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R0033-17141 · q-20261014-advance-20261102-r02-linear-map-kernel-0091
Use this same exact integer matrix A=[[1, 16], [2, 32]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=1. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).
G001 — Use this same exact integer matrix A=[[1, 16], [2, 32]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=1. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).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
real map determinant diagonal product := mul(1, 32)
After
real map determinant diagonal product = 32
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001
Before
real map determinant off-diagonal product := mul(16, 2)
After
real map determinant off-diagonal product = 32
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
real map exact determinant := sub(32, 32)
After
real map exact determinant = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
same ordered integer matrix=[[1, 16], [2, 32]];usual real map T:R2 to R2;original columns=[[1, 2], [16, 32]]
After
exact determinant=0;rank=1
Conditions: none
A nonzero determinant gives rank2.With determinant0,distinguish the zero matrix from a nonzero singular matrix to get rank0 or1.Keep the same matrix and real vector spaces;do not replace columns by rows or assume the image is the entire codomain.
Rule: real_map_same_matrix_det_and_nonzero_rank_cases · Uses: G001, S003
Before
primitive kernel first coordinate := div(16, 1)
After
primitive kernel first coordinate = 16
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
primitive kernel second coordinate := div(-1, 1)
After
primitive kernel second coordinate = -1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
primitive kernel canonical sign coordinate0 := mul(1, 16)
After
primitive kernel canonical sign coordinate0 = 16
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
primitive kernel canonical sign coordinate1 := mul(1, -1)
After
primitive kernel canonical sign coordinate1 = -1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
other row dependence factor := div(2, 1)
After
other row dependence factor = 2
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
other row dependence coordinate0 := mul(2, 1)
After
other row dependence coordinate0 = 2
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
other row dependence coordinate1 := mul(2, 16)
After
other row dependence coordinate1 = 32
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
FIRST nonzero original row index=0;row=[1, 16];gcd=1;raw integer kernel direction=[16, -1];canonical sign=1
After
primitive kernel basis=[[16, -1]];other original row=[2, 32]=(2)*selected row
Conditions: none
Use the first nonzero original row to solve u*x+v*y=0.Divide by the gcd for a primitive integer direction and make its first nonzero coordinate positive.The other row is verified to be a multiple,so both equations reduce to this one rather than a guess from determinant0.
Rule: real_map_rank1_kernel_canonical_row_and_full_equation · Uses: G001, S011
Before
canonical kernel basis vector row0 first term := mul(1, 16)
After
canonical kernel basis vector row0 first term = 16
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
canonical kernel basis vector row0 second term := mul(16, -1)
After
canonical kernel basis vector row0 second term = -16
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
canonical kernel basis vector image coordinate0 := sum(16, -16)
After
canonical kernel basis vector image coordinate0 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
canonical kernel basis vector row1 first term := mul(2, 16)
After
canonical kernel basis vector row1 first term = 32
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
canonical kernel basis vector row1 second term := mul(32, -1)
After
canonical kernel basis vector row1 second term = -32
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
canonical kernel basis vector image coordinate1 := sum(32, -32)
After
canonical kernel basis vector image coordinate1 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
one nonzero row equation(1)*x+(16)*y=0;nonzero verified kernel vector=[16, -1];chosen nonzero vector coordinate=0
After
EVERY real solution is lambda*[16, -1];choose lambda=solution_coordinate0/(16);conversely every real lambda multiple maps to0
Conditions: none
If w0 is nonzero,then v is nonzero.Set λ=x/w0;subtract the two zero equations to get v(y−λw1)=0,hence y=λw1.If w0=0,then v=0 and nonzero u forces x=0 while y is arbitrary.Conversely linearity sends every real multiple to zero.This describes the full real kernel,not an integer grid.
Rule: real_map_rank1_kernel_all_real_solutions_not_one_test · Uses: G001, S018
Before
other column dependence factor := div(16, 1)
After
other column dependence factor = 16
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
other column dependence coordinate0 := mul(16, 1)
After
other column dependence coordinate0 = 16
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
other column dependence coordinate1 := mul(16, 2)
After
other column dependence coordinate1 = 32
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
FIRST nonzero ORIGINAL column index=0;basis vector kept unnormalized=[1, 2];other original column=[16, 32]=(16)*basis
After
for ANY real(s,t),M(s,t)=((1)*s+(16)*t)*basis;EVERY lambda*basis obtained from input lambda*e0
Conditions: none
Every output is a linear combination of the original columns,now verified to lie in this one column's span.Conversely place λ in that original column's input coordinate to obtain λ times the basis.So the image is the full real span.Do not normalize the original column or substitute a row.
