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Question 12626: Evaluate a Named l1 Norm Exactly

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0024-12626 · q-20261013-advance-20261026-r02-real-normed-vector-space-0001

Original question and givens

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. Integer vector=[-12, -8, -9, -2]. Give (vector,norm_name,value).

  • 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. Integer vector=[-12, -8, -9, -2]. Give (vector,norm_name,value).

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.

Step-by-step derivation

S001 · implies

Before

given vector:coordinate=0;signed value=-12

After

absolute value=12

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.

Step identifiers and dependencies

Rule: named_norm_each_coordinate_absolute_value · Uses: G001

S002 · implies

Before

given vector:coordinate=1;signed value=-8

After

absolute value=8

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.

Step identifiers and dependencies

Rule: named_norm_each_coordinate_absolute_value · Uses: G001, S001

S003 · implies

Before

given vector:coordinate=2;signed value=-9

After

absolute value=9

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.

Step identifiers and dependencies

Rule: named_norm_each_coordinate_absolute_value · Uses: G001, S002

S004 · implies

Before

given vector:coordinate=3;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.

Step identifiers and dependencies

Rule: named_norm_each_coordinate_absolute_value · Uses: G001, S003

S005 · calculation

Before

given vector l1 sum := sum(12, 8, 9, 2)

After

given vector l1 sum = 31

Conditions: none

Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

Reverse checks

  • S005 · premise_and_result_audit — Recompute given vector l1 sum with exact rational arithmetic, not rounded decimals.
  • 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([-12, -8, -9, -2], l1, 31)

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: 60922d4e28d40e14613c6d6d70669a3d6221e01ffb7173e90dd55254766f9208
  • candidate record SHA-256: 1aca2e7cca9781c7de3761affc1511d98d807765d3f26efeaca091c1b45aab4d
  • semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35
  • question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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