Graph Definition — Implementation Status, Test Strategy, Glossary, and Build (Appendices B–F)

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Compendium: §§1–5 Core · §§6–10 Context · §§11.1–11.4 Units · §11.5 Progression · §12 Planning · §13 Implicit Mastery · Appendices B–F · Appendix A (Rules & Ampel)

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Appendix B — Implementation Status and Boundaries of Claims

Status of this edition. This appendix records the repository states against which the specification's claims were audited and documents where the inspected implementation diverges from the normative model (GD-D01). This unified Single-Source edition consolidates the 4 November concept correction, the 8 November atomic normal-form mandate and milestone mapping, the 9 November projection contract clarification, the 10 November closed Personal Curriculum learning scope, and the 11 November 2026 decisions on non-compilable learning contexts and on learning-unit validity across Web and PDF. Since 11 November 2026, CQR-001 also enforces G17 for the canonical Gymnasium catalog. Both backend and WebGUI now evaluate the fixed atomic prerequisite closure; the closure compiler PersonalCurriculumLearningScope.java reuses canonical normalization. Local regression tests cover learnability, planning, focus, reference errors, and cross-subject prerequisite delivery. Full native, curriculum, and remote-CI acceptance as well as alignment of applicability and migration validators remain separate steps. Global application recognition (\(\Meff\)) is not yet implemented. Readable documentation evidence or a green document build proves neither runtime conformance nor production deployment.

Binding Product Owner decisions.

• 4 November 2026: Implicit mastery through passed assessments of a valid local contract demonstrates independent application and carries global effect (\(\Meff\)); transitive minimality remains mandatory. Assessment and memory goals remain protected: for memory goals, verified completion recall alone establishes goal completion, and subsequent card due state never revokes it (Sections 3, 8, and 13; GD-D03).
• 7 November 2026: Every reachable prerequisite of a visible goal requires an explicit projection role (target or prerequisiteOnly) in subject composition views; at the Personal Curriculum compile boundary, the 10 November 2026 closure rule supersedes blanket manual per-atom view entries with derived target roles (Section 7.2).
• 8 November 2026: Atomic normal form \(R\subseteq A\times A\) as the binding canonical target state without cluster exceptions, mandatory ordered migration sequence, and mapping of all 143 rule groups to validation areas and milestones (Sections 2 and 3, Appendix A).
• 9 November 2026: Projections preserve full prerequisite closure; a canonical cluster never becomes an atomic runtime goal through a composition view (Sections 5 and 6.3).
• 10 November 2026: The Level 2 Personal Curriculum learning scope is \(L=\Closure(S)\) on the unpruned canonical skill graph, fixed independently of mastery, assigning derived target roles to all necessary prerequisite atoms (Sections 5 and 7).
• 11 November 2026 (learning context): Hard closure errors belong to validation, curriculum QA, and native acceptance, not to learner reads. When the canonical graph of a learning context is not strictly compilable, the runtime keeps the reviewed seed scope without derived closure, claims no learning basis, offers no goal with broken demand, and logs the failure once per curriculum and error code; seed-selection errors stay hard (Section 7.2).
• 11 November 2026 (learning units): The recognition rule of 4 November stays: an own ordinary goal is recognized once no open own assessment or memory goal of its unit still requires it (if assessments \(\nd{Q_1}\) and \(\nd{Q_2}\) of the same unit both require \(\nd{A}\), \(\nd{A}\) stays open until both are passed). The alternative recorded on the archived branch of pull request #83 as a “6 November 2026 target decision” – immediate recognition from the qualified coverage list of a single passed assessment, including proof by a parent or foreign unit – was reviewed and is not adopted. Recognition is sound only inside units that satisfy rules R1–R5: R1, R3, and R4 as hard validity conditions from the first authored unit, R2 at M5 with the current per-goal binding at M7, and R5 as a runtime precondition before recognition is enabled in production; a unit's purpose is a secure foundation before planned learning builds on it (Sections 11 and 13.3).

Audited repository states. The technical findings in this appendix and in the rule registry are anchored to five explicitly named repository states; none of them substitutes for a fresh full-system acceptance run on a later HEAD:

• 616324e857 – Historical implementation audit underlying the runtime divergence rows in the comparison table below; not a fresh full runtime acceptance run.
• 3b71ee9370 – Baseline gate-reach audit in Appendix A against main.
• 68db5bf499 – Dedicated projection audit of 9 November 2026 regarding direct goalEntry references to canonical clusters.
• cacf323919 – main integration of the Personal Curriculum prerequisite-closure compiler \(L=\Closure(S)\) (10 November 2026 decision) and baseline for the traffic-light (Ampel) reconciliation and this Single-Source edition.
• 44c5b06b77 – main state of this edition (11 November 2026) with the standard G17 milestone gate (15f74ef564, pull request #119), the repaired protected subject data (pull request #121), and the seed-scope fallback (b511654efb); basis of the 11 November rule reconciliation. The runtime findings in the table below were not re-audited at this state.

