Question 13904: One Passing Vector Does Not Certify a Uniform Bound

Original question and givens

Use the explicit rectangular integer matrix A=[[-9, 0, 4], [1, 9, -4]] 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=linfinity, vector=[-1, 1, 0], 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=[[-9, 0, 4], [1, 9, -4]] 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=linfinity, vector=[-1, 1, 0], 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=[[-9, 0, 4], [1, 9, -4]];input=[-1, 1, 0]

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

provided input image row0 column0 product := mul(-9, -1)

provided input image row0 column0 product = 9

provided input image row0 column1 product := mul(0, 1)

provided input image row0 column1 product = 0

provided input image row0 column2 product := mul(4, 0)

provided input image row0 column2 product = 0

provided input image row0 sum := sum(9, 0, 0)

provided input image row0 sum = 9

provided input image row1 column0 product := mul(1, -1)

provided input image row1 column0 product = -1

provided input image row1 column1 product := mul(9, 1)

provided input image row1 column1 product = 9

provided input image row1 column2 product := mul(-4, 0)

provided input image row1 column2 product = 0

provided input image row1 sum := sum(-1, 9, 0)

provided input image row1 sum = 8

provided input:signed vector=[-1, 1, 0];norm=linfinity

coordinate absolute values=[1, 1, 0]

provided input:all absolute coordinates=[1, 1, 0]

maximum=1;first attaining coordinate=0

provided output:signed vector=[9, 8];norm=linfinity

coordinate absolute values=[9, 8]

provided output:all absolute coordinates=[9, 8]

maximum=9;first attaining coordinate=0

proposed C times input norm := mul(14, 1)

proposed C times input norm = 14

provided input=[-1, 1, 0];output norm=9;C=14;input norm=1;rhs=14

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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

(linfinity, [-1, 1, 0], [9, 8], 1, 9, 14, 14, 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.