Question 13906: One Passing Vector Does Not Certify a Uniform Bound

Original question and givens

Use the explicit rectangular integer matrix A=[[-7, 2, -1], [-2, 7, 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=linfinity, vector=[1, -2, 2], proposed C=17. 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=[[-7, 2, -1], [-2, 7, 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=linfinity, vector=[1, -2, 2], proposed C=17. 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=[[-7, 2, -1], [-2, 7, 6]];input=[1, -2, 2]

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

provided input image row0 column0 product := mul(-7, 1)

provided input image row0 column0 product = -7

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

provided input image row0 column1 product = -4

provided input image row0 column2 product := mul(-1, 2)

provided input image row0 column2 product = -2

provided input image row0 sum := sum(-7, -4, -2)

provided input image row0 sum = -13

provided input image row1 column0 product := mul(-2, 1)

provided input image row1 column0 product = -2

provided input image row1 column1 product := mul(7, -2)

provided input image row1 column1 product = -14

provided input image row1 column2 product := mul(6, 2)

provided input image row1 column2 product = 12

provided input image row1 sum := sum(-2, -14, 12)

provided input image row1 sum = -4

provided input:signed vector=[1, -2, 2];norm=linfinity

coordinate absolute values=[1, 2, 2]

provided input:all absolute coordinates=[1, 2, 2]

maximum=2;first attaining coordinate=1

provided output:signed vector=[-13, -4];norm=linfinity

coordinate absolute values=[13, 4]

provided output:all absolute coordinates=[13, 4]

maximum=13;first attaining coordinate=0

proposed C times input norm := mul(17, 2)

proposed C times input norm = 34

provided input=[1, -2, 2];output norm=13;C=17;input norm=2;rhs=34

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, -2, 2], [-13, -4], 2, 13, 17, 34, 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.