Singapore School Mathematics Operating Manual · Chapter 46
Diagnose empty feasible sets and conflicting requirements, isolate the contradiction, and quantify the smallest justified change that restores feasibility.
This chapter extends the operating manual as a cross-topic control layer. It is designed to complement, not replace, existing BTT owners for constraints, optimisation, error diagnosis, modelling, proof and examination craft.
1. Infeasible means the feasible set is empty
The key discipline is to make infeasible means the feasible set is empty explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
2. Contradictory equations expose infeasibility
The key discipline is to make contradictory equations expose infeasibility explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
3. Interval constraints can conflict
The key discipline is to make interval constraints can conflict explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
4. Geometry can be infeasible
The key discipline is to make geometry can be infeasible explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
5. Capacity can make schedules infeasible
The key discipline is to make capacity can make schedules infeasible explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
6. Diagnose infeasibility before optimisation
The key discipline is to make diagnose infeasibility before optimisation explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
7. Find a conflicting core
The key discipline is to make find a conflicting core explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
8. Redundant constraints are not causes
The key discipline is to make redundant constraints are not causes explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
9. Minimal relaxation changes as little as possible
The key discipline is to make minimal relaxation changes as little as possible explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
10. Relaxation needs authority
The key discipline is to make relaxation needs authority explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
11. Relax one bound and recompute
The key discipline is to make relax one bound and recompute explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
12. Slack becomes deficit under violation
The key discipline is to make slack becomes deficit under violation explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
13. Multiple relaxations can be compared
The key discipline is to make multiple relaxations can be compared explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
14. Penalty models do not create feasibility
The key discipline is to make penalty models do not create feasibility explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
15. Exact contradiction is stronger than numerical difficulty
The key discipline is to make exact contradiction is stronger than numerical difficulty explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
16. Rounding can create false infeasibility
The key discipline is to make rounding can create false infeasibility explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
17. Model assumptions can create artificial conflict
The key discipline is to make model assumptions can create artificial conflict explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
18. Repair should preserve provenance
The key discipline is to make repair should preserve provenance explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
19. Re-optimise after relaxation
The key discipline is to make re-optimise after relaxation explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
20. An infeasibility audit
Ask what the problem requires, what can fail, what evidence is decisive, and which earlier operating-manual control should be invoked before calculation continues.
21. Worked mini-cases
Case A. A candidate satisfies every preference but violates a domain restriction. The candidate remains inadmissible because validity is checked before preference.
Case B. Two calculations disagree. Return to the last trusted common input, separate the branches, and identify whether the discrepancy comes from a shared dependency or a branch-specific step.
Case C. A result lies close to a threshold. Preserve precision and compare the uncertainty interval with the decision margin before classifying it.
22. Independent practice
1. Identify the admissibility condition in a problem of your choice.
2. Give one independent check that could catch a wrong intermediate result.
3. Explain why a mathematically attractive candidate may still be invalid.
4. Describe one situation where uncertainty changes the strength of a conclusion.
5. State what evidence would justify stopping the calculation.
23. Worked answers
1. The admissibility condition is the rule every candidate must satisfy before comparison.
2. Examples include substitution, a unit check, a bound, an alternative representation or an independent method.
3. Objective performance cannot compensate for violation of a hard mathematical condition.
4. A rounded or measured value near a threshold may have a plausible interval crossing both decision regimes.
5. A complete proof, verified candidate set, guaranteed bound, tolerance certificate or exhausted feasible search can justify stopping.
24. Continue through Batch 12
Return to the BTT Mathematics Hub for Batch 12 and use the neighbouring chapters as a connected control system.