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R0024-12605 · q-20261013-advance-20261026-r02-real-normed-vector-space-0080
Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-8, 1, -4], right=[8, 1, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).
G001 — Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-8, 1, -4], right=[8, 1, -2] with equal dimension. 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
distance coordinate0 left-minus-right := sub(-8, 8)
After
distance coordinate0 left-minus-right = -16
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001
Before
distance coordinate1 left-minus-right := sub(1, 1)
After
distance coordinate1 left-minus-right = 0
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
distance coordinate2 left-minus-right := sub(-4, -2)
After
distance coordinate2 left-minus-right = -2
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
named certified norm=l1;left=[-8, 1, -4];right=[8, 1, -2]
After
ordered difference=[-16, 0, -2];distance=norm(left-right)
Conditions: none
Use the same named norm on the same coordinate space and retain the difference direction.This does not assert every metric is norm-induced,or add completeness or inner-product claims.
Rule: fixed_norm_induced_distance_scope · Uses: G001, S003
Before
induced distance:coordinate=0;signed value=-16
After
absolute value=16
Conditions: none
Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector.
Rule: named_norm_each_coordinate_absolute_value · Uses: G001, S004
Before
induced distance:coordinate=1;signed value=0
After
absolute value=0
Conditions: none
Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector.
Rule: named_norm_each_coordinate_absolute_value · Uses: G001, S005
Before
induced distance:coordinate=2;signed value=-2
After
absolute value=2
Conditions: none
Take the absolute value of each entire coordinate.Negative coordinates are not summed or maximized as signed values;retain the original signed vector.
Rule: named_norm_each_coordinate_absolute_value · Uses: G001, S006
Before
induced distance l1 sum := sum(16, 0, 2)
After
induced distance l1 sum = 18
Conditions: none
Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S007
S008 · premise_and_result_audit — Recompute induced 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 distance coordinate2 left-minus-right with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute distance coordinate1 left-minus-right with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Recompute distance coordinate0 left-minus-right with exact rational arithmetic, not rounded decimals.(l1, [-8, 1, -4], [8, 1, -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: 1be4a5893e8d21292c90c841c38b760dd2bb8e97d660144f8cc6580990647d8bcandidate record SHA-256: c150b7d1915d65227b092f7e208df3c1e39b57508b4d73736d94ab68dec3f70asemantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8Lean: NOT_RUN · human review: NOT_PERFORMED