# Question 17150: A Bounded Integer Box Is Not the Full Real Kernel

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0033-17150 · q-20261014-advance-20261102-r02-linear-map-kernel-0100



## Original question and givens

Use this same exact integer matrix A=[[1, 25], [2, 50]] 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=5. 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, 25], [2, 50]] 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=5. 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).





## 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-kernel](https://ocme.evemisslab.com/en/mko/mko-linear-map-kernel/)





## Step-by-step derivation

S001 · calculationBefore

real map determinant diagonal product := mul(1, 50)

After

real map determinant diagonal product = 50

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001





S002 · calculationBefore

real map determinant off-diagonal product := mul(25, 2)

After

real map determinant off-diagonal product = 50

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

real map exact determinant := sub(50, 50)

After

real map exact determinant = 0

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

same ordered integer matrix=[[1, 25], [2, 50]];usual real map T:R2 to R2;original columns=[[1, 2], [25, 50]]

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.

Step identifiers and dependenciesRule: real_map_same_matrix_det_and_nonzero_rank_cases · Uses: G001, S003





S005 · calculationBefore

primitive kernel first coordinate := div(25, 1)

After

primitive kernel first coordinate = 25

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

primitive kernel canonical sign coordinate0 := mul(1, 25)

After

primitive kernel canonical sign coordinate0 = 25

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

other row dependence coordinate1 := mul(2, 25)

After

other row dependence coordinate1 = 50

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · impliesBefore

FIRST nonzero original row index=0;row=[1, 25];gcd=1;raw integer kernel direction=[25, -1];canonical sign=1

After

primitive kernel basis=[[25, -1]];other original row=[2, 50]=(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.

Step identifiers and dependenciesRule: real_map_rank1_kernel_canonical_row_and_full_equation · Uses: G001, S011





S013 · calculationBefore

canonical kernel basis vector row0 first term := mul(1, 25)

After

canonical kernel basis vector row0 first term = 25

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

canonical kernel basis vector row0 second term := mul(25, -1)

After

canonical kernel basis vector row0 second term = -25

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · calculationBefore

canonical kernel basis vector image coordinate0 := sum(25, -25)

After

canonical kernel basis vector image coordinate0 = 0

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

canonical kernel basis vector row1 first term := mul(2, 25)

After

canonical kernel basis vector row1 first term = 50

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

canonical kernel basis vector row1 second term := mul(50, -1)

After

canonical kernel basis vector row1 second term = -50

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · calculationBefore

canonical kernel basis vector image coordinate1 := sum(50, -50)

After

canonical kernel basis vector image coordinate1 = 0

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · impliesBefore

one nonzero row equation(1)*x+(25)*y=0;nonzero verified kernel vector=[25, -1];chosen nonzero vector coordinate=0

After

EVERY real solution is lambda*[25, -1];choose lambda=solution_coordinate0/(25);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.

Step identifiers and dependenciesRule: real_map_rank1_kernel_all_real_solutions_not_one_test · Uses: G001, S018





S020 · calculationBefore

other column dependence factor := div(25, 1)

After

other column dependence factor = 25

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · calculationBefore

other column dependence coordinate0 := mul(25, 1)

After

other column dependence coordinate0 = 25

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · calculationBefore

other column dependence coordinate1 := mul(25, 2)

After

other column dependence coordinate1 = 50

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021





S023 · impliesBefore

FIRST nonzero ORIGINAL column index=0;basis vector kept unnormalized=[1, 2];other original column=[25, 50]=(25)*basis

After

for ANY real(s,t),M(s,t)=((1)*s+(25)*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.

Step identifiers and dependenciesRule: real_map_rank1_image_both_inclusions_original_column · Uses: G001, S022





S024 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · impliesBefore

same real map rank=1;full kernel basis=[[25, -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.

Step identifiers and dependenciesRule: real_map_full_space_basis_counts_not_grid_counts · Uses: G001, S024





S026 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-5, -5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-130, -260];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S025





S027 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-5, -4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-105, -210];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S026





S028 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-5, -3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-80, -160];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S027





S029 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-5, -2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-55, -110];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S028





S030 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-5, -1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-30, -60];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S029





S031 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-5, 0];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-5, -10];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S030





S032 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-5, 1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[20, 40];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S031





S033 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-5, 2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[45, 90];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S032





S034 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-5, 3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[70, 140];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S033





S035 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-5, 4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[95, 190];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S034





S036 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-5, 5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[120, 240];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S035





S037 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-4, -5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-129, -258];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S036





S038 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-4, -4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-104, -208];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S037





S039 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-4, -3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-79, -158];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S038





S040 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-4, -2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-54, -108];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S039





S041 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-4, -1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-29, -58];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S040





S042 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-4, 0];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-4, -8];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S041





S043 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-4, 1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[21, 42];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S042





S044 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-4, 2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[46, 92];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S043





S045 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-4, 3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[71, 142];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S044





S046 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-4, 4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[96, 192];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S045





S047 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-4, 5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[121, 242];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S046





S048 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-3, -5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-128, -256];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S047





S049 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-3, -4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-103, -206];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S048





S050 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-3, -3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-78, -156];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S049





S051 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-3, -2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-53, -106];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S050





S052 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-3, -1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-28, -56];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S051





S053 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-3, 0];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-3, -6];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S052





S054 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-3, 1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[22, 44];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S053





S055 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-3, 2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[47, 94];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S054





S056 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-3, 3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[72, 144];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S055





S057 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-3, 4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[97, 194];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S056





S058 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-3, 5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[122, 244];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S057





S059 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-2, -5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-127, -254];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S058





