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Topic "{topic}". A previous step produced a flat list of learning blocks that is TOO FINE-GRAINED — several blocks are **constituent sub-definitions / notation of ONE larger definition or model** and should become a single umbrella block. Find these groups. A good run finds several genuine umbrellas AND leaves most blocks standalone; judge each candidate on its merits.
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Topic "{topic}". A previous step produced a flat list of learning items that is TOO FINE-GRAINED for a table of contents — single environment variables, single CLI flags, single config fields, near-synonyms. Your job is **card sorting**: group the items into a small set of **theme blocks**, so that EVERY item lands in exactly one theme. Nothing is discarded — grouping never loses an item; it only organises them.
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**Propose real umbrellas, judge each on its merits.** Merge when the members are constituent parts of one whole (test 3); keep a member separate when it stands as a full, usable unit on its own. Do NOT withhold a genuine merge merely because the members are lexically dissimilar (facets of one whole routinely are) or because you are unsure of the parent's exact name.
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Think of the result as chapters: ~15–25 themes for a whole topic, each a coherent learning block that a learner meets as one unit. A theme with only one fitting item is fine (a genuinely standalone concept), but prefer pulling related items together over leaving many singletons.
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CANDIDATES (your starting point):
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CANDIDATES you may group (your starting point — a pre-clustered neighbourhood):
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{candidates}
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FULL BLOCK LIST (you may pull in ANY numbers below that are constituents of the same definition):
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FULL ITEM LIST (you may pull in ANY numbers below that share the theme):
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{list}
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## Merge test — propose an umbrella when ALL THREE hold
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1. **One parent.** The members are constituent parts/facets of ONE named parent (a model, system, setup, or definition) — each member PRESUPPOSES that parent: you cannot introduce it without first invoking the parent, and it has no purpose outside it.
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2. **Studied together.** A learner meets them together as one unit.
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3. **No standalone unit among them — autonomy test (decisive).** Remove the OTHER members, then ask of each: does it still stand as a COMPLETE, usable unit with its own goal? YES → autonomous → a SIBLING, keep separate (dropping the others took nothing from it). NO → alone it serves no goal and exists only to build the shared parent → a constituent PART, merge it. Structural containment („X is a file/step/field of Y") is NOT the test — autonomy is: own goal, complete without the siblings. This holds in every domain.
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## How to form a theme
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- **One coherent subject.** The members belong together because a learner would study them as one block — the parts/facets/instances of one model, system, setup, tool family, or category. Examples of the *kind* of grouping (not domain rules): many individual settings/variables/flags of one configuration → one „…-Konfiguration" theme; the parts of one setup → one „…-Setup" theme; a base concept plus its variants/instances → one theme.
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- **Lexical dissimilarity is NORMAL.** Facets of one theme often read very differently — that is not a reason to split them.
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- **Title:** name the theme after the shared subject. A real self-contained heading; must NOT contain „ — " (reserved separator) — use „(…)" or „:".
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- **Description:** name EVERY member explicitly (the next step recovers them as sub-points). E.g. „Environment-Konfiguration: APP_ENV, APP_SECRET, DATABASE_URL, MAILER_DSN, …".
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- **Members:** the item NUMBERS from the full list. Each number goes into exactly ONE theme; do not repeat a number across themes.
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*Note on test 1: „can this be defined at all?" is the WRONG question — Alphabet Σ and DTM CAN be stated in isolation, yet in THIS topic they are parts of the Turing-machine model and belong together. The question is whether the member PRESUPPOSES the shared parent, not whether a standalone sentence exists.*
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## Examples
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DO NOT MERGE — distinct named units that merely share a topic:
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- „Greedy-Algorithmus GA" + „ModifiedGreedy" + „Multiple-Choice-Knapsack" → two algorithms + a problem, each standalone (test 3 fails). Keep separate.
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- „Plugin" + „Plugin-Konfiguration (config.xml)" + „Plugin lifecycle" → „Plugin-Konfiguration" is its OWN unit (authoring config fields, snippets, its own tasks), a SIBLING of „Plugin", not a facet (test 3 fails). Structural containment — the file belongs to the plugin — is NOT the merge test; being studied as one definition is. Keep „Plugin-Konfiguration" standalone.
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MERGE — one definition decomposed (the canonical cases — end here so this is your default lens):
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- „Alphabet Σ" + „NDTM" + „DTM" + „Akzeptierende Berechnung" + „Folgekonfiguration" → ONE umbrella **„Turingmaschine (Modell)"**. (The members are lexically very different from each other — that is EXPECTED for facets of one model and is NOT a reason to keep them apart.)
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- „Klausel" + „Boolesche Variable" + „Erfüllende Belegung" + „KNF" → ONE umbrella **„Aussagenlogik & KNF"**.
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## Synthesize each umbrella
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- `title`: the parent concept's name (e.g. „Turingmaschine (Modell)"). A real self-contained definition; must NOT contain „ — " (a reserved separator) — use „(…)" or „:".
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- `description`: **name EVERY merged child explicitly** — the next step recovers the children as sub-points from the source. E.g. „Formales TM-Modell: Konfiguration, Übergangsfunktion δ, Alphabet Σ, Akzeptierende Berechnung, Folgekonfiguration."
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- `members`: the block NUMBERS (from the full list) folded in. At least 2 per umbrella; each number appears in at most one umbrella.
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Assign as many items as you confidently can. Items you are unsure about you may leave out — a later completion step and a singleton fallback catch them; never force a bad fit.
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Write ONLY the JSON file to: {out_path}
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Format (`umbrellas` may be empty):
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{{"umbrellas": [{{"title": "Turingmaschine (Modell)", "description": "Formales TM-Modell: Konfiguration, Übergangsfunktion δ, Alphabet Σ, Akzeptierende Berechnung, Folgekonfiguration.", "members": [1, 2, 5, 12, 34]}}]}}
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Format (`umbrellas` = the themes; may be empty if nothing fits together):
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{{"umbrellas": [{{"title": "Environment-Konfiguration", "description": "Konfiguration über Umgebungsvariablen: APP_ENV, APP_SECRET, DATABASE_URL, MAILER_DSN, .env-Dateien.", "members": [1, 2, 5, 12, 34]}}]}}
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