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R0009-05427 · q-20261008-advance-20261012-r01-mathematical-induction-0052
N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=3, increment d=2; candidate c(k)=(2)k+(1), n=5. Give signed integers (c(0)−s, c(n+1)−c(n)−d).
G001 — N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=3, increment d=2; candidate c(k)=(2)k+(1), n=5. Give signed integers (c(0)−s, c(n+1)−c(n)−d).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
successor target index := sum(5, 1)
After
successor target index = 6
Conditions: none
Add1 to the current natural index to obtain the successor target.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
candidate c(0) product := mul(2, 0)
After
candidate c(0) product = 0
Conditions: none
Substitute the specified index into the candidate affine product term.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
candidate c(0) value := sum(0, 1)
After
candidate c(0) value = 1
Conditions: none
Then add the candidate constant b.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
candidate c(5) product := mul(2, 5)
After
candidate c(5) product = 10
Conditions: none
Substitute the specified index into the candidate affine product term.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
candidate c(5) value := sum(10, 1)
After
candidate c(5) value = 11
Conditions: none
Then add the candidate constant b.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
candidate c(6) product := mul(2, 6)
After
candidate c(6) product = 12
Conditions: none
Substitute the specified index into the candidate affine product term.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
candidate c(6) value := sum(12, 1)
After
candidate c(6) value = 13
Conditions: none
Then add the candidate constant b.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
candidate base defect := sub(1, 3)
After
candidate base defect = -2
Conditions: none
The base defect is c(0)−s,distinct from the step defect.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
candidate adjacent difference := sub(13, 11)
After
candidate adjacent difference = 2
Conditions: none
First subtract adjacent candidate values c(n+1)−c(n).
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
candidate step defect := sub(2, 2)
After
candidate step defect = 0
Conditions: none
Then subtract the required increment d,retaining the signed defect.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
c(k)=2k+1;specified n=5
After
base defect=-2;step defect=0
Conditions: none
Report the specified substitutions.A separate symbolic identity gives base b−s and step a−d for this affine model;that differs from extrapolating a general proposition from finitely many samples.
Rule: affine_candidate_base_step_separation · 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 candidate step defect with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute candidate adjacent difference with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute candidate base defect with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute candidate c(6) value with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute candidate c(6) product with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute candidate c(5) value with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute candidate c(5) product with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute candidate c(0) value with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute candidate c(0) product with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute successor target index 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.(-2, 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: 7c00897a72b3dcd52fcf66bcf5814f35f13ed69136408de9ab854c594a3e082bcandidate record SHA-256: 62766ad77a42d528db0b18c52ece88a69c78c2b2e0119b651eebfa78a2723574semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462Lean: NOT_RUN · human review: NOT_PERFORMED