Canonical Gymnasium Mathematics: Sek II Analytic Geometry / Vectors Audit

Snapshot: 2026-04-05

Purpose:

• review the current canonical Sek II Analytische Geometrie / Vektoren inventory after the packaging passes for lower-secondary topics, analysis, and stochastics
• use already reviewed upper-secondary state evidence to test whether the current canonical space/vector cuts are pedagogically stable
• define the next canonical work packages for Sek II Analytische Geometrie / Vektoren

Scope

In scope:

• the canonical upper-secondary analytic-geometry / vector topic surface in
• curricula/DE/Gymnasium/canonical/DE_DEU_S_GYM_CANONICAL_MATHEMATIK.de.json
• the already reviewed upper-secondary source and mapping evidence from
• BB
• BE
• BY
• HB
• HH
• HE
• MV
• RP
• SN
• ST
• SH

Out of scope:

• direct canonical JSON refactoring in this step
• learner-facing composition views
• the matrix / transition-process branch except where the current Q2 overview packaging still mixes with it

Reviewed source evidence

Hessen (HE)

Observed signal:

• HE remains the mature donor baseline for the upper-secondary space/vector branch
• the Hessen lane confirms that the canonical graph already has real material for vector basics, scalar-product reasoning, line/plane equations, positional relations, distances, angles, and reflections

Bremen (HB)

Observed signal:

• HB already contributes reviewed corridors for:
• LA1 vector basics
• AG5 reflections, distances, and angle reasoning
• Bremen therefore pressure-tests whether the canonical graph separates
• vector entry,
• line/plane relations,
• metric reasoning,
• and reflection/application work clearly enough

Hamburg (HH)

Observed signal:

• HH already contributes reviewed corridors for:
• Modul 3 vector and coordinate basics
• Modul 6 lines and planes in space
• Hamburg increases pressure on whether the current visible split between
• vector coordination and scalar-product basics
• line/plane representations and positional relations
• distances, angles, and reflections is already stable enough

Schleswig-Holstein (SH)

Observed signal:

• SH already contributes reviewed geometry corridors for:
• E vector basics, line descriptions, and line relations
• Q1 scalar product, planes, vector product, line-plane relations, and distances
• Schleswig-Holstein confirms that the frozen AGV surface already absorbs both an earlier entry corridor and a later mixed geometry corridor without forcing another visible bridge package

Sachsen (SN)

Observed signal:

• SN now adds a first reviewed upper-secondary geometry corridor directly on the broad upper snapshot
• the Sachsen Grundkurs split cleanly separates:
• point/vector entry and elementary vector operations
• line and plane equation forms
• positional relations and intersections
• scalar/vector-product-based metric foundations
• distances and intersection angles
• Sachsen therefore supports the tightened visible AGV1-AGV5 surface without forcing another package-level bridge

Sachsen-Anhalt (ST)

Observed signal:

• ST now adds a first reviewed upper-secondary geometry corridor on top of the newly mapped GA / EA snapshot lane
• the Sachsen-Anhalt source separates both demand levels into:
• Geraden und Ebenen
• Kreise
• these reviewed corridors are still broad and do not expose the tighter AGV3-AGV5 split directly, but they fit the frozen visible space-geometry surface without forcing another bridge package

Rheinland-Pfalz (RP)

Observed signal:

• RP now adds a first reviewed upper-secondary Raum und Form split directly on the Grundfach / Leistungsfach lane
• the Rheinland-Pfalz source explicitly separates
• matrix/application work in A1
• broad Geraden und Ebenen im Raum work in A2
• this makes RP a useful pressure test for whether the frozen visible space-geometry surface can absorb broad geometry overview corridors without forcing another package-level bridge

Mecklenburg-Vorpommern (MV)

Observed signal:

• MV now adds a first reviewed broad upper-secondary geometry corridor on the active qualification-phase snapshot lane
• the current Mecklenburg-Vorpommern source still exposes the space branch only through the mixed bands Vektoren und Matrizen and Geraden und Ebenen
• MV therefore confirms that the frozen visible shared space-geometry surface can absorb a reviewed broad corridor without forcing a sharper AGV3-AGV5 bridge package

Additional reviewed close-out evidence (NI, NW)

Observed signal:

• NI upper-secondary geometry residue is fully closed on source-goal level across the retained geometry / coordinate / representation leaves, and the lane is now explicitly classified as a stable broad reviewed comparison lane
• NW already carries the first explicit upper-secondary geometry / linear-algebra entry strip plus multiple later reviewed space-geometry strips, and the retained plane-form, point/vector, line/plane, reflection, and distance/application residues now sit on the matching canonical space-geometry surfaces instead of broad parent leftovers

Judgment:

• together with the already reviewed BB / BE realignment pass already recorded below, these lanes remove the remaining n/s debt for the upper-secondary analytic-geometry row
• they do not force a new visible package beyond the current AGV1-AGV5 surface

Current canonical analytic-geometry inventory

The canonical graph is already materially seeded here.

