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R0025-13175 · q-20261014-advance-20261027-r02-norm-induced-metric-0050
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=[12, 0, 1], right=[-12, -3, -2]. 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=[12, 0, 1], right=[-12, -3, -2]. Give (norm_name,d_xy,d_yx,equal?).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
forward distance difference coordinate0 := sub(12, -12)
After
forward distance difference coordinate0 = 24
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001
Before
forward distance difference coordinate1 := sub(0, -3)
After
forward distance difference coordinate1 = 3
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
forward distance difference coordinate2 := sub(1, -2)
After
forward distance difference coordinate2 = 3
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
forward distance:norm=l1;left=[12, 0, 1];right=[-12, -3, -2]
After
signed left-right=[24, 3, 3];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.
Rule: norm_distance_same_space_and_direction · Uses: G001, S003
Before
forward distance:coordinate=0;signed value=24
After
absolute value=24
Conditions: none
Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.
Rule: named_norm_coordinate_absolute_value · Uses: G001, S004
Before
forward distance:coordinate=1;signed value=3
After
absolute value=3
Conditions: none
Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.
Rule: named_norm_coordinate_absolute_value · Uses: G001, S005
Before
forward distance:coordinate=2;signed value=3
After
absolute value=3
Conditions: none
Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.
Rule: named_norm_coordinate_absolute_value · Uses: G001, S006
Before
forward distance l1 sum := sum(24, 3, 3)
After
forward distance l1 sum = 30
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
reverse distance difference coordinate0 := sub(-12, 12)
After
reverse distance difference coordinate0 = -24
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
reverse distance difference coordinate1 := sub(-3, 0)
After
reverse distance difference coordinate1 = -3
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
reverse distance difference coordinate2 := sub(-2, 1)
After
reverse distance difference coordinate2 = -3
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
reverse distance:norm=l1;left=[-12, -3, -2];right=[12, 0, 1]
After
signed left-right=[-24, -3, -3];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.
Rule: norm_distance_same_space_and_direction · Uses: G001, S011
Before
reverse distance:coordinate=0;signed value=-24
After
absolute value=24
Conditions: none
Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.
Rule: named_norm_coordinate_absolute_value · Uses: G001, S012
Before
reverse distance:coordinate=1;signed value=-3
After
absolute value=3
Conditions: none
Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.
Rule: named_norm_coordinate_absolute_value · Uses: G001, S013
Before
reverse distance:coordinate=2;signed value=-3
After
absolute value=3
Conditions: none
Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.
Rule: named_norm_coordinate_absolute_value · Uses: G001, S014
Before
reverse distance l1 sum := sum(24, 3, 3)
After
reverse distance l1 sum = 30
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
forward=30;reverse=30;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.
Rule: numeric_symmetry_with_universal_norm_reason · Uses: G001, S016
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.(l1, 30, 30, Yes)
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: c77760f0c57b6a511a5dca0696537d6633da9ceea46f6ff143d40ed7d1272cfdcandidate record SHA-256: bca3e6fb9a20e55107507ab296b1fcd7fe870251864867a955e31a66932f7544semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8aLean: NOT_RUN · human review: NOT_PERFORMED