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R0024-12690 · q-20261013-advance-20261026-r02-real-normed-vector-space-0065
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=15. 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=15. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).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.
Before
arbitrary x in R^15;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.
Rule: named_norm_universal_positivity_separation · Uses: G001
Before
arbitrary real scalar a and x in R^15;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.
Rule: named_norm_all_real_scalars_homogeneity · Uses: G001, S001
Before
arbitrary x,y in R^15;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.
Rule: named_norm_universal_triangle_not_samples · Uses: G001, S002
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.(l1, 15, 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.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 45bbda713db384384662733162034d7ed90fadd8c27ded53bae0a2ae473af8a6candidate record SHA-256: 5706ca049b95bd3fca683f30f6a7a2456f2b8e4787149fe04a308a3deb74ce2dsemantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8Lean: NOT_RUN · human review: NOT_PERFORMED