# Question 2440: Feasible Indices in Canonical Candidate-Grid Order

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0003-02440 · q-20261013-advance-20261021-r06-linear-programming-0040



## Original question and givens

Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 4], [2, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.


- G001 — Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 2], [2, 4], [2, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical constraints,every [index,x,y,constraint truth list,all constraints hold?,null] row,feasible canonical indices). Here null records that no objective was supplied.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-linear-programming](https://ocme.evemisslab.com/en/mko/mko-linear-programming/)





## Step-by-step derivation

S001 · impliesBefore

explicit finite candidate grid and linear rows a·x+b·y≤c

After

keep candidate-grid optimization separate from continuous LP

Conditions: none

Audit only the listed finite candidates;without a complete-vertex or full-feasible-domain certificate,this is not a general continuous-LP optimum claim.

Step identifiers and dependenciesRule: finite_LP_scope · Uses: G001





S002 · impliesBefore

[[0, 0], [1, 2], [2, 4], [2, 2]]

After

deduplicated canonical indexed points=[[0, 0, 0], [1, 1, 2], [2, 2, 2], [3, 2, 4]]

Conditions: none

Keep the original data;output indices follow the source contract's sorted distinct points,not original input positions.

Step identifiers and dependenciesRule: candidate_index_normalization · Uses: G001, S001





S003 · calculationBefore

point0 row0 a·x := mul(-1, 0)

After

point0 row0 a·x = 0

Conditions: none

Multiply the first coordinate by its row coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

point0 row0 b·y := mul(0, 0)

After

point0 row0 b·y = 0

Conditions: none

Compute the second coordinate in the same declared order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

point0 row0 LHS := sum(0, 0)

After

point0 row0 LHS = 0

Conditions: none

Add before comparing the complete row inequality.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

row0: 0≤0

After

satisfied=Yes

Conditions: none

Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality.

Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S005





S007 · calculationBefore

point0 row1 a·x := mul(0, 0)

After

point0 row1 a·x = 0

Conditions: none

Multiply the first coordinate by its row coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

point0 row1 b·y := mul(-1, 0)

After

point0 row1 b·y = 0

Conditions: none

Compute the second coordinate in the same declared order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

point0 row1 LHS := sum(0, 0)

After

point0 row1 LHS = 0

Conditions: none

Add before comparing the complete row inequality.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

row1: 0≤0

After

satisfied=Yes

Conditions: none

Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality.

Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S009





S011 · calculationBefore

point0 row2 a·x := mul(1, 0)

After

point0 row2 a·x = 0

Conditions: none

Multiply the first coordinate by its row coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

point0 row2 b·y := mul(1, 0)

After

point0 row2 b·y = 0

Conditions: none

Compute the second coordinate in the same declared order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

point0 row2 LHS := sum(0, 0)

After

point0 row2 LHS = 0

Conditions: none

Add before comparing the complete row inequality.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · impliesBefore

row2: 0≤3

After

satisfied=Yes

Conditions: none

Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality.

Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S013





S015 · calculationBefore

point1 row0 a·x := mul(-1, 1)

After

point1 row0 a·x = -1

Conditions: none

Multiply the first coordinate by its row coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

point1 row0 b·y := mul(0, 2)

After

point1 row0 b·y = 0

Conditions: none

Compute the second coordinate in the same declared order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

point1 row0 LHS := sum(-1, 0)

After

point1 row0 LHS = -1

Conditions: none

Add before comparing the complete row inequality.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · impliesBefore

row0: -1≤0

After

satisfied=Yes

Conditions: none

Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality.

Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S017





S019 · calculationBefore

point1 row1 a·x := mul(0, 1)

After

point1 row1 a·x = 0

Conditions: none

Multiply the first coordinate by its row coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · calculationBefore

point1 row1 b·y := mul(-1, 2)

After

point1 row1 b·y = -2

Conditions: none

Compute the second coordinate in the same declared order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · calculationBefore

point1 row1 LHS := sum(0, -2)

After

point1 row1 LHS = -2

Conditions: none

Add before comparing the complete row inequality.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · impliesBefore

row1: -2≤0

After

satisfied=Yes

Conditions: none

Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality.

Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S021





S023 · calculationBefore

point1 row2 a·x := mul(1, 1)

After

point1 row2 a·x = 1

Conditions: none

Multiply the first coordinate by its row coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · calculationBefore

point1 row2 b·y := mul(1, 2)

After

point1 row2 b·y = 2

Conditions: none

Compute the second coordinate in the same declared order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · calculationBefore

point1 row2 LHS := sum(1, 2)

After

point1 row2 LHS = 3

Conditions: none

Add before comparing the complete row inequality.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · impliesBefore

row2: 3≤3

After

satisfied=Yes

Conditions: none

Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality.

Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S025





S027 · calculationBefore

point2 row0 a·x := mul(-1, 2)

After

point2 row0 a·x = -2

Conditions: none

Multiply the first coordinate by its row coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · calculationBefore

point2 row0 b·y := mul(0, 2)

After

point2 row0 b·y = 0

Conditions: none

Compute the second coordinate in the same declared order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · calculationBefore

point2 row0 LHS := sum(-2, 0)

After

point2 row0 LHS = -2

Conditions: none

Add before comparing the complete row inequality.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · impliesBefore

row0: -2≤0

After

satisfied=Yes

Conditions: none

Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality.

Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S029





S031 · calculationBefore

point2 row1 a·x := mul(0, 2)

After

point2 row1 a·x = 0

Conditions: none

Multiply the first coordinate by its row coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · calculationBefore

point2 row1 b·y := mul(-1, 2)

After

point2 row1 b·y = -2

Conditions: none

Compute the second coordinate in the same declared order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031





S033 · calculationBefore

point2 row1 LHS := sum(0, -2)

After

point2 row1 LHS = -2

Conditions: none

Add before comparing the complete row inequality.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032





S034 · impliesBefore

row1: -2≤0

After

satisfied=Yes

Conditions: none

Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality.

Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S033





S035 · calculationBefore

point2 row2 a·x := mul(1, 2)

After

point2 row2 a·x = 2

Conditions: none

Multiply the first coordinate by its row coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · calculationBefore

point2 row2 b·y := mul(1, 2)

After

point2 row2 b·y = 2

Conditions: none

Compute the second coordinate in the same declared order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035





S037 · calculationBefore

point2 row2 LHS := sum(2, 2)

After

point2 row2 LHS = 4

Conditions: none

Add before comparing the complete row inequality.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · impliesBefore

row2: 4≤3

After

satisfied=No

Conditions: none

Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality.

Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S037





S039 · calculationBefore

point3 row0 a·x := mul(-1, 2)

After

point3 row0 a·x = -2

Conditions: none

Multiply the first coordinate by its row coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038





S040 · calculationBefore

point3 row0 b·y := mul(0, 4)

After

point3 row0 b·y = 0

Conditions: none

Compute the second coordinate in the same declared order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039





S041 · calculationBefore

point3 row0 LHS := sum(-2, 0)

After

point3 row0 LHS = -2

Conditions: none

Add before comparing the complete row inequality.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040





S042 · impliesBefore

row0: -2≤0

After

satisfied=Yes

Conditions: none

Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality.

Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S041





S043 · calculationBefore

point3 row1 a·x := mul(0, 2)

After

point3 row1 a·x = 0

Conditions: none

Multiply the first coordinate by its row coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042





S044 · calculationBefore

point3 row1 b·y := mul(-1, 4)

After

point3 row1 b·y = -4

Conditions: none

Compute the second coordinate in the same declared order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043





S045 · calculationBefore

point3 row1 LHS := sum(0, -4)

After

point3 row1 LHS = -4

Conditions: none

Add before comparing the complete row inequality.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044





S046 · impliesBefore

row1: -4≤0

After

satisfied=Yes

Conditions: none

Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality.

Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S045





S047 · calculationBefore

point3 row2 a·x := mul(1, 2)

After

point3 row2 a·x = 2

Conditions: none

Multiply the first coordinate by its row coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046





S048 · calculationBefore

point3 row2 b·y := mul(1, 4)

After

point3 row2 b·y = 4

Conditions: none

Compute the second coordinate in the same declared order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047





S049 · calculationBefore

point3 row2 LHS := sum(2, 4)

After

point3 row2 LHS = 6

Conditions: none

Add before comparing the complete row inequality.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048





S050 · impliesBefore

row2: 6≤3

After

satisfied=No

Conditions: none

Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality.

Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S049





S051 · impliesBefore

[[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 2, 2, [Yes, Yes, No], No, null], [3, 2, 4, [Yes, Yes, No], No, null]]

After

all feasible listed indices=[0, 1]

Conditions: none

Scan every listed point and retain every feasible one;they need not all be optimal.

Step identifiers and dependenciesRule: feasible_candidate_inventory · Uses: G001, S050





S052 · impliesBefore

finite candidates have been completely audited

After

no unlisted-point or general-LP optimality claim

Conditions: none

A best value in the finite list does not exclude a better unlisted feasible point;script PASS does not lift this boundary.

