Owned Math M8 — The freshness index: claim re-verification is Age-of-Information scheduling

TL;DR

The M5 stretch goal ("Whittle-index form of claim re-verification") resolves by isomorphism: each ledger claim is an arm of the Age-of-Information restless bandit with a verification price — state = paid age τ (steps since evidence was last attributed to the claim; M7's vocabulary), passive cost c_e·s(τ) with staleness s(τ) = 1−(1−ρ)^τ (M2's flip rate), audit = pay c_v, reset. Indexability for nondecreasing age costs is settled in the AoI literature (Tripathi & Modiano) — per the ADOPT rule we import it and derive our variant's closed form: W(τ) = c_e·[τ·s(τ) − S(τ−1)] − c_v, verified against exact bisection to 7.8e-13 and against exact policy iteration. What the isomorphism unlocks: (1) M2's EOQ cadence is the index's zero-crossing — and the shipped 30-minute tick, read at the ledger's measured ρ̂, implies verification is priced at 10.8% of an error-step (de-burst: 1.6%): the magic constant becomes a falsifiable economic claim; (2) a binding audit budget enters as a uniform surcharge on the verification price (crossing(λ) ≈ √(2(c_v+λ)/(c_e ρ))) — scarcity delays everyone, it never starves high-risk claims; (3) the index is measurable w.r.t. paid age alone — computed on expressed freshness it reproduces M7's audit starvation, and on paid age it wins the policy contest under a binding budget (1.784 < EOQ-overdue 1.793 < round-robin 2.020 < expressed-confidence gate 4.470). M3 said when to ground is not optional; M7 said whose claims must be keyed by paid evidence; M8 gives the priority order in closed form — one law, all three.

1. The isomorphism

The mapping is exact, not analogical: dynamics, costs and constraint all carry over. The c_v term is what the AoI form lacks and the inspection/maintenance tradition supplies — and it is precisely the term that makes the index meet M2's EOQ economics (below).

2. The index

For the λ-subsidy single-arm problem, every stationary deterministic policy's recurrent class is the cycle 1…A for A = its smallest audit age — so average cost is g_λ(A) = [c_v + c_e·S(A−1) − λ(A−1)]/A and threshold optimality is immediate (one line, no VI needed). Indifference between adjacent thresholds at state τ gives

W(τ) = c_e·[τ·s(τ) − S(τ−1)] − c_v ,          S(k) = Σ_{u=1..k} s(u)

     = c_e·[1 − τβ^τ + (β−β^τ)/(1−β)] − c_v    (geometric, β = 1−ρ)

strictly increasing in τ (increments c_e(τ+1)[s(τ+1)−s(τ)] > 0), bounded above by c_e/ρ − c_v (the budget shadow price's feasibility ceiling). Monotone index ⇒ the passive set grows monotonically in λ ⇒ indexable, and W is the Whittle index. Under a budget of M audits per step: audit the M claims of highest positive index — the Whittle policy (asymptotically optimal in the Weber–Weiss regime; adopted, not re-proven).

Machine checks (artifact, all exact): indifference-λ recovered by bisection on the exact renewal scan matches W(τ) to 7.8e-13 across a (ρ, c_e, c_v) × τ grid; passive-set monotonicity witnessed over 200-point λ sweeps; exact policy iteration over all stationary policies recovers the same thresholds. (Method note kept honestly: relative value iteration is the wrong tool here — the optimal chain is a periodic deterministic cycle and synchronous relative VI oscillates, which produced an off-by-one in the first run; PI with closed-form cycle evaluation is finite and exact.)

3. Corollary— M2's cadence is the index's zero-crossing, and the tick has implied economics

Small-ρ expansion: W(τ) ≈ c_e·ρτ²/2 − c_v, so the λ=0 crossing is **T\* = √(2(c_v/c_e)/ρ)** — exactly M2's EOQ law, now derived from optimality rather than a renewal approximation. Measured convergence: relative error 0.41 → 0.007 as ρ falls 0.1 → 0.0003. Under a binding budget the shadow price λ shifts the crossing to √(2((c_v+λ)/c_e)/ρ) (measured rel. err ≤ 2.1%): budget scarcity is a uniform surcharge on the verification price — it delays every claim by the same economics; it never selectively starves the high-ρ claims (contrast M7's confidence gate, which starves exactly them).

