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Canonical Gymnasium Mathematics: Sek I Functions / Mappings Audit

Snapshot: 2026-04-05

Purpose:

  • review the current canonical Sek I Funktionen / Zuordnungen inventory after the lower-secondary breadth sweep
  • use reviewed lower-secondary state evidence to test whether the current canonical function cuts are pedagogically stable
  • decide whether the nationwide Sek I Funktionen / Zuordnungen row can be treated as closed for now

Scope

In scope:

  • the canonical Sek-I function and mapping topic surface in
  • curricula/DE/Gymnasium/canonical/DE_DEU_S_GYM_CANONICAL_MATHEMATIK.de.json
  • the reviewed lower-secondary source and mapping evidence across all 16 Bundeslaender

Out of scope:

  • learner-facing composition views
  • Sek-II differential calculus packaging
  • the geometry-owned trigonometric bridge except where it explicitly turns into late Sek-I function corridors

Reviewed source evidence

Hessen (HE)

Observed signal:

  • HE remains the mature baseline for early mappings, proportional reasoning, linear functions, and the first clear algebra/function boundary
  • the Hessen lane confirms that function work can stay visible without pulling core term and equation packaging back into the function topic

Brandenburg and Berlin (BB, BE)

Observed signal:

  • the currently reviewed BB and BE lanes already anchor the shared linear-function core
  • they validate the existence of a stable lower-secondary function spine, but they do not yet force many additional visible subpackages on their own

Bremen (HB)

Observed signal:

  • HB still contributes mostly broad mixed lower-secondary function/algebra corridors
  • Bremen is therefore useful as a packaging pressure test, but not yet as a strong driver for new function atoms

Hamburg (HH)

Observed signal:

  • HH is currently the strongest reviewed late Sek-I function lane
  • the reviewed Hamburg corridors explicitly surface:
  • linear and inverse-proportional relationships
  • representation changes
  • linear functions in applications
  • quadratic functions and optimisation
  • periodic processes
  • exponential growth
  • exponential equations and logarithms
  • change rates and derivative entry

Sachsen (SN)

Observed signal:

  • SN now contributes five explicit function-side evidence lanes:
  • a broad K10 Lernbereich 1 Wachstumsvorgaenge und periodische Vorgaenge strip
  • a late J10 strip inside K10 Lernbereich 4 Funktionale Zusammenhaenge
  • a later K9 Funktionen und Potenzen strip
  • a middle-stage K8 Funktionen und lineare Gleichungssysteme strip
  • an early K6 Zuordnungen in der Umwelt strip
  • the broad Sachsen K10 Lernbereich 1 strip already fits the visible J10 trigonometric/exponential corridor without forcing a separate shared growth-model or periodic-model package
  • the late Sachsen lane mixes periodic / trigonometric / exponential / logarithmic continuation with sequence, limit, and parametrized-curve residue
  • the later Sachsen K9 lane confirms the current quadratic boundary and keeps powers / roots on the existing late number-extension side instead of forcing a separate shared power-function package
  • the middle Sachsen lane confirms the current F3 surface for linear functions and regression while keeping its linear-system residue on the algebra side instead of forcing a new shared mixed package
  • the early Sachsen lane confirms the existing early F1/F2 surface for mappings, representations, and proportional assignments without forcing a separate package for the attached frequency-interpretation residue
  • together these five Sachsen lanes widen the reviewed evidence across the early, middle, later, and late Sek-I function topic without forcing a new shared regression package, a new mixed function-equation package, a new shared power-function package, a separate shared growth/periodicity package, or a new visible sequence / parametric-curve subpackage

Sachsen-Anhalt (ST)

Observed signal:

  • ST now contributes a first reviewed early lower-secondary function lane through JG 5/6
  • the lane splits the broad Sachsen-Anhalt source anchor into an early correspondence strip and a broad direct/inverse-proportionality strip
  • the correspondence strip fits the existing early F1/F2 surface
  • the direct/inverse mix stays acceptable on the existing broad proportional bridge without forcing a separate visible inverse-proportional package
  • together this keeps the canonical early function surface stable and shifts the remaining pressure from packaging to coverage