Step identifiers and dependenciesRule: finite_grid_boundary · Uses: G001, S051









## Reverse checks


- S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S049 · premise_and_result_audit — Recompute point3 row2 LHS with exact rational arithmetic, not rounded decimals.
- S048 · premise_and_result_audit — Recompute point3 row2 b·y with exact rational arithmetic, not rounded decimals.
- S047 · premise_and_result_audit — Recompute point3 row2 a·x with exact rational arithmetic, not rounded decimals.
- S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S045 · premise_and_result_audit — Recompute point3 row1 LHS with exact rational arithmetic, not rounded decimals.
- S044 · premise_and_result_audit — Recompute point3 row1 b·y with exact rational arithmetic, not rounded decimals.
- S043 · premise_and_result_audit — Recompute point3 row1 a·x with exact rational arithmetic, not rounded decimals.
- S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S041 · premise_and_result_audit — Recompute point3 row0 LHS with exact rational arithmetic, not rounded decimals.
- S040 · premise_and_result_audit — Recompute point3 row0 b·y with exact rational arithmetic, not rounded decimals.
- S039 · premise_and_result_audit — Recompute point3 row0 a·x with exact rational arithmetic, not rounded decimals.
- S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S037 · premise_and_result_audit — Recompute point2 row2 LHS with exact rational arithmetic, not rounded decimals.
- S036 · premise_and_result_audit — Recompute point2 row2 b·y with exact rational arithmetic, not rounded decimals.
- S035 · premise_and_result_audit — Recompute point2 row2 a·x with exact rational arithmetic, not rounded decimals.
- S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S033 · premise_and_result_audit — Recompute point2 row1 LHS with exact rational arithmetic, not rounded decimals.
- S032 · premise_and_result_audit — Recompute point2 row1 b·y with exact rational arithmetic, not rounded decimals.
- S031 · premise_and_result_audit — Recompute point2 row1 a·x with exact rational arithmetic, not rounded decimals.
- S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S029 · premise_and_result_audit — Recompute point2 row0 LHS with exact rational arithmetic, not rounded decimals.
- S028 · premise_and_result_audit — Recompute point2 row0 b·y with exact rational arithmetic, not rounded decimals.
- S027 · premise_and_result_audit — Recompute point2 row0 a·x with exact rational arithmetic, not rounded decimals.
- S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S025 · premise_and_result_audit — Recompute point1 row2 LHS with exact rational arithmetic, not rounded decimals.
- S024 · premise_and_result_audit — Recompute point1 row2 b·y with exact rational arithmetic, not rounded decimals.
- S023 · premise_and_result_audit — Recompute point1 row2 a·x with exact rational arithmetic, not rounded decimals.
- S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S021 · premise_and_result_audit — Recompute point1 row1 LHS with exact rational arithmetic, not rounded decimals.
- S020 · premise_and_result_audit — Recompute point1 row1 b·y with exact rational arithmetic, not rounded decimals.
- S019 · premise_and_result_audit — Recompute point1 row1 a·x with exact rational arithmetic, not rounded decimals.
- S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S017 · premise_and_result_audit — Recompute point1 row0 LHS with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Recompute point1 row0 b·y with exact rational arithmetic, not rounded decimals.
- S015 · premise_and_result_audit — Recompute point1 row0 a·x with exact rational arithmetic, not rounded decimals.
- S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S013 · premise_and_result_audit — Recompute point0 row2 LHS with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute point0 row2 b·y with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Recompute point0 row2 a·x with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Recompute point0 row1 LHS with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Recompute point0 row1 b·y with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute point0 row1 a·x with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Recompute point0 row0 LHS with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute point0 row0 b·y with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute point0 row0 a·x with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

([[-1, 0, 0], [0, -1, 0], [1, 1, 3]], [[0, 0, 0, [Yes, Yes, Yes], Yes, null], [1, 1, 2, [Yes, Yes, Yes], Yes, null], [2, 2, 2, [Yes, Yes, No], No, null], [3, 2, 4, [Yes, Yes, No], No, null]], [0, 1])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: f7e2d59e9a30e9bab933b7022c14058f9fef4c69ad2331b12a29acb12663affc
- candidate record SHA-256: 9660290ca505b2d352758198d58fc05a713a98cd6b14ffca0652e56832588c10
- semantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f
- question-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r06-linear-programming-0040.json)