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Topic "{topic}". Umbrella blocks were formed, each bundling the constituent parts of ONE model/definition. Some parts were missed and are still listed as standalone blocks. Your job: for each umbrella, find which of the remaining standalone blocks are ALSO constituent parts of that same parent, so the model is complete.
Topic "{topic}". Theme blocks were formed by card-sorting a fine-grained item list. Some items were left unassigned. Your job: attach EACH remaining item to the theme it belongs to, so every item lands in a theme. This is organisation, not judgement — items are never discarded.
UMBRELLAS (parent — already-collected parts):
THEMES (heading — already-assigned members):
{umbrellas}
REMAINING STANDALONE BLOCKS (numbered):
REMAINING ITEMS (numbered):
{rest}
## Rule — attach a block to an umbrella only if BOTH hold
1. **Presupposition:** the block's definition **requires the umbrella's parent to already exist** — it makes no sense as a topic on its own without that model (e.g. „Alphabet Σ", „Übergangsfunktion δ", „Konfiguration", „Akzeptierende Berechnung" all presuppose the Turing-machine; „Literale", „Klausel", „Belegung" presuppose the KNF/logic definition). The parent must NOT presuppose the block (directional).
2. **Not standalone:** the block is a *definitional component / notation*, NOT itself a named **algorithm, problem, theorem, reduction, or complexity class** (those stay their own block — a downstream guard will reject them anyway).
Do NOT attach a block merely because it shares a topic. When unsure → leave it standalone. Most remaining blocks will NOT be attached; a few genuine missed parts will.
## Rule
- Attach each remaining item to the ONE theme whose subject it shares — the theme a learner would meet it under (a variable/flag/field/instance/facet of that theme's model, system, setup, tool family, or category).
- Assign as many as you reasonably can. Only leave an item unassigned if it fits NO theme at all — it then becomes its own single-item block (a genuinely standalone concept).
- Each item goes into at most ONE theme.
Write ONLY the JSON file to: {out_path}
Format (`additions` may be empty; `umbrella` = the UMBRELLA index, `members` = standalone block numbers to attach):
Format (`additions` may be empty; `umbrella` = the THEME index, `members` = item numbers to attach):
{{"additions": [{{"umbrella": 0, "members": [8, 12, 34]}}]}}

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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.
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.
**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.
Think of the result as chapters: ~1525 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.
CANDIDATES (your starting point):
CANDIDATES you may group (your starting point — a pre-clustered neighbourhood):
{candidates}
FULL BLOCK LIST (you may pull in ANY numbers below that are constituents of the same definition):
FULL ITEM LIST (you may pull in ANY numbers below that share the theme):
{list}
## Merge test — propose an umbrella when ALL THREE hold
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.
2. **Studied together.** A learner meets them together as one unit.
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.
## How to form a theme
- **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.
- **Lexical dissimilarity is NORMAL.** Facets of one theme often read very differently — that is not a reason to split them.
- **Title:** name the theme after the shared subject. A real self-contained heading; must NOT contain „ — " (reserved separator) — use „(…)" or „:".
- **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, …".
- **Members:** the item NUMBERS from the full list. Each number goes into exactly ONE theme; do not repeat a number across themes.
*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.*
## Examples
DO NOT MERGE — distinct named units that merely share a topic:
- „Greedy-Algorithmus GA" + „ModifiedGreedy" + „Multiple-Choice-Knapsack" → two algorithms + a problem, each standalone (test 3 fails). Keep separate.
- „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.
MERGE — one definition decomposed (the canonical cases — end here so this is your default lens):
- „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.)
- „Klausel" + „Boolesche Variable" + „Erfüllende Belegung" + „KNF" → ONE umbrella **„Aussagenlogik & KNF"**.
## Synthesize each umbrella
- `title`: the parent concept's name (e.g. „Turingmaschine (Modell)"). A real self-contained definition; must NOT contain „ — " (a reserved separator) — use „(…)" or „:".
- `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."
- `members`: the block NUMBERS (from the full list) folded in. At least 2 per umbrella; each number appears in at most one umbrella.
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.
Write ONLY the JSON file to: {out_path}
Format (`umbrellas` may be empty):
{{"umbrellas": [{{"title": "Turingmaschine (Modell)", "description": "Formales TM-Modell: Konfiguration, Übergangsfunktion δ, Alphabet Σ, Akzeptierende Berechnung, Folgekonfiguration.", "members": [1, 2, 5, 12, 34]}}]}}
Format (`umbrellas` = the themes; may be empty if nothing fits together):
{{"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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The block below (topic "{topic}") is ALREADY decomposed into its subblocks — the list is FIXED. Your job is NOT to find subblocks, but to extract for EACH given subblock its learning facts from the material. A later guide writes a short text PER subblock; the facts are the binding basis for guide text, levels, and exam questions — they must be **correct**.