Dedicated projection finding of 9 November 2026. At audited state 68db5bf499, CompletionCanonicalGraphResolver loads prerequisite chains recursively. However, view projection in LearnerService and the resolver still treat direct goalEntry references to canonical clusters as childless runtime atomic goals. Historical evidence for the package-backed runtime catalog, offerings, and composition views (work package DPK-006c-c) documents that legacy implementation rather than proving preservation of the canonical atomic basis now explicitly required. The code fix and native verification remain open. This targeted read audit does not refresh the remaining historical runtime findings in the table below.

Area	Binding specification here	Audited repository status
Completion traversal (transitivity)	Full transitive open demand \(N_U=\operatorname{Reach}_{\Rat}^{*}(P_U\setminus M)\) without stopping at mastered intermediate goals.	CompletionEvaluator.evaluateAutomaticGraph stops at mastered intermediate goals: in \(\nd{Q}\to\nd{B}\to\nd{A}\) with only \(\nd{B}\) mastered, \(\nd{A}\) is released there, whereas the new formula keeps \(\nd{A}\) needed for open assessment \(\nd{Q}\). The legacy profile path in the same service is a different algorithm and likewise does not implement the new contract.
Protected goals and local coverage	The protected set \(P_U=Q_U\cup A_{\mathrm{mem},U}\) protects all own assessment and memory goals; every own content goal requires local assessment coverage within \(\Own(U)\), without turning pure aggregates or parent syntheses into artificial contracts over descendants.	The existing automatic path starts solely from open assessment goals; joint protection of selected memory goals and own-unit assessment coverage are not yet implemented in runtime evaluation. The unit validity rules R1–R5 of 11 November 2026 have no validator yet.
Global recognition (\(\Meff\))	Person- and revision-bound global union \(\Meff=M\cup\bigcup_{U\in\mathcal U_{\mathrm{valid}}}I_U\) acts durably across downstream frontiers, focus switches, and cumulative progress without writing synthetic Mastery records.	Not yet implemented. Context-local legacy exemption is not the durable contract-bound set \(\Meff\); downstream frontier, cumulative progress, contract revision binding, and historical meaning must be implemented and verified together.
Atomic normal form and minimality (G17)	Complete atomic normal form \(R\subseteq A\times A\), ordered migration (materialize atomic prerequisites, remove cluster edges, reduce transitively), and transitive minimality as mandatory baseline before maturity level M1.	Since 15f74ef564 (on main with pull requests #119 and #121, 11 November 2026), CQR-001 runs the G17 check (atomic normal form, transitive minimality, explicit types, including required cross-subject goals) for every root of the complete canonical Gymnasium catalog and caps a failing subject at M0; hard in CI via the ten protected floors. In the status generated on 11 November 2026, the ten protected subjects plus Italian, Polish, Russian, and Czech pass; Chinese, English, French, Greek, Music, Spanish, and the Gymnasium overview root fail and report M0. Non-canonical roots have no G17 gate; the migration order is evidenced only through the prerequisite-reduction proof registry (Appendix A).
Cluster satisfaction (\(\Sat\))	The normative core requires all atomic descendants \(\Atoms(g)\subseteq\Meff\).	Audited backend paths filter recursively via private helper isCoreForPrereqs: children with no tags or tag GK take precedence over other children when present; otherwise all children count. No boolean core field is read there. That differs from normative full-basis aggregation.
Memorization and card due state	Verified completion recall (RecallPassed) establishes memory-goal completion; subsequent SRS card due dates never revoke completed mastery, and ordinary goals carry no review-due expiry (GD-D03).	Legacy due-date target formulas are superseded; historical derivation from recall flags in useSrsMastery.ts alone does not prove full runtime conformance across all flows.
Recursive learning units (LearningUnit)	Recursive target and child hierarchy with unique direct ownership \(\Own(U)\), derived completion \(\Complete(U)\), and separate optional calendar/budget planning; technical unit locks and direct assessment entry under them remain undecided.	Binding domain-structure specification; not yet a claimed runtime feature or productive unit validator. R1, R3, and R4 must be checked fail-closed from the first authored unit (decision of 11 November 2026).
Direct cluster references in views	A composition view never turns a canonical cluster into an atom; \(\Atoms(g)\) is preserved (GD-X09).	At audited state 68db5bf499, LearnerService and CompletionCanonicalGraphResolver still treat direct goalEntry references to canonical clusters as childless runtime atoms; fix and native verification remain open.
Personal Curriculum scope (\(L=\Closure(S)\))	Valid seed targets \(S\) are closed on the unpruned canonical graph (\(L=\Closure(S)\)), fixed independently of mastery, and assigned derived target roles for necessary prerequisite atoms.	Implemented in backend (PersonalCurriculumLearningScope.java) and WebGUI (cacf323919) with local regressions; full native, curriculum, and remote-CI acceptance plus alignment of applicability/migration validators remain separate. Since b511654efb (11 November 2026), a learning context whose canonical graph is not strictly compilable keeps its reviewed seed scope without derived closure and claims no learning basis.
Focus and filter modes (normal vs. strict)	Level 3 focus narrows candidate goals while keeping Level 2 denominator \(L\) fixed; normal mode checks prerequisites inside focus, strict mode checks globally; focus widening is offered only after actual focus completion.	Implemented in learner runtime and WebGUI over explicit \(M\); does not yet evaluate \(\Meff\) and remains subject to the isCoreForPrereqs and goalEntry boundaries above.
Didactic routes (\(O_k\to a\to Z_k\))	Every non-exempt atomic goal lies on a direct atomic prerequisite path from an orientation anchor to a terminal autonomy goal within its route profile.	Evaluated in twelve configured QA-scope profiles across ten Gymnasium subjects (CQR-101–CQR-104) and hard-gated via GVR-012 in Mathematics; curricula without route profiles cap at M2.
Assessment tasks and application coverage	Releasing assessments must require independent application of every recognized essential competence under every passing score combination (GD-A08).	CQR-201–CQR-203 enforce examData presence, non-placeholder readiness, and metadata on protected floors, without proving live session grading or non-compensability of every passing rubric combination. Only for terminal assessments of canonical Economics, CQR-104 binds every coveredGoalIds entry to a visible prerequisite path (related to R3).
Optional acquisition and time planning	Optional calendar periods verify structural completion witnesses over \(B_\pi\) and \(\Work+\Reserve\le\Budget\) only under explicitly declared estimation assumptions \(\Theta_\pi\).	Optional planning profile (GD-P01–GD-P11); no productive curriculum or runtime budget gate is claimed, and weight never denotes minutes.
Cumulative progress vs. period completions	Cumulative plan progress counts \(\lvert T_{\mathrm{Plan}}\cap\Meff\rvert\) (with an optional clearly labelled explicit-only plan counter \(\lvert T_{\mathrm{Plan}}\cap M\rvert\)), while active periods count only reliably dated explicit completions.	Current Cockpit and velocity views count explicit completion events; segmented explicit/implicit accounting awaits implementation of \(\Meff\).
Maturity levels M0–M7 and quality floors	Automated phase-1 curriculum QA M0–M7 remains strictly separated from human sign-off, learner trials, and runtime conformance.	CI protects ten Gymnasium subjects via checkCurriculumMaturityFloors.ts (5 at M7, 5 at M6) plus status-file diffing; unfloored experimental landscapes remain reportable at M0, and docs/**-only changes are exempted in ci.yml.

