Question 22521: Shift the Full State and Append Each Exact New Scalar

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0044-22521 · q-20261014-advance-20261110-r05-recurrence-relation-0046

Original question and givens

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=-2 and initial_values=[20, 21, 22, 23], 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=7. Give (order, initial_values, target_n, prefix, target_value, certificate).

G001 — Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=-2 and initial_values=[20, 21, 22, 23], 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=7. Give (order, initial_values, target_n, prefix, target_value, 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

Step-by-step derivation

S001 · implies

Before

source operation=recurrence-forward-generation;original snapshot and mathematics retained

After

explicit ordered source output fields=[order, initial_values, target_n, prefix, target_value, 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 dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

recurrence ORIGINAL:ORIGINAL order=4;oldest-to-newest coefficients=[1, 2, 3, 4];constant=-2;ALL initial a_0..a_(r-1)=[20, 21, 22, 23]

After

for EVERY n>=4,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=4;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 dependencies

Rule: recurrence_complete_initial_state_and_total_single_value_rule · Uses: G001, S001

S003 · calculation

Before

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 := mul(1, 20)

After

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 20

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(-2, 20)

After

recurrence ORIGINAL n=4 partial sum after coefficient0 = 18

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(2, 21)

After

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 42

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(18, 42)

After

recurrence ORIGINAL n=4 partial sum after coefficient1 = 60

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(3, 22)

After

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 66

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(60, 66)

After

recurrence ORIGINAL n=4 partial sum after coefficient2 = 126

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(4, 23)

After

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 92

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(126, 92)

After

recurrence ORIGINAL n=4 partial sum after coefficient3 = 218

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[20, 21, 22, 23];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 20, 20], [1, 1, 2, 21, 42], [2, 2, 3, 22, 66], [3, 3, 4, 23, 92]]

After

new a_4=218;next state=[21, 22, 23, 218];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 22, 23, 218]

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 dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S010

S012 · calculation

Before

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(1, 21)

After

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 21

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(-2, 21)

After

recurrence ORIGINAL n=5 partial sum after coefficient0 = 19

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(2, 22)

After

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 44

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(19, 44)

After

recurrence ORIGINAL n=5 partial sum after coefficient1 = 63

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 := mul(3, 23)

After

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 69

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(63, 69)

After

recurrence ORIGINAL n=5 partial sum after coefficient2 = 132

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(4, 218)

After

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 872

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(132, 872)

After

recurrence ORIGINAL n=5 partial sum after coefficient3 = 1004

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[21, 22, 23, 218];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 21, 21], [1, 2, 2, 22, 44], [2, 3, 3, 23, 69], [3, 4, 4, 218, 872]]

After

new a_5=1004;next state=[22, 23, 218, 1004];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 22, 23, 218, 1004]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S019

S021 · calculation

Before

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(1, 22)

After

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 22

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(-2, 22)

After

recurrence ORIGINAL n=6 partial sum after coefficient0 = 20

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(2, 23)

After

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 46

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(20, 46)

After

recurrence ORIGINAL n=6 partial sum after coefficient1 = 66

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(3, 218)

After

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 654

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(66, 654)

After

recurrence ORIGINAL n=6 partial sum after coefficient2 = 720

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(4, 1004)

After

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 4016

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(720, 4016)

After

recurrence ORIGINAL n=6 partial sum after coefficient3 = 4736

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[22, 23, 218, 1004];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 22, 22], [1, 3, 2, 23, 46], [2, 4, 3, 218, 654], [3, 5, 4, 1004, 4016]]

After

new a_6=4736;next state=[23, 218, 1004, 4736];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 22, 23, 218, 1004, 4736]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S028

S030 · calculation

Before

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(1, 23)

After

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 23

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(-2, 23)

After

recurrence ORIGINAL n=7 partial sum after coefficient0 = 21

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(2, 218)

After

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 436

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(21, 436)

After

recurrence ORIGINAL n=7 partial sum after coefficient1 = 457

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(3, 1004)

After

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 3012

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(457, 3012)

After

recurrence ORIGINAL n=7 partial sum after coefficient2 = 3469

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(4, 4736)

After

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 18944

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(3469, 18944)

After

recurrence ORIGINAL n=7 partial sum after coefficient3 = 22413

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[23, 218, 1004, 4736];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 23, 23], [1, 4, 2, 218, 436], [2, 5, 3, 1004, 3012], [3, 6, 4, 4736, 18944]]

After

new a_7=22413;next state=[218, 1004, 4736, 22413];drop ONLY oldest state entry then append new scalar;prefix now=[20, 21, 22, 23, 218, 1004, 4736, 22413]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S037

S039 · implies

Before

recurrence ORIGINAL:complete ORIGINAL initial state=[20, 21, 22, 23];actual generated prefix=[20, 21, 22, 23, 218, 1004, 4736, 22413]

After

ANY two sequences with SAME r initials and SAME total rule agree at n=4 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 dependencies

Rule: recurrence_forward_uniqueness_induction_not_analytic_claim · Uses: G001, S038

Reverse checks

S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S037 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S028 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S019 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S010 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 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

(4, [20, 21, 22, 23], 7, [20, 21, 22, 23, 218, 1004, 4736, 22413], 22413, exact_forward_application_only)

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.

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