the annuity — per-epoch accrual of the yield stream

property #28 in cyber/launch.md (the phase-1 tracker). date 2026-09-21, revision 39e0caf (base master), rs/annuity.rs this branch.

what this closes

specs/rewards.md §11–12 define the annuity as a continuous integral, R_{i→j}(T) = ∫₀ᵀ ω(t)·Δφ*_j(t) dt, and describe it as self-correcting: a link later falsified stops drawing it, with no audit reaching backward (§12). Neither had code. rs/annuity.rs discretizes the integral as a per-epoch Riemann sum (§12: "the present re-scores itself" every epoch, so a discrete sum over epoch samples is the right shape, not a closed-form integral): accrue returns the total through epoch T, accrue_running exposes the running total after each epoch for a caller that pays out per epoch rather than as one lump sum.

ω(t) (current ICBS price × karma) and Δφ*_j(t) are computed upstream by the market and focusing passes; this module takes them as an EpochSample and does not compute them, matching how allocation.rs (#27) leaves c_sec/r_atk as inputs rather than re-deriving them.

verified properties

• an empty history accrues nothing.
• a single epoch's contribution is exactly ω·Δφ* for growth (the integrand). the integrand is read as the directed impulse of §2, ω·[Δφ*]₊: an epoch in which the target's focus fell draws zero, never a negative amount — a paid epoch is final (§12), so a claw-back through a negative term would be the backward re-grading the spec forbids. (review 2026-09-22: the first cut multiplied the raw signed Δφ*, under which the "never drops" claim below was false for negative growth.)
• accrual sums correctly across epochs and is commutative for the final total, but the intermediate running totals are not — a payer must feed samples in epoch order, matching "the present re-scores itself" rather than a global re-sort.
• a falsified link (ω → 0) stops drawing from that epoch on, and the running total never drops — through zero-growth and negative-growth epochs alike; past accrual is never reversed, per §12.

remains

this is the accrual primitive only. "per-epoch re-scoring on real graph" (the registry's stated evidence bar) needs ω(t) and Δφ*_j(t) wired to the actual ICBS price and focusing output over pussy-rc or bostrom genesis data — unclaimed. the discovery leak (§12, §5) is unrelated and stays an accepted open frontier for phase 1 per the registry.