Inspectable code anchors. PersonalCurriculumLearningScope.java; CompletionEvaluator.java; LearnerService.java; useSrsMastery.ts; validateGraph.ts; Mastery.java. The linked main branch continues to evolve. The generated build manifest binds the exact input files read during each document build via SHA-256.

Sources: AGENTS.md, §2; completion-semantics-clarification-2026-11-04.md; dual-curriculum-package-implementation-status.md; PersonalCurriculumLearningScope.java; CompletionEvaluator.java.

Appendix C — Verification Strategy for Completion Semantics

The accompanying script check-assessment-model.py is a mathematical reference check written with the Python standard library alone. It verifies the formulas from Section 13 across small finite models, reads no learner data, and mutates no runtime state (GD-D04).

• Finite model checking. Every topologically ordered DAG of up to four vertices under every assignment of explicit mastery and protected sets where local content goals have complete local assessment coverage: \(5\,222\) valid cases. Graph traversal and static transitive closure yield the identical implicit set \(I_U\).
• Minimality. Transitive reduction is minimal and preserves closure. Redundant edges of the normalized relation do not alter recognition. The older inheritance counterexample diagnoses transitional data; this bounded model test does not by itself verify the new atomic migration priority (GD-D05).
• Global monotonicity. Under fixed valid contracts, additional explicit mastery never reopens effectively mastered work. Assessment and memory goals are never implicitly passed (GD-D06).
• Recursive units. \(128\) state assignments of the Mechanics example verify completion \(\Complete(U)\), own recognition \(I_U\), and explicit child obligations. A pure aggregate requires no own assessment; a pure parent synthesis requires no new content goals.
• Unit block grouping. \(948\) grouping cases compare the unit-block DAG against a complete block sequence. One counterexample exhibits an acyclic atomic DAG whose coarse block summary forms a cycle; no technical entry-lock policy is derived from it.
• Concrete counterexamples. No stopping barrier at mastered intermediate goals; local recognition preserved despite an open later assessment; durable union across contracts, future prerequisites, intersection with planned target sets, and external support outside \(\Own(U)\).
• Invalid inputs. Cycles, unknown references, and uncovered local content goals fail closed as errors rather than defaulting to automatic mastery.

Diagram generation in generate-completion-diagrams.py verifies the concrete sets displayed in each figure and compares freshly rendered PNG bytes in --check mode (GD-D07). These model and diagram checks verify neither real task quality, grading reliability, identity verification, nor pedagogical effectiveness. Required independent subject-matter content review remains a separate step; runtime integration and its security, concurrency, and UI tests remain separate engineering work.

Sources: check-assessment-model.py; generate-completion-diagrams.py.

Appendix D — Glossary and Symbols

D.1 Terms

The first table defines the terms in the order in which this specification introduces them and gives the corresponding German terms. The second table lists the mathematical symbols. None of these definitions introduces an additional stored outcome kind: only the binary outcome \(m\) is persisted per atomic goal.

