Question 17050: Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Original question and givens

Use this same exact integer matrix A=[[2, 0], [0, 2]] 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. Target=[49, 1], nonnegative bound=5. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

• G001 — Use this same exact integer matrix A=[[2, 0], [0, 2]] 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. Target=[49, 1], nonnegative bound=5. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

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-linear-map-image

Step-by-step derivation

real map determinant diagonal product := mul(2, 2)

real map determinant diagonal product = 4

real map determinant off-diagonal product := mul(0, 0)

real map determinant off-diagonal product = 0

real map exact determinant := sub(4, 0)

real map exact determinant = 4

same ordered integer matrix=[[2, 0], [0, 2]];usual real map T:R2 to R2;original columns=[[2, 0], [0, 2]]

exact determinant=4;rank=2

first adjugate column row0 first term := mul(2, 2)

first adjugate column row0 first term = 4

first adjugate column row0 second term := mul(0, 0)

first adjugate column row0 second term = 0

first adjugate column image coordinate0 := sum(4, 0)

first adjugate column image coordinate0 = 4

first adjugate column row1 first term := mul(0, 2)

first adjugate column row1 first term = 0

first adjugate column row1 second term := mul(2, 0)

first adjugate column row1 second term = 0

first adjugate column image coordinate1 := sum(0, 0)

first adjugate column image coordinate1 = 0

second adjugate column row0 first term := mul(2, 0)

second adjugate column row0 first term = 0

second adjugate column row0 second term := mul(0, 2)

second adjugate column row0 second term = 0

second adjugate column image coordinate0 := sum(0, 0)

second adjugate column image coordinate0 = 0

second adjugate column row1 first term := mul(0, 0)

second adjugate column row1 first term = 0

second adjugate column row1 second term := mul(2, 2)

second adjugate column row1 second term = 4

second adjugate column image coordinate1 := sum(0, 4)

second adjugate column image coordinate1 = 4

same matrix=[[2, 0], [0, 2]];adjugate columns=[[2, 0], [0, 2]];verified products=[[4, 0], [0, 4]]

M*adjugate=det*I;det=4 nonzero;inverse=adjugate/det;kernel basis=[];image basis ORIGINAL columns=[[2, 0], [0, 2]]

real kernel source dimension minus rank := sub(2, 2)

real kernel source dimension minus rank = 0

same real map rank=2;full kernel basis=[];full image basis=[[2, 0], [0, 2]]

nullity=0;image basis cardinality=2;rank plus nullity=2

image Cramer first diagonal target term := mul(2, 49)

image Cramer first diagonal target term = 98

image Cramer first off-diagonal target term := mul(0, 1)

image Cramer first off-diagonal target term = 0

image Cramer first numerator := sub(98, 0)

image Cramer first numerator = 98

image Cramer first real coordinate := div(98, 4)

image Cramer first real coordinate = 49/2

image Cramer second signed target term := mul(0, 49)

image Cramer second signed target term = 0

image Cramer second diagonal target term := mul(2, 1)

image Cramer second diagonal target term = 2

image Cramer second numerator := sum(0, 2)

image Cramer second numerator = 2

image Cramer second real coordinate := div(2, 4)

image Cramer second real coordinate = 1/2

image Cramer substitution row0 first term := mul(2, 49/2)

image Cramer substitution row0 first term = 49

image Cramer substitution row0 second term := mul(0, 1/2)

image Cramer substitution row0 second term = 0

image Cramer substitution image coordinate0 := sum(49, 0)

image Cramer substitution image coordinate0 = 49

image Cramer substitution row1 first term := mul(0, 49/2)

image Cramer substitution row1 first term = 0

image Cramer substitution row1 second term := mul(2, 1/2)

image Cramer substitution row1 second term = 1

image Cramer substitution image coordinate1 := sum(0, 1)

image Cramer substitution image coordinate1 = 1

original target=[49, 1];same invertible real matrix=[[2, 0], [0, 2]];exact solved coordinates=[49/2, 1/2]

substitution equals target;unique real preimage;integer coordinates NOT required

image bounded search:integer point in x-outer/y-inner order=[-5, -5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-10, -10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-5, -4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-10, -8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-5, -3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-10, -6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-5, -2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-10, -4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-5, -1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-10, -2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-5, 0];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-10, 0];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-5, 1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-10, 2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-5, 2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-10, 4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-5, 3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-10, 6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-5, 4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-10, 8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-5, 5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-10, 10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-4, -5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-8, -10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-4, -4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-8, -8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-4, -3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-8, -6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-4, -2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-8, -4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-4, -1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-8, -2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-4, 0];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-8, 0];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-4, 1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-8, 2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-4, 2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-8, 4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-4, 3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-8, 6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-4, 4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-8, 8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-4, 5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-8, 10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-3, -5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-6, -10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-3, -4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-6, -8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-3, -3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-6, -6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-3, -2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-6, -4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-3, -1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-6, -2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-3, 0];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-6, 0];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-3, 1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-6, 2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-3, 2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-6, 4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-3, 3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-6, 6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-3, 4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-6, 8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-3, 5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-6, 10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-2, -5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-4, -10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-2, -4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-4, -8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-2, -3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-4, -6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-2, -2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-4, -4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-2, -1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-4, -2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-2, 0];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-4, 0];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-2, 1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-4, 2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-2, 2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-4, 4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-2, 3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-4, 6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-2, 4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-4, 8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-2, 5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-4, 10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-1, -5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-2, -10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-1, -4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-2, -8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-1, -3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-2, -6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-1, -2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-2, -4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-1, -1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-2, -2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-1, 0];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-2, 0];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-1, 1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-2, 2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-1, 2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-2, 4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-1, 3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-2, 6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-1, 4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-2, 8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[-1, 5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[-2, 10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[0, -5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[0, -10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[0, -4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[0, -8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[0, -3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[0, -6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[0, -2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[0, -4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[0, -1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[0, -2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[0, 0];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[0, 0];equals target=No

