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R0033-17031 · q-20261014-advance-20261102-r03-linear-map-image-0081
Use this same exact integer matrix A=[[2, 0], [0, 2]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Target=[11, 1], nonnegative bound=1. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).
G001 — Use this same exact integer matrix A=[[2, 0], [0, 2]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Target=[11, 1], nonnegative bound=1. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).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.
Before
real map determinant diagonal product := mul(2, 2)
After
real map determinant diagonal product = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001
Before
real map determinant off-diagonal product := mul(0, 0)
After
real map determinant off-diagonal product = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
real map exact determinant := sub(4, 0)
After
real map exact determinant = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
same ordered integer matrix=[[2, 0], [0, 2]];usual real map T:R2 to R2;original columns=[[2, 0], [0, 2]]
After
exact determinant=4;rank=2
Conditions: none
A nonzero determinant gives rank2.With determinant0,distinguish the zero matrix from a nonzero singular matrix to get rank0 or1.Keep the same matrix and real vector spaces;do not replace columns by rows or assume the image is the entire codomain.
Rule: real_map_same_matrix_det_and_nonzero_rank_cases · Uses: G001, S003
Before
first adjugate column row0 first term := mul(2, 2)
After
first adjugate column row0 first term = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
first adjugate column row0 second term := mul(0, 0)
After
first adjugate column row0 second term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
first adjugate column image coordinate0 := sum(4, 0)
After
first adjugate column image coordinate0 = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
first adjugate column row1 first term := mul(0, 2)
After
first adjugate column row1 first term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
first adjugate column row1 second term := mul(2, 0)
After
first adjugate column row1 second term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
first adjugate column image coordinate1 := sum(0, 0)
After
first adjugate column image coordinate1 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
second adjugate column row0 first term := mul(2, 0)
After
second adjugate column row0 first term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
second adjugate column row0 second term := mul(0, 2)
After
second adjugate column row0 second term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
second adjugate column image coordinate0 := sum(0, 0)
After
second adjugate column image coordinate0 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
second adjugate column row1 first term := mul(0, 0)
After
second adjugate column row1 first term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
second adjugate column row1 second term := mul(2, 2)
After
second adjugate column row1 second term = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
second adjugate column image coordinate1 := sum(0, 4)
After
second adjugate column image coordinate1 = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
same matrix=[[2, 0], [0, 2]];adjugate columns=[[2, 0], [0, 2]];verified products=[[4, 0], [0, 4]]
After
M*adjugate=det*I;det=4 nonzero;inverse=adjugate/det;kernel basis=[];image basis ORIGINAL columns=[[2, 0], [0, 2]]
Conditions: none
The two products give det times the standard basis.Dividing by nonzero det gives an inverse.Hence Mx=0 has only the zero solution,and every real target has a preimage.The original columns form an image basis.An empty kernel basis spans the zero space;the kernel itself is not empty.
Rule: real_map_full_rank_inverse_both_kernel_and_image · Uses: G001, S016
Before
real kernel source dimension minus rank := sub(2, 2)
After
real kernel source dimension minus rank = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
same real map rank=2;full kernel basis=[];full image basis=[[2, 0], [0, 2]]
After
nullity=0;image basis cardinality=2;rank plus nullity=2
Conditions: none
Count bases of the established full real spaces,not searched grid points.Keep rank,nullity and image-basis count distinct;the same source dimension is2.
Rule: real_map_full_space_basis_counts_not_grid_counts · Uses: G001, S018
Before
image Cramer first diagonal target term := mul(2, 11)
After
image Cramer first diagonal target term = 22
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
image Cramer first off-diagonal target term := mul(0, 1)
After
image Cramer first off-diagonal target term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
image Cramer first numerator := sub(22, 0)
After
image Cramer first numerator = 22
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
image Cramer first real coordinate := div(22, 4)
After
image Cramer first real coordinate = 11/2
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
image Cramer second signed target term := mul(0, 11)
After
image Cramer second signed target term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
image Cramer second diagonal target term := mul(2, 1)
After
image Cramer second diagonal target term = 2
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
image Cramer second numerator := sum(0, 2)
After
image Cramer second numerator = 2
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
image Cramer second real coordinate := div(2, 4)
After
image Cramer second real coordinate = 1/2
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
image Cramer substitution row0 first term := mul(2, 11/2)
After
image Cramer substitution row0 first term = 11
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
image Cramer substitution row0 second term := mul(0, 1/2)
After
image Cramer substitution row0 second term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
image Cramer substitution image coordinate0 := sum(11, 0)
After
image Cramer substitution image coordinate0 = 11
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
image Cramer substitution row1 first term := mul(0, 11/2)
After
image Cramer substitution row1 first term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
image Cramer substitution row1 second term := mul(2, 1/2)
After
image Cramer substitution row1 second term = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
image Cramer substitution image coordinate1 := sum(0, 1)
After
image Cramer substitution image coordinate1 = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
original target=[11, 1];same invertible real matrix=[[2, 0], [0, 2]];exact solved coordinates=[11/2, 1/2]
After
substitution equals target;unique real preimage;integer coordinates NOT required
Conditions: none
Solve with Cramer fractions using the same determinant and substitute in the original matrix.The difference of two solutions lies in the zero kernel,hence uniqueness.Rational coordinates are legal real preimages;do not add an integer-only condition.