Term	Meaning
goal (Lernziel, Ziel)	Node of the skill graph with a stable UUID identity.
atomic goal, atom (atomares Ziel, Atom)	Goal without contains children. Only atoms carry a stored learner outcome.
cluster goal, cluster (Cluster)	Goal with contains children; structures and aggregates content rather than being mastered directly.
containment (Containment)	The hierarchy formed by authored contains edges.
requirement, prerequisite (Anforderung, Voraussetzung)	The edge \((g,p)\in R\) is a requirement of \(g\); goal \(p\) is a prerequisite of \(g\).
authored, derived (verfasst, abgeleitet)	Only the stored edges in \(C\) and \(R\) are authored. Everything else is computed from them.
inherited, effective requirement (geerbte, effektive Anforderung)	An inherited requirement is one that a goal acquires from a contains ancestor. Effective requirements combine a goal's own and inherited requirements.
atomic basis (atomare Basis)	For an atom, the goal itself; for a cluster, its atomic descendants.
atomic demand (atomarer Bedarf)	All atoms that a goal directly or transitively requires, independent of learner state.
atomic normal form (atomare Normalform)	The property \(R\subseteq A\times A\): all authored requirements connect atomic goals.
target state, transitional data (Zielzustand, Übergangsbestand)	Target state is the intended state of canonical data: atomic normal form and transitive minimality. Transitional data denotes inventories that still carry cluster requirements.
transitive minimality (transitive Minimalität)	No authored requirement is already reachable through the remaining requirements.
motivation anchor, terminal autonomy goal (Orientierungsanker, terminales Autonomieziel)	Starting and ending endpoints of a didactic route.
explicit mastery, Mastery (Mastery)	The stored explicit outcome of an atomic goal: mastered or not yet mastered.
implicit mastery (implizite Beherrschung)	Mastery derived from passed assessments of a valid local recognition contract; for content goals it demonstrates independent application.
effective mastery, effectively mastered (effektiv beherrscht)	Explicitly mastered or implicitly mastered. Both carry equivalent global effect.
satisfaction (Erfüllung)	A goal or cluster is satisfied when its atomic basis is non-empty and effectively mastered.
frontier (Frontier)	The next available atomic goals: themselves not yet mastered, with all required atoms satisfied.
filter, focus (Filter, Fokus)	A filter selects a subset of the skill graph. Focus is a learner's temporary working scope (Level 3).
normal (optimistic) mode, strict (pessimistic) mode (Normalmodus, strikter Modus)	Normal (optimistic) mode checks prerequisites inside the selected scope; strict mode checks all prerequisites globally.
level of personalization (Personalisierungsstufe, Level)	Four narrowing levels: base curriculum, Personal Curriculum, focus with one active goal, and global mastery.
Personal Curriculum (persönlicher Lehrplan)	Committed personal goal selection including its necessary atomic prerequisites at Level 2; fixed independently of mastery under a given configuration. Existing valid recognition counts as satisfied.
projection, composition view (Projektion, Kompositionssicht)	A projection derives the visible tree for a resolved scope. A composition view explicitly defines that scope's reviewed upper tree structure.
projection role (Projektionsrolle)	target: selectable learning target of the view. prerequisiteOnly: subject-local prerequisite support; necessary atoms still receive an acquisition target in the full Personal Curriculum.
applicability (applicability, Anwendbarkeit)	Compiled dimensional metadata specifying in which scopes a goal is visible.
program unit, goal placement (programUnits, goalPlacements)	Organizational container such as a school year or semester; entry attaching an unchanged goal to such a unit.
goal type, node type (Zieltyp)	Pedagogical kind of an atomic goal: content goal, orientation goal, memory goal, or assessment goal.
