Explore related knowledge and questions →
← Question derivation index · Original question batch
R0004-02604 · q-20261013-advance-20261021-r01-mathematical-representation-0004
Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=-9, inputs=[0, 1, 3]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).
G001 — Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=-9, inputs=[0, 1, 3]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).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
declared symbolic rule or complete finite table
After
keep the stated domain and representation distinct
Conditions: none
Use only the declared rule or table;agreement on a finite table alone does not establish function equality on an unlisted domain.
Rule: representation_scope · Uses: G001
Before
[0, 1, 3]
After
[0, 1, 3]
Conditions: none
Deduplicate and numerically order the domain;repeating an input creates no additional function value.
Rule: distinct_numeric_domain · Uses: G001, S001
Before
x=0 ax := mul(0, 0)
After
x=0 ax = 0
Conditions: none
Evaluate a·x first;do not replace the rule by a(x+b).
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
x=0 ax+b := sum(0, -9)
After
x=0 ax+b = -9
Conditions: none
Then add the intercept b,fully substituting this input.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
x=0; ax+b=-9
After
[0, -9]
Conditions: none
Each input is paired with its calculated output;the ordered pair lists input before output.
Rule: representation_table_row · Uses: G001, S004
Before
x=1 ax := mul(0, 1)
After
x=1 ax = 0
Conditions: none
Evaluate a·x first;do not replace the rule by a(x+b).
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
x=1 ax+b := sum(0, -9)
After
x=1 ax+b = -9
Conditions: none
Then add the intercept b,fully substituting this input.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
x=1; ax+b=-9
After
[1, -9]
Conditions: none
Each input is paired with its calculated output;the ordered pair lists input before output.
Rule: representation_table_row · Uses: G001, S007
Before
x=3 ax := mul(0, 3)
After
x=3 ax = 0
Conditions: none
Evaluate a·x first;do not replace the rule by a(x+b).
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
x=3 ax+b := sum(0, -9)
After
x=3 ax+b = -9
Conditions: none
Then add the intercept b,fully substituting this input.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
x=3; ax+b=-9
After
[3, -9]
Conditions: none
Each input is paired with its calculated output;the ordered pair lists input before output.
Rule: representation_table_row · 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 x=3 ax+b with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute x=3 ax with exact rational arithmetic, not rounded decimals.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 x=1 ax+b with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute x=1 ax 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 x=0 ax+b with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute x=0 ax 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.(0, -9, [0, 1, 3], [[0, -9], [1, -9], [3, -9]])
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: 9487f662fab0dfab33f9de091f76a8ffcf961a4b89fc71fe7c7742e77f6a4104candidate record SHA-256: e3dd06e453ee9562302edbf2a024fde1c6bb8690b90679b4ea19e1975af13595semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED