# Question 29828: Finite Minimum Value and Every Original Minimizer Index

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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.

R0058-29828 · q-20261013-advance-20261020-r04-optimization-problem-0028



## Original question and givens

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-8, -2, 10, 2], feasible=[true, true, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).


- G001 — Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-8, -2, 10, 2], feasible=[true, true, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).





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





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-app20-optimization-argmin, optimization-argmin]

After

derived = [[values, mask, min, best, arg_indices]]

Conditions: none

Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention.

Step identifiers and dependenciesRule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001





S002 · impliesBefore

source roles = [[-8, -2, 10, 2], [Yes, Yes, Yes, Yes]]

After

derived = [[0, 1, 2, 3]]

Conditions: none

Require an ACTUAL Boolean mask aligned to original slots and retain zero-based indices with at least one eligible candidate.A better objective cannot change False to True,and0/1 are not Booleans.

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





S003 · impliesBefore

source roles = [explicit Boolean mask, [-8, -2, 10, 2], [0, 1, 2, 3], min]

After

derived = [-8, [0], [[0, -8, Yes, Yes], [1, -2, Yes, No], [2, 10, Yes, No], [3, 2, Yes, No]]]

Conditions: none

Check the corresponding bound for EVERY feasible ORIGINAL index and retain ALL equality indices.The optimal value differs from its arg set.Ignore better infeasible values and do not assume uniqueness.The conclusion is ONLY for this complete declared finite set.

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









## Reverse checks


- S003 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
- S002 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
- S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.





## Result

([-8, -2, 10, 2], [Yes, Yes, Yes, Yes], min, -8, [0])

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: 02a661d31a6e85022b97e0562535638d08d1591dd79f71361c159a23edd56180
- candidate record SHA-256: ff2da6aa7591f94834d54fb65745e45b7b2262d9b2be1d1d939e2e338bb9bf64
- semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df
- question-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261020-r04-optimization-problem-0028.json)