NODE.ID: MATHOS-00
NODE.TYPE: canonical_supply_lane
SUBOS.ID: MATHOS
SUBOS.NAME: Mathematics OS
PARENT.OS: EDUOS
SUBOS.CLASS: capability_supply_line
SUBOS.PURPOSE: regenerate_quantitative_reasoning_for_civilisation_functions
AXIS: time
VERSION: v0.1

DEF.MATHOS:
  Mathematics OS is the education-supplied capability pipeline that regenerates
  quantitative reasoning, symbolic manipulation, and formal modelling needed to
  operate and upgrade civilisation systems (trade, engineering, science, finance,
  governance, logistics, and prediction) across generations.
RULE:
  mathematics is not primarily a school subject; it is a civilisation supply line.

MATHOS.ORIGIN.DRIVERS:
  - counting_and_allocation (food, resources, fairness)
  - measurement (land, tools, construction, time)
  - trade_and_exchange (value, equivalence)
  - navigation_and_planning (space, routes)
  - error_control (repeatable procedures, verification)
EMERGENCE_CHAIN:
  survival_pressure -> quantification -> representation -> standardisation -> transfer

MATHOS.FUNC.1: quantify (count, measure, compare)
MATHOS.FUNC.2: model (represent relationships)
MATHOS.FUNC.3: transform (symbolic manipulation)
MATHOS.FUNC.4: verify (proof, checks, error detection)
MATHOS.FUNC.5: optimise (efficiency, constraints, trade-offs)
MATHOS.FUNC.6: predict (trend, dynamics, uncertainty bounds)

DEP.LANE.1: GOVOS (budgets, allocation, auditing, planning)
DEP.LANE.2: FINOS (risk, interest, pricing, accounting)
DEP.LANE.3: TROS  (routing, scheduling, capacity planning)
DEP.LANE.4: HEAOS (dosage, statistics, epidemiology, diagnostics)
DEP.LANE.5: ENEOS (energy systems, grids, optimisation)
DEP.LANE.6: SECOS (logistics, operations, verification)
DEP.LANE.7: SCIOS (measurement, modelling, inference)

weak_mathos -> downstream_lanes become noisy, brittle, and error-prone

MATHOS.STAGES:
  S0: number_sense + basic_counting (allocation readiness)
  S1: arithmetic_fluency (transaction readiness)
  S2: ratios_fractions_percentages (scaling readiness)
  S3: algebraic_symbol_system (abstraction readiness)
  S4: functions_graphs (relationship modelling readiness)
  S5: geometry_trigonometry (space + structure readiness)
  S6: statistics_probability (uncertainty readiness)
  S7: calculus_like_thinking (change + optimisation readiness)

S3 is the key phase-transition for "secondary mathematics" in most systems.

MATHOS.PHASES:
  P0: collapse (cannot execute reliably; avoidance/shutdown)
  P1: unstable (high variance; fragile retention)
  P2: functional_leaky (works but fails under time/novelty)
  P3: stable_mastery (durable, transferable, low variance)
KEY:
  MATH capability is defined by reliability under load, not exposure to content.

FAIL.M1: fragile_fluency (slow recall -> time collapse)
FAIL.M2: symbol_confusion (algebra breaks -> downstream modelling breaks)
FAIL.M3: working_memory_overflow (multi-step chains snap)
FAIL.M4: no_error_checking (careless cascades)
FAIL.M5: cramming_without_consolidation (short-lived competence)
FAIL.M6: transfer_failure (can do drills, fails on novel questions)

SENSOR.MS1: time_to_first_step (latency)
SENSOR.MS2: error_recurrence_rate (same mistake repeats)
SENSOR.MS3: variance_under_time_pressure
SENSOR.MS4: multi_step_chain_integrity (step loss rate)
SENSOR.MS5: checking_rate (verification behavior frequency)
SENSOR.MS6: transfer_rate_to_novel_problems

low_latency + low_variance + high_transfer => P3-like stability

IF MATHOS degrades at population scale THEN:
  - auditing fails
  - resource allocation errors rise
  - engineering reliability drops
  - risk mispricing rises
  - prediction quality falls
  -> coordination entropy increases
  -> HRL thins
  -> Civλ effective increases

MATHOS.IMPL:
  - school_math
  - self_study
  - apprenticeship_math
  - tuition_math (intervention_route)
RULE:
  implementation != root

LINK.PREV.1: EDUOS-00 (pillar)
LINK.PREV.2: EDUOS-02 (anti-decay)
LINK.NEXT.1: LANGOS-00 (language as coordination supply line)
LINK.NEXT.2: SCIOS-00  (science as prediction/control supply line)
LINK.NEXT.3: FINOS-bridge (optional: finance depends on math)
LINK.NEXT.4: EDUOS-IMPL-02 (tuition as intervention router)

LOCK:
  Mathematics is a civilisation capability supply line.
  "Secondary mathematics" is the algebraic-abstraction phase transition.
  Success is reliability under load (P0–P3), not mere syllabus coverage.

