# Question 33236: Candidate Coordinate Lists and the Conditional Uniqueness Obligation [Return to catalogue page 167](https://ocme.evemisslab.com/en/questions/page/167/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261025-r04-linear-independence-basis-foundations/#q-20261013-advance-20261025-r04-linear-independence-basis-0086) 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. R0065-33236 · q-20261013-advance-20261025-r04-linear-independence-basis-0086 ## Original question and givens Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -2], [0, 1]], target=[-2, 4], first integer coordinates=[-2, 0], second=[-2, 0]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?). - G001 — Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -2], [0, 1]], target=[-2, 4], first integer coordinates=[-2, 0], second=[-2, 0]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?). ## 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-independence-basis](https://ocme.evemisslab.com/en/mko/mko-linear-independence-basis/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-mod25-basisdeep-r2-coordinate-uniqueness-audit, basisdeep-r2-coordinate-uniqueness-audit, R2_basis_density_without_dimension_extension, []] After derived = [[basis, target, D, first, reconstruction1, match1, second, reconstruction2, match2, implication]] Conditions: none MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status. Step identifiers and dependenciesRule: R65_SOURCE_bank_keys_operation_domain_ordered_output_roles · Uses: G001 S002 · impliesBefore source roles = [ordered R2 basis COLUMNS vector0, [1, -2]] After derived = [[1, -2]] Conditions: none RAW vectors are nonempty with1..6 components bounded±1000.ONLY the R² coordinate task adds dimension2;do not infer an extra2D gate elsewhere from a domain label. Step identifiers and dependenciesRule: Basis_RAW_vector_1to6_components_bound1000 · Uses: G001, S001 S003 · impliesBefore source roles = [ordered R2 basis COLUMNS vector1, [0, 1]] After derived = [[0, 1]] Conditions: none RAW vectors are nonempty with1..6 components bounded±1000.ONLY the R² coordinate task adds dimension2;do not infer an extra2D gate elsewhere from a domain label. Step identifiers and dependenciesRule: Basis_RAW_vector_1to6_components_bound1000 · Uses: G001, S002 S004 · impliesBefore source roles = [R2 target, [-2, 4]] After derived = [[-2, 4]] Conditions: none RAW vectors are nonempty with1..6 components bounded±1000.ONLY the R² coordinate task adds dimension2;do not infer an extra2D gate elsewhere from a domain label. Step identifiers and dependenciesRule: Basis_RAW_vector_1to6_components_bound1000 · Uses: G001, S003 S005 · calculationBefore R2 ORIGINAL b1x times b2y := mul(1, 1) After R2 ORIGINAL b1x times b2y = 1 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore R2 ORIGINAL b2x times b1y := mul(0, -2) After R2 ORIGINAL b2x times b1y = 0 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore R2 ORIGINAL signed column determinant := sub(1, 0) After R2 ORIGINAL signed column determinant = 1 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore R2 true coordinate ORIGINAL targetx times b2y := mul(-2, 1) After R2 true coordinate ORIGINAL targetx times b2y = -2 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore R2 true coordinate ORIGINAL b2x times targety := mul(0, 4) After R2 true coordinate ORIGINAL b2x times targety = 0 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore R2 true coordinate ORIGINAL first numerator := sub(-2, 0) After R2 true coordinate ORIGINAL first numerator = -2 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore R2 true coordinate ORIGINAL b1x times targety := mul(1, 4) After R2 true coordinate ORIGINAL b1x times targety = 4 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore R2 true coordinate ORIGINAL targetx times b1y := mul(-2, -2) After R2 true coordinate ORIGINAL targetx times b1y = 4 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore R2 true coordinate ORIGINAL second numerator := sub(4, 4) After R2 true coordinate ORIGINAL second numerator = 0 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore source roles = [R2 source helper TRUEcoordinate0, RAW-pair, [-2, 1], [], -2, 1] After derived = [-2, 1, -2] Conditions: none This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer. Step identifiers and dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S013 S015 · calculationBefore R2 source helper TRUEcoordinate0 ORIGINAL raw numerator over nonzero denominator := div(-2, 1) After R2 source helper TRUEcoordinate0 ORIGINAL raw numerator over nonzero denominator = -2 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore