# Question 7666: Three 1-Based Matrix Queries with Eligibility Guards

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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.

R0014-07666 · q-20261012-advance-20261016-r02-matrix-0066



## Original question and givens

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[3], [4], [5], [6]], ordered (i,j) candidates=[[1, 1], [4, 1], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.


- G001 — Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[3], [4], [5], [6]], ordered (i,j) candidates=[[1, 1], [4, 1], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.





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





## Step-by-step derivation

S001 · impliesBefore

M=[[3], [4], [5], [6]];shape=[4, 1]

After

external indices start at1;transpose is not inverse

Conditions: none

First confirm a nonempty rectangular shape and complete integer entries.Transposition is defined for rectangular matrices without invertibility and is not matrix inversion.

Step identifiers and dependenciesRule: rectangular_matrix_scope · Uses: G001





S002 · impliesBefore

query0=(1,1);shape=[4, 1]

After

qualified=Yes;value/status=3

Conditions: none

Both external indices must lie from1 through the corresponding size before reading the internal(i−1,j−1) position.Do not fill an invalid query using negative or wrapping indices.

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





S003 · impliesBefore

query1=(4,1);shape=[4, 1]

After

qualified=Yes;value/status=6

Conditions: none

Both external indices must lie from1 through the corresponding size before reading the internal(i−1,j−1) position.Do not fill an invalid query using negative or wrapping indices.

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





S004 · impliesBefore

query2=(0,1);shape=[4, 1]

After

qualified=No;value/status=invalid index

Conditions: none

Both external indices must lie from1 through the corresponding size before reading the internal(i−1,j−1) position.Do not fill an invalid query using negative or wrapping indices.

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









## Reverse checks


- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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, 1], [Yes, Yes, No], [3, 6, 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: 88fd80b131be8aba9e29e002bf57988044e7222c81e2f67b6b8957cb915d9895
- candidate record SHA-256: ae588f6293680ac74e8c09ebff9c486d44edbeb4c3c9f58b9b7ac8e92caa3b24
- semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
- question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261016-r02-matrix-0066.json)