 0.30 % / 0.30 % (0.07 % on ±30 % weight changes), against 21.7 % for the same cases without the ground. focus_is_continuous_at_a_first_out_axon: an ε-weight first out-axon moves φ* by O(ε) with the ground, by more than 10⁻² without it.

6.2 what the ground does to φ*

ℓ1 between φ* with and without the ground, and the overlap of the 1,000 highest-focus particles:

Most of the shift is already there at 1/8: it is the cost of having a ground at all (in-heavy particles stop hoarding focus), and rises slowly with γ. J falls from 3.831 to 3.734 nats on bostrom (power-law).

6.3 bostrom, re-measured (EpochParams::default(): ground 1/4, 4 nodes)

Bit-identical across 1/4/12 threads and with the input reversed; the debug and release builds agree on the re-blessed vectors (φ digest e0ff7999…8fc8 under hemera 0.4).