# Question 33209: A Nonzero Proportional-Pair Witness Must Actually Match [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-0059) 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. R0065-33209 · q-20261013-advance-20261025-r04-linear-independence-basis-0059 ## 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. nonzero left=[9, 10], right=[36, 40], nonzero integer scalar=4. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?). - 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. nonzero left=[9, 10], right=[36, 40], nonzero integer scalar=4. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?). ## 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-proportional-pair-dependence-certificate, basisdeep-proportional-pair-dependence-certificate, R2_basis_density_without_dimension_extension, []] After derived = [[left, right, scalar, scaled, relation, THIS_certificate]] 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 = [proportional left, [9, 10]] After derived = [[9, 10]] 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 = [proportional right, [36, 40]] After derived = [[36, 40]] 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 · calculationBefore Proportional ORIGINAL scalar times left coordinate0 := mul(4, 9) After Proportional ORIGINAL scalar times left coordinate0 = 36 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, S003 S005 · calculationBefore Proportional ORIGINAL scaled minus right coordinate0 := sub(36, 36) After Proportional ORIGINAL scaled minus right coordinate0 = 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, S004 S006 · calculationBefore Proportional ORIGINAL scalar times left coordinate1 := mul(4, 10) After Proportional ORIGINAL scalar times left coordinate1 = 40 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 Proportional ORIGINAL scaled minus right coordinate1 := sub(40, 40) After Proportional ORIGINAL scaled minus right coordinate1 = 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, S006 S008 · impliesBefore source roles = [[9, 10], [36, 40], 4] After derived = [[36, 40], [0, 0], Yes, [4, -1]] Conditions: none FIRST require nonzero left/scalar,THEN audit EVERY coordinate of scalar*left−right.ONLY all-zero residuals validate the nontrivial(scalar,−1)relation.No does not establish pair independence;another scalar may fit. Step identifiers and dependenciesRule: Basis_PROPORTIONAL_nonzero_premises_ALL_coordinates_relation_not_assumed · Uses: G001, S007 ## Reverse checks - S008 · 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. - S007 · premise_and_result_audit — Recompute Proportional ORIGINAL scaled minus right coordinate1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Proportional ORIGINAL scalar times left coordinate1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Proportional ORIGINAL scaled minus right coordinate0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Proportional ORIGINAL scalar times left coordinate0 with exact rational arithmetic, not rounded decimals. - 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 ([9, 10], [36, 40], 4, [36, 40], [0, 0], 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: 634a0628748f7aba94225e93143aa6ea437b403d7215d77c193ee4b6e92208bc - candidate record SHA-256: b2dd5783a56a2a8c18fe54dbd4ff1b5a5d3ad9829139dd70a74c5c45b904c048 - 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-0059.json)