# Question 12694: A Named-Formula Certificate Is Universal, Not Sampled

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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.

R0024-12694 · q-20261013-advance-20261026-r02-real-normed-vector-space-0069



## 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. norm_name=l1, dimension=19. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).


- 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. norm_name=l1, dimension=19. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).





## 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

arbitrary x in R^19;named l1 formula=sum_i abs(x_i)

After

norm(x)>=0;norm(x)=0 iff every coordinate is0

Conditions: none

Every absolute value is nonnegative.A nonnegative sum or maximum is0 exactly when every term is0,that is,the zero vector.Nonempty dimension makes the maximum defined.

Step identifiers and dependenciesRule: named_norm_universal_positivity_separation · Uses: G001





S002 · impliesBefore

arbitrary real scalar a and x in R^19;abs(a*x_i)=abs(a)*abs(x_i)

After

l1(a*x)=abs(a)*l1(x)

Conditions: none

Pull the common nonnegative factor|a| through the finite sum or maximum,including a=0 and negative a.This covers all real scalars,not sampled integer multiples.

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





S003 · impliesBefore

arbitrary x,y in R^19;coordinatewise abs(x_i+y_i)<=abs(x_i)+abs(y_i)

After

sum_i abs(x_i+y_i)<=sum_i abs(x_i)+sum_i abs(y_i);l1(x+y)<=l1(x)+l1(y)

Conditions: none

Apply the absolute-value triangle inequality coordinatewise.For l1 sum;for linfinity bound each coordinate by the two global maxima,then take a maximum.This is a universal algebraic certificate for all vectors.

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









## Reverse checks


- 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

(l1, 19, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

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: a99be97d3c17482286ea21584b1e886281afdb28b2334b44866d404c682547b9
- candidate record SHA-256: d042bda239da1806773cfc20ac05367b4f020e41425e6b564b90b3aa49fdf2d3
- 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-0069.json)