# Question 7194: Finite Sequence Query Bounds before Evaluation

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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.

R0013-07194 · q-20261011-advance-20261015-r05-sequence-0019



## 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. L=4; only 0≤n<L is allowed for a_n=6+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).


- 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. L=4; only 0≤n<L is allowed for a_n=6+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).





## 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 · calculationBefore

query0 d times index := mul(-1, 0)

After

query0 d times index = 0

Conditions: none

Substitute into the term formula only inside the permitted finite index domain.

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





S003 · calculationBefore

query0 legal sequence value := sum(6, 0)

After

query0 legal sequence value = 6

Conditions: none

Add the initial value to signed increment times index;negative term values are allowed.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

candidate index=0;0≤index<4=Yes

After

value/status=6

Conditions: none

An outside candidate is not an already supplied term,even if a number could be inserted into an extrapolated formula.L=0 has no legal indices.

Step identifiers and dependenciesRule: finite_sequence_index_guard · Uses: G001, S003





S005 · calculationBefore

query1 d times index := mul(-1, 3)

After

query1 d times index = -3

Conditions: none

Substitute into the term formula only inside the permitted finite index domain.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

query1 legal sequence value := sum(6, -3)

After

query1 legal sequence value = 3

Conditions: none

Add the initial value to signed increment times index;negative term values are allowed.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

candidate index=3;0≤index<4=Yes

After

value/status=3

Conditions: none

An outside candidate is not an already supplied term,even if a number could be inserted into an extrapolated formula.L=0 has no legal indices.

Step identifiers and dependenciesRule: finite_sequence_index_guard · Uses: G001, S006





S008 · impliesBefore

candidate index=4;0≤index<4=No

After

value/status=invalid index

Conditions: none

An outside candidate is not an already supplied term,even if a number could be inserted into an extrapolated formula.L=0 has no legal indices.

Step identifiers and dependenciesRule: finite_sequence_index_guard · Uses: G001, S007









## Reverse checks


- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 query1 legal sequence value with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute query1 d times index with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute query0 legal sequence value with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute query0 d times index with exact rational arithmetic, not rounded decimals.
- 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, [Yes, Yes, No], [6, 3, invalid-index])

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: 79b95eed272630f9ad236a555016244292110e4169bbc0bc6063beb4cb75dd72
- candidate record SHA-256: 261b60f9822c92e8bf9c6e744b424132befa4328feb6727d52fb3043244d5ced
- 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-0019.json)