# Question 18647: External Direct-Sum Coordinates Live on a Product Carrier [Return to catalogue page 94](https://ocme.evemisslab.com/en/questions/page/94/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261105-r01-direct-sum-foundations/#q-20261014-advance-20261105-r01-direct-sum-0097) 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. R0036-18647 · q-20261014-advance-20261105-r01-direct-sum-0097 ## Original question and givens Use the pair (left,right) with left=[21, 22], right=[23], another pair other_left=[24, 43], other_right=[25], and scalar=22. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, boundary). - G001 — Use the pair (left,right) with left=[21, 22], right=[23], another pair other_left=[24, 43], other_right=[25], and scalar=22. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pair, sum, scaled_pair, 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-direct-sum](https://ocme.evemisslab.com/en/mko/mko-direct-sum/) ## Step-by-step derivation S001 · impliesBefore source operation=directsum-external-coordinate;original snapshot and mathematics retained After explicit ordered source output fields=[pair, sum, scaled_pair, boundary] Conditions: none Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore external left coordinate0 sum := sum(21, 24) After external left coordinate0 sum = 45 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore external left coordinate1 sum := sum(22, 43) After external left coordinate1 sum = 65 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore external left coordinate0 scalar product := mul(22, 21) After external left coordinate0 scalar product = 462 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore external left coordinate1 scalar product := mul(22, 22) After external left coordinate1 scalar product = 484 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore external right coordinate0 sum := sum(23, 25) After external right coordinate0 sum = 48 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore external right coordinate0 scalar product := mul(22, 23) After external right coordinate0 scalar product = 506 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore original pair=([21, 22],[23]);block lengths=[2, 1];other pair=([24, 43],[25]);scalar=22 After coordinatewise pair sum=[[45, 65], [48]];scaled pair=[[462, 484], [506]];new PRODUCT carrier,not assumed internal subspaces Conditions: none Retain both labeled coordinate blocks,including valid empty blocks.The two-term external direct sum uses coordinatewise operations on a pair carrier;matching lengths do not prove an internal decomposition of another ambient space. Step identifiers and dependenciesRule: directsum_external_pair_not_ambient_internal_identification · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute external right coordinate0 scalar product with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute external right coordinate0 sum with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute external left coordinate1 scalar product with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute external left coordinate0 scalar product with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute external left coordinate1 sum with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute external left coordinate0 sum with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([[21, 22], [23]], [[45, 65], [48]], [[462, 484], [506]], external_product_carrier_not_internal_subspaces) 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: 098b57617adc513bb786ebee32124bae6f70b26203f234087fd7f7f3085c1fe8 - candidate record SHA-256: fec1db31b2a4dc10bfcca42953ef741fbee1e79d452e3aea1e34ac636aa153a8 - semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301 - question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r01-direct-sum-0097.json)