source roles = [R2 source helper TRUEcoordinate1, RAW-pair, [0, 1], [], 0, 1] After derived = [0, 1, 0] Conditions: none This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer. Step identifiers and dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S015 S017 · calculationBefore R2 source helper TRUEcoordinate1 ORIGINAL raw numerator over nonzero denominator := div(0, 1) After R2 source helper TRUEcoordinate1 ORIGINAL raw numerator over nonzero denominator = 0 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore R2 FIRST integer coordinate list ORIGINAL vector0 coordinate0 weighted term := mul(-2, 1) After R2 FIRST integer coordinate list ORIGINAL vector0 coordinate0 weighted term = -2 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore R2 FIRST integer coordinate list ORIGINAL vector1 coordinate0 weighted term := mul(0, 0) After R2 FIRST integer coordinate list ORIGINAL vector1 coordinate0 weighted term = 0 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore R2 FIRST integer coordinate list ORIGINAL coordinate0 all terms sum := sum(-2, 0) After R2 FIRST integer coordinate list ORIGINAL coordinate0 all terms sum = -2 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore source roles = [R2 FIRST integer coordinate list, 0, [[1, -2], [0, 1]], [-2, 0]] After derived = [[-2, 0], -2] Conditions: none For EACH coordinate retain EVERY coefficient*vector component before summing.Do not recap derived reconstructions as RAW vector/coefficient inputs. Step identifiers and dependenciesRule: Basis_EACH_coordinate_complete_linear_combination_no_derived_raw_cap · Uses: G001, S020 S022 · calculationBefore R2 FIRST integer coordinate list ORIGINAL vector0 coordinate1 weighted term := mul(-2, -2) After R2 FIRST integer coordinate list ORIGINAL vector0 coordinate1 weighted term = 4 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore R2 FIRST integer coordinate list ORIGINAL vector1 coordinate1 weighted term := mul(0, 1) After R2 FIRST integer coordinate list ORIGINAL vector1 coordinate1 weighted term = 0 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore R2 FIRST integer coordinate list ORIGINAL coordinate1 all terms sum := sum(4, 0) After R2 FIRST integer coordinate list ORIGINAL coordinate1 all terms sum = 4 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore source roles = [R2 FIRST integer coordinate list, 1, [[1, -2], [0, 1]], [-2, 0]] After derived = [[4, 0], 4] Conditions: none For EACH coordinate retain EVERY coefficient*vector component before summing.Do not recap derived reconstructions as RAW vector/coefficient inputs. Step identifiers and dependenciesRule: Basis_EACH_coordinate_complete_linear_combination_no_derived_raw_cap · Uses: G001, S024 S026 · calculationBefore R2 SECOND integer coordinate list ORIGINAL vector0 coordinate0 weighted term := mul(-2, 1) After R2 SECOND integer coordinate list ORIGINAL vector0 coordinate0 weighted term = -2 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore R2 SECOND integer coordinate list ORIGINAL vector1 coordinate0 weighted term := mul(0, 0) After R2 SECOND integer coordinate list ORIGINAL vector1 coordinate0 weighted term = 0 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore R2 SECOND integer coordinate list ORIGINAL coordinate0 all terms sum := sum(-2, 0) After R2 SECOND integer coordinate list ORIGINAL coordinate0 all terms sum = -2 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore source roles = [R2 SECOND integer coordinate list, 0, [[1, -2], [0, 1]], [-2, 0]] After derived = [[-2, 0], -2] Conditions: none For EACH coordinate retain EVERY coefficient*vector component before summing.Do not recap derived reconstructions as RAW vector/coefficient inputs. Step identifiers and dependenciesRule: Basis_EACH_coordinate_complete_linear_combination_no_derived_raw_cap · Uses: G001, S028 S030 · calculationBefore R2 SECOND integer coordinate list ORIGINAL vector0 coordinate1 weighted term := mul(-2, -2) After R2 SECOND integer coordinate list ORIGINAL vector0 coordinate1 weighted term = 4 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore R2 SECOND integer coordinate list ORIGINAL vector1 coordinate1 weighted term := mul(0, 1) After R2 SECOND integer coordinate list ORIGINAL vector1 coordinate1 weighted term = 0 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore R2 SECOND integer coordinate list ORIGINAL coordinate1 all terms sum := sum(4, 0) After R2 SECOND integer coordinate list ORIGINAL coordinate1 all terms sum = 4 Conditions: none Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · impliesBefore source roles = [R2 SECOND integer coordinate list, 