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R0020-10874 · q-20261013-advance-20261023-r02-linear-combination-span-0024
Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[11, -1, 1]. Give (original vectors,original coefficients,the coordinate sum).
G001 — Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[11, -1, 1]. Give (original vectors,original coefficients,the coordinate sum).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.
Before
fixed real coordinate space;span uses all finite real linear combinations
After
given coefficient witness and existence of some coefficients are separate
Conditions: none
Span quantifies all finite real linear combinations.These integer/rational instances are declared slices;a failed designated witness does not show every coefficient choice fails.
Rule: span_quantifier_and_witness_scope · Uses: G001
Before
original ordered generators=[[1, 0], [0, 1], [1, 1]];aligned coefficients=[11, -1, 1]
After
fixed coordinate dimension=2;generator duplicates/order retained
Conditions: none
Pair each coefficient with its own generator,retaining duplicates and order.Do not sort/deduplicate by semantic identity or replace a finite combination by an infinite series.
Rule: span_generator_coefficient_pairing · Uses: G001, S001
Before
generator0 coordinate0 contribution := mul(11, 1)
After
generator0 coordinate0 contribution = 11
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
generator1 coordinate0 contribution := mul(-1, 0)
After
generator1 coordinate0 contribution = 0
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
generator2 coordinate0 contribution := mul(1, 1)
After
generator2 coordinate0 contribution = 1
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
weighted coordinate0 complete sum := sum(11, 0, 1)
After
weighted coordinate0 complete sum = 12
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
generator0 coordinate1 contribution := mul(11, 0)
After
generator0 coordinate1 contribution = 0
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
generator1 coordinate1 contribution := mul(-1, 1)
After
generator1 coordinate1 contribution = -1
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
generator2 coordinate1 contribution := mul(1, 1)
After
generator2 coordinate1 contribution = 1
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
weighted coordinate1 complete sum := sum(0, -1, 1)
After
weighted coordinate1 complete sum = 0
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
all ordered weighted coordinates=[12, 0]
After
finite combination=[12, 0]
Conditions: none
Add every generator contribution at the same coordinate;an inner product or norm is not the requested vector.
Rule: span_complete_componentwise_reconstruction · Uses: G001, S010
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Recompute weighted coordinate1 complete sum with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute generator2 coordinate1 contribution with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute generator1 coordinate1 contribution with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute generator0 coordinate1 contribution with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute weighted coordinate0 complete sum with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute generator2 coordinate0 contribution with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute generator1 coordinate0 contribution with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute generator0 coordinate0 contribution with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0])
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: c372aab80a290809dc1eea79d6af5c22e6ddc9ff32b8eea8f1f9947e36858744candidate record SHA-256: 698794cabf200a20c55bf5aa44e18e413d1b9561aed139e16599a7c566a20585semantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598fquestion-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4Lean: NOT_RUN · human review: NOT_PERFORMED