# Question 17031: Failure to Find an Integer Preimage Does Not Establish Real Nonmembership [Return to catalogue page 86](https://ocme.evemisslab.com/en/questions/page/86/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261102-r03-linear-map-image-foundations/#q-20261014-advance-20261102-r03-linear-map-image-0081) 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. R0033-17031 · q-20261014-advance-20261102-r03-linear-map-image-0081 ## Original question and givens 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). ## 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-map-image](https://ocme.evemisslab.com/en/mko/mko-linear-map-image/) ## Step-by-step derivation S001 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_same_matrix_det_and_nonzero_rank_cases · Uses: G001, S003 S005 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_full_rank_inverse_both_kernel_and_image · Uses: G001, S016 S018 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_full_space_basis_counts_not_grid_counts · Uses: G001, S018 S020 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · impliesBefore 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. Step identifiers and dependenciesRule: image_full_rank_real_solution_substitution_and_uniqueness · Uses: G001, S033 S035 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S034 S036 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S035 S037 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S036 S038 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S037 S039 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S038 S040 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S039 S041 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S040 S042 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S041 S043 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S042 S044 · impliesBefore 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. Step identifiers and dependenciesRule: real_map_complete_bounded_list_not_unbounded_search · Uses: G001, S043 S045 · impliesBefore 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. Step identifiers and dependenciesRule: image_bounded_failure_not_global_real_nonmembership · Uses: G001, S044 ## Reverse checks - 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. ## Result ([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. ## 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: f1ae3c0ae04dd8aac64c4bb234bd251a325803103e53c008daec642bdd65417e - candidate record SHA-256: abc2aab3778fd2703b9dfced8bf8ebe85a3f39b1cae540d7db0f0affa0a4c961 - semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3 - question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261102-r03-linear-map-image-0081.json)