Rheinland-Pfalz (RP)

Observed signal:

  • RP now contributes a first reviewed lower-secondary function corridor on the explicit Klassenstufen 7 und 8 lane
  • the Rheinland-Pfalz source separates
  • Proportionale und antiproportionale Zuordnungen
  • Lineare Funktionen
  • this makes RP a useful pressure test for whether the visible middle Sek-I function surface can absorb a broad proportional bridge next to a cleaner linear-function strip without forcing another package boundary

Mecklenburg-Vorpommern (MV)

Observed signal:

  • MV now contributes a first reviewed lower-secondary function corridor on the explicit Klasse 7 lane
  • the Mecklenburg-Vorpommern source separates
  • Zuordnungen
  • proportionale und umgekehrt proportionale Zuordnungen
  • this makes MV a useful early-lane pressure test for whether the visible early Sek-I function surface can absorb a broad direct/inverse-proportionality bridge next to a cleaner mappings strip without forcing a separate inverse-proportional package

Current canonical function inventory

The canonical graph is already materially seeded here.

Important current package questions:

  1. early mappings, function concept, and representation change
  2. proportional and inverse-proportional bridges after the algebra A3 decision
  3. linear functions, linear calculations, and the remaining algebra residue inside the J8 calculation strip
  4. quadratic functions versus the equation boundary
  5. late J10 function continuations: periodic, exponential, logarithmic, and derivative-entry corridors

Audit judgment

The canonical Sek-I function topic is not missing a usable backbone. The main remaining risk was packaging, not first missing content.

After the algebra cleanup, and with the new Mecklenburg-Vorpommern early lane connected, the decisive nationwide close-out questions sat in three places:

  1. the J8 linear calculation strip
  2. the quadratic/equation boundary
  3. the late J10 function continuations

Canonical design step executed (2026-04-01)

The first canonical packaging pass is now in place.

Inserted visible package surface in the canonical graph:

  1. F3 linear functions, calculations, and applications
  2. F4 quadratic functions and the equation boundary
  3. F5 late J10 function continuations

Boundary decisions executed:

  • the J8 linear calculation strip was cleaned of the explicit linear-equation residue
  • the quadratic representation corridor was narrowed so that the explicit quadratic-equation branch is visible instead of being hidden inside the same subcorridor
  • late J10 periodic, exponential, logarithmic, and derivative-entry continuations are now grouped as an explicit visible continuation area of the Sek-I function topic
  • F1 and the basic F2 bridge remain materially seeded and were therefore not repackaged again in this step

Recommendation

Do not widen more bundesland mappings on Sek-I functions unless a later reviewed lane or validator finding forces a genuinely new shared package beyond the current surface.

  1. keep the current F1-F5 function surface frozen and use later function evidence only for residue control
  2. move the active lower-secondary breadth wave to Sek I Geometrie / Raum

Exit criteria for this audit

This topic audit is complete when:

Thueringen (TH)

Observed signal:

  • TH now contributes a first reviewed lower-secondary functions corridor on the explicit Klassenstufen 5/6 lane
  • the Thueringen source currently separates
  • Zuordnungen
  • this makes TH a useful pressure test for whether the visible early mapping surface can absorb a clean 5/6 correspondence lane without forcing a broader early function split
  • TH now also contributes a first reviewed broad Klassenstufen 7/8 functions corridor
  • the archived 7/8 source still exposes only one shared Funktionen strip
  • this confirms that the shared broad Sek-I function surface can absorb a reviewed wide 7/8 lane without forcing a sharper canonical split from source granularity alone
  • TH now also contributes a first reviewed broad Klassenstufen 9/10 functions corridor
  • the archived 9/10 source still exposes only one shared Funktionen strip

  • this confirms that the same shared broad Sek-I function surface can absorb a reviewed wide 9/10 lane, while the strip remains deliberately broad because it crosses the visible F4 / late-J10 boundary too broadly for a narrower reviewed split

