UK A-Level Maths OS — Almost-Code (v1.1)

SYSTEM: Education OS
SUBJECT MODULE: UK A-Level Maths OS
PLUGS INTO:
  - UK Verification OS: A-Level OS (Module B)
  - UK Subject OS Plug-In Template
ZOOM LINK: Z6 (UK gate) ↔ Z5 (city execution) ↔ Z0 (student execution)
REFERENCE: CivOS (Phase P0–P3)
PRIMARY FUNCTION:
  Convert abstract mathematical structures into stable multi-step reasoning,
  transferable problem solving, and timed endurance across full papers.

P3 (A-LEVEL MATHS):
  The student can:
    - retrieve core methods without cues (P-S1)
    - sustain long reasoning chains without breaking (AL-S1)
    - transfer across novel wrappers and mixed-topic synthesis (AL-S2, P-S4)
    - execute under time with low volatility (P-S5)
    - maintain endurance (late-paper error density stays low) (AL-S3)
    - recover after mistakes using a recovery protocol (R4)

CORE SPINE (A-LEVEL MATHS):
  SP1: Algebra primitives (manipulation, equations, inequalities, functions)
  SP2: Graphs + transformations + interpretation
  SP3: Calculus (differentiation/integration) as mechanism, not recipe
  SP4: Sequences/series and proof-like reasoning where relevant
  SP5: Trigonometry as structure (identities, equations, modelling)
  SP6: Modelling + multi-topic synthesis (the exam signature)

FAILURE_MODES:
  AM-F01: Silent prerequisite gaps (GCSE success masked missing spine)
  AM-F02: Chain-break instability (multi-step reasoning collapses)
  AM-F03: Transfer collapse on synthesis questions (GF9)
  AM-F04: Timed endurance collapse (late-paper fatigue → errors spike)
  AM-F05: Speed-chasing (fast practice, weak concept spine) (GF2→GF9)
  AM-F06: Error-repeat density (same mistakes recur) (no closure loop)
  AM-F07: Late repair trap (repair starts near exam season) (GF7)
  AM-F08: Topic partitioning (can do topics alone, fails when mixed) (GF9)
  AM-F09: Autonomy scheduling collapse (missed weeks → ramp panic)
  AM-F10: Narrative substitution (“work harder”) replaces sensors (GF10)

PHASE SENSORS:
  P-S1 retrieval reliability (methods without cues)
  P-S2 error histogram (concept vs execution vs attention)
  P-S3 time-to-solve distribution (median + worst-case)
  P-S4 transfer check (wrapper variation)
  P-S5 timed stability variance (volatility across full papers)

A-LEVEL GATE SENSORS:
  AL-S1 chain-length stability
  AL-S2 synthesis transfer score
  AL-S3 endurance stability (late-paper error density vs early)
  AL-S5 repair latency vs exam horizon
  AL-S6 self-scheduling stability (weekly plan variance)

MATHS SUBJECT SENSORS:
  AMS-S1: spine integrity score (SP1–SP6 coverage completeness)
  AMS-S2: chain-break location map (where steps fail)
  AMS-S3: mixed-paper selection accuracy (choosing correct method under mix)
  AMS-S4: modelling robustness (worded/graphical → equation → solve)
  AMS-S5: checking discipline rate (errors caught before final)

REPAIR LOOPS (A-LEVEL MATHS):
GR1 Buffer Build Loop (spine rebuild):
  - rebuild SP1–SP2 first if weak
  - then calculus as mechanism
  - spaced retrieval until stable
GR3 Transfer Loop (synthesis training):
  - same concept across wrappers
  - then mixed-topic synthesis sets
  - weekly synthesis audit (AL-S2)
GR4 Time-Pressure Loop (exam execution):
  - timed micro-sets
  - recovery protocol after mistakes
  - speed floor + accuracy floor
GR2 Error-Log Loop (closure):
  - tag chain-break errors (AMS-S2)
  - micro-drill root cause
  - timed retest until repeat rate collapses
GR6 Continuity Loop (autonomy stability):
  - weekly minimums locked
  - consolidation sessions
  - prevent “missed month” collapse
GR8 Early-Warning Thresholds:
  - trigger repairs before exam season
GR11 Pressure-Safety Scheduling (optional but powerful):
  - cap load below recovery capacity
  - protect consolidation time to prevent burnout tail

EARLY WARNING (A-LEVEL MATHS):
  Trigger repair immediately if any are true:
    - AL-S1 chain-length stability drops (can’t sustain multi-step)
    - AL-S2 synthesis transfer score below threshold
    - AL-S3 late-paper error density spikes
    - P-S5 volatility high across full papers
    - AL-S6 weekly plan variance high (autonomy drift)
    - AMS-S1 spine integrity score incomplete (SP1–SP2 not stable)

FAILURE TRACE:
  AM-F01 prerequisite gaps →
  AM-F02 chain-break instability →
  AM-F08 topic partitioning →
  AM-F03 synthesis transfer collapse →
  AM-F04 timed endurance collapse →
  AM-F07 late repair trap →
  A-Level gate failure
REPAIR TRACE:
  GR1 spine rebuild (SP1–SP2 → SP3...) →
  GR3 synthesis transfer audits →
  GR4 timed stability + recovery protocol →
  GR2 error closure loops →
  GR6 continuity stabilisation + (GR11 pressure safety) →
  P3 reliability across full papers

PROOF OF STABILITY (P3):
  - AL-S1 chain-length stable at target depth
  - AL-S2 synthesis transfer above threshold across 3 wrapper families
  - AL-S3 late-paper error density does not spike
  - P-S5 volatility low across 3 full timed papers
  - AMS-S3 method-selection accuracy high on mixed papers
  - AMS-S5 checking discipline catches a meaningful fraction of errors

PORTS:
  - UK Verification OS: A-Level OS (Module B)
  - UK Verification OS: GCSE OS (Module A) (prereq reference)
  - UK Subject OS Plug-in Template
  - London City Education OS (Z5)
  - W4 Global Failure Library (GF*)
  - W5 Global Repair Library (GR*