PAGE: EKS.SECMATH.DIR.INDEX.v0_1TITLE: eduKateSG — Secondary Mathematics Directory IndexCONTENT:- EKS.SECMATH.DIR.LANE.v0_1- EKS.SEC1MATH.DIR.LANE.v0_1- EKS.SEC2MATH.DIR.LANE.v0_1- EKS.EMATH.DIR.LANE.v0_1- EKS.AMATH.DIR.LANE.v0_1- EKS.SECMATH.DIR.CORE_SKILLS.v0_1- EKS.SECMATH.DIR.TESTS.v0_1- EKS.SECMATH.DIR.BINDS.v0_1OUTPUT:- EKS.SECMATH.CLAIM.CANONICAL.v0_1

PAGE: EKS.SECMATH.DIR.LANE.v0_1TITLE: Secondary Mathematics (Sec 1–4) — Lane Family DirectoryMISSION:- produce P3 execution under exam load across Z0–Z3- prevent false competence (P2-looking → P0 snap)LEVELS:- SEC1MATH, SEC2MATH, EMATH, AMATHOUTPUT:- EKS.SECMATH.Z3.P3.NODE.EXAM_STABILITY.v0_1

DIR: EKS.SECMATH.DIR.CORE_SKILLS.v0_1CORE_SKILLS:- EKS.SECMATH.Z0.NODE.ALGBRA_SYMBOL_SENSE.v0_1- EKS.SECMATH.Z0.NODE.ARITHMETIC_ACCURACY.v0_1- EKS.SECMATH.Z0.NODE.FRACTIONS_RATIO_RATE.v0_1- EKS.SECMATH.Z0.NODE.EQUATIONS_INEQUALITIES.v0_1- EKS.SECMATH.Z0.NODE.GRAPHS_FUNCTIONS.v0_1- EKS.SECMATH.Z0.NODE.GEOMETRY_ANGLES.v0_1- EKS.SECMATH.Z0.NODE.TRIG_FUNDAMENTALS.v0_1- EKS.SECMATH.Z0.NODE.PROB_STATS.v0_1- EKS.SECMATH.Z0.NODE.CHECKING_ERROR_CONTROL.v0_1- EKS.SECMATH.Z0.NODE.SPEED_UNDER_TIME.v0_1RULE:These Z0 nodes are reused across all sub-lanes as dependencies.

TEST: EKS.SECMATH.TEST.P_SCORE.v0_1P0: cannot solve independently; collapses under time/noveltyP1: solves with prompts; dependency; fragile confidenceP2: solves standard formats; breaks under variation/speedP3: solves independently under time + variation; bounded error tail

SENSOR: EKS.SECMATH.SENSOR.EXECUTION.v0_1MEASURES:- independent_success_rate (no hints)- time_to_solve_tail (slow tail kills grades)- recurring_error_types (same mistake repeats)- transfer_rate (new form, same concept)- careless_rate (often not careless: weak checking)

LOOP: EKS.SECMATH.LOOP.TRUNCATE_STITCH.v0_1TRUNCATE:- stop repeated error loops early (same mistake 3×)- cut dependency (remove hints, force retrieval)STITCH:- rebuild the missing Z0 pocket- re-run under time and variationGOAL:- push P1/P2 → P3 and prevent snap collapse at exams

PAGE: EKS.SEC1MATH.DIR.LANE.v0_1TITLE: Secondary 1 Mathematics — Lane DirectoryFOCUS:- algebra entry + real numbers + foundations for all future mathZ0_NODES:- EKS.SEC1MATH.Z0.NODE.REAL_NUMBERS.v0_1- EKS.SEC1MATH.Z0.NODE.ALGEBRA_BASICS.v0_1- EKS.SEC1MATH.Z0.NODE.LINEAR_EXPRESSIONS.v0_1- EKS.SEC1MATH.Z0.NODE.BASIC_GEOMETRY.v0_1- EKS.SEC1MATH.Z0.NODE.INTRO_GRAPHS.v0_1Z1_LOOPS:- EKS.SEC1MATH.Z1.LOOP.HW_REPAIR.v0_1- EKS.SEC1MATH.Z1.LOOP.ERROR_NOTEBOOK.v0_1Z2_CONTROL:- EKS.SEC1MATH.Z2.NODE.MASTERY_SEQUENCING.v0_1Z3_OUTPUT:- EKS.SEC1MATH.Z3.P3.NODE.SEC1_FOUNDATION_LOCK.v0_1