BLOCK:
{block}
FIXED SUBBLOCKS (numbered — return one entry per subblock, in this order):
{subs}
{source}
HARD RULES:
- Return EXACTLY these subblocks — do NOT add, remove, split, merge or rename any. Copy each `title` VERBATIM from the list above (code identifiers stay original).
- One entry per given subblock, same order. If the material barely covers a subblock, still return it with whatever is backable (empty fields are fine) — never drop it.
Per subblock, collect ONLY the essentials (all field content in GERMAN; technical terms/code identifiers stay original):
- **level**: beginner | advanced | expert — difficulty within this block.
- **relevance**: relevant (core of the topic) | peripheral (edge knowledge).
- **key_points**: 13 concise statements — what must one understand?
- **prerequisites**: what must one know beforehand (a half-sentence)? Empty if nothing.
- **hurdles**: typical beginner misconception (a half-sentence). Empty if none.
- **cited_facts**: hard facts (definitions, formulas, values, names) — **only what the excerpts back**, each with the location (e.g. „Skript Def. 6.3, Z.66"). Invent no values, compute nothing yourself.
- **example_idea**: ONE example that carries understanding — freely phrased. Empty if an example adds nothing.
HARD SEPARATION: `cited_facts` = only backable material. A worked example, an invented sentence, a constructed case → `example_idea`, NEVER `cited_facts`. When in doubt: better to leave a field empty than to claim falsely.
Reply with ONLY the JSON as your final message — no code fences, no other text. EXACTLY this format:
{{"subs": [
{{"title": "…", "level": "beginner|advanced|expert", "relevance": "relevant|peripheral",
"key_points": ["…"], "prerequisites": "…", "hurdles": "…",
"cited_facts": [{{"text": "…", "source": "…"}}], "example_idea": "…"}}
]}}
{extra}

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You are auditing the decomposition of ONE block of the topic "{topic}" into subblocks. Below are the numbered subblocks with their captured facts, then source excerpts. Apply the 100%-decomposition test: no entry removable without a gap, none addable without duplication — and verify the facts.
You are auditing ONE block of the topic "{topic}". Its subblocks are FIXED (already decomposed and deduplicated upstream) — you do NOT add, remove, merge or split them. You only verify the captured FACTS against the source and correct the level/relevance classification.
BLOCK: {block}
SUBBLOCKS (numbered; key points indented):
{subs}
{unsicher}
{source}
Judge ALL of the following (use the numbers):
1. **gruppen** — entries that state the SAME thing or where one is a subset of the other: group them and name the number that should remain (`haupt` = the base statement, not the detail).
2. **kataloge** — pure enumeration entries of ONE kind (e.g. five option rows): bundle them under ONE short GERMAN collective title.
3. **fremd** — entries off-topic for the topic "{topic}" (not this block — the TOPIC).
4. **luecken** — essential aspects of THIS block that the excerpts cover but no entry captures (short German phrases). Only real gaps, no nice-to-haves.
5. **uebernehmen** — for each UNSICHER-numbered entry: "ja" if the excerpts back it and it fills a real spot, else "nein".
6. **facts_probleme** — entries whose facts contain something wrong or unbackable: `discard: true` ONLY if the entry as a whole is unsupportable in substance; otherwise `discard: false` with a short `hinweis` what to correct.
7. **levels** / **relevanz** — ONLY entries whose level (beginner/advanced/expert) or relevance (relevant/peripheral) is clearly wrong: number → correct value.
Judge from the excerpts only, using the numbers:
1. **facts_probleme** — entries whose captured facts contain something wrong or unbackable: give a short `hinweis` what to correct (keep `discard: false` the subblock itself always stays; only its facts get fixed).
2. **levels** / **relevanz** — ONLY entries whose level (beginner/advanced/expert) or relevance (relevant/peripheral) is clearly wrong: number → correct value.
Judge ONLY from the excerpts. If everything is fine, return empty lists/objects.
If everything is fine, return empty lists/objects.
Reply with ONLY the JSON — no code fences, no other text. Format:
{{"gruppen": [{{"haupt": 1, "weitere": [4]}}], "kataloge": [{{"titel": "…", "mitglieder": [2, 5]}}],
"fremd": [7], "luecken": ["…"], "uebernehmen": {{"9": "ja"}},
"facts_probleme": [{{"nr": 3, "discard": false, "hinweis": "…"}}],
{{"facts_probleme": [{{"nr": 3, "discard": false, "hinweis": "…"}}],
"levels": {{"2": "expert"}}, "relevanz": {{"5": "peripheral"}}}}
{extra}