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R0026-13926 · q-20261014-advance-20261028-r05-bounded-linear-operator-0076
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, proposed C=14. 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=[[-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, proposed C=14. 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=[-12, -2]
After
absolute entries=[12, 2]
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(12, 2)
After
operator column0 absolute sum = 14
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=[-3, 12]
After
absolute entries=[3, 12]
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(3, 12)
After
operator column1 absolute sum = 15
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=[-5, -6]
After
absolute entries=[5, 6]
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(5, 6)
After
operator column2 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, S005
Before
all absolute column sums=[14, 15, 11]
After
maximum=15;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<=15
After
norm1(Ax)<=sum_j(sum_i abs(Aij))*abs(xj)<=15*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=[[-12, -3, -5], [-2, 12, -6]];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(-12, 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(-3, 1)
After
attaining unit input image row0 column1 product = -3
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(-5, 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, -3, 0)
After
attaining unit input image row0 sum = -3
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(-2, 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(12, 1)
After
attaining unit input image row1 column1 product = 12
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(-6, 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, 12, 0)
After
attaining unit input image row1 sum = 12
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=[-3, 12];norm=l1
After
coordinate absolute values=[3, 12]
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(3, 12)
After
attaining output l1 sum = 15
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=[-3, 12];output norm=15
After
uniform upper bound=15 and attained ratio=15;exact induced norm=15
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=15;proposed nonnegative C=14;attaining unit input=[0, 1, 0]
After
uniform bound=No;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, 15, 14, No, No, bound_too_small)
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: 1b9348f58d6f9ac2062d37f304a909236fd8ddf4974ead5b0839fafb58a9e8d3candidate record SHA-256: f0beb9407b5a2e149135c4624825b7cf6cec2dddfcfaace10c86bb4e54889858semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77Lean: NOT_RUN · human review: NOT_PERFORMED