# Question 7107: Ordered Recurrence Window and Adjacent Differences [Return to catalogue page 36](https://ocme.evemisslab.com/en/questions/page/36/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261011-advance-20261015-r05-sequence-foundations/#q-20261011-advance-20261015-r05-sequence-0032) 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. R0013-07107 · q-20261011-advance-20261015-r05-sequence-0032 ## 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=-6, a_(n+1)=a_n+(-1), start=4, length=3. 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=-6, a_(n+1)=a_n+(-1), start=4, length=3. 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 a(0)=-6;a(n+1)=a(n)+(-1) After a(n)=-6+n·(-1) by this supplied affine recurrence Conditions: none The base n=0 holds;if a(n)=s+nd,adding d gives s+(n+1)d.This follows from the explicitly given recurrence,not guessing an infinite rule from a few observations. Step identifiers and dependenciesRule: affine_recurrence_closed_form_derivation · Uses: G001, S001 S003 · calculationBefore window start increment := mul(4, -1) After window start increment = -4 Conditions: none Use the derived form for the accumulated increment at the window start. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore window start value := sum(-6, -4) After window start value = -10 Conditions: none Add the increment to the initial value to obtain the first window term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore window recurrence term1 := sum(-10, -1) After window recurrence term1 = -11 Conditions: none Inside the window,add d according to the same recurrence. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore window recurrence term2 := sum(-11, -1) After window recurrence term2 = -12 Conditions: none Inside the window,add d according to the same recurrence. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore window adjacent difference0 := sub(-11, -10) After window adjacent difference0 = -1 Conditions: none Subtract actual neighboring window values to audit the increment. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore window adjacent difference1 := sub(-12, -11) After window adjacent difference1 = -1 Conditions: none Subtract actual neighboring window values to audit the increment. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore window indices=[4, 5, 6];values=[-10, -11, -12] After adjacent deltas=[-1, -1] Conditions: none Align each index with its term;length=1 has no adjacent differences.Do not deduplicate term values or require them to be natural numbers. Step identifiers and dependenciesRule: ordered_recurrence_window_audit · Uses: G001, S008 ## Reverse checks - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute window adjacent difference1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute window adjacent difference0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute window recurrence term2 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute window recurrence term1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute window start value with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute window start increment with exact rational arithmetic, not rounded decimals. - 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 ([4, 5, 6], [-10, -11, -12], [-1, -1]) 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: 4272c4a0c940cd78926779fab5c4fc033320edb1cb3ebd03f61bede618c887a3 - candidate record SHA-256: e89fc11a267fc7a40d558719f77dcc8e7f22fadfb8f5221bc67789cbad275fa8 - 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-0032.json)