image bounded search:integer point in x-outer/y-inner order=[0, 1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[0, 2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[0, 2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[0, 4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[0, 3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[0, 6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[0, 4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[0, 8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[0, 5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[0, 10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[1, -5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[2, -10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[1, -4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[2, -8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[1, -3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[2, -6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[1, -2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[2, -4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[1, -1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[2, -2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[1, 0];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[2, 0];equals target=No

image bounded search:integer point in x-outer/y-inner order=[1, 1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[2, 2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[1, 2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[2, 4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[1, 3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[2, 6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[1, 4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[2, 8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[1, 5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[2, 10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[2, -5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[4, -10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[2, -4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[4, -8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[2, -3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[4, -6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[2, -2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[4, -4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[2, -1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[4, -2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[2, 0];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[4, 0];equals target=No

image bounded search:integer point in x-outer/y-inner order=[2, 1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[4, 2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[2, 2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[4, 4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[2, 3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[4, 6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[2, 4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[4, 8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[2, 5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[4, 10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[3, -5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[6, -10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[3, -4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[6, -8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[3, -3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[6, -6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[3, -2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[6, -4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[3, -1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[6, -2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[3, 0];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[6, 0];equals target=No

image bounded search:integer point in x-outer/y-inner order=[3, 1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[6, 2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[3, 2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[6, 4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[3, 3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[6, 6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[3, 4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[6, 8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[3, 5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[6, 10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[4, -5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[8, -10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[4, -4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[8, -8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[4, -3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[8, -6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[4, -2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[8, -4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[4, -1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[8, -2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[4, 0];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[8, 0];equals target=No

image bounded search:integer point in x-outer/y-inner order=[4, 1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[8, 2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[4, 2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[8, 4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[4, 3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[8, 6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[4, 4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[8, 8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[4, 5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[8, 10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[5, -5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[10, -10];equals target=No

image bounded search:integer point in x-outer/y-inner order=[5, -4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[10, -8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[5, -3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[10, -6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[5, -2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[10, -4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[5, -1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[10, -2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[5, 0];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[10, 0];equals target=No

image bounded search:integer point in x-outer/y-inner order=[5, 1];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[10, 2];equals target=No

image bounded search:integer point in x-outer/y-inner order=[5, 2];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[10, 4];equals target=No

image bounded search:integer point in x-outer/y-inner order=[5, 3];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[10, 6];equals target=No

image bounded search:integer point in x-outer/y-inner order=[5, 4];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[10, 8];equals target=No

image bounded search:integer point in x-outer/y-inner order=[5, 5];same matrix=[[2, 0], [0, 2]];target=[49, 1]

exact row-dot image=[10, 10];equals target=No

image bounded search:bound=5;complete integer box pair count=121

all hits in original scan order=[]

bounded integer preimages in original scan order=[];SEPARATE full-real membership=Yes;global certificate=[unique_solution, [49/2, 1/2]]

boundary=bounded_failure_not_global_nonmembership

Reverse checks

• S157 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S154 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S149 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute image Cramer substitution image coordinate1 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute image Cramer substitution row1 second term with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute image Cramer substitution row1 first term with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute image Cramer substitution image coordinate0 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute image Cramer substitution row0 second term with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute image Cramer substitution row0 first term with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute image Cramer second real coordinate with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute image Cramer second numerator with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute image Cramer second diagonal target term with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute image Cramer second signed target term with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute image Cramer first real coordinate with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute image Cramer first numerator with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute image Cramer first off-diagonal target term with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute image Cramer first diagonal target term 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 real kernel source dimension minus rank with exact rational arithmetic, not rounded decimals.
• 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 — Recompute second adjugate column image coordinate1 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute second adjugate column row1 second term with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute second adjugate column row1 first term with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute second adjugate column image coordinate0 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute second adjugate column row0 second term with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute second adjugate column row0 first term with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute first adjugate column image coordinate1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute first adjugate column row1 second term with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute first adjugate column row1 first term with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute first adjugate column image coordinate0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute first adjugate column row0 second term with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute first adjugate column row0 first term 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.

Result

([49, 1], 5, [], Yes, bounded_failure_not_global_nonmembership)

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.