# Question 13416: Keep the Metric Binding and Completeness Source Separate

[Return to catalogue page 68](https://ocme.evemisslab.com/en/questions/page/68/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261028-r03-real-banach-space-foundations/#q-20261014-advance-20261028-r03-real-banach-space-0041)

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.

R0025-13416 · q-20261014-advance-20261028-r03-real-banach-space-0041



## Original question and givens

Use the explicit real coordinate space R^n of dimension=16, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=unrelated_metric, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.


- G001 — Use the explicit real coordinate space R^n of dimension=16, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. metric_source=unrelated_metric, completeness_source=finite_dimensional_theorem. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.





## 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-real-banach-space](https://ocme.evemisslab.com/en/mko/mko-real-banach-space/)





## Step-by-step derivation

S001 · impliesBefore

record dimension=16;norm=linfinity;metric_source=unrelated_metric;completeness_source=finite_dimensional_theorem

After

metric binding matches=No;completeness source matches=Yes;record qualified=No

Conditions: none

Each flag records an exact source-label match to same_norm_induced or finite_dimensional_theorem,then conjunction.A missing binding makes this record unqualified;it does not refute the finite-dimensional completeness theorem.

Step identifiers and dependenciesRule: Banach_checklist_record_matches_not_space_refutation · Uses: G001









## Reverse checks


- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(16, linfinity, unrelated_metric, finite_dimensional_theorem, No, Yes, No)

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: 3644199f5415b7b75dede58f66cb94d8544b47b723ce8fe76e2fb3f124c45546
- candidate record SHA-256: 100e02af1fbf7ba58b5d71fb9ca259aa81c67d10a1a68741897b59730988bb1b
- semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5
- question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261028-r03-real-banach-space-0041.json)