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R0009-05421 · q-20261008-advance-20261012-r01-mathematical-induction-0046
N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=8, a(k+1)=a(k)+(-2), through=5. List a(0) through a(through), inclusive.
G001 — N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=8, a(k+1)=a(k)+(-2), through=5. List a(0) through a(through), inclusive.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
natural indices start at0;finite induction protocol
After
index bookkeeping and finite values are not proofs of all P(n)
Conditions: none
Natural indices do not require nonnegative sequence values.Checking a finite range or a successor index does not supply proposition truth or an induction proof over all naturals.
Rule: finite_induction_claim_boundary · Uses: G001
Before
a(0)=8;through=5
After
[8]
Conditions: none
Start with the supplied initial value;through=0 retains only it.
Rule: recurrence_prefix_base · Uses: G001, S001
Before
recurrence next value1 := sum(8, -2)
After
recurrence next value1 = 6
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
a(0)=8;d=-2
After
a(1)=6
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S003
Before
recurrence next value2 := sum(6, -2)
After
recurrence next value2 = 4
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
a(1)=6;d=-2
After
a(2)=4
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S005
Before
recurrence next value3 := sum(4, -2)
After
recurrence next value3 = 2
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
a(2)=4;d=-2
After
a(3)=2
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S007
Before
recurrence next value4 := sum(2, -2)
After
recurrence next value4 = 0
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
a(3)=2;d=-2
After
a(4)=0
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S009
Before
recurrence next value5 := sum(0, -2)
After
recurrence next value5 = -2
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
a(4)=0;d=-2
After
a(5)=-2
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S011
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Recompute recurrence next value5 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Recompute recurrence next value4 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 recurrence next value3 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Recompute recurrence next value2 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute recurrence next value1 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.[8, 6, 4, 2, 0, -2]
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: 4d324f33575fd0773ed86eb07fae926206943db88972235417585c2f746fad88candidate record SHA-256: 66d7e754cb5b7a0900c253c5906e72d5f440bfd1b98f4f7adf0c1b19aa44fcc9semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462Lean: NOT_RUN · human review: NOT_PERFORMED