1, [[1, -2], [0, 1]], [-2, 0]] After derived = [[4, 0], 4] Conditions: none For EACH coordinate retain EVERY coefficient*vector component before summing.Do not recap derived reconstructions as RAW vector/coefficient inputs. Step identifiers and dependenciesRule: Basis_EACH_coordinate_complete_linear_combination_no_derived_raw_cap · Uses: G001, S032 S034 · impliesBefore source roles = [[[1, -2], [0, 1]], [-2, 4], 1] After derived = [[-2, 0], [-2, 0], [-2, 4], Yes, [-2, 0], [-2, 4], Yes, Yes] Conditions: none The ordered vectors are COLUMNS;D≠0 qualifies unique coordinates.The SOURCE true-coordinate helper must pass its rational envelope EVEN when not in the output.Submitted lists are INTEGER coordinates;reconstruct each.The final BOTH-valid⇒equal implication may be true with an invalid list and does not certify both. Step identifiers and dependenciesRule: Basis_R2_nonzero_column_basis_source_TRUEcoordinate_gate_and_conditional_uniqueness · Uses: G001, S033 ## Reverse checks - S034 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S033 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S032 · premise_and_result_audit — Recompute R2 SECOND integer coordinate list ORIGINAL coordinate1 all terms sum with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute R2 SECOND integer coordinate list ORIGINAL vector1 coordinate1 weighted term with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute R2 SECOND integer coordinate list ORIGINAL vector0 coordinate1 weighted term with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S028 · premise_and_result_audit — Recompute R2 SECOND integer coordinate list ORIGINAL coordinate0 all terms sum with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute R2 SECOND integer coordinate list ORIGINAL vector1 coordinate0 weighted term with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute R2 SECOND integer coordinate list ORIGINAL vector0 coordinate0 weighted term with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S024 · premise_and_result_audit — Recompute R2 FIRST integer coordinate list ORIGINAL coordinate1 all terms sum with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute R2 FIRST integer coordinate list ORIGINAL vector1 coordinate1 weighted term with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute R2 FIRST integer coordinate list ORIGINAL vector0 coordinate1 weighted term with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S020 · premise_and_result_audit — Recompute R2 FIRST integer coordinate list ORIGINAL coordinate0 all terms sum with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute R2 FIRST integer coordinate list ORIGINAL vector1 coordinate0 weighted term with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute R2 FIRST integer coordinate list ORIGINAL vector0 coordinate0 weighted term with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute R2 source helper TRUEcoordinate1 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S015 · premise_and_result_audit — Recompute R2 source helper TRUEcoordinate0 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S013 · premise_and_result_audit — Recompute R2 true coordinate ORIGINAL second numerator with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute R2 true coordinate ORIGINAL targetx times b1y with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute R2 true coordinate ORIGINAL b1x times targety with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute R2 true coordinate ORIGINAL first numerator with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute R2 true coordinate ORIGINAL b2x times targety with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute R2 true coordinate ORIGINAL targetx times b2y with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute R2 ORIGINAL signed column determinant with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute R2 ORIGINAL b2x times b1y with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute R2 ORIGINAL b1x times b2y with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S003 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S002 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. ## Result ([[1, -2], [0, 1]], [-2, 4], 1, [-2, 0], [-2, 4], Yes, [-2, 0], [-2, 4], Yes, Yes) 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: 410b8ed05ee0afdd2f80d0d590e63b64e02a70e3afea2799ec762e5625f90638 - candidate record SHA-256: 11b0f36c9e7f8a8290041ed35b72a594c834a9d65feb8add76ec92758c3048c3 - semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6 - question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261025-r04-linear-independence-basis-0086.json)