content goal, ordinary subject goal (Inhaltsziel)	An ordinary subject-matter learning goal; labelled “ordinary” in figures.
verified recall, completion recall (verifizierter Abruf, Abschlussabruf)	Controlled unaided recall of a card without prompt hints. Verified completion recall across all required cards completes a memory goal.
direct assessment access (direkter Prüfungszugang)	A deliberately chosen assessment attempt is not blocked by unmastered preparation.
learning unit, child unit, required child (Lerneinheit, Kindeinheit, Pflichtkind)	Authorized educational unit with a stated learning outcome; its subunits; the mandatory subunits among them.
aggregate, synthesis (Aggregat, Synthese)	An aggregate bundles child units without adding its own assessment. A synthesis unit additionally assesses their integration.
local recognition contract (lokaler Vertrag)	Binds a learning unit's own goals, its curriculum revision, and the subject-reviewed application meaning of its assessments; also called an acquisition contract.
unit validity rules R1–R5 (Gültigkeitsregeln)	The five conditions of 11 November 2026 without which implicit recognition is unsound: an assessment path for every own ordinary goal (R1), reviewed coverage evidence (R2), coverage only within the assessment's prerequisites (R3), exactly one directly owning unit (R4), and a versioned contract (R5).
protected goal (geschütztes Ziel)	Assessment or memory goal; never implicitly passed.
open demand (offene Nachfrage)	A learning unit's still-open protected goals together with their full transitive atomic demand.
consumer (Verbraucher)	A goal that requires another goal, viewed from that prerequisite.
whole-unit prerequisite, entry lock (Einheitsvoraussetzung, Einheitssperre)	Explicitly authorized prerequisite between whole learning units; a hard technical access lock derived from it. Whether such locks exist remains undecided.
planning period, entry basis, completion witness (Planungsstufe, Eintrittsbasis, Abschlusszeuge)	Calendar interval of an optional schedule; competencies assumed present at entry; ordering that proves a period's targets can be acquired step by step.
planned target, actual count, cumulative progress (Plan-Soll, IST, Gesamt-IST)	Planned target is the scheduled progress of a named scope; actual count is achieved progress. Cumulative actual progress (Gesamt-IST) counts a plan's effectively mastered goals.
profile (Profil)	Named set of additional constraints that applies only where explicitly selected, such as a route, validator, or planning profile.
scope, QA scope (Scope, QA-Scope)	The named validation scope of a rule or route profile.
gate (Gate)	Automated check whose failure blocks a milestone or build.
maturity level, milestone (Reifegrad, Meilenstein)	Level M0 to M7 of automated curriculum QA; Appendix A refers to them as milestones. M7 is not human effectiveness proof.

D.2 Symbols

The following table summarizes all mathematical symbols across the specification (graph core in Sections 1 to 3, learner state and filters in Sections 4 and 5, Personal Curriculum in Section 7, goal types in Section 8, learning units and implicit mastery in Sections 11 and 13, planning in Section 12).

Note on unified notation. Earlier standalone English Markdown drafts used \(D(g)\) for atomic demand, \(U(T)\) for the prerequisite closure of a subset, \(M_A\) for the explicit mastery set, \(M_k\) and \(T_k\) (or \(M\) and \(T\)) for orientation anchors and terminal autonomy goals of a route profile, and \(K\) for memory goals. The unified Single-Source specification standardizes these symbols across both Web and PDF to \(\ReqAtoms(g)\), \(\Closure(T)\), \(M\), \(O_k\) and \(Z_k\) (or \(O\) and \(Z\)), and \(A_{\mathrm{mem}}\), keeping \(K=G\setminus A\) unambiguous for structural clusters and \(M\) unambiguous for explicit mastery. The unchanged static original illustration 06-available-next-goals.png (Figure 8) still shows the historical label \(M_A\), as noted in its caption.

