C.29:4.2b - Discover a candidate from the working cue

Choose the row that fits the problem, or use a closer domain construction. The menu is informative; when a listed construction is used, its named mathematical elements and conditions must be supplied. The shared test is :4.1, not membership in this list.

Working cue	Candidate structure and first mathematical work	Conditions that can change the use
Waiting, backlog, throughput	Queue or flow network: identify stations, routing, arrivals, service and waiting; derive a capacity bound or waiting relation.	Check discipline, batching, rework, finite buffers and station availability before applying the result.
State change, trajectory, stabilization or control	State space, Markov model, ODE or control model: name state, transition law and constraints; add an observation map when the receiving use needs it.	A.3.3 supplies dynamics semantics; C.27.TA and C.27 apply to the temporal aspect and temporal-use claim being made.
Conditional independence or computational-boundary cue	Probabilistic graph, Markov blanket or active-inference model: state variables, conditional-independence assumptions, observation/action partition and model boundary.	Recover a separately claimed physical interface, component or agency condition through its direct pattern.
Dependency, interface, composition or change of algebra	Graph, hypergraph, category, operad, optic or semiring: state edge/slot meanings, objects, morphisms, identities, interface conditions and composition laws; test the required composition or transform.	Classical, tropical, Fourier–Laplace or Legendre transforms can change the retained law. F.9 supplies cross-context semantic correspondence when needed.
Local-to-global flow or balance	Boundary operator, exterior derivative, divergence or Stokes-like construction: name domain, boundary, field/form/flow, local operator, boundary conditions and source or conservation balance.	The chosen domain law and regularity assumptions must support the global inference.
Local rule with no global extension	Cohomology, closed/exact distinction or another obstruction: identify the cycle/cocycle, equivalence class, local closure and failed global witness.	Use the obstruction to delimit this transfer; the failure does not select the rival model or identify a cause.
Sameness under transformations	Group action, symmetry, invariant or equivariant representation: identify transformations, action on variables and preserved quantity; derive a conservation link only under its theorem’s assumptions.	State coordinate details and distinctions lost; a physical conservation claim needs its domain basis.
Extremum, trade-off, potential or dual view	Variational, Lagrangian/Hamiltonian, action, energy, free-energy, loss, value or entropy functional; constrained optimization or Legendre/convex duality. Name variation space, constraints, boundary conditions and stationarity/extremum or dual transform.	A system’s actually following that extremum is a separate dynamics or causal question.
Similarity, distribution shift, population or shape movement	Metric, topology, order, embedding, coupling or optimal transport: define neighborhood/distance or transport plan, conserved mass and cost; calculate the comparison.	State what is transported and lost. C.16 supplies a used measurement/comparability construction; a policy effect or fairness claim needs its own argument.
Scale transition, coarse behavior, universality or knee	Coarse-graining, RG or fixed-point view: name scale variable/window, coarse-graining rule, fixed point or attractor, basin/regularity assumptions and invariant or exponent.	Return to the microdescription when an omitted distinction matters. C.18.1, C.19.1 and C.31.ASAP supply the separately claimed scale-law, method preference and architecture preference results.
Self-reference or a universal evaluator	Diagonal, self-application or fixed-point construction: specify encoding, evaluator/self-map, tested universal claim and exact obstruction.	A recursive-looking loop alone does not establish a no-go result.
Uncertainty, missing observation or next sample	Probability/information measure, Bayesian workflow, BED/OED, active learning or Bayesian optimization: state variables, priors/likelihood, utility or information criterion, design/acquisition variable and estimation method.	Check prior-data conflict, predictive mismatch, estimation cost, uncertainty and robustness to model/noise error; use the result within its validation boundary.
Recoverable structure under a resource bound	MDL, epiplexity or another code/measure: name source episteme/trace, observer, admissible model/coding scheme and execution bound; distinguish selected structure from residual description.	Apply the correspondence and observation/postulate boundary in :4.2c; return to source when the discarded structure matters.
Learning update, curvature or optimization trajectory	Information geometry or a learning-dynamics model: specify update variables, metric/noise relation and the property being calculated.	Establish the mapping to the particular learning process before using the result beyond the formal model.
Nonlinear dynamics needing a tractable observable	Koopman/operator, DMD or system-identification construction: select observables, operator and approximation; derive the forecast or diagnostic consequence.	Finite closure and predictive/control validity need checking; A.3.3 and C.27 supply the actual dynamics and temporal-use conditions.
Learned scientific representation or surrogate solver	Neural operator, latent representation, embedding or world model: specify function/state/field mapping, observation map, training or simulation regime and resolution policy.	Apply :4.5a’s learned-lens and validation conditions, including generalization scope and approximation loss.
Intervention, policy effect or counterfactual	SCM, causal graph or micro-to-macro causal abstraction: specify assignment/intervention, outcome and preserved or approximated intervention/counterfactual structure.	C.28 supplies identification and causal-use justification; an associative graph or latent manifold may not answer this question.
Probe, order or context effect with incompatible frames	Quantum-like or contextual-probability model: identify the contextual obstruction that still changes inference or action after the ordinary subject patterns.	Apply C.26’s adequacy conditions. A physical quantum claim additionally needs the relevant physics and observations.
Storage, computational or realizability limit	Count actual represented objects and operations; apply a resource bound or constructive/impossibility argument.	Recompute for the actual alternative representation. A valid rejection of one implementation does not yet supply a feasible replacement.

For a constrained extremum or stationary construction, construct or reuse a family of allowed candidates and calculate the resulting change in the target quantity. Distinguish an improving candidate, a necessary stationary condition and a justified optimum; carry the resulting conclusion and its assumptions into the working question.

For a symmetry argument, follow the transformation through the problem’s data, conditions and required answer. Derive the transferred solution, restriction or obstruction to selection, and say which problem that consequence answers.

When a balance must be constructed or its boundary changes, C.29.BB identifies the additive quantity, included stores and crossing transfers. It combines compatible accounts and returns the total, a bound or the missing contribution.

For a first candidate, compare with ordinary prose, direct observation or the accepted domain model before a broader survey. When the question is a tradition-scale source synthesis, use G.2; C.29 needs only the candidate or rival relevant to this working question.

When expected information gain determines which observation to obtain, choose a feasible way to estimate it. If density approximations are used, allocate samples between fitting them and computing the expected gain. In high dimensions, consider reducing the parameter or observation space and account for the information lost by that reduction. Compare estimation cost and error with the distinction needed for the choice. Li, Baptista and Marzouk (2026) develop these sample-allocation and dimension-reduction choices.