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R0010-05716 · q-20261008-advance-20261012-r03-proposition-0016
In a classical integer context, ordered pairs=[[3, 0], [-14, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
G001 — In a classical integer context, ordered pairs=[[3, 0], [-14, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).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
classical integer formulas with explicit binding/data
After
a truth verdict is not a delivered general proof object
Conditions: none
A false closed statement is still a proposition.These finite substitutions and comparison witnesses do not promote a truth label to a general formal proof or completeness of arbitrary inference rules.
Rule: proposition_truth_and_proof_scope · Uses: G001
Before
integer comparison difference0 := sub(3, 0)
After
integer comparison difference0 = 3
Conditions: none
Compute a−b from the ordered pair,retaining zero and either sign.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
row=0;a=3;b=0;a−b=3
After
statement=a≤b;truth=False
Conditions: none
Equality requires difference0 and ≤ requires a nonpositive difference;the signed difference supplies the exact comparison witness,not a candidate report.
Rule: closed_integer_statement_row · Uses: G001, S002
Before
integer comparison difference1 := sub(-14, 2)
After
integer comparison difference1 = -16
Conditions: none
Compute a−b from the ordered pair,retaining zero and either sign.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
row=1;a=-14;b=2;a−b=-16
After
statement=a≤b;truth=True
Conditions: none
Equality requires difference0 and ≤ requires a nonpositive difference;the signed difference supplies the exact comparison witness,not a candidate report.
Rule: closed_integer_statement_row · Uses: G001, S004
Before
integer comparison difference2 := sub(-1, 0)
After
integer comparison difference2 = -1
Conditions: none
Compute a−b from the ordered pair,retaining zero and either sign.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
row=2;a=-1;b=0;a−b=-1
After
statement=a≤b;truth=True
Conditions: none
Equality requires difference0 and ≤ requires a nonpositive difference;the signed difference supplies the exact comparison witness,not a candidate report.
Rule: closed_integer_statement_row · Uses: G001, S006
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 — Recompute integer comparison difference2 with exact rational arithmetic, not rounded decimals.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 — Recompute integer comparison difference1 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Recompute integer comparison difference0 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.([False, True, True], [3, -16, -1])
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: 44b2b41834dc828e5e4cb0ad42247ab6f3f7827f2d6b876427552689dd87c22acandidate record SHA-256: 6c33a3e2bfc7475679d181601fc603613b28216f138d15decf739e282feb118asemantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06fquestion-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fcLean: NOT_RUN · human review: NOT_PERFORMED