Symbol	Meaning
\(G\), \(A\), \(K\)	Goals, structural atoms, clusters; \(K=G\setminus A\).
\(C\), \(R\)	Authored edges: containment (contains) and direct requirements (requires).
\(X^+\), \(X^{-1}\)	Non-empty transitive closure and inverse relation of a relation \(X\).
\(\Anc(g)\), \(\Desc(g)\)	Ancestors and descendants along contains.
\(\Atoms(g)\)	Atomic basis; the singleton \(\{g\}\) if \(g\) is atomic, otherwise its atomic descendants.
\(\Reff\), \(R_{\mathrm{inherited}}\)	Effective requirements and their inherited subset; \(\Pre_{\mathrm{eff}}(g)\) denotes the effective prerequisites of \(g\).
\(\Rat\)	Fully atomically expanded effective requirement relation, also used in the completion formulas.
\(R_A\)	Directly authored atomic requirements \(R\cap(A\times A)\).
\(B(g)\), \(\ReqAtoms(g)\)	Atomic basis of the effective prerequisites of \(g\) and full atomic demand (written \(D(g)\) in earlier standalone English drafts).
\(O\), \(Z\), \(E_{\mathrm{route}}\) (\(O_k\), \(Z_k\))	Orientation (motivation) anchors, terminal autonomy goals, and exempt support goals (or per route profile \(k\); written \(M\) and \(T\) or \(M_k\) and \(T_k\) in earlier standalone drafts).
\(m\), \(M\)	Stored binary state per atomic goal and the set of its explicit successes. \(m(a)=0\) does not rule out implicit mastery. Earlier standalone drafts and static Figures 8 and 9 write \(M_A\).
\(\Meff\)	Global union of explicit mastery and validly derived implicit mastery.
\(\Sat(g,\Meff)\)	Satisfaction of an atomic goal or cluster.
\(\Frontier\), \(\Frontier_K\)	Next available atomic goals; derived cluster availability.
\(P(S)\), \(w(a)\)	Weighted progress across a goal set \(S\); progress weight of an atomic goal \(a\).
\(F\), \(G_F\), \(A_F\)	Filter predicate, selected goals, selected atomic goals.
\(\Applicability\), \(D\), \(V_d\)	Applicability mapping, set of filter dimensions, and value vocabulary of dimension \(d\). In Section 5, \(Q\) denotes the active query selection.
\(\Closure(T)\)	Selected atomic goals \(T\) together with their full atomic prerequisite demand (written \(U(T)\) in earlier standalone drafts).
\(S\), \(L\)	Union of valid atomic seed targets across the selected reviewed subject scopes and the resulting closed Personal Curriculum learning scope \(L=\Closure(S)\) at Level 2.
\(\Frontier_{\mathrm{opt}}\), \(\Frontier_{\mathrm{strict}}\)	Frontier in normal (optimistic) mode and in strict mode.
\(\Missing\), \(\MissingAtoms\)	Unsatisfied effective prerequisite nodes; missing atomic goals of the demand.
\(\operatorname{Cards}(g)\)	Fully resolved valid set of required cards of memory goal \(g\).
\(\operatorname{RecallPassed}(g)\)	Explicitly verified successful completion recall across \(\operatorname{Cards}(g)\).
\(\Score\), \(\Passed\)	Score and pass result of an assessment goal; \(\MaxPoints\) and \(\PassingPoints\) come from the scoring rubric.
\(U\), \(\Own(U)\), \(T(U)\)	Learning unit, directly owned atomic acquisition targets, and total recursive scope including child units; \(\Children(U)\) denotes its required child units.
\(\Complete(U)\)	Recursively derived unit completion; never an additionally persisted unit-mastery state.
\(Q\), \(A_{\mathrm{mem}}\)	In Section 13, the sets of atomic assessment goals and atomic memory goals (written \(K\) for memory goals in earlier standalone drafts).
\(Q_U\), \(A_{\mathrm{mem},U}\), \(P_U\)	Directly owned assessment, memory, and combined protected goal sets; \(A_{\mathrm{mem},U}=\Own(U)\cap A_{\mathrm{mem}}\).
\(\operatorname{Reach}_{\Rat}^{*}(X)\)	The set \(X\) together with all atomic goals its elements directly or transitively require.
\(N_U\)	Open demand: full transitive demand of the unit's still-open own protected goals, without stopping at mastered intermediate goals.
\(I_U\)	Implicit set: directly owned unprotected goals outside \(M\) and \(N_U\). For content goals, it witnesses locally derived independent application. Not an additional stored outcome kind.
\(\operatorname{Cov}(q)\)	Reviewed coverage claim of an assessment \(q\): the goals whose independent application its tasks and normal grading test; under rule R3 never larger than \(\{b\in A\mid(q,b)\in\Rat^+\}\). Grants no recognition by itself.
\(\mathcal U_{\mathrm{valid}}\)	Person- and revision-bound, durably applicable authoritative local recognition contracts; not merely currently visible units or their latest revisions.
\(T_{\mathrm{Plan}}\), \(H_{\mathrm{gesamt}}\)	Planned target goal set and count of its effectively mastered goals.
\(T_\pi\), \(B_\pi\), \(\ell_\pi\)	Planned acquisition targets, entry basis, and optional calendar-bound planning periods; \(n_\pi\) is the number of periods and \(S_i\) is the target set of period \(i\).
\(\Rat^{\pi}\), \(\Frontier_\pi\), \(\ReqAtoms_\pi\)	Plan requirement relation excluding outgoing requirements of the entry basis; corresponding frontier and atomic demand.
\(W^{(r)}\)	Hypothetical mastery state of the witness verification algorithm after \(r\) steps.
\(\Theta_\pi\), \(\widehat t_h(\Theta_\pi)\)	Declared estimation assumptions of the plan and the non-negative duration of activity \(h\) estimated under them; \(\mathcal H_i\) denotes the activities of period \(i\).
\(\Work\), \(\Reserve\), \(\Budget\)	Scheduled workload, protected reserve, and available time budget of a planning period.