PAGE: EKS.SEC2MATH.DIR.LANE.v0_1TITLE: Secondary 2 Mathematics — Lane DirectoryFOCUS:- algebra expansion + functions/graphs + probability/stats; pre-O-level rampZ0_NODES:- EKS.SEC2MATH.Z0.NODE.ALGEBRA_EXPANSION_FACTORISATION.v0_1- EKS.SEC2MATH.Z0.NODE.FUNCTIONS_GRAPHS.v0_1- EKS.SEC2MATH.Z0.NODE.RATIO_RATE_SPEED.v0_1- EKS.SEC2MATH.Z0.NODE.PROB_STATS_CORE.v0_1- EKS.SEC2MATH.Z0.NODE.GEOMETRY_ADVANCE.v0_1Z1_LOOPS:- EKS.SEC2MATH.Z1.LOOP.TOPICAL_VARIATION.v0_1- EKS.SEC2MATH.Z1.LOOP.SPEED_BUILD.v0_1Z2_CONTROL:- EKS.SEC2MATH.Z2.NODE.EXAM_FORMAT_TRANSFER.v0_1Z3_OUTPUT:- EKS.SEC2MATH.Z3.P3.NODE.SEC2_STABILITY_LOCK.v0_1

PAGE: EKS.EMATH.DIR.LANE.v0_1TITLE: E-Mathematics — O-Level Lane DirectoryFOCUS:- full-syllabus execution + exam strategy + speed + checkingZ0_NODES:- EKS.EMATH.Z0.NODE.ALGEBRA_SYSTEMS.v0_1- EKS.EMATH.Z0.NODE.GRAPHS_FUNCTIONS.v0_1- EKS.EMATH.Z0.NODE.GEOMETRY_TRIG.v0_1- EKS.EMATH.Z0.NODE.MENSURATION.v0_1- EKS.EMATH.Z0.NODE.PROB_STATS.v0_1- EKS.EMATH.Z0.NODE.MODELLING_WORD_PROBLEMS.v0_1Z1_LOOPS:- EKS.EMATH.Z1.LOOP.TEN_YEAR_SERIES.v0_1- EKS.EMATH.Z1.LOOP.CARELESSNESS_ZEROING.v0_1Z2_CONTROL:- EKS.EMATH.Z2.NODE.PAPER_ROUTING.v0_1  (Paper 1 vs Paper 2 tactics)Z3_OUTPUT:- EKS.EMATH.Z3.P3.NODE.OLEVEL_A1_STABILITY.v0_1

PAGE: EKS.AMATH.DIR.LANE.v0_1TITLE: A-Mathematics — O-Level Lane DirectoryFOCUS:- algebraic power + calculus + trig identities; high-precision executionZ0_NODES:- EKS.AMATH.Z0.NODE.ALGEBRA_TECHNIQUE.v0_1- EKS.AMATH.Z0.NODE.TRIG_IDENTITIES_EQUATIONS.v0_1- EKS.AMATH.Z0.NODE.LOGS_EXPONENTIALS.v0_1- EKS.AMATH.Z0.NODE.CALCULUS_DIFF.v0_1- EKS.AMATH.Z0.NODE.CALCULUS_INTEGRATION.v0_1- EKS.AMATH.Z0.NODE.PROOF_CHAINING.v0_1Z1_LOOPS:- EKS.AMATH.Z1.LOOP.SKILL_DRILLS_TO_VARIATION.v0_1- EKS.AMATH.Z1.LOOP.EXAM_SPEED_PRECISION.v0_1Z2_CONTROL:- EKS.AMATH.Z2.NODE.TOPIC_DEPENDENCY_ROUTER.v0_1Z3_OUTPUT:- EKS.AMATH.Z3.P3.NODE.OLEVEL_AMATH_A1_STABILITY.v0_1

DIR: EKS.SECMATH.DIR.TESTS.v0_1TESTS:- EKS.SECMATH.TEST.P_SCORE.v0_1- EKS.SECMATH.TEST.INDEPENDENCE.v0_1- EKS.SECMATH.TEST.SPEED_TAIL.v0_1- EKS.SECMATH.TEST.TRANSFER.v0_1- EKS.SECMATH.TEST.ERROR_REPEAT.v0_1

TEST: EKS.SECMATH.TEST.INDEPENDENCE.v0_1PASS: ≥80% correct with zero hints on mixed setFAIL: needs prompts/rescues or only works on “same-format” questions

TEST: EKS.SECMATH.TEST.SPEED_TAIL.v0_1PASS: tail time bounded (no time sink questions)FAIL: a few questions consume most time → paper collapses

DIR: EKS.SECMATH.DIR.BINDS.v0_1BINDS:- EKS.SECMATH.BIND.EDU_CORE.v0_1      TO: EDU.Z3.P3.NODE.CAPABILITY_STABILITY.v0_1- EKS.SECMATH.BIND.FAM_LOAD.v0_1      TO: FAM.Z0.NODE.HOMEWORK_SUPPORT.v0_1- EKS.SECMATH.BIND.HLT_STRESS.v0_1    TO: HLT.Z0.NODE.PATIENT_MONITORING.v0_1CLAIM:Secondary Maths stability reduces household load and prevents P0 education collapse.

CLAIM: EKS.SECMATH.CLAIM.CANONICAL.v0_1Secondary Mathematics works when Z0 execution becomes P3 under time + variation,and repair loops prevent false competence from snapping into exam collapse.