A-Math OS — Z0–Z3 Directory Table (Skills → Failure Modes → Sensors → Repairs)

Unified Almost-Code Directory Spec (Registry-Clean) v0.2

PAGE-ID: EDK-DIR-AMTH-Z0Z3-v0.2
Role: DIR.TopicMap + SkillGraph + FailureAtlas + SensorPack + RepairLoops + TrapGovernance
Purpose: publishable reference page that makes Secondary A-Math executable: algebra kernel → chains → traps → timed stability.

0) Definition Lock

A-Math OS (Secondary) := the capability system for transformation chain reliability under time pressure, with explicit trap governance.

Z-ladder meaning (LOCK):

  • Z0 = algebra kernel atoms (transforms must be fast + clean)
  • Z1 = method signals + constraints (choose + set domains early)
  • Z2 = transfer + multi-step chains (variation survivability)
  • Z3 = timed stability + trap audit (exam reliability)

1) Directory Table Schema (copy-paste standard)

NodeID | Z | Skill / Topic Family | Typical Failures | Primary Sensors | Repairs (Labs + Rules)

2) A-Math OS Directory (Z0–Z3)

Z0 — Algebra Kernel (Non-Negotiables)

Node: Algebra Manipulation Kernel (Expand / Factorise / Simplify)

  • NodeID: AMTH-Z0-KRN-ALG
  • Failures: sign slips; messy expansion; wrong factorisation family; algebra explosion (B2/B5)
  • Sensors: Foundation Fluency; Messiness/Hesitation; Error Recurrence
  • Repairs: A-Math Kernel Daily 15 + Chain Discipline format (one step/line) + retest ≤ 7 days

Node: Algebraic Fractions & Restrictions

  • NodeID: AMTH-Z0-KRN-FRAC
  • Failures: common denominator errors; cancels illegally; forgets denom ≠ 0 (TR2) (B2/TR2)
  • Sensors: Foundation Fluency; Trap activation rate
  • Repairs: Kernel micro-set + “Restriction first” rule + Trap Lab (domain)

Node: Indices / Surds / Rationalisation (Exactness)

  • NodeID: AMTH-Z0-KRN-IND-SURD
  • Failures: wrong index law; surd simplification errors; premature decimals; rationalisation mistakes (B2/B5/TR4)
  • Sensors: Foundation Fluency; Error Recurrence
  • Repairs: Fluency Lab + Exactness checklist (TR4) + retest

Z1 — Method Signals + Constraints (Choose + Fence)

Node: Quadratic Structure (Recognise + Choose Tool)

  • NodeID: AMTH-Z1-QUAD-SIGNAL
  • Failures: wrong tool choice; cannot see factorable form; wastes time expanding (B3/B2)
  • Sensors: Method Selection; Bandwidth (structure recognition)
  • Repairs: Method Signal Lab (no solving) + Variation Ladder (forms) + “simplify before expand” rule

Node: Domain / Constraint Setting (Global Trap Control)

  • NodeID: AMTH-Z1-CONSTRAINT-FENCE
  • Failures: ignores denom ≠ 0; loses valid range; extraneous roots survive (TR2/TR3)
  • Sensors: DriftRisk; Trap activation frequency
  • Repairs: Trap Lab + “Constraints written before transforms” rule + substitution validation

Node: Trig Identities & General Solutions (Readiness)

  • NodeID: AMTH-Z1-TRIG-READY
  • Failures: wrong identity; missing general solutions; wrong domain; algebra errors in trig (TR4/TR2)
  • Sensors: Method Selection; Constraint sensor; Error Recurrence
  • Repairs: Identity signal drills + constraint-first + mini timed sets

Z2 — Transfer + Multi-Step Chains (Chain Survivability)

Node: Transformation Chain Discipline (Multi-Step Reliability)

  • NodeID: AMTH-Z2-CHAIN-DISC
  • Failures: chain break; loses equal signs; skips steps; messy working makes errors invisible (B2→B5)
  • Sensors: Messiness; Error Recurrence; Speed decay
  • Repairs: Chain Discipline Lab (rewrite messy solutions) + one operation per line + retest

Node: Variation Ladder (A-Math Transfer Engine)

  • NodeID: AMTH-Z2-TRANSFER-VAR
  • Failures: perfect on drills, fails on changed form; freezes when question is unfamiliar (B3 hidden B1)
  • Sensors: Transfer/Variation Sensor
  • Repairs: Variation Ladder Lab (V0→V4) + weekly mixed retest

Node: Differentiation Pipeline (Algebra + Method)

  • NodeID: AMTH-Z2-DIFF-PIPE
  • Failures: algebra swamp; wrong differentiation step; cannot interpret gradient/stationary; chain errors cascade (B2/B3/TR1)
  • Sensors: Fluency; Method Selection; Error Recurrence
  • Repairs: simplify-first rule + differentiation mini chains + timed stability minis

Z3 — Timed Stability + Trap Audit (Exam Reliability)

Node: Timed Mini-Paper Stability (Pace Budget)

  • NodeID: AMTH-Z3-PACE-STABLE
  • Failures: slow transforms; over-expands; unfinished paper; late careless spike (B6→B5)
  • Sensors: Pace Budget; DriftRisk
  • Repairs: Timed Stability Lab (smaller scope) + skip/return policy + reserved final audit time

Node: Trap Audit Routine (End-of-Question Validation)

  • NodeID: AMTH-Z3-TRAP-AUDIT
  • Failures: extraneous roots; invalid domains; missed restrictions; wrong final form (TR2/TR3/TR4)
  • Sensors: Trap frequency; DriftRisk
  • Repairs: Trap Lab + mandatory audit checklist:
  • domain check
  • substitution check
  • sign scan
  • final form check

Node: O-Level A-Math Wrapper Readiness

  • NodeID: AMTH-Z3-OLEVEL-WRP
  • Failures: collapse only in full paper even if topics “known” (hidden drift)
  • Sensors: Math Gauge + Monitoring Math Health (trend)
  • Repairs: O-Level A-Math Wrapper cycle (kernel + traps + timed minis + transfer ladder)

3) Trap Taxonomy (baked into A-Math OS)

Every A-Math topic must bind at least one trap drill:

  • TR1 Sign trap
  • TR2 Constraint/domain trap
  • TR3 Extraneous root trap
  • TR4 Form/exactness trap

A-Math without trap training produces high DriftRisk.

4) Minimal Sensor Pack (A-Math footer)

Every A-Math topic page should link to:

  • Foundation Fluency
  • Method Selection
  • Representation (when word problems / modelling appear)
  • Error Recurrence
  • Pace Budget
  • Transfer/Variation
  • Trap Audit checklist

5) Why this Directory is Important (eduKateSG “completion” logic)

This directory makes A-Math teachably reliable by:

  1. making the algebra kernel explicit (the real bottleneck)
  2. enforcing chain discipline (transformation reliability)
  3. enforcing trap governance (silent-failure prevention)
  4. connecting everything to timed stability (exam reality)

That’s the difference between “topic coverage” and an operating system.

Start here if you want the full sequence:

Vocabulary OS Series Index:
https://edukatesg.com/vocabulary-os-series-index/

Fence English Learning System: 

eduKateSG Learning Systems: 

Recommended Internal Links (Spine)

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