# Question 12630: Evaluate a Named l1 Norm Exactly

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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.

R0024-12630 · q-20261013-advance-20261026-r02-real-normed-vector-space-0005



## Original question and givens

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-8, 8, 0, 2]. Give (vector,norm_name,value).


- G001 — Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-8, 8, 0, 2]. Give (vector,norm_name,value).





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





## Step-by-step derivation

S001 · impliesBefore

given vector:coordinate=0;signed value=-8

After

absolute value=8

Conditions: none

Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector.

Step identifiers and dependenciesRule: named_norm_each_coordinate_absolute_value · Uses: G001





S002 · impliesBefore

given vector:coordinate=1;signed value=8

After

absolute value=8

Conditions: none

Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector.

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





S003 · impliesBefore

given vector:coordinate=2;signed value=0

After

absolute value=0

Conditions: none

Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector.

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





S004 · impliesBefore

given vector:coordinate=3;signed value=2

After

absolute value=2

Conditions: none

Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector.

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





S005 · calculationBefore

given vector l1 sum := sum(8, 8, 0, 2)

After

given vector l1 sum = 18

Conditions: none

Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.

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









## Reverse checks


- S005 · premise_and_result_audit — Recompute given vector l1 sum 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 — 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

([-8, 8, 0, 2], l1, 18)

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: 091db1d9f8dcf4dc6cf1002acd519d5d6d07f28ce7d76b06f1bc959e59a2b6b1
- candidate record SHA-256: eaccf263f5ac1ab073beed031814f430e6f4fdc803a4aac2f103e177d4efebf2
- semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35
- question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261026-r02-real-normed-vector-space-0005.json)