# Question 7101: Ordered Recurrence Window and Adjacent Differences

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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.

R0013-07101 · q-20261011-advance-20261015-r05-sequence-0026



## Original question and givens

N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=0, a_(n+1)=a_n+(1), start=0, length=0. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.


- G001 — N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=0, a_(n+1)=a_n+(1), start=0, length=0. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.





## 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-sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/)





## Step-by-step derivation

S001 · impliesBefore

N starts at0;ordered finite indices and repeated values retained

After

given recurrences differ from guessing an infinite rule from observations

Conditions: none

Sequences are ordered and may repeat.A finite observed prefix alone does not determine a unique infinite rule;when a recurrence is explicitly supplied,use that rule rather than inventing one.

Step identifiers and dependenciesRule: finite_ordered_sequence_scope · Uses: G001





S002 · impliesBefore

start=0;length=0

After

([], [], [])

Conditions: none

An empty window has no recorded terms or adjacent differences;do not invent entries for the empty output.

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









## Reverse checks


- 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.





## Result

([], [], [])

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: 233e62b8efdda49aa136bc22533885ddb7aee173bb8b558b90f1ed9587fea77d
- candidate record SHA-256: 95ddbbd74099eee9806cb77e415f1161cc7f1aeec1e6ce46bb3a6289daddc250
- semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59
- question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261011-advance-20261015-r05-sequence-0026.json)