Appendix E — Normative Source Mapping and Illustrations

The bilingual chapters under docs/concept/skill-graph/source/ form the sole authored Single-Source baseline for both the MkDocs web pages and the PDF compendia (source/README.md) and connect them to the specialized domain contracts in the repository (GD-D09). Cross-references identify domain ownership rather than claiming that all optional or conceptual parts are already deployed in production.

Part	Normative sources	Role
Formal core and filters (§§1–5)	source/chapters/00-frontmatter.{de,en}.md, source/chapters/01-core.{de,en}.md (spec-manifest.json).	Canonical single-source definition of goals, contains, requires, inheritance, atomic expansion, validity, frontier, and filter modes for Web and PDF.
Layers and migration (§6)	source/chapters/02-context.{de,en}.md; General Goal System and Migration; Canonical Gymnasium Rollout.	Separation of goal, program, placement, competency-axis, and composition-view layers plus rollout policy for canonical skill landscapes.
Views and personal selection (§7)	source/chapters/02-context.{de,en}.md; View Projection and Goal Placement; Levels of Personalization.	Projection contract, projectionRole, personalization Levels 1–4, closed learning scope \(L=\Closure(S)\), and focus switching.
Goal types and mastery evidence (§8)	source/chapters/02-context.{de,en}.md; Node Types; AGENTS.md.	Pedagogical goal types (content, orientation, memory, and assessment goals), verified recall, and binary outcome semantics.
Recursive learning units (§11)	source/chapters/03-learning-units.{de,en}.md (source/README.md); Learning-Unit Design Criteria and Examples.	Canonical single-source definition of recursive LearningUnits, unique direct ownership \(\Own(U)\), derived completion \(\Complete(U)\), and undecided unit-entry policy; design criteria and worked candidates.
Optional time and budget planning (§§9, 12)	source/chapters/04-planning.{de,en}.md; Curriculum Time Axis and Pacing; Unified Learning Plan Status.	Optional calendar periods, entry basis \(B_\pi\), structural completion witness, and workload budget under declared estimation assumptions \(\Theta_\pi\).
Implicit mastery and local contracts (§13)	source/chapters/05-assessment.{de,en}.md; Completion Semantics Clarification (2026-11-04); Exam Preparation Planning.	Canonical single-source specification of local recognition contracts, transitive open demand \(N_U\), implicit set \(I_U\), and global set \(\Meff\) for Web and PDF.
Provenance and learning materials (§10)	Source and Resource Links; Content Integration; Human-Readable Source Rationales.	Strict separation between official curriculum provenance (sourceRef), optional external content packages, and human-readable rationales.
Visualizations and goal books (§10)	Atomic Goal Visualizations; Learning-Goal Book and Evidence-Review Pipeline.	Governance of atomic goal visualizations and the D/P/A/M/V evidence and review pipeline up to maturity level M7.
Validation and curriculum QA (§§3, 10, Appendix A)	Graph Validation Rules; Curriculum Quality Maturity and Routes; Semantic Atomicity Review.	Structural GVR-* rules, content-quality CQR-* rules, promotion chain M0–M7, protected maturity floors, and semantic atomicity review.
Package interchange and licensing (§10)	Dual Curriculum Package Releases; MEM/FWU Roundtrip Plan; LICENSING.md.	Dual JSON and ontology release packages, roundtrip interchange boundaries, and licensing/privacy governance (Apache-2.0 / CC-BY-4.0).
Rule registry, glossary, and build (Appendices A–F)	source/rules-and-gates.json, glossary-and-symbols.json, implementation-status.json, figures.json, Glossary.	145 GD-* rules with traffic-light status, glossary and symbol reference, dated implementation audit, figure attribution, and single-source build contract.