Rule: image_full_rank_real_solution_substitution_and_uniqueness · Uses: G001, S033
Before
image bounded search:integer point in x-outer/y-inner order=[-1, -1];same matrix=[[2, 0], [0, 2]];target=[11, 1]
After
exact row-dot image=[-2, -2];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S034
Before
image bounded search:integer point in x-outer/y-inner order=[-1, 0];same matrix=[[2, 0], [0, 2]];target=[11, 1]
After
exact row-dot image=[-2, 0];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S035
Before
image bounded search:integer point in x-outer/y-inner order=[-1, 1];same matrix=[[2, 0], [0, 2]];target=[11, 1]
After
exact row-dot image=[-2, 2];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S036
Before
image bounded search:integer point in x-outer/y-inner order=[0, -1];same matrix=[[2, 0], [0, 2]];target=[11, 1]
After
exact row-dot image=[0, -2];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S037
Before
image bounded search:integer point in x-outer/y-inner order=[0, 0];same matrix=[[2, 0], [0, 2]];target=[11, 1]
After
exact row-dot image=[0, 0];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S038
Before
image bounded search:integer point in x-outer/y-inner order=[0, 1];same matrix=[[2, 0], [0, 2]];target=[11, 1]
After
exact row-dot image=[0, 2];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S039
Before
image bounded search:integer point in x-outer/y-inner order=[1, -1];same matrix=[[2, 0], [0, 2]];target=[11, 1]
After
exact row-dot image=[2, -2];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S040
Before
image bounded search:integer point in x-outer/y-inner order=[1, 0];same matrix=[[2, 0], [0, 2]];target=[11, 1]
After
exact row-dot image=[2, 0];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S041
Before
image bounded search:integer point in x-outer/y-inner order=[1, 1];same matrix=[[2, 0], [0, 2]];target=[11, 1]
After
exact row-dot image=[2, 2];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S042
Before
image bounded search:bound=1;complete integer box pair count=9
After
all hits in original scan order=[]
Conditions: none
Both endpoints are included,giving(2b+1)² pairs.Every pair was checked;completeness of the list is relative to this box only.
Rule: real_map_complete_bounded_list_not_unbounded_search · Uses: G001, S043
Before
bounded integer preimages in original scan order=[];SEPARATE full-real membership=Yes;global certificate=[unique_solution, [11/2, 1/2]]
After
boundary=bounded_failure_not_global_nonmembership
Conditions: none
No integer preimage in the box can coexist with a real preimage;noninteger solutions may be missed by every integer box.Global nonmembership must likewise come from the separate full-real certificate,not this search failure alone.
Rule: image_bounded_failure_not_global_real_nonmembership · Uses: G001, S044
S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 image Cramer substitution image coordinate1 with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute image Cramer substitution row1 second term with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute image Cramer substitution row1 first term with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute image Cramer substitution image coordinate0 with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute image Cramer substitution row0 second term with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute image Cramer substitution row0 first term with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute image Cramer second real coordinate with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute image Cramer second numerator with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute image Cramer second diagonal target term with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute image Cramer second signed target term with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute image Cramer first real coordinate with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute image Cramer first numerator with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute image Cramer first off-diagonal target term with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute image Cramer first diagonal target term with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Recompute real kernel source dimension minus rank with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Recompute second adjugate column image coordinate1 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute second adjugate column row1 second term with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute second adjugate column row1 first term with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute second adjugate column image coordinate0 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute second adjugate column row0 second term with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute second adjugate column row0 first term with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute first adjugate column image coordinate1 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute first adjugate column row1 second term with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute first adjugate column row1 first term with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute first adjugate column image coordinate0 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute first adjugate column row0 second term with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute first adjugate column row0 first term with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute real map exact determinant with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute real map determinant off-diagonal product with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Recompute real map determinant diagonal product with exact rational arithmetic, not rounded decimals.([11, 1], 1, [], Yes, bounded_failure_not_global_nonmembership)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: f1ae3c0ae04dd8aac64c4bb234bd251a325803103e53c008daec642bdd65417ecandidate record SHA-256: abc2aab3778fd2703b9dfced8bf8ebe85a3f39b1cae540d7db0f0affa0a4c961semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93Lean: NOT_RUN · human review: NOT_PERFORMED