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R0025-13121 · q-20261014-advance-20261027-r02-norm-induced-metric-0021
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=[8, 3, 0], right=[-8, 3, -2]. Give (norm_name,left,right,left−right,distance).
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=[8, 3, 0], right=[-8, 3, -2]. Give (norm_name,left,right,left−right,distance).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
requested distance difference coordinate0 := sub(8, -8)
After
requested distance difference coordinate0 = 16
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001
Before
requested distance difference coordinate1 := sub(3, 3)
After
requested distance difference coordinate1 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
requested distance difference coordinate2 := sub(0, -2)
After
requested distance difference coordinate2 = 2
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
requested distance:norm=l1;left=[8, 3, 0];right=[-8, 3, -2]
After
signed left-right=[16, 0, 2];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
requested distance:coordinate=0;signed value=16
After
absolute value=16
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
requested distance:coordinate=1;signed value=0
After
absolute value=0
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
requested distance:coordinate=2;signed value=2
After
absolute value=2
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
requested distance l1 sum := sum(16, 0, 2)
After
requested distance l1 sum = 18
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S007
S008 · premise_and_result_audit — Recompute requested 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 requested distance difference coordinate2 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute requested distance difference coordinate1 with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Recompute requested distance difference coordinate0 with exact rational arithmetic, not rounded decimals.(l1, [8, 3, 0], [-8, 3, -2], [16, 0, 2], 18)
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: 219540392fde6c61a0314117e153ac26c75558772e780a221e191521ad324228candidate record SHA-256: 8bbba7a783220dfc4a4891bd933f1f3b3186574e3789f70e563c92966041979dsemantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8aLean: NOT_RUN · human review: NOT_PERFORMED