C.23:5 - Archetypal Grounding (Tell–Show–Show)

(Plain register for pedagogy; Core remains notation‑independent per E.10/E.8.)

Show‑1 - Continuous dynamics (ODE task). S2 excerpt. DataShape=ODE; stiff?=unknown; Size≈10^3; Objective={↓error@ratio, ↑throughput@ratio}; Constraints={safety_gate@ordinal}; Jacobian_sparsity=high; Missingness=MAR. Families. Implicit‑BDF vs Explicit‑RK vs Symplectic. Rules.

• Implicit‑BDF: Eligibility tolerates stiff?=unknown if Jacobian_sparsity=high (guarded precondition); MaturityCard=L3 (replicated & benchmarked). Outcome: Admit.
• Explicit‑RK: requires stiff?=false; with unknown ⇒ Degrade(sandbox) (probe).
• Symplectic: eligible only when Hamiltonian=true; here ⇒ Abstain. Didactic anchor. This mirrors C.22’s typed‑signature discipline and CHR legality (no ordinal means; unit alignment for ratio).

The cited ecosystem examples of these families (post‑2015) are organised in DifferentialEquations.jl, which exposes multiple solver families under one call surface—precisely the pattern G.5 expects. (Journal of Open Research Software)

Show‑2 - Planning/scheduling (MIP task). S2 excerpt. DataShape=MIP; NoiseModel=deterministic; Objective={↓cost@ratio, ↑service_level@ordinal}; Size≈10^5 vars; convex_relaxation=available. Families. MILP (branch‑and‑bound), Constraint‑Programming, Heuristic meta‑search. Rules.

• MILP: Eligibility requires convex_relaxation=available; the cited L3 MaturityCard edition names the registered family, evidence profile, benchmark basis, claim scope, qualification window, and intended selector use ⇒ Admit.
• Constraint‑Programming: MaturityCard=L2; Acceptance demands service_level≥B (ordinal predicate). With B met but baseline parity unknown ⇒ Degrade(scope‑narrow).
• Heuristic meta‑search: MaturityCard=L1 ⇒ Degrade(sandbox) or Abstain depending on RSCR parity policy. Didactic anchor. Selector returns a Pareto set (no cross‑ordinal weighting), as required by G.5.

The cited “single call / many solvers” packaging that motivates MethodFamily rows is exemplified by JuMP (2017–2022), which cleanly separates model description from solver choice. (Miles Lubin)

• DifferentialEquations.jl illustrates family‑based solver packaging (multi‑method under one interface), 2017–2024 ecosystem. (Journal of Open Research Software)
• JuMP illustrates model/solver separation and registry‑like selection (2021–2022 papers, site). (Miles Lubin)
• Science of Science review (2018) supports the emphasis on replication/benchmarks in maturity assessment. (Science)

Show‑3 - QD archive (policy search). S2 excerpt. PortfolioMode=Archive; CharacteristicSpaceRef(d=2); ArchiveConfig(CVT, res=1k cells, K=1, DistanceDefRef.edition=v2, InsertionPolicyRef=dyn‑elite); EmitterPolicyRef=v3; DominanceRegime=ParetoOnly. Rules. After Admit, G.5 returns an archive; illumination reported; changes to DistanceDef/Emitter editioned in SCR; dominance remains ParetoOnly.

Show‑4 - Open‑ended GeneratorFamily (POET‑class). S2 excerpt. GeneratorIntent{GeneratorFamilyRef=GF‑01, EnvironmentValidityRegion=EVR‑A, TransferRulesRef=TR‑A, CoverageMetric=…}; PortfolioMode=Archive. Rules. After Admit, G.5 returns declared sets over {environment, method}; Degrade(scope‑narrow) if TransferRules=unknown; telemetry publishes coverage/regret and IlluminationSummary with edition/policy‑id on improvements.