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R0025-13119 · q-20261014-advance-20261027-r02-norm-induced-metric-0019
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=[6, 1, 5], right=[-6, 6, 0]. 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=[6, 1, 5], right=[-6, 6, 0]. 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(6, -6)
After
requested distance difference coordinate0 = 12
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(1, 6)
After
requested distance difference coordinate1 = -5
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(5, 0)
After
requested distance difference coordinate2 = 5
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=[6, 1, 5];right=[-6, 6, 0]
After
signed left-right=[12, -5, 5];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=12
After
absolute value=12
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=-5
After
absolute value=5
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=5
After
absolute value=5
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(12, 5, 5)
After
requested distance l1 sum = 22
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, [6, 1, 5], [-6, 6, 0], [12, -5, 5], 22)
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: 5ffd9c21572383021c2ce5b7226d09cf682b7154dfa8d7cdf6ccb4ae4ec70620candidate record SHA-256: e437520bb5528640cfcf0a122ee2627ca7d896476caf526e6981147c50f1d55bsemantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8aLean: NOT_RUN · human review: NOT_PERFORMED