Important current package surfaces:

1. Analytische Geometrie, lineare Algebra und vertiefte Analysis
2. Orientieren und Bewegen im Raum
3. Vektoren im Raum beschreiben und darstellen
4. Skalarprodukt geometrisch deuten und berechnen
5. Geraden und Ebenen im Raum
6. Schnittwinkel zwischen geometrischen Objekten berechnen
7. Punkte an Ebenen spiegeln

Audit judgment

The canonical Sek-II analytic-geometry topic is not missing a backbone. The main risk is packaging, not first missing content.

The most relevant tensions are:

1. vector entry and metric/skalar-product reasoning may still sit too close together in the visible surface
2. line/plane representation work and positional-relation work may still be bundled more broadly than ideal
3. metric continuations such as distances, angles, and reflections need an explicit check for whether they already sit at the right visible depth
4. the broad Q2 overview node still mixes analytic geometry with matrices and advanced analysis more strongly than the final steady-state view should

Findings

1. Vector basics should stay visible as a separate package

The reviewed state evidence, now also including the broad Mecklenburg-Vorpommern corridor, supports a stable common entry corridor for:

• points and vectors in space
• coordinate representations
• elementary vector operations
• first geometric interpretations of vectors

This package surface should remain explicit and should not be absorbed into later line/plane or reflection corridors.

2. Scalar-product reasoning is a real bridge package

The reviewed lanes support a shared visible bridge for:

• orthogonality
• lengths and projections
• angle reasoning based on scalar products

This bridge is pedagogically important because it sits between vector basics and later metric geometry in space.

3. Line/plane representation and relation work need a deliberate boundary check

The current graph already has material for:

• line equations
• plane equations
• intersections and positional relations

The open question is whether the visible package surface separates these ideas cleanly enough, or whether one broader corridor still hides too much structure.

4. Distances, angles, and reflections should likely remain explicit late continuations

The current graph already exposes late space-geometry surfaces for:

• distance calculations
• intersection angles
• reflections across planes

That separation is useful and should be treated as materially real unless reviewed residue proves otherwise.

Proposed canonical work packages

For Sek II Analytische Geometrie / Vektoren, use these work packages:

1. AGV1 Points, vectors, coordinates, and elementary vector operations
2. points and vectors in space
3. coordinate representations

4. elementary vector operations and first geometric interpretation

5. AGV2 Scalar product, orthogonality, lengths, and first metric reasoning

6. scalar product
7. angles, lengths, projections

8. metric bridge from vector basics into later space geometry

9. AGV3 Lines, planes, and equation forms

10. parametric line equations
11. plane equation forms

12. converting and interpreting representations

13. AGV4 Positional relations and intersections in space

14. line-line, line-plane, and plane-plane relations

15. intersections and classification of spatial configurations

16. AGV5 Distances, intersection angles, reflections, and applications

17. distances in space
18. angles between geometric objects
19. reflections across planes
20. later application-oriented interpretation

Design step executed

The first canonical Sek-II analytic-geometry packaging pass is now in place.

Accepted package surface:

1. AGV1 is now tightened as Punkte, Vektoren und Bewegungen im Raum.
2. AGV2 is now visible as Skalarprodukt, Orthogonalitaet und metrische Grundlagen im Raum.
3. AGV3 is now visible as Geraden, Ebenen und Gleichungsformen im Raum.
4. AGV4 is now visible as Lagebeziehungen und Schnittmengen im Raum.
5. AGV5 is now visible as Abstaende, Schnittwinkel, Spiegelungen und Anwendungen im Raum.

Important boundary decision:

1. the former mixed space/vector corridor has been tightened instead of widened further
2. the former broad Geraden und Ebenen corridor now carries representation work only
3. late metric and application work is now kept separate from general relation work
4. no new atomic wave was created; only visible package surfaces were clarified

First reviewed realignment judgment

The reviewed BB / BE / BY / HB / HH / HE / RP / SH / SN / ST pressure test does not force another canonical package split.