Read at the calibration ledger's measured staleness (M2): with raw ρ̂ (48 h half-life) the crossing at c_v/c_e = 0.1 is 29 minutes — the shipped 30-minute GROUNDING_TICK is the index's zero-crossing at an implied verification price of 10.8% of an error-step (de-burst ρ̂,h: implied 1.6%). The magic constant is now an economic statement the M2 spaced-probe instrumentation (#2787) can falsify per topic.

4. Corollary— attribution is inside the index (M7 ⊂ M8)

The arm's state is paid age, full stop: the transition and cost kernels do not read expressed confidence, so the index is measurable w.r.t. the paid-evidence history alone — free confidence has index weight zero. Computing the "same" index on expressed freshness re-creates audit starvation: a laundered claim (expressed age ≈forever) gets W = −c_v <and is never scheduled. Policy contest under a binding budget (8 arms, two of them laundered, contention 1.8×, exact expected costs, no RNG):

The confidence gate pays 2.5× — the starvation cost, now inside a scheduling benchmark. Honest reading of the thin Whittle-vs-EOQ margin (0.5% at this contention): EOQ-overdue is a strong heuristic; the index's edge is priority under contention and heterogeneity, and if per-topic parameters never materialize (#2787), "EOQ-overdue is near-optimal" is the recorded fallback verdict (pre-stated kill criterion in #2926).

5. One law (the arc completed)

• M3: re-grounding must happen on a hard external cadence — when is not optional.
• M7: the audit must land on claims keyed by paid evidence — whose is not optional.
• M8: given both, the order is argmax W(τ_paid) with the budget as a price

surcharge — and the shipped tick is this law's zero-crossing at measured parameters.

Shipped: whittleFreshnessIndex() in grounding-policy.js — pure, exported,unit tests, unwired (the hard tick stays the floor; the index is a priority on top, never a replacement — M3 forbids the replacement reading). Wiring behind GROUNDING_PRIORITY=whittle is specified in #2926 and blocked, correctly, on the same instrumentation as M2.

6. Prior art, named — and what is ours

Adopted: Whittle (1988); Tripathi & Modiano (arXiv:1908.10438) — indexability + closed-form indices for nondecreasing AoI costs (our reliable-channel, priced variant sits in the maintenance/inspection-index tradition, e.g. Glazebrook et al.); Weber & Weiss asymptotic optimality; Rothschild (1974) incomplete learning (the M7 lineage). Ours: the claim-verification instantiation with the verification-price term; the EOQ-crossing identity tying the index to shipped M2 economics (including the tick's implied c_v/c_e from measured ρ̂); the budget-as-price-surcharge reading; the attribution corollary (index weight zero for free confidence) closing the M7 loop; the exact artifact and the tested product function.

7. Honest scope & kill criteria

• Geometric staleness (constant flip hazard) is the model; per-topic ρ is instrumentation-

blocked (#2787) — same limit as M2, inherited knowingly. Non-monotone "self-healing" claims break threshold structure and are out of scope (the index requires monotone deterioration).

• Whittle optimality under a hard per-step budget is asymptotic (Weber–Weiss), not exact

for N = 8; the contest is evidence, not proof, and the EOQ-overdue baseline is reported at full strength.

• Audit = perfect repair here (audit reveals and fixes). Imperfect audits compose with

M7's ρ_catch analysis; not re-derived.

• Kill criteria: closed form dies on any indifference-gap cell above tolerance; the

refinement claim dies if Whittle stops beating EOQ-overdue under binding budgets as heterogeneity grows; the economics reading dies if per-topic ρ̂ (#2787) moves the implied c_v/c_e outside a defensible range.