# Question 13143: Distinguish a Universal Norm Certificate from Finite Samples [Return to catalogue page 66](https://ocme.evemisslab.com/en/questions/page/66/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261027-r02-norm-induced-metric-foundations/#q-20261014-advance-20261027-r02-norm-induced-metric-0093) 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. R0025-13143 · q-20261014-advance-20261027-r02-norm-induced-metric-0093 ## Original question and givens Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=18, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric. - G001 — Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=18, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric. ## 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-norm-induced-metric](https://ocme.evemisslab.com/en/mko/mko-norm-induced-metric/) ## Step-by-step derivation S001 · impliesBefore certified named norm=linfinity;fixed real space R^18;d(x,y)=norm(x-y) After d>=0;d(x,y)=0 iff x-y=0 iff x=y Conditions: none The certified l1 or linfinity norm axioms give nonnegativity and separation at the zero vector.Substitute x−y to obtain metric separation. Step identifiers and dependenciesRule: norm_metric_positivity_and_separation · Uses: G001 S002 · impliesBefore arbitrary x,y in R^18;y-x=-(x-y) After norm(y-x)=abs(-1)*norm(x-y)=norm(x-y) Conditions: none Apply absolute homogeneity at scalar−1,not an inference from one pair of equal numeric distances. Step identifiers and dependenciesRule: norm_metric_symmetry_absolute_homogeneity · Uses: G001, S001 S003 · impliesBefore arbitrary x,y,z in R^18;x-z=(x-y)+(y-z) After d(x,z)<=d(x,y)+d(y,z) by norm triangle inequality Conditions: none The vector-difference identity and universal norm triangle inequality apply to every triple.This does not assert every metric is norm-induced or establish completeness. Step identifiers and dependenciesRule: norm_metric_triangle_universal_decomposition · Uses: G001, S002 S004 · impliesBefore norm=linfinity;dimension=18;selected evidence=named_formula_certificate After evidence sufficient=Yes;sample failure does not deny named formula metricity Conditions: none A universal named-norm certificate suffices;finite distance samples do not cover all points of the metric axioms.No judges the evidence insufficient,not the known named formula nonmetric. Step identifiers and dependenciesRule: induced_metric_evidence_mode_not_property_negation · Uses: G001, S003 ## Reverse checks - 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 (linfinity, 18, named_formula_certificate, Yes, universal_from_norm_axioms) 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: 8250356029999c5166e7d8e2b676258bd5426561b3d4788ed48d2e2ea1e626ae - candidate record SHA-256: 5c4648306b3e9c8c20d244805f72c6d05ff58ef4796d94c47ec936758c688fdf - 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-20261027-r02-norm-induced-metric-0093.json)