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=[21, 22], right=[21, 22], nonzero integer scalar=1. 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=[21, 22], right=[21, 22], nonzero integer scalar=1. 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.
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.
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.
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.
Proportional ORIGINAL scalar times left coordinate0 := mul(1, 21)
After
Proportional ORIGINAL scalar times left coordinate0 = 21
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.
Proportional ORIGINAL scaled minus right coordinate0 := sub(21, 21)
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.
Proportional ORIGINAL scalar times left coordinate1 := mul(1, 22)
After
Proportional ORIGINAL scalar times left coordinate1 = 22
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.
Proportional ORIGINAL scaled minus right coordinate1 := sub(22, 22)
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.
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.
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
([21, 22], [21, 22], 1, [21, 22], [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.