Question 13901: One Passing Vector Does Not Certify a Uniform Bound

Original question and givens

Use the explicit rectangular integer matrix A=[[-12, -3, -5], [-2, 12, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-4, -2, -3], proposed C=14. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).

• G001 — Use the explicit rectangular integer matrix A=[[-12, -3, -5], [-2, 12, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-4, -2, -3], proposed C=14. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).

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.

• mko-bounded-linear-operator

Step-by-step derivation

provided input image:matrix=[[-12, -3, -5], [-2, 12, -6]];input=[-4, -2, -3]

input dimension=3;output dimension=2;rows dot input

provided input image row0 column0 product := mul(-12, -4)

provided input image row0 column0 product = 48

provided input image row0 column1 product := mul(-3, -2)

provided input image row0 column1 product = 6

provided input image row0 column2 product := mul(-5, -3)

provided input image row0 column2 product = 15

provided input image row0 sum := sum(48, 6, 15)

provided input image row0 sum = 69

provided input image row1 column0 product := mul(-2, -4)

provided input image row1 column0 product = 8

provided input image row1 column1 product := mul(12, -2)

provided input image row1 column1 product = -24

provided input image row1 column2 product := mul(-6, -3)

provided input image row1 column2 product = 18

provided input image row1 sum := sum(8, -24, 18)

provided input image row1 sum = 2

provided input:signed vector=[-4, -2, -3];norm=l1

coordinate absolute values=[4, 2, 3]

provided input l1 sum := sum(4, 2, 3)

provided input l1 sum = 9

provided output:signed vector=[69, 2];norm=l1

coordinate absolute values=[69, 2]

provided output l1 sum := sum(69, 2)

provided output l1 sum = 71

proposed C times input norm := mul(14, 9)

proposed C times input norm = 126

provided input=[-4, -2, -3];output norm=71;C=14;input norm=9;rhs=126

this instance passes=Yes;instance_only

Reverse checks

• 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 — Recompute proposed C times input norm with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute provided output l1 sum with exact rational arithmetic, not rounded decimals.
• 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 provided input l1 sum with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute provided input image row1 sum with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute provided input image row1 column2 product with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute provided input image row1 column1 product with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute provided input image row1 column0 product with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute provided input image row0 sum with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute provided input image row0 column2 product with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute provided input image row0 column1 product with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute provided input image row0 column0 product with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(l1, [-4, -2, -3], [69, 2], 9, 71, 14, 126, Yes, instance_only)

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.