S060 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-2, -4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-102, -204];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S059





S061 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-2, -3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-77, -154];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S060





S062 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-2, -2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-52, -104];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S061





S063 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-2, -1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-27, -54];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S062





S064 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-2, 0];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-2, -4];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S063





S065 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-2, 1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[23, 46];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S064





S066 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-2, 2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[48, 96];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S065





S067 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-2, 3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[73, 146];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S066





S068 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-2, 4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[98, 196];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S067





S069 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-2, 5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[123, 246];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S068





S070 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-1, -5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-126, -252];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S069





S071 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-1, -4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-101, -202];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S070





S072 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-1, -3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-76, -152];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S071





S073 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-1, -2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-51, -102];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S072





S074 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-1, -1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-26, -52];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S073





S075 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-1, 0];same matrix=[[1, 25], [2, 50]];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S074





S076 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-1, 1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[24, 48];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S075





S077 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-1, 2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[49, 98];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S076





S078 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-1, 3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[74, 148];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S077





S079 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-1, 4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[99, 198];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S078





S080 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[-1, 5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[124, 248];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S079





S081 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[0, -5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-125, -250];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S080





S082 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[0, -4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-100, -200];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S081





S083 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[0, -3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-75, -150];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S082





S084 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[0, -2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-50, -100];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S083





S085 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[0, -1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-25, -50];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S084





S086 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[0, 0];same matrix=[[1, 25], [2, 50]];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S085





S087 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[0, 1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[25, 50];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S086





S088 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[0, 2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[50, 100];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S087





S089 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[0, 3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[75, 150];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S088





S090 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[0, 4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[100, 200];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S089





S091 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[0, 5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[125, 250];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S090





S092 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[1, -5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-124, -248];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S091





S093 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[1, -4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-99, -198];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S092





S094 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[1, -3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-74, -148];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S093





S095 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[1, -2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-49, -98];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S094





S096 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[1, -1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-24, -48];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S095





S097 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[1, 0];same matrix=[[1, 25], [2, 50]];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S096





S098 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[1, 1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[26, 52];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S097





S099 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[1, 2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[51, 102];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S098





S100 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[1, 3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[76, 152];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S099





S101 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[1, 4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[101, 202];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S100





S102 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[1, 5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[126, 252];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S101





S103 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[2, -5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-123, -246];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S102





S104 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[2, -4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-98, -196];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S103





S105 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[2, -3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-73, -146];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S104





S106 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[2, -2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-48, -96];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S105





S107 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[2, -1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-23, -46];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S106





S108 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[2, 0];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[2, 4];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S107





S109 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[2, 1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[27, 54];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S108





S110 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[2, 2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[52, 104];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S109





S111 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[2, 3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[77, 154];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S110





S112 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[2, 4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[102, 204];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S111





S113 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[2, 5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[127, 254];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S112





S114 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[3, -5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-122, -244];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S113





S115 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[3, -4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-97, -194];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S114





S116 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[3, -3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-72, -144];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S115





S117 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[3, -2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-47, -94];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S116





S118 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[3, -1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-22, -44];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S117





S119 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[3, 0];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[3, 6];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S118





S120 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[3, 1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[28, 56];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S119





S121 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[3, 2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[53, 106];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S120





S122 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[3, 3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[78, 156];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S121





S123 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[3, 4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[103, 206];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S122





S124 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[3, 5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[128, 256];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S123





S125 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[4, -5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-121, -242];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S124





S126 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[4, -4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-96, -192];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S125





S127 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[4, -3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-71, -142];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S126





S128 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[4, -2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-46, -92];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S127





S129 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[4, -1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-21, -42];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S128





S130 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[4, 0];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[4, 8];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S129





S131 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[4, 1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[29, 58];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S130





S132 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[4, 2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[54, 108];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S131





S133 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[4, 3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[79, 158];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S132





S134 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[4, 4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[104, 208];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S133





S135 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[4, 5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[129, 258];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S134





S136 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[5, -5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-120, -240];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S135





S137 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[5, -4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-95, -190];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S136





S138 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[5, -3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-70, -140];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S137





S139 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[5, -2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-45, -90];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S138





S140 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[5, -1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[-20, -40];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S139





S141 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[5, 0];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[5, 10];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S140





S142 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[5, 1];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[30, 60];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S141





S143 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[5, 2];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[55, 110];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S142





S144 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[5, 3];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[80, 160];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S143





S145 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[5, 4];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[105, 210];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S144





S146 · impliesBefore

kernel bounded search:integer point in x-outer/y-inner order=[5, 5];same matrix=[[1, 25], [2, 50]];target=[0, 0]

After

exact row-dot image=[130, 260];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.

Step identifiers and dependenciesRule: real_map_every_bounded_integer_point_exact_image · Uses: G001, S145





S147 · impliesBefore

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

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.

Step identifiers and dependenciesRule: real_map_complete_bounded_list_not_unbounded_search · Uses: G001, S146





S148 · impliesBefore

bounded integer kernel points=[[0, 0]];SEPARATE symbolic full-real basis=[[25, -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.

Step identifiers and dependenciesRule: kernel_box_scope_not_denial_of_symbolic_or_special_equality · Uses: G001, S147









## Reverse checks


- 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 — 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.





## Result

(5, [[0, 0]], [[25, -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.





## 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 87894ffc43deb2ac1f25e76a456c7be08f35a52f8635a532460d92c36dd6a39b
- candidate record SHA-256: 572895ba8288a64a6fcbe1e90dcba8879b14e65ac964c5a8beab959170b3ed0c
- semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3
- question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0100.json)