Question 13170: Reverse the Difference, Not the Norm Definition

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0025-13170 · q-20261014-advance-20261027-r02-norm-induced-metric-0045

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, left=[7, 2, -3], right=[-7, -2, 1]. 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=linfinity, left=[7, 2, -3], right=[-7, -2, 1]. 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

S001 · calculation

Before

forward distance difference coordinate0 := sub(7, -7)

After

forward distance difference coordinate0 = 14

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001

S002 · calculation

Before

forward distance difference coordinate1 := sub(2, -2)

After

forward distance difference coordinate1 = 4

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

forward distance difference coordinate2 := sub(-3, 1)

After

forward distance difference coordinate2 = -4

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

forward distance:norm=linfinity;left=[7, 2, -3];right=[-7, -2, 1]

After

signed left-right=[14, 4, -4];use same named norm

Conditions: none

Use the same stated norm on the same real coordinate space.The derived difference can exceed original input coordinate bounds and must not be clipped or rejected as a new raw input.

Step identifiers and dependencies

Rule: norm_distance_same_space_and_direction · Uses: G001, S003

S005 · implies

Before

forward distance:coordinate=0;signed value=14

After

absolute value=14

Conditions: none

Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.

Step identifiers and dependencies

Rule: named_norm_coordinate_absolute_value · Uses: G001, S004

S006 · implies

Before

forward distance:coordinate=1;signed value=4

After

absolute value=4

Conditions: none

Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.

Step identifiers and dependencies

Rule: named_norm_coordinate_absolute_value · Uses: G001, S005

S007 · implies

Before

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

After

absolute value=4

Conditions: none

Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.

Step identifiers and dependencies

Rule: named_norm_coordinate_absolute_value · Uses: G001, S006

S008 · implies

Before

forward distance:all absolute coordinates=[14, 4, 4]

After

maximum=14;first attaining coordinate=0

Conditions: none

The maximum must bound every entry and be attained at one entry,as guaranteed for nonempty finite coordinates.

Step identifiers and dependencies

Rule: linfinity_full_maximum_witness · Uses: G001, S007

S009 · calculation

Before

reverse distance difference coordinate0 := sub(-7, 7)

After

reverse distance difference coordinate0 = -14

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

reverse distance difference coordinate1 := sub(-2, 2)

After

reverse distance difference coordinate1 = -4

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

reverse distance difference coordinate2 := sub(1, -3)

After

reverse distance difference coordinate2 = 4

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

reverse distance:norm=linfinity;left=[-7, -2, 1];right=[7, 2, -3]

After

signed left-right=[-14, -4, 4];use same named norm

Conditions: none

Use the same stated norm on the same real coordinate space.The derived difference can exceed original input coordinate bounds and must not be clipped or rejected as a new raw input.

Step identifiers and dependencies

Rule: norm_distance_same_space_and_direction · Uses: G001, S011

S013 · implies

Before

reverse distance:coordinate=0;signed value=-14

After

absolute value=14

Conditions: none

Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.

Step identifiers and dependencies

Rule: named_norm_coordinate_absolute_value · Uses: G001, S012

S014 · implies

Before

reverse distance:coordinate=1;signed value=-4

After

absolute value=4

Conditions: none

Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.

Step identifiers and dependencies

Rule: named_norm_coordinate_absolute_value · Uses: G001, S013

S015 · implies

Before

reverse distance:coordinate=2;signed value=4

After

absolute value=4

Conditions: none

Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.

Step identifiers and dependencies

Rule: named_norm_coordinate_absolute_value · Uses: G001, S014

S016 · implies

Before

reverse distance:all absolute coordinates=[14, 4, 4]

After

maximum=14;first attaining coordinate=0

Conditions: none

The maximum must bound every entry and be attained at one entry,as guaranteed for nonempty finite coordinates.

Step identifiers and dependencies

Rule: linfinity_full_maximum_witness · Uses: G001, S015

S017 · implies

Before

forward=14;reverse=14;opposite difference vectors

After

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

Conditions: none

Check both numeric directions;the general reason is absolute homogeneity at−1,not loss of sign when the original vector roles are reversed.

Step identifiers and dependencies

Rule: numeric_symmetry_with_universal_norm_reason · Uses: G001, S016

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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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

(linfinity, 14, 14, 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.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: f67923c4aeb941728a56ef56b0c5ff5dd636d86e650be0b8edda5e0d2052d3a5
candidate record SHA-256: 0169538073758ba8d87798692c51bd12f8f643bed8506e0c621a6153d7cb0fa3
semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5
question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a
Lean: NOT_RUN · human review: NOT_PERFORMED

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