Rule: real_map_rank1_image_both_inclusions_original_column · Uses: G001, S022
Before
real kernel source dimension minus rank := sub(2, 1)
After
real kernel source dimension minus rank = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
same real map rank=1;full kernel basis=[[16, -1]];full image basis=[[1, 2]]
After
nullity=1;image basis cardinality=1;rank plus nullity=2
Conditions: none
Count bases of the established full real spaces,not searched grid points.Keep rank,nullity and image-basis count distinct;the same source dimension is2.
Rule: real_map_full_space_basis_counts_not_grid_counts · Uses: G001, S024
Before
kernel bounded search:integer point in x-outer/y-inner order=[-1, -1];same matrix=[[1, 16], [2, 32]];target=[0, 0]
After
exact row-dot image=[-17, -34];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S025
Before
kernel bounded search:integer point in x-outer/y-inner order=[-1, 0];same matrix=[[1, 16], [2, 32]];target=[0, 0]
After
exact row-dot image=[-1, -2];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S026
Before
kernel bounded search:integer point in x-outer/y-inner order=[-1, 1];same matrix=[[1, 16], [2, 32]];target=[0, 0]
After
exact row-dot image=[15, 30];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S027
Before
kernel bounded search:integer point in x-outer/y-inner order=[0, -1];same matrix=[[1, 16], [2, 32]];target=[0, 0]
After
exact row-dot image=[-16, -32];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S028
Before
kernel bounded search:integer point in x-outer/y-inner order=[0, 0];same matrix=[[1, 16], [2, 32]];target=[0, 0]
After
exact row-dot image=[0, 0];equals target=Yes
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S029
Before
kernel bounded search:integer point in x-outer/y-inner order=[0, 1];same matrix=[[1, 16], [2, 32]];target=[0, 0]
After
exact row-dot image=[16, 32];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S030
Before
kernel bounded search:integer point in x-outer/y-inner order=[1, -1];same matrix=[[1, 16], [2, 32]];target=[0, 0]
After
exact row-dot image=[-15, -30];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S031
Before
kernel bounded search:integer point in x-outer/y-inner order=[1, 0];same matrix=[[1, 16], [2, 32]];target=[0, 0]
After
exact row-dot image=[1, 2];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S032
Before
kernel bounded search:integer point in x-outer/y-inner order=[1, 1];same matrix=[[1, 16], [2, 32]];target=[0, 0]
After
exact row-dot image=[17, 34];equals target=No
Conditions: none
Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs.
Rule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S033
Before
kernel bounded search:bound=1;complete integer box pair count=9
After
all hits in original scan order=[[0, 0]]
Conditions: none
Both endpoints are included,giving(2b+1)² pairs.Every pair was checked;completeness of the list is relative to this box only.
Rule: real_map_complete_bounded_list_not_unbounded_search · Uses: G001, S034
Before
bounded integer kernel points=[[0, 0]];SEPARATE symbolic full-real basis=[[16, -1]]
After
general completeness inference from box alone=No;bounded_integer_search_not_full_real_kernel
Conditions: none
The No limits inference from an integer box alone to the full real kernel.It does not deny the separate symbolic basis.In special cases the list may even equal the kernel;the general evidence boundary is not a claim of unequal sets in every instance.
Rule: kernel_box_scope_not_denial_of_symbolic_or_special_equality · Uses: G001, S035
S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S024 · premise_and_result_audit — Recompute real kernel source dimension minus rank with exact rational arithmetic, not rounded decimals.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 — Recompute other column dependence coordinate1 with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute other column dependence coordinate0 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute other column dependence factor with exact rational arithmetic, not rounded decimals.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 — Recompute canonical kernel basis vector image coordinate1 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute canonical kernel basis vector row1 second term with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute canonical kernel basis vector row1 first term with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute canonical kernel basis vector image coordinate0 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute canonical kernel basis vector row0 second term with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute canonical kernel basis vector row0 first term with exact rational arithmetic, not rounded decimals.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 — Recompute other row dependence coordinate1 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute other row dependence coordinate0 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute other row dependence factor with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute primitive kernel canonical sign coordinate1 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute primitive kernel canonical sign coordinate0 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute primitive kernel second coordinate with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute primitive kernel first coordinate with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute real map exact determinant with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute real map determinant off-diagonal product with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Recompute real map determinant diagonal product with exact rational arithmetic, not rounded decimals.(1, [[0, 0]], [[16, -1]], No, bounded_integer_search_not_full_real_kernel)
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: 1cfc1195c693474839065e913d5dc29afc3d023bbabef0e6d992fc82d9e2191fcandidate record SHA-256: ed4eca0b4691716f9816e61110685d65de062e76666be86a14418d7b2d7eab71semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93Lean: NOT_RUN · human review: NOT_PERFORMED