  • the canonical subpackage boundaries above are either accepted or revised

  • the canonical graph has a stable Sek-I function packaging after the algebra cleanup
  • reviewed nationwide evidence can be described as aligned to the revised packaging or intentionally broader because of source granularity
  • it is clear whether the remaining residue at late J10 depth or around the quadratic/equation boundary requires further canonical restructuring

First reviewed mapping realignment connected (2026-04-01)

  • HH J10 Strukturen und funktionaler Zusammenhang now points to the visible F5 continuation package instead of the older trig/exponential-only trunk
  • HH lineare Funktionen in Anwendungen now points to F3, which better reflects application-heavy linear function work than the older single linear-function atom
  • BB, BE, and HE keep their more precise atomic or already adequate subcorridor targets
  • the remaining broad HB and HH mixed lower-secondary function corridors intentionally stay broader because they still span algebra and function packaging at source level

Sachsen K10, K9, K8, and K6 function-side corridors connected (2026-04-03)

  • SN K10 Lernbereich 1 Wachstumsvorgaenge und periodische Vorgaenge now also points to the visible J10 trigonometric/exponential corridor surface
  • SN K10 Lernbereich 4 Funktionale Zusammenhaenge now points to the visible F5 package surface
  • the split Sachsen source strips on periodic / trigonometric / exponential graphs, logarithmic and exponential-equation work, and the broader sequences / parametrized-curves residue make SN the next explicit late-J10 evidence lane after HH
  • SN K9 Lernbereich 1 Funktionen und Potenzen now also contributes a reviewed later lane: its powers / roots strip stays on the late number-extension package, its power-function strip stays on the existing power-function surface, and its quadratic-function / equation strips confirm the current F4 boundary without forcing another shared package
  • SN K8 Lernbereich 3 Funktionen und lineare Gleichungssysteme now also contributes a reviewed middle-stage lane: its function-concept strip points to the shared early representation surface, its linear-function strip points to the visible F3 surface, its regression strip intentionally stays broad on F3, and its linear-system strip stays on the algebra side
  • SN K6 Lernbereich 2 Zuordnungen in der Umwelt now also points to the early shared function surface via mappings / representations and proportional-assignment work
  • the Sachsen frequency-related residue inside K6 intentionally stays on the broad early function surface instead of forcing a separate shared package at this stage
  • the Sachsen broad K10 Lernbereich 1 growth-and-periodicity anchor intentionally stays on the visible J10 trigonometric/exponential corridor instead of being over-split into separate exponential and periodic subtargets without reviewed source strips
  • the Sachsen late-J10 strip intentionally stays broad on F5, because the current canonical Sek-I function surface does not yet expose a narrower shared package for sequences, limits, and parametrized curves

Sachsen-Anhalt JG 5/6 correspondence / proportionality corridor connected (2026-04-03)

  • the broad ST JG 5/6 source anchor Zuordnungen, direkte und indirekte Proportionalitaet now sits on the visible early function surface via an explicit reviewed corridor
  • its Zuordnungen strip points to the early mapping / representation surface
  • its broad proportionality strip intentionally stays on the existing proportional bridge instead of forcing a separate visible inverse-proportional package just from one mixed source lane
  • ST therefore confirms the early F1/F2 surface without reopening the current Sek-I function packaging

Sachsen-Anhalt JG 7/8 linear-functions corridor connected (2026-04-03)

  • the ST JG 7/8 strip Lineare Funktionen now points to the visible F3 linear-functions surface
  • the reviewed ST lane therefore confirms the current F3 cut without pulling algebra residue back into the function topic

Sachsen-Anhalt JG 9 quadratics corridor connected (2026-04-03)

  • the ST JG 9 strip Quadratische Gleichungen und quadratische Funktionen now points to the visible F4 quadratics surface
  • the reviewed ST lane therefore confirms the current F4 boundary without forcing a separate mixed quadratics package beyond the existing function/equation cut

Sachsen-Anhalt JG 10 function-classes corridor connected (2026-04-03)

  • the ST JG10 strip Funktionsklassen now points to the broad visible J10 trig/exponential corridor
  • the reviewed ST lane therefore confirms the existing late-J10 continuation surface without forcing a separate mixed function-classes package

