Question 13153: 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=[-10, -1, 1], right=[10, 4, 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=[-10, -1, 1], right=[10, 4, 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(-10, 10)

forward distance difference coordinate0 = -20

forward distance difference coordinate1 := sub(-1, 4)

forward distance difference coordinate1 = -5

forward distance difference coordinate2 := sub(1, 0)

forward distance difference coordinate2 = 1

forward distance:norm=l1;left=[-10, -1, 1];right=[10, 4, 0]

signed left-right=[-20, -5, 1];use same named norm

forward distance:coordinate=0;signed value=-20

absolute value=20

forward distance:coordinate=1;signed value=-5

absolute value=5

forward distance:coordinate=2;signed value=1

absolute value=1

forward distance l1 sum := sum(20, 5, 1)

forward distance l1 sum = 26

reverse distance difference coordinate0 := sub(10, -10)

reverse distance difference coordinate0 = 20

reverse distance difference coordinate1 := sub(4, -1)

reverse distance difference coordinate1 = 5

reverse distance difference coordinate2 := sub(0, 1)

reverse distance difference coordinate2 = -1

reverse distance:norm=l1;left=[10, 4, 0];right=[-10, -1, 1]

signed left-right=[20, 5, -1];use same named norm

reverse distance:coordinate=0;signed value=20

absolute value=20

reverse distance:coordinate=1;signed value=5

absolute value=5

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

absolute value=1

reverse distance l1 sum := sum(20, 5, 1)

reverse distance l1 sum = 26

forward=26;reverse=26;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, 26, 26, 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.