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R0026-13928 · q-20261014-advance-20261028-r05-bounded-linear-operator-0078
Use the explicit rectangular integer matrix A=[[-10, -1, 1], [0, 10, 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=l1, proposed C=13. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
G001 — Use the explicit rectangular integer matrix A=[[-10, -1, 1], [0, 10, 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=l1, proposed C=13. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.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
original column index=0;ordered entries=[-10, 0]
After
absolute entries=[10, 0]
Conditions: none
For l1 sum down each column;for linfinity sum across each row.Preserve axis order and do not transpose rectangular roles.
Rule: operator_correct_axis_absolute_sum · Uses: G001
Before
operator column0 absolute sum := sum(10, 0)
After
operator column0 absolute sum = 10
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
original column index=1;ordered entries=[-1, 10]
After
absolute entries=[1, 10]
Conditions: none
For l1 sum down each column;for linfinity sum across each row.Preserve axis order and do not transpose rectangular roles.
Rule: operator_correct_axis_absolute_sum · Uses: G001, S002
Before
operator column1 absolute sum := sum(1, 10)
After
operator column1 absolute sum = 11
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
original column index=2;ordered entries=[1, 4]
After
absolute entries=[1, 4]
Conditions: none
For l1 sum down each column;for linfinity sum across each row.Preserve axis order and do not transpose rectangular roles.
Rule: operator_correct_axis_absolute_sum · Uses: G001, S004
Before
operator column2 absolute sum := sum(1, 4)
After
operator column2 absolute sum = 5
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
all absolute column sums=[10, 11, 5]
After
maximum=11;first maximizing column=1
Conditions: none
Compute every required axis sum before taking a maximum.The value must have both a universal upper bound and an attaining unit input.
Rule: operator_maximum_axis_sum · Uses: G001, S006
Before
arbitrary x in R^3;all column sums<=11
After
norm1(Ax)<=sum_j(sum_i abs(Aij))*abs(xj)<=11*norm1(x)
Conditions: none
Apply the triangle inequality to each output,exchange finite sums,and bound every column coefficient by L.This yields one constant for every real input,not samples.
Rule: operator_l1_universal_column_bound · Uses: G001, S007
Before
attaining unit input image:matrix=[[-10, -1, 1], [0, 10, 4]];input=[0, 1, 0]
After
input dimension=3;output dimension=2;rows dot input
Conditions: none
Retain column-input/row-output roles for rectangular matrices.Computed values can exceed raw input bounds and are not clipped.
Rule: matrix_row_column_roles · Uses: G001, S008
Before
attaining unit input image row0 column0 product := mul(-10, 0)
After
attaining unit input image row0 column0 product = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
attaining unit input image row0 column1 product := mul(-1, 1)
After
attaining unit input image row0 column1 product = -1
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
attaining unit input image row0 column2 product := mul(1, 0)
After
attaining unit input image row0 column2 product = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
attaining unit input image row0 sum := sum(0, -1, 0)
After
attaining unit input image row0 sum = -1
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
attaining unit input image row1 column0 product := mul(0, 0)
After
attaining unit input image row1 column0 product = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
attaining unit input image row1 column1 product := mul(10, 1)
After
attaining unit input image row1 column1 product = 10
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
attaining unit input image row1 column2 product := mul(4, 0)
After
attaining unit input image row1 column2 product = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
attaining unit input image row1 sum := sum(0, 10, 0)
After
attaining unit input image row1 sum = 10
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
attaining input:signed vector=[0, 1, 0];norm=l1
After
coordinate absolute values=[0, 1, 0]
Conditions: none
Take absolute values of whole coordinates without changing their pairing.This also accepts computed matrix images without clipping to raw input bounds.
Rule: operator_named_norm_original_signs · Uses: G001, S017
Before
attaining input l1 sum := sum(0, 1, 0)
After
attaining input l1 sum = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
attaining output:signed vector=[-1, 10];norm=l1
After
coordinate absolute values=[1, 10]
Conditions: none
Take absolute values of whole coordinates without changing their pairing.This also accepts computed matrix images without clipping to raw input bounds.
Rule: operator_named_norm_original_signs · Uses: G001, S019
Before
attaining output l1 sum := sum(1, 10)
After
attaining output l1 sum = 11
Conditions: none
Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
explicit input=[0, 1, 0];input norm=1;image=[-1, 10];output norm=11
After
uniform upper bound=11 and attained ratio=11;exact induced norm=11
Conditions: none
For l1 take the maximizing column's basis vector;for linfinity take signs of a maximizing row.For a zero matrix still choose a nonzero unit vector.The universal bound L and an attained unit ratio L exclude every smaller constant,establishing the exact norm.
Rule: operator_attainment_proves_least_uniform_constant · Uses: G001, S021
Before
exact induced norm=11;proposed nonnegative C=13;attaining unit input=[0, 1, 0]
After
uniform bound=Yes;optimal bound=No
Conditions: none
For C≥L,nonnegativity of the input norm permits relaxing the universal bound to C.For C<L,the attaining unit input is a counterexample.Only equality is optimal;a usable larger C is not the exact norm.
Rule: operator_usable_bound_distinct_from_exact_optimum · Uses: G001, S022
S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S021 · premise_and_result_audit — Recompute attaining output l1 sum with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Recompute attaining input l1 sum with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Recompute attaining unit input image row1 sum with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute attaining unit input image row1 column2 product with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute attaining unit input image row1 column1 product with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute attaining unit input image row1 column0 product with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute attaining unit input image row0 sum with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute attaining unit input image row0 column2 product with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute attaining unit input image row0 column1 product with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute attaining unit input image row0 column0 product with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Recompute operator column2 absolute sum with exact rational arithmetic, not rounded decimals.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 — Recompute operator column1 absolute sum with exact rational arithmetic, not rounded decimals.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 — Recompute operator column0 absolute sum 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.(l1, 11, 13, Yes, No, uniform_bound_certificate)
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: 93c545ff887fafa4175406903723d9fd85ef07d89bd074883162be55a09ab5cbcandidate record SHA-256: e5b0c7376029889a8f69e8d90753a05e4e941ad40d2b31967c0ea386f106dc31semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77Lean: NOT_RUN · human review: NOT_PERFORMED