The seven static English-labelled didactic graph illustrations (01-skill-graph-overview.png through 07-filter-modes.png: Figures 1, 2, 3, 4, 6, 8, and 9) originate from docs/concept/skill-graph/assets/graph-definition/ (SkillPilot: Graph Definition, edition of 29 October 2026) and are embedded unchanged with bilingual captions from figures.json. The ten additional bilingual diagrams (00-title-skill-graph.{de,en}.png for the title page and 08-transitive-demand.{de,en}.png through 16-recursive-learning-unit.{de,en}.png covering transitive demand, local recognition, protected support, plan period accounting, calendar planning, atomic expansion, inherited redundancy, shared local assessments, and recursive learning units) are rendered deterministically with Matplotlib as PNG bitmaps by generate-completion-diagrams.py. The formulas and captions are authoritative; the illustrations assert no additional edges or requirements (GD-D10). Technical prose, mathematical specifications, structured source datasets, and build scripts are licensed under Apache-2.0; own didactic illustrations are licensed under CC-BY-4.0 where applicable rights exist. Full license texts and boundaries: LICENSE, LICENSES/CC-BY-4.0.txt, and LICENSING.md. Attribution for the didactic illustrations: SkillPilot, enpasos – Enterprise Patterns & Solutions GmbH, Graph Definition (static original illustrations 01–07: edition of 29 October 2026; bilingual generated diagrams 00, 08–16 and single-source integration: this edition, CC-BY-4.0).

Appendix F — Building and Verifying the Specification

All bilingual web pages and PDF editions are compiled from the sole authored source under docs/concept/skill-graph/source/ (source/README.md, spec-manifest.json). The compiler docs/concept/skill-graph/build_spec.py verifies 1:1 structural parity between German (DE) and English (EN), translates chapters and structured JSON datasets into MkDocs Markdown and LuaLaTeX sources, and embeds both the static original illustrations (01–07) and the deterministically generated bilingual bitmap PNGs (00, 08–16) from docs/concept/skill-graph/assets/graph-definition/. From the repository root:

• python3 docs/concept/skill-graph/build_spec.py --web-and-tex – uses only the Python standard library to verify DE/EN AST parity and generate all bilingual MkDocs web pages plus intermediate LaTeX files under tmp/skill-graph-document/.
• python3 docs/concept/skill-graph/latex/check-assessment-model.py – runs the finite mathematical reference checks for Section 13.
• python3 docs/concept/skill-graph/latex/generate-completion-diagrams.py --check – verifies deterministic bilingual PNG diagram rendering without modifying files.
• python3 docs/concept/skill-graph/build_spec.py --all – runs the parity check, formula model verification, bilingual PNG rendering, Web Markdown emission, and multi-pass LuaLaTeX builds for both the German (skillpilot-graph-definition.pdf, skillpilot-graph-definition.de.pdf) and English (skillpilot-graph-definition.en.pdf) PDFs together with .build.json manifests (GD-D11).

Python 3 alone suffices for --web-and-tex and the formula verifier; full --all runs additionally require Matplotlib and LuaLaTeX with German (ngerman) and English (english) Babel support plus preamble packages. Multi-pass LaTeX compilation resolves cross-references, and generated .build.json manifests bind input files, SOURCE_DATE_EPOCH, tool versions, and output PDF digests via SHA-256. Byte-for-byte PDF identity across different TeX Live versions is not guaranteed. All generated Web Markdown pages, generated diagram PNGs (00, 08–16), intermediate .tex files, PDFs, and build manifests are .gitignored build artifacts and must never be edited by hand or committed to Git.

Complete document verification requires an error-free build, resolved cross-references, inspection for overfull or underfull boxes, and visual inspection of both generated PDFs (GD-D12). Technical specification prose, schemas, and build scripts are licensed under Apache-2.0 (LICENSE); own didactic illustrations are licensed under CC-BY-4.0 (LICENSES/CC-BY-4.0.txt, LICENSING.md). Build and formula checks complement document QA and replace neither runtime tests nor independent subject-matter and empirical validation.

Sources: source/README.md; spec-manifest.json; check-assessment-model.py; LICENSING.md.