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Question 18632: External Direct-Sum Coordinates Live on a Product Carrier On this page Original question and givens Definition and scope Step-by-step derivation Reverse checks Result This question review and source binding Return to catalogue page 94
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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.
R0036-18632 · q-20261014-advance-20261105-r01-direct-sum-0082
Original question and givens Use the pair (left,right) with left=[6, 7], right=[8], another pair other_left=[9, 13], other_right=[10], and scalar=7. 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=[6, 7], right=[8], another pair other_left=[9, 13], other_right=[10], and scalar=7. 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.
Step-by-step derivation S001 · implies Before
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 dependencies Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · calculation Before
external left coordinate0 sum := sum(6, 9) After
external left coordinate0 sum = 15 Conditions: none
Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.
Step identifiers and dependencies Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation Before
external left coordinate1 sum := sum(7, 13) After
external left coordinate1 sum = 20 Conditions: none
Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.
Step identifiers and dependencies Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation Before
external left coordinate0 scalar product := mul(7, 6) After
external left coordinate0 scalar product = 42 Conditions: none
Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.
Step identifiers and dependencies Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation Before
external left coordinate1 scalar product := mul(7, 7) After
external left coordinate1 scalar product = 49 Conditions: none
Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.
Step identifiers and dependencies Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation Before
external right coordinate0 sum := sum(8, 10) After
external right coordinate0 sum = 18 Conditions: none
Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.
Step identifiers and dependencies Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation Before
external right coordinate0 scalar product := mul(7, 8) After
external right coordinate0 scalar product = 56 Conditions: none
Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.
Step identifiers and dependencies Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies Before
original pair=([6, 7],[8]);block lengths=[2, 1];other pair=([9, 13],[10]);scalar=7 After
coordinatewise pair sum=[[15, 20], [18]];scaled pair=[[42, 49], [56]];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 dependencies Rule: 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 ([[6, 7], [8]], [[15, 20], [18]], [[42, 49], [56]], 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: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: bf03055edffe7e935044319859116fed27aac6c3af54ac123df031eac5eb0c00candidate record SHA-256: 4324121bb8a53c7f4ee771fcbd6e6b77a53d10533bc4b9df0bfd434dc087f3c7semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2Lean: NOT_RUN · human review: NOT_PERFORMEDSame-source full JSON