Mecklenburg-Vorpommern Klasse 8 linear-functions corridor connected (2026-04-04)

  • the MV Klasse 8 lane now contributes a reviewed middle-stage function corridor with separate strips for the broad Funktionen concept work and for Lineare und konstante Funktionen
  • the concept strip now points to the visible early function-concept surface
  • the linear strip now points to the visible F3 linear-functions surface
  • the MV lane therefore confirms the current middle Sek-I function cut without pulling algebra residue back into the function topic

Mecklenburg-Vorpommern Klasse 9 quadratics corridor connected (2026-04-04)

  • the MV Klasse 9 lane now contributes a reviewed explicit quadratics corridor with separate strips for Quadratische Funktionen and Quadratische Gleichungen
  • the function strip now points to the visible late Sek-I quadratic representations surface
  • the equations strip now points to the explicit quadratic-equations branch
  • the MV lane therefore confirms the current F4 boundary without forcing a separate mixed quadratics package beyond the existing function/equation cut

Mecklenburg-Vorpommern Klasse 10 late function corridor connected (2026-04-04)

  • the MV Klasse 10 lane now contributes a reviewed late function corridor with separate strips for Exponential- und Logarithmusfunktionen and Sinusfunktionen
  • both reviewed strips intentionally stay on the broad visible J10 trig/exponential continuation surface
  • the MV lane therefore confirms the existing late-J10 continuation surface without forcing a separate mixed late-function-classes package

Rheinland-Pfalz Klassenstufen 7 und 8 function corridor connected (2026-04-03)

  • the broad RP Klassenstufen 7 und 8: Funktionaler Zusammenhang anchor now points to the visible F3 surface instead of the early general function surface
  • its reviewed Proportionale und antiproportionale Zuordnungen strip points to the broad proportional bridge
  • its reviewed Lineare Funktionen strip points to the explicit linear-function surface
  • the RP lane therefore confirms the visible middle Sek-I function cut without forcing a new package boundary between proportional assignments and linear functions

Saarland (SL)

Observed signal:

  • SL now contributes a first reviewed lower-secondary function corridor on the explicit Klassenstufe 7 lane
  • the Saarland source separates
  • Zuordnungen
  • Proportionale Zuordnungen
  • Umgekehrt proportionale Zuordnungen
  • this keeps the broad Klassenstufe 7 function parent on the visible early function corridor, places the Zuordnungen strip on the early mapping / representation surface, and keeps both proportionality strips on the broad proportional bridge without forcing a separate visible inverse-proportional package
  • the Saarland lane therefore confirms the early Sek-I function surface without reopening the current packaging

Saarland Klassenstufe 8 broad functions corridor connected (2026-04-04)

  • the broad SL Klassenstufe 8: Strukturen und funktionaler Zusammenhang anchor now carries a first reviewed corridor on the shared Sek-I function-foundations surface
  • the current Saarland grade-8 source still exposes only one shared functions strip
  • this confirms the shared broad Sek-I function-foundations surface can absorb an explicit reviewed class-8 lane without forcing a sharper state-local split

Saarland Klassenstufe 9 broad functions corridor connected (2026-04-04)

  • the broad SL Klassenstufe 9: Strukturen und funktionaler Zusammenhang anchor now carries a first reviewed corridor on the shared broad Sek-I function surface
  • the current Saarland grade-9 source still exposes only one shared functions strip
  • this confirms the shared broad Sek-I function surface can absorb an explicit reviewed class-9 lane without forcing a sharper state-local split

Saarland Klassenstufe 10 broad functions corridor connected (2026-04-04)

  • the broad SL Klassenstufe 10: Strukturen und funktionaler Zusammenhang anchor now carries a first reviewed corridor on the shared broad Sek-I function-foundations surface
  • the current Saarland grade-10 source still exposes only one shared functions strip
  • this confirms the shared broad Sek-I function-foundations surface can absorb an explicit reviewed class-10 lane without forcing a sharper state-local split

Additional reviewed close-out evidence (BW, BY, NI, NW, SH)

Baden-Wuerttemberg (BW)

Observed signal:

