Question 13165: Reverse the Difference, Not the Norm Definition

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=l1, left=[2, -3, 4], right=[-2, -1, 0]. Give (norm_name,d_xy,d_yx,equal?).

• 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=l1, left=[2, -3, 4], right=[-2, -1, 0]. Give (norm_name,d_xy,d_yx,equal?).

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

Step-by-step derivation

forward distance difference coordinate0 := sub(2, -2)

forward distance difference coordinate0 = 4

forward distance difference coordinate1 := sub(-3, -1)

forward distance difference coordinate1 = -2

forward distance difference coordinate2 := sub(4, 0)

forward distance difference coordinate2 = 4

forward distance:norm=l1;left=[2, -3, 4];right=[-2, -1, 0]

signed left-right=[4, -2, 4];use same named norm

forward distance:coordinate=0;signed value=4

absolute value=4

forward distance:coordinate=1;signed value=-2

absolute value=2

forward distance:coordinate=2;signed value=4

absolute value=4

forward distance l1 sum := sum(4, 2, 4)

forward distance l1 sum = 10

reverse distance difference coordinate0 := sub(-2, 2)

reverse distance difference coordinate0 = -4

reverse distance difference coordinate1 := sub(-1, -3)

reverse distance difference coordinate1 = 2

reverse distance difference coordinate2 := sub(0, 4)

reverse distance difference coordinate2 = -4

reverse distance:norm=l1;left=[-2, -1, 0];right=[2, -3, 4]

signed left-right=[-4, 2, -4];use same named norm

reverse distance:coordinate=0;signed value=-4

absolute value=4

reverse distance:coordinate=1;signed value=2

absolute value=2

reverse distance:coordinate=2;signed value=-4

absolute value=4

reverse distance l1 sum := sum(4, 2, 4)

reverse distance l1 sum = 10

forward=10;reverse=10;opposite difference vectors

equal=Yes by norm(-v)=abs(-1)*norm(v)=norm(v)

Reverse checks

• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Recompute reverse distance l1 sum with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute reverse distance difference coordinate2 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute reverse distance difference coordinate1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute reverse distance difference coordinate0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute forward distance l1 sum with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute forward distance difference coordinate2 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute forward distance difference coordinate1 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute forward distance difference coordinate0 with exact rational arithmetic, not rounded decimals.

Result

(l1, 10, 10, Yes)

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.