# Question 5480: Induction-Hypothesis and Successor Indices

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0009-05480 · q-20261008-advance-20261012-r01-mathematical-induction-0005



## Original question and givens

N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=4. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.


- G001 — N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=4. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.





## Definition and scope

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.


- [mko-mathematical-induction](https://ocme.evemisslab.com/en/mko/mko-mathematical-induction/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: finite_induction_claim_boundary · Uses: G001





S002 · calculationBefore

successor target index := sum(4, 1)

After

successor target index = 5

Conditions: none

Add1 to the current natural index to obtain the successor target.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · impliesBefore

hypothesis index=4

After

target index=5

Conditions: none

P(n) marks the induction hypothesis and P(n+1) the target;listing their indices does not assert either proposition.

Step identifiers and dependenciesRule: hypothesis_vs_successor_target · Uses: G001, S002









## Reverse checks


- 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 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.





## Result

(4, 5)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 36c9f816a0104d276144178e86ab2cb5117ce24718e6f8bc2cc224a1ee19ce20
- candidate record SHA-256: becf107bb39da7370e746fe183049fe2c6a25a6388b8792b100c4192a7b6c776
- semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
- question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r01-mathematical-induction-0005.json)