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R0026-13937 · q-20261014-advance-20261028-r05-bounded-linear-operator-0087
Use the explicit rectangular integer matrix A=[[-1, 1, -5], [-1, 1, -3]] 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, proposed C=9. 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=[[-1, 1, -5], [-1, 1, -3]] 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, proposed C=9. 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 row index=0;ordered entries=[-1, 1, -5]
After
absolute entries=[1, 1, 5]
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 row0 absolute sum := sum(1, 1, 5)
After
operator row0 absolute sum = 7
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 row index=1;ordered entries=[-1, 1, -3]
After
absolute entries=[1, 1, 3]
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 row1 absolute sum := sum(1, 1, 3)
After
operator row1 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, S003
Before
all absolute row sums=[7, 5]
After
maximum=7;first maximizing row=0
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, S004
Before
arbitrary x in R^3;all row sums<=7
After
each abs((Ax)i)<=sum_j abs(Aij)*normInfinity(x)<=7*normInfinity(x);take maximum_i
Conditions: none
Each input coordinate is bounded by its maximum norm.Bound each row and then maximize over outputs to obtain a universal uniform bound.
Rule: operator_linfinity_universal_row_bound · Uses: G001, S005
Before
attaining unit input image:matrix=[[-1, 1, -5], [-1, 1, -3]];input=[-1, 1, -1]
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, S006
Before
attaining unit input image row0 column0 product := mul(-1, -1)
After
attaining unit input image row0 column0 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, S007
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, S008
Before
attaining unit input image row0 column2 product := mul(-5, -1)
After
attaining unit input image row0 column2 product = 5
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 sum := sum(1, 1, 5)
After
attaining unit input image row0 sum = 7
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 row1 column0 product := mul(-1, -1)
After
attaining unit input image row1 column0 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, S011
Before
attaining unit input image row1 column1 product := mul(1, 1)
After
attaining unit input image row1 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, S012
Before
attaining unit input image row1 column2 product := mul(-3, -1)
After
attaining unit input image row1 column2 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, S013
Before
attaining unit input image row1 sum := sum(1, 1, 3)
After
attaining unit input image row1 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, S014
Before
attaining input:signed vector=[-1, 1, -1];norm=linfinity
After
coordinate absolute values=[1, 1, 1]
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, S015
Before
attaining input:all absolute coordinates=[1, 1, 1]
After
maximum=1;first attaining coordinate=0
Conditions: none
The finite nonempty maximum must bound every value and be attained,not be the largest signed coordinate.
Rule: operator_linfinity_full_maximum · Uses: G001, S016
Before
attaining output:signed vector=[7, 5];norm=linfinity
After
coordinate absolute values=[7, 5]
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 output:all absolute coordinates=[7, 5]
After
maximum=7;first attaining coordinate=0
Conditions: none
The finite nonempty maximum must bound every value and be attained,not be the largest signed coordinate.
Rule: operator_linfinity_full_maximum · Uses: G001, S018
Before
explicit input=[-1, 1, -1];input norm=1;image=[7, 5];output norm=7
After
uniform upper bound=7 and attained ratio=7;exact induced norm=7
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, S019
Before
exact induced norm=7;proposed nonnegative C=9;attaining unit input=[-1, 1, -1]
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, S020
S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Recompute attaining unit input image row1 sum with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute attaining unit input image row1 column2 product with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute attaining unit input image row1 column1 product with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute attaining unit input image row1 column0 product with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute attaining unit input image row0 sum with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute attaining unit input image row0 column2 product with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute attaining unit input image row0 column1 product with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute attaining unit input image row0 column0 product with exact rational arithmetic, not rounded decimals.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 row1 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 row0 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.(linfinity, 7, 9, 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: 827db40caac0124cca3819d8953d3ceb3a5edff8fad4c80ebf20bfaa7fbb08d8candidate record SHA-256: e50b3d514cdcd9a764a6a26d33a3397b4901545c787487603265cd3f79dd8f3csemantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77Lean: NOT_RUN · human review: NOT_PERFORMED