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R0004-02605 · q-20261013-advance-20261021-r01-mathematical-representation-0005
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=1, b=-8, inputs=[1, 2, 4]; 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=1, b=-8, inputs=[1, 2, 4]; 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
[1, 2, 4]
After
[1, 2, 4]
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=1 ax := mul(1, 1)
After
x=1 ax = 1
Conditions: none
Evaluate a·x first;do not replace the rule by a(x+b).
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
x=1 ax+b := sum(1, -8)
After
x=1 ax+b = -7
Conditions: none
Then add the intercept b,fully substituting this input.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
x=1; ax+b=-7
After
[1, -7]
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=2 ax := mul(1, 2)
After
x=2 ax = 2
Conditions: none
Evaluate a·x first;do not replace the rule by a(x+b).
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
x=2 ax+b := sum(2, -8)
After
x=2 ax+b = -6
Conditions: none
Then add the intercept b,fully substituting this input.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
x=2; ax+b=-6
After
[2, -6]
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=4 ax := mul(1, 4)
After
x=4 ax = 4
Conditions: none
Evaluate a·x first;do not replace the rule by a(x+b).
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
x=4 ax+b := sum(4, -8)
After
x=4 ax+b = -4
Conditions: none
Then add the intercept b,fully substituting this input.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
x=4; ax+b=-4
After
[4, -4]
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=4 ax+b with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute x=4 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=2 ax+b with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute x=2 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=1 ax+b with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute x=1 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.(1, -8, [1, 2, 4], [[1, -7], [2, -6], [4, -4]])
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: 5694f859a1e3d2b4444e2e6e793298c50939da00e195cc5fa4ab6576e3871ecbcandidate record SHA-256: 8e5e835ea4bcb3c7776c37b90300a1c2ee3f929ae9137c4dd76386dd7efb28fcsemantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED