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R0024-12687 · q-20261013-advance-20261026-r02-real-normed-vector-space-0062
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=linfinity, dimension=12. 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=linfinity, dimension=12. 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^12;named linfinity formula=max_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^12;abs(a*x_i)=abs(a)*abs(x_i)
After
linfinity(a*x)=abs(a)*linfinity(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^12;coordinatewise abs(x_i+y_i)<=abs(x_i)+abs(y_i)
After
each abs(x_i+y_i)<=max_j abs(x_j)+max_j abs(y_j),then take maximum;linfinity(x+y)<=linfinity(x)+linfinity(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.(linfinity, 12, 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: 739aca7668922fd1aa5f206061d50981e92738925ef3c6680d3957c3b38a08b8candidate record SHA-256: e2516b7b2c3e3af35065b276ac5976be2b8537775573791c7db8fe55e155c021semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8Lean: NOT_RUN · human review: NOT_PERFORMED