Case C - Homotopy Theory Explanation

Use: graph-heavy and structure-heavy mathematical theory rendered into sequential explanatory narrative for learners.

Patterns opened: NSTD.1, NSTD.2, NSTD.3, NSTD.5, NSTD.6, and NSTD.8 if taught as a series.

Source structures that must survive: definitions, dependency order, examples, counterexamples, proof-status boundaries, theorem prerequisites, and source return to formal statements.

Temporal posture and rendering mediation mode: retrospective or atemporal explanatory route over existing mathematical source structures; direct source-structure mode unless a teaching architecture or knowledge-graph architecture is explicitly used as mediation.

Ordering rule: didactic dependency order with source-return to formal proof order.

Construction route:

1. NSTD.1: select the learner use and source structures: definitions, dependency graph, examples, counterexamples, theorem prerequisites, proof-status boundaries, and formal-source return.
2. NSTD.2: choose didactic dependency order; explicitly mark where it differs from formal proof order, historical order, or publication order.
3. NSTD.3: define the reconstruction target for each step: what the learner must be able to state, distinguish, use in an example, or return to as formal proof status.
4. NSTD.5: introduce an analogy, question or visual example where it helps the learner. Supply a condition before the learner’s next inference needs it; distinguish an intuitive image from a formal result.
5. NSTD.8: if the explanation becomes a series, turn dependencies into session anchors and reconstruction tasks rather than a sequence of entertaining analogies.
6. NSTD.6: evaluate whether learners can recover definitions, dependency order, example boundaries, proof-status boundaries, and source-return points; low values repair source selection and ordering before adding more metaphor.

Intentionally lost or deferred: full proof detail, every lemma, historical order, and advanced generalizations not needed for declared learner use.

Related work: C.29 supports a mathematical lens when one is being constructed; the mathematical source supplies the definition or proof. A.10 applies to an evidence claim and E.17 to the explanation’s publication form.

Low NSTD.6 repair: if learners can retell an analogy but cannot state definitions, dependency order, example boundaries, or proof status, repair NSTD.1, NSTD.2, and NSTD.3; do not raise engagement alone.