Observed result:

1. HB now aligns more cleanly because the reviewed AG5 corridor no longer hangs on the representation package and instead lands on the late metric/application package
2. HH now aligns more cleanly because
3. Modul 3 sits on the tightened vector-entry package
4. the broad Modul 6 parent no longer hangs on the line/plane representation package
5. late Modul 6 residue is distributed over the reviewed relation and metric/application packages
6. HE remains exact enough that no package-level correction is needed
7. SN now fits the frozen visible AGV surface cleanly enough on a newly opened GK corridor:
8. vector entry lands on AGV1
9. line/plane forms land on AGV3
10. positional relations land on AGV4
11. scalar-product and orthogonality work land on AGV2
12. distances and angles land on AGV5
13. ST now fits the frozen visible AGV surface cleanly enough on newly opened GA/EA overview corridors:
14. broad Geraden und Ebenen and Kreise parents can stay on the shared upper-secondary space corridor

15. the source still does not force a sharper package-level bridge between analytic circles and the existing AGV3-AGV5 surface

16. RP now fits the frozen visible AGV surface cleanly enough on newly opened GF/LF Raum und Form splits:

17. the broad Geraden und Ebenen im Raum children can stay on the shared upper-secondary space corridor
18. the source explicitly keeps the parallel matrix/application work outside the geometry child lanes
19. the reviewed split still does not force a sharper package-level bridge between the broad overview corridor and AGV3-AGV5

Interpretation:

The revised visible surface survives the first reviewed realignment pass without forcing a new shared canonical package. The remaining tension is now mainly source-granularity residue, not an obvious canonical gap.

Recommendation

Do not widen more bundesland mappings on Sek-II analytic geometry unless another reviewed lane exposes a concrete shared gap.

Open questions now concentrated here:

1. Is AGV2 sufficient as the metric bridge, or do later reviewed lanes still mix it too strongly with AGV5 residue?
2. Are AGV4 and AGV5 now stable enough as separate visible continuations, or will broader multi-state residue force another cut?
3. Does another reviewed lane require a clearer visible bridge between broad space-geometry overview nodes and the now tighter late packages?

Recommended next concrete step

Freeze package churn here unless another reviewed lane exposes a concrete shared gap.

If the next upper-secondary topic is opened, prefer:

1. Sek II Lineare Algebra / Matrizen

Keep AGV1 stable unless later reviewed residue shows that the tightened entry package still leaks too much later metric or relation material.

Close-out judgment (2026-04-05)

The current Sek-II analytic-geometry / vector sweep can now be treated as closed.

Why this is now strong enough:

1. the accepted visible package surface stays at AGV1-AGV5
2. the corresponding workboard row now has no remaining n/s cells
3. the reviewed BB / BE / BY / HB / HH / HE / MV / NI / NW / RP / SH / SL / SN / ST / TH evidence does not force a new shared canonical package
4. the remaining tension is documented as source-granularity residue rather than an open package blocker

Operational consequence:

1. keep Sek II Analytische Geometrie / Vektoren in residue-control mode only
2. do not schedule another dedicated package pass for this topic
3. reopen only if a later reviewed lane or validator finding exposes a real shared canonical gap

Exit criteria for this audit

This topic audit is complete when:

1. the canonical subpackage boundaries above are either accepted or revised
2. the canonical graph has a stable Sek-II analytic-geometry packaging
3. reviewed state evidence across the resolved upper-secondary analytic-geometry lanes can be described as aligned to the revised packaging or intentionally broader because of source granularity
4. it is clear whether distances, angles, and reflections require further visible separation from general line/plane relations

Saarland (SL)

Observed signal:

• SL now adds a first reviewed upper-secondary geometry corridor on top of the newly mapped Einführungsphase, Hauptphase G-Kurs, and Hauptphase Leistungskurs snapshot lane
• the Saarland source separates the corridor explicitly into
• G-Kurs: Vektoren and G-Kurs: Vektorielle Untersuchung geometrischer Strukturen
• Leistungskurs: Vektoren and Leistungskurs: Vektorielle Untersuchung geometrischer Strukturen
• the separate Kreis und Kugel Pflichtbereich remains outside this first reviewed corridor, so Saarland currently pressure-tests the shared vector / line-plane / positional-relation surface rather than a broader optional geometry extension
• Saarland therefore supports the frozen visible upper-secondary space-geometry surface without forcing another package-level bridge

Baden-Wuerttemberg upper-secondary geometry parents connected (2026-04-04)

• the broad BW course-stage geometry / spatial-model corridor now sits on the shared Sek-II space/matrix-model surface
• the BW Basisfach and Leistungsfach geometry parents now also sit on that same broad surface
• the two BW application parents on planes, reflections, motions, and volumes deliberately stay on the shared surface instead of forcing a sharper package-level bridge