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R0013-07125 · q-20261011-advance-20261015-r05-sequence-0050
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=12, a_(n+1)=a_n+(-1), start=22, 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=12, a_(n+1)=a_n+(-1), start=22, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.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.
Before
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.
Rule: finite_ordered_sequence_scope · Uses: G001
Before
a(0)=12;a(n+1)=a(n)+(-1)
After
a(n)=12+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.
Rule: affine_recurrence_closed_form_derivation · Uses: G001, S001
Before
window start increment := mul(22, -1)
After
window start increment = -22
Conditions: none
Use the derived form for the accumulated increment at the window start.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
window start value := sum(12, -22)
After
window start value = -10
Conditions: none
Add the increment to the initial value to obtain the first window term.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
window recurrence term1 := sum(-10, -1)
After
window recurrence term1 = -11
Conditions: none
Inside the window,add d according to the same recurrence.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
window recurrence term2 := sum(-11, -1)
After
window recurrence term2 = -12
Conditions: none
Inside the window,add d according to the same recurrence.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
window adjacent difference0 := sub(-11, -10)
After
window adjacent difference0 = -1
Conditions: none
Subtract actual neighboring window values to audit the increment.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
window adjacent difference1 := sub(-12, -11)
After
window adjacent difference1 = -1
Conditions: none
Subtract actual neighboring window values to audit the increment.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
window indices=[22, 23, 24];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.
Rule: ordered_recurrence_window_audit · Uses: G001, S008
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.([22, 23, 24], [-10, -11, -12], [-1, -1])
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 77d0382638e0cbf849077cff25676085c9ac12d0d7f78dbd2477e1dacde2cb2bcandidate record SHA-256: 49370e7b74466fb2f6042a3475206cbae8e3530a021fcaf36f68fead91763243semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED