Original question and givens
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[8], c=[1]. Candidate x=[9], y=[9/8]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
G001 — Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[8], c=[1]. Candidate x=[9], y=[9/8]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
Step-by-step derivation
S001 · implies
Before
source roles = [lpdual-strong-boundary]
After
derived = [[mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary]]
Conditions: none
List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding.
Step identifiers and dependencies
Rule: source_ordered_fields_without_JS · Uses: G001
S002 · implies
Before
source roles = [[[1]], [8], [1]]
After
derived = [1, 1, [[1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes]
Conditions: none
The fixed max/<=/x>=0 convention pairs with min/>=/y>=0.The dual has m variables and n constraints.Transpose every position;either coefficient sign is legal and dimensions do not imply feasibility.
Step identifiers and dependencies
Rule: LPdual_fixed_conventions_complete_transpose_and_dimensions · Uses: G001, S001
S003 · calculation
Before
LPdual ORIGINAL Ax row0 product0 := mul(1, 9)
After
LPdual ORIGINAL Ax row0 product0 = 9
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
LPdual ORIGINAL Ax row0 sum := sum(9)
After
LPdual ORIGINAL Ax row0 sum = 9
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
LPdual ORIGINAL ATy row0 product0 := mul(1, 9/8)
After
LPdual ORIGINAL ATy row0 product0 = 9/8
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
LPdual ORIGINAL ATy row0 sum := sum(9/8)
After
LPdual ORIGINAL ATy row0 sum = 9/8
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
LPdual ORIGINAL primal objective product0 := mul(1, 9)
After
LPdual ORIGINAL primal objective product0 = 9
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
LPdual ORIGINAL primal objective sum := sum(9)
After
LPdual ORIGINAL primal objective sum = 9
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
LPdual ORIGINAL middle yTAx product0 := mul(9/8, 9)
After
LPdual ORIGINAL middle yTAx product0 = 81/8
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
LPdual ORIGINAL middle yTAx sum := sum(81/8)
After
LPdual ORIGINAL middle yTAx sum = 81/8
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
LPdual ORIGINAL dual objective product0 := mul(8, 9/8)
After
LPdual ORIGINAL dual objective product0 = 9
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · calculation
Before
LPdual ORIGINAL dual objective sum := sum(9)
After
LPdual ORIGINAL dual objective sum = 9
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · implies
Before
source roles = [[9], [9/8], [9], [9/8], [8], [1]]
After
derived = [No, Yes, 9, 81/8, 9, [[0, 0, 9/8, 1, 9, 81/8]], 81/8, [[0, 9, 1/8, 9/8]], 9/8, [[0, 9/8, -1, -9/8]], -9/8, Yes, No, Yes]
Conditions: none
Check both feasibilities separately.Expanding the double sum proves(ATy)Tx=yTAx.The first gap is sum x_j((ATy)_j−c_j),the second sum y_i(b_i−(Ax)_i).Only with both feasibility premises do nonnegative factors establish the weak chain.With missing premises,numerical inequalities may be accidental.Teaching residuals impose no new source numeric bounds.
Step identifiers and dependencies
Rule: LPdual_ALL_feasibilities_exact_pairings_residuals_and_conditional_weak_chain · Uses: G001, S012
S014 · implies
Before
source roles = [primal_infeasible_equal_objective, No, Yes, 9, 9, Yes]
After
derived = [No]
Conditions: none
Primal-infeasible mode does not additionally require dual feasibility;dual-infeasible mode requires primal feasibility;finite-grid mode requires both but may have nonzero gap.No rejects unsupported optimality/general strong-duality inference,not actual optimality of a particular feasible zero-gap pair.
Step identifiers and dependencies
Rule: LPdual_original_boundary_mode_premises_NOT_bare_equality_general_proof · Uses: G001, S013
Result
(primal_infeasible_equal_objective, [9], [9/8], [9], [9/8], No, Yes, 9, 81/8, 9, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.