# Question 22538: A Complete Initial State Gives a Unique Forward Prefix [Return to catalogue page 113](https://ocme.evemisslab.com/en/questions/page/113/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261110-r05-recurrence-relation-foundations/#q-20261014-advance-20261110-r05-recurrence-relation-0063) 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. R0044-22538 · q-20261014-advance-20261110-r05-recurrence-relation-0063 ## Original question and givens Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=0 and initial_values=[12, 13], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_count, target_n, target_value, unique_prefix, certificate). - G001 — Use order r=2, coefficients=[1, 2] in oldest-to-newest order, constant=0 and initial_values=[12, 13], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=5. Give (order, initial_count, target_n, target_value, unique_prefix, certificate). ## 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-recurrence-relation](https://ocme.evemisslab.com/en/mko/mko-recurrence-relation/) ## Step-by-step derivation S001 · impliesBefore source operation=recurrence-initial-value-uniqueness;original snapshot and mathematics retained After explicit ordered source output fields=[order, initial_count, target_n, target_value, unique_prefix, certificate] Conditions: none Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore recurrence ORIGINAL:ORIGINAL order=2;oldest-to-newest coefficients=[1, 2];constant=0;ALL initial a_0..a_(r-1)=[12, 13] After for EVERY n>=2,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=2;forward rule is total and single-valued Conditions: none Fix all r initials and the oldest-to-newest coefficient roles.Given the preceding r values,the next value is uniquely determined.This is a forward definition,not a closed form or convergence claim. Step identifiers and dependenciesRule: recurrence_complete_initial_state_and_total_single_value_rule · Uses: G001, S001 S003 · calculationBefore recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 := mul(1, 12) After recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 = 12 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore recurrence ORIGINAL n=2 partial sum after coefficient0 := sum(0, 12) After recurrence ORIGINAL n=2 partial sum after coefficient0 = 12 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 := mul(2, 13) After recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 = 26 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore recurrence ORIGINAL n=2 partial sum after coefficient1 := sum(12, 26) After recurrence ORIGINAL n=2 partial sum after coefficient1 = 38 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore recurrence ORIGINAL:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[12, 13];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 12, 12], [1, 1, 2, 13, 26]] After new a_2=38;next state=[13, 38];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 38] Conditions: none Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update. Step identifiers and dependenciesRule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S006 S008 · calculationBefore recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 := mul(1, 13) After recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 = 13 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(0, 13) After recurrence ORIGINAL n=3 partial sum after coefficient0 = 13 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 := mul(2, 38) After recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 = 76 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore recurrence ORIGINAL n=3 partial sum after coefficient1 := sum(13, 76) After recurrence ORIGINAL n=3 partial sum after coefficient1 = 89 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[13, 38];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 13, 13], [1, 2, 2, 38, 76]] After new a_3=89;next state=[38, 89];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 38, 89] Conditions: none Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update. Step identifiers and dependenciesRule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S011 S013 · calculationBefore recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 := mul(1, 38) After recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 = 38 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(0, 38) After recurrence ORIGINAL n=4 partial sum after coefficient0 = 38 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 := mul(2, 89) After recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 = 178 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(38, 178) After recurrence ORIGINAL n=4 partial sum after coefficient1 = 216 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[38, 89];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 38, 38], [1, 3, 2, 89, 178]] After new a_4=216;next state=[89, 216];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 38, 89, 216] Conditions: none Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update. Step identifiers and dependenciesRule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S016 S018 · calculationBefore recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 := mul(1, 89) After recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 = 89 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(0, 89) After recurrence ORIGINAL n=5 partial sum after coefficient0 = 89 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 := mul(2, 216) After recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 = 432 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(89, 432) After recurrence ORIGINAL n=5 partial sum after coefficient1 = 521 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · impliesBefore recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[89, 216];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 89, 89], [1, 4, 2, 216, 432]] After new a_5=521;next state=[216, 521];drop ONLY oldest state entry then append new scalar;prefix now=[12, 13, 38, 89, 216, 521] Conditions: none Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update. Step identifiers and dependenciesRule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S021 S023 · impliesBefore recurrence ORIGINAL:complete ORIGINAL initial state=[12, 13];actual generated prefix=[12, 13, 38, 89, 216, 521] After ANY two sequences with SAME r initials and SAME total rule agree at n=2 by substitution,then at every later n by induction;this determines the requested prefix only Conditions: none Induct from equality of all r initials:the same single-valued rule gives equality of the next term,and hence every later prefix term.Finite generation proves no closed form,asymptotic behavior or analyticity. Step identifiers and dependenciesRule: recurrence_forward_uniqueness_induction_not_analytic_claim · Uses: G001, S022 ## Reverse checks - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_4 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_3 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_3 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_2 with exact rational arithmetic, not rounded decimals. - 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 recurrence ORIGINAL n=3 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute recurrence ORIGINAL n=3 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute recurrence ORIGINAL n=2 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute recurrence ORIGINAL n=2 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute recurrence ORIGINAL n=2 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute recurrence ORIGINAL n=2 ORIGINAL coefficient0 times a_0 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 (2, 2, 5, 521, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix) 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: 00e8c019b751162915b6452e94ea70c2170c3b8378f2469f94a958764261f2e4 - candidate record SHA-256: 0cd1ea61a9b516c2c8be1d29ff3bb66c1d34b5594e92d645063f0ad65c1bde76 - semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81 - question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261110-r05-recurrence-relation-0063.json)