Item 30: A learner with sublinear Φ_law-regret, and what it retires

The controlling question of the track.

Given the frozen finite comparator and replay environment, determine whether a specified online learner achieves sublinear Φ_law-regret, and whether that guarantee implies retirement of every positive-rate uniformly remediable failure pattern.

Both halves. The expected-loss consequence is one line given the first — total mixed-action mass Ω(T) on source labels with a fixed admitted repair saving δ > 0 forces R_T(φ) ≥ ρδT − B. A density claim about one sampled path needs an additional sampling argument. The content is entirely in the hypotheses, three of which the environment supplies and one of which is the bound.

Use the fixed labels and nine programs in projects/normativity/rounds/2026-08-11-phi-regret-bridge/. Implement Blum--Mansour Theorem 18's row-conditioned weights over eight source labels and nine programs, and its stationary distribution. Measure expected mixed-action charge first. Use the sqrt(8 log 9) dependence; do not use plain exponential weights over nine transformations or report a sqrt(log 9) bound. A sampled trajectory requires a separately stated sampling result. Report whether the implementation is horizon-tuned or supplies a proved anytime schedule. If it retains the workspace's exact-rational execution discipline, also state how the source's optimized real parameter and stationary distribution are represented without silently changing the bound.

Result: projects/normativity/rounds/2026-08-11-phi-regret-learner/ implements the Theorem 18 construction with 8 source rows and 9 program weights per row, the transformation-weighted matrix, a deterministic exact stationary selector for the represented numerical weights, and the source row update. The real exponential update uses controlled Decimal arithmetic; transition and stationarity calculations are exact for those finite decimal weights. Declared experiments show decreasing maximum regret per round on the adversarial impediment fixture, and the cited theorem plus the item-29 bridge supplies the expected mixed-loss guarantee. The recurrent-failure consequence therefore retires positive asymptotic expected mass on represented uniformly saving repairs.

The learning computation is not yet integrated into the bounded answerability architecture. Sampled canonical responses produce faithful, non-erasing, response-service-feasible records, but ServiceCosts has no coordinate for the 72 weight updates or stationary solve, and the learner state is not recorded as an answerable artifact. Item 30 remains open only on that explicit integration interface. Also open: sampled-path and anytime guarantees, exact-real executable identity, and comparator coverage beyond the weak nine-program class.

**Report |Φ_law| and its contents alongside any regret number.** Sublinear regret against a nine-element class is a weak statement, and a report without the class is not a result.

Deliverable shape: enumeration-verified over the declared finite environment at the four declared horizons, or lean-proved for the bound; the consequence as lean-proved with an inhabitation witness. Acceptance check: python3 -m checkers.run accepts the registered entry; or the lean gate is green. Context: PHI_REGRET_TEST_SPEC.md §§1–6 fixes the environment, the three baselines, the measured quantities and the order of work; projects/normativity/rounds/2026-08-11-phi-regret-learner/ is the controlling item-30 result. A solution ships: the bound or its absence, at each horizon, against the declared class; and an honest statement of whether the successful learner is still answerable and inside its declared service work, which the spec calls S4 and expects to be where the round spends its time.