NSTD.2:5 - Archetypal Grounding

NSTD.2:5.1 - Worked example: choosing a path through a graph

A knowledge graph can connect concepts, dependencies, counterexamples, evidence and practice routes. To introduce part of it, choose a traversal that helps the reader encounter the necessary dependencies. The optional example note below retains those choices for a homotopy introduction.

NarrativeOrderingRule@HomotopyIntro:
  narrativeRenderingRef: HomotopyIntroNarrative@v1
  sourceStructureRefs:
    - topological space
    - path
    - deformation under constraints
    - invariant
    - example and counterexample
    - proof-status boundary
  orderingRuleKind: didactic prerequisite plus analogy-first cue
  orderRationale: learner needs visual intuition before formal return, but formal dependency must remain recoverable
  preservedSourceRelations: prerequisite relation; analogy-to-definition return; example-to-counterexample contrast
  foregroundedSourceRelations: deformation intuition and invariant question
  coarsenedOrLostSourceRelations: full proof order; advanced generality; categorical reformulation
  sourceReturnCondition: use the formal statement and proof when deciding a theorem, equivalence or proof claim
  additionalClaimReferences: `A.6.3.CSC`, `E.17.EFP`, `A.10`, `NSTD.6` when the corresponding claim is made

The introduction can begin with a picture of paths being deformed, then return to the formal definition. For example: “First picture a loop being deformed while its endpoints stay fixed. The next definition specifies which deformations are allowed.” This connects the intuitive sequence to its missing mathematical condition.

NSTD.2:5.2 - Before and after repair: architecture order

Before:

We tried several designs and eventually found the architecture that solved the coordination problem.

Failure: “eventually” supplies a chronology, while “solved” asserts a successful result. The account does not say which coordination problem was addressed, what trade-off was accepted or what observation supports success.

After:

Begin with the coordination problem, compare the candidate module splits, then explain the chosen trade-off and remaining interface exceptions. Consult the decision record for why that split was selected and observations of the implemented system for whether it reduced coordination work.

What changed: the explanation connects the choice to its problem, alternatives and remaining cost. A reader seeking the implementation sequence or an observed outcome has a different source to consult.

NSTD.2:5.3 - Before and after repair: live commentary

Before:

The midfield is late, so the press has failed, and that is why the match is turning.

Failure: live chronological observation is mixed with causal explanation and outcome projection. The source at that moment may support late movement and lost possession, but not a settled causal account.

After:

The midfield line arrived after the forwards began pressing, leaving space between them. That gap may explain the lost possession; the replay can help distinguish this explanation from a missed individual challenge.

What changed: the observed sequence remains distinct from the provisional explanation, with a specific question for later review.

NSTD.2:5.4 - Comparing candidate orders

Use NSTD.6 when a qualified quality result is needed. The number of worked examples alone does not determine an ordering-quality value.

NSTD.2:5.5 - SoTA-to-action translation

Narratology distinguishes selected material, story, discourse and presentation. Here those distinctions help the author separate what is included from the order in which it is encountered. NLG likewise distinguishes content selection, document planning and wording. A smooth sentence can still present a dependency in a misleading place.

A homotopy-theory explanation orders material by learner dependency: spaces, paths, homotopy, fundamental group, examples, proof-status boundaries. This is not the historical discovery order and not the formal proof order of a research monograph. The narrative must state that definitions and proof obligations remain in source-return formal statements.

Consider this constructed decision source: two teams maintain the same validation rule in separate modules. They compare a shared library with a shared service, then choose the service so that a rule change can be deployed once. They accept an extra service call and client handling of version mismatches. These are design expectations; runtime effects have not been measured.

Two short orders preserve the same source:

• Cost first: “The chosen service adds a call and requires version-mismatch handling. The teams accepted these costs to deploy a shared validation rule once, after comparing a library with a service. The runtime effect is not yet measured.”
• Decision first: “To deploy a shared validation rule once, the teams chose a service over a shared library. They accepted the extra call and version-mismatch handling; the runtime effect is not yet measured.”

For a new maintainer asking why the service was selected, the second order foregrounds the reason. For a reader investigating the added call, the first order foregrounds the relevant cost. Neither order supplies an observed improvement. Use the decision source for further alternative detail and measurements for a claim about the realized effect.

A franchise-continuation outline may order scenes by reveal and tension. That does not mean reveal order is causal order. The underlying causal and continuity order must still name premise constraint, character motivation, event cause, consequence, and canon-return point. If a later twist works only because a source constraint was hidden from the writer as well as from the reader, the order is not a legitimate reveal; it is a source-selection failure.

A scientific storytelling case may order experiments by tension: failed attempt, surprising measurement, revised hypothesis, unresolved conflict. That order can be useful because it mirrors inquiry, but it cannot make unresolved tension into evidence closure. The sequence must name which calculations, mechanisms, and experimental facts remain source-return points.

Worked repair sequence:

1. Initial symptom: “The team first tried A, then B, and finally discovered the right architecture.”
2. Hidden-order diagnosis: chronology is being read as decision quality and finality.
3. Source topology repair: name candidate structures, quality characteristics, rejected alternatives, and telemetry or decision basis.
4. Ordering repair: present the problem, alternatives and trade-off; distinguish that explanation from the implementation sequence.
5. Loss repair: state what the narrative omits, such as lower-level exceptions or candidates excluded for scope.
6. Source-return repair: link to architecture description, decision record, or candidate comparison when a reader needs authority.

Second worked repair sequence:

1. Initial symptom: “We introduce homotopy with loops, then groups, then examples, because that is the natural story.”
2. Hidden-order diagnosis: “natural story” hides the didactic prerequisite order and may be mistaken for formal proof order.
3. Source topology repair: list definitions, examples, theorem prerequisites, proof-status boundaries, and formal-source returns.
4. Ordering rule repair: choose didactic dependency order and explicitly state where it diverges from formal order.
5. Loss repair: record which formal details are deferred and where they return.
6. Evaluation repair: NSTD.6 checks whether learners can reconstruct dependency and proof boundaries, not whether they enjoyed the story.

Case matrix for ordering work:

Use the matrix when comparing possible orders. It can guide a first draft or diagnose an existing passage. Reconstructing an earlier author’s choices is useful for repair; judging the current sequence does not require establishing how it was originally written.

When two orders conflict, preserve both as named layers. A learning route may teach prerequisites first and later show historical discovery. A mystery-like story may reveal effects before causes while keeping an internal causal order. A live recap may reorder events by causal explanation after first recording live chronology. The reader must be told which layer they are following.