# Question 22733: Negative Indices Are Outside This Catalan Domain

[Return to catalogue page 114](https://ocme.evemisslab.com/en/questions/page/114/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261111-r05-catalan-numbers-foundations/#q-20261014-advance-20261111-r05-catalan-numbers-0083)

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.

R0044-22733 · q-20261014-advance-20261111-r05-catalan-numbers-0083



## Original question and givens

Use negative integer n=-8. Give (n, in_domain, value_or_null, boundary).


- G001 — Use negative integer n=-8. Give (n, in_domain, value_or_null, boundary).





## 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-catalan-numbers](https://ocme.evemisslab.com/en/mko/mko-catalan-numbers/)





## Step-by-step derivation

S001 · impliesBefore

source operation=catalan-negative-index-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[n, in_domain, value_or_null, boundary]

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

ORIGINAL negative n=-8

After

in_domain=No;value_or_null=null;this declared sequence startsC0=1,not a negative-index extension

Conditions: none

This sequence starts at natural index0.Negative indices remain null without silently adopting an extension or shifting C0 to C1.

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









## Reverse checks


- 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

(-8, No, null, Catalan_domain_is_natural_numbers_including_zero)

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: 26f092ce884fd47dc2e7ab6eb982002c677bc32f05240ae76baf5994e1d812a7
- candidate record SHA-256: e9a8359854db1573eacf53046d78d67d645ebe89f88d13b6f6fe8bd54ce83071
- 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-20261111-r05-catalan-numbers-0083.json)