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R0009-05452 · q-20261008-advance-20261012-r01-mathematical-induction-0077
N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(1000) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).
G001 — N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(1000) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).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
covered inclusive index count := sum(1000, 1)
After
covered inclusive index count = 1001
Conditions: none
P(0) through P(through) includes both endpoints,so there are through+1 indices.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
covered indices0,...,1000
After
1000 adjacent steps;next uncovered target=1001
Conditions: none
The adjacent-step count equals the final index and the next target is through+1.This counts declared coverage,not independent verification of each P.
Rule: finite_range_obligation_counts · Uses: G001, S002
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 covered inclusive index count 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.(1001, 1000, 1001)
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: 5d2cda00f61e422eec3964d3658c1f6c12f1e9b7984b5061aba94fb634407c6acandidate record SHA-256: 6e270588ac86243384077cbed0479d511adc97f9af2e4c8f3b0336e0db8ea3ebsemantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462Lean: NOT_RUN · human review: NOT_PERFORMED