# Question 12621: The Norm of an Ordered Vector Difference [Return to catalogue page 64](https://ocme.evemisslab.com/en/questions/page/64/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations/#q-20261013-advance-20261026-r02-real-normed-vector-space-0096) 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. R0024-12621 · q-20261013-advance-20261026-r02-real-normed-vector-space-0096 ## Original question and givens Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[8, 3, 0], right=[-8, 3, -2] with equal dimension. Give (norm_name,left,right,left−right,distance). - G001 — Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[8, 3, 0], right=[-8, 3, -2] with equal dimension. Give (norm_name,left,right,left−right,distance). ## 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-real-normed-vector-space](https://ocme.evemisslab.com/en/mko/mko-real-normed-vector-space/) ## Step-by-step derivation S001 · calculationBefore distance coordinate0 left-minus-right := sub(8, -8) After distance coordinate0 left-minus-right = 16 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore distance coordinate1 left-minus-right := sub(3, 3) After distance coordinate1 left-minus-right = 0 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore distance coordinate2 left-minus-right := sub(0, -2) After distance coordinate2 left-minus-right = 2 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore named certified norm=l1;left=[8, 3, 0];right=[-8, 3, -2] After ordered difference=[16, 0, 2];distance=norm(left-right) Conditions: none Use the same named norm on the same coordinate space and retain the difference direction.This does not assert every metric is norm-induced,or add completeness or inner-product claims. Step identifiers and dependenciesRule: fixed_norm_induced_distance_scope · Uses: G001, S003 S005 · impliesBefore induced distance:coordinate=0;signed value=16 After absolute value=16 Conditions: none Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector. Step identifiers and dependenciesRule: named_norm_each_coordinate_absolute_value · Uses: G001, S004 S006 · impliesBefore induced distance:coordinate=1;signed value=0 After absolute value=0 Conditions: none Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector. Step identifiers and dependenciesRule: named_norm_each_coordinate_absolute_value · Uses: G001, S005 S007 · impliesBefore induced distance:coordinate=2;signed value=2 After absolute value=2 Conditions: none Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector. Step identifiers and dependenciesRule: named_norm_each_coordinate_absolute_value · Uses: G001, S006 S008 · calculationBefore induced distance l1 sum := sum(16, 0, 2) After induced distance l1 sum = 18 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Recompute induced distance l1 sum with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 distance coordinate2 left-minus-right with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute distance coordinate1 left-minus-right with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute distance coordinate0 left-minus-right with exact rational arithmetic, not rounded decimals. ## Result (l1, [8, 3, 0], [-8, 3, -2], [16, 0, 2], 18) 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: feba4c3d4edcb663df6704e5f296b542cd86f3487cee7e1cb417c4135f13881a - candidate record SHA-256: 3b97c18ad4a0b0c4671313735e6148351da27943855ceda2cdd794c94c2a621c - semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35 - question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261026-r02-real-normed-vector-space-0096.json)