# Question 5440: Separate Base and Successor-Step Defects

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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.

R0009-05440 · q-20261008-advance-20261012-r01-mathematical-induction-0065



## Original question and givens

N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. Required initial value s=0, increment d=1; candidate c(k)=(3)k+(15), n=2. 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=0, increment d=1; candidate c(k)=(3)k+(15), n=2. Give signed integers (c(0)−s, c(n+1)−c(n)−d).





## 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(2, 1)

After

successor target index = 3

Conditions: none

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

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





S003 · calculationBefore

candidate c(0) product := mul(3, 0)

After

candidate c(0) product = 0

Conditions: none

Substitute the specified index into the candidate affine product term.

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





S004 · calculationBefore

candidate c(0) value := sum(0, 15)

After

candidate c(0) value = 15

Conditions: none

Then add the candidate constant b.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

candidate c(2) product := mul(3, 2)

After

candidate c(2) product = 6

Conditions: none

Substitute the specified index into the candidate affine product term.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

candidate c(2) value := sum(6, 15)

After

candidate c(2) value = 21

Conditions: none

Then add the candidate constant b.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

candidate c(3) product := mul(3, 3)

After

candidate c(3) product = 9

Conditions: none

Substitute the specified index into the candidate affine product term.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

candidate c(3) value := sum(9, 15)

After

candidate c(3) value = 24

Conditions: none

Then add the candidate constant b.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

candidate base defect := sub(15, 0)

After

candidate base defect = 15

Conditions: none

The base defect is c(0)−s,distinct from the step defect.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

candidate adjacent difference := sub(24, 21)

After

candidate adjacent difference = 3

Conditions: none

First subtract adjacent candidate values c(n+1)−c(n).

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

candidate step defect := sub(3, 1)

After

candidate step defect = 2

Conditions: none

Then subtract the required increment d,retaining the signed defect.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · impliesBefore

c(k)=3k+15;specified n=2

After

base defect=15;step defect=2

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.

Step identifiers and dependenciesRule: affine_candidate_base_step_separation · Uses: G001, S011









## Reverse checks


- 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(3) value with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute candidate c(3) product with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute candidate c(2) value with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute candidate c(2) 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.





## Result

(15, 2)

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: 10ec5f5292f6ceeaeba9997f760b5b782977e2a6a33c73d56355474bef761283
- candidate record SHA-256: 334f8d212951ea713e349ef8662ef2fd721f315c2e0c7e18f3b496a5fc3ccd82
- 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-0065.json)