  • BW now contributes a reviewed entry corridor plus a reviewed JG7/8 linear follow-on corridor on the shared Sek-I function spine
  • the first Baden-Wuerttemberg cut already reopens the shared mappings, representation, proportionality, and function-concept surface with one explicit retained prerequisite bridge instead of forcing a new canonical package
  • the adjacent JG7/8 cut adds reviewed graph-reading, linear-description, line-equation, and rate-of-change interpretation strips, while direct function-value and variable-term prerequisites remain explicit retained bridges
  • together this confirms the current early and middle Sek-I F1/F2/F3 surface without forcing a Baden-Wuerttemberg-specific function split

Bayern (BY)

Observed signal:

  • the reviewed Bavaria M8-M10 tranches now cover introductory function work, linear functions, quadratics, exponential/logarithmic continuation, and sinus/cosinus continuation on the existing shared J8-J10 function surface
  • M8 1 Funktion und Term and M8 2 Lineare Funktionen intentionally stay conservative broad bridges where the current canonical function corridor is still coarser than the Bavaria source wording, while the explicit function-concept and remaining linear-function leaves already align to the existing shared surface
  • M9 2 Quadratische Funktionen confirms the current F4 quadratic/equation boundary through mixed corridor-level and leaf-level attachments
  • M10 confirms the existing late J10 exponential, logarithmic, and trigonometric continuation corridor without forcing another visible split beyond the current late function surface

Niedersachsen (NI)

Observed signal:

  • NI now contributes a clean prerequisite-backed lower-secondary function corridor on the shared proportional and linear spine
  • Zuordnungen analysieren is already exact, while proportionality, representation work, and linear-function follow-ons are reviewed partial bridges on the same shared surface
  • the remaining Niedersachsen breadth mismatch is documented as source-granularity residue rather than as evidence for a missing canonical atom
  • together this confirms the current early and middle Sek-I function surface for Niedersachsen without reopening packaging

Nordrhein-Westfalen (NW)

Observed signal:

  • the NRW prerequisite widening removed the hidden-entry blocker, and the first reviewed exact bridges now activate both mappings and function-concept work on real lower-secondary source support
  • reviewed follow-on bridges now also cover function values, proportionality, line description, and linear-parameter interpretation without introducing new applicability overrides
  • the remaining Nordrhein-Westfalen mismatch is explicitly retained as source breadth, not treated as evidence for a missing shared package
  • together this confirms the current F1/F2/F3 surface and keeps the next NRW work out of lower-secondary function packaging

Schleswig-Holstein (SH)

Observed signal:

  • the Schleswig-Holstein lower-secondary source snapshot is now refined across all official year bands instead of relying only on one broad parent lane
  • the reviewed SH function-side mappings now cover simple mappings, representation change, proportional and inverse-proportional work, linear functions, quadratics, trigonometric functions, exponential growth, and logarithms
  • the late J10 SH strips intentionally stay on the broad visible trigonometric/exponential continuation corridor rather than forcing a sharper state-local split
  • there are now no remaining unmapped SH Sek-I source atoms or source clusters in the active lane, and the remaining SH function residue is broad-source residue rather than a missing canonical split

Close-out judgment (2026-04-05)

  • with BW, BY, NI, NW, and SH now folded into the reviewed evidence base, the Sek I Funktionen / Zuordnungen row can be treated as coverage-closed for the current nationwide sweep
  • the visible F1-F5 surface absorbs all 16 state lanes without forcing a new shared regression package, a standalone inverse-proportional package, a mixed function/equation package, a separate power-function package, or a sharper late-J10 split
  • the remaining broad HB/HH overview lanes, the Sachsen sequence / parametrized-curve residue, and similar wide state-local strips are better described as accepted source-granularity residue than as open canonical gaps
  • the active canonical corridor SEK1.J10.FUNCTION_FAMILIES can therefore move from active to completed; the remaining late-function breadth is accepted source-granularity residue and should reopen only if a later reviewed lane or validator finding exposes a real shared gap beyond the current F1-F5 surface
  • the next lower-secondary breadth step should therefore move to Sek I Geometrie / Raum, not back into function package churn