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R0022-11772 · q-20261013-advance-20261025-r01-boundary-value-problem-0072
Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-23, 23, -22, 24]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=3; supplied [edge,coefficients] records=[[left, [1090, 20, -1]], [right, [1044, -26, -1]], [bottom, [-409, 21, 2]], [top, [-639, -25, 2]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.
G001 — Use the quadratic candidate coefficients=[9, -1, -3, 2, -1, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-23, 23, -22, 24]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Interior source=3; supplied [edge,coefficients] records=[[left, [1090, 20, -1]], [right, [1044, -26, -1]], [bottom, [-409, 21, 2]], [top, [-639, -25, 2]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
original coefficients=[9, -1, -3, 2, -1, -1];basis=[1,x,y,x^2,xy,y^2]
After
u(x,y)=(9)+(-1)*x+(-3)*y+(2)*x^2+(-1)*x*y+(-1)*y^2
Conditions: none
Read the six coefficients in their fixed monomial order,not the descending[a,b,c] order of a univariate quadratic.
Rule: bivariate_scalar_polynomial_basis_roles · Uses: G001
Before
u_xx constant := mul(2, 2)
After
u_xx constant = 4
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
u_yy constant := mul(2, -1)
After
u_yy constant = -2
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
u coefficients=[9, -1, -3, 2, -1, -1];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y
After
u_xx=4;u_xy=-1;u_yy=-2
Conditions: none
Differentiate each monomial by fixed-coordinate slice rules.The linear x term differentiates to a constant,and xy contributes c4 to u_xy.These are polynomial identities,not an assumed general mixed-partial exchange theorem.
Rule: polynomial_second_partials_explicit_monomials · Uses: G001, S003
Before
interior Laplacian := sum(4, -2)
After
interior Laplacian = 2
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
interior constant signed residual := sub(2, 3)
After
interior constant signed residual = -1
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
u coefficients=[9, -1, -3, 2, -1, -1];constant Laplacian=2;source=3
After
constant residual=-1;interior polynomial identity=No
Conditions: none
For this candidate u_xx+u_yy is constant.A zero constant residual is the declared symbolic identity.No boundary data have been checked here,nor is this a general Poisson existence/uniqueness proof.
Rule: interior_identity_separate_boundary_obligation · Uses: G001, S006
Before
ordered bounds=[-23, 23, -22, 24];supplied edge order=[left, right, bottom, top]
After
all four distinct edges present;preserve original record order
Conditions: none
Each of the four edges appears once;duplicates do not substitute for missing edges.Preserve supplied record order and pairing.Degenerate rectangles still use the declared polynomial-certificate protocol;do not invent a nonempty open interior.
Rule: four_boundary_records_complete_not_sorted_away · Uses: G001, S007
Before
boundary record0 fixed coordinate squared := mul(-23, -23)
After
boundary record0 fixed coordinate squared = 529
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
boundary record0 fixed linear term := mul(-1, -23)
After
boundary record0 fixed linear term = 23
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
boundary record0 fixed square term := mul(2, 529)
After
boundary record0 fixed square term = 1058
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
boundary record0 constant coefficient := sum(9, 23, 1058)
After
boundary record0 constant coefficient = 1090
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
boundary record0 cross contribution := mul(-1, -23)
After
boundary record0 cross contribution = 23
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
boundary record0 linear coefficient := sum(-3, 23)
After
boundary record0 linear coefficient = 20
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
boundary record0:vary y;fix x=-23;original coefficients=[9, -1, -3, 2, -1, -1]
After
ascending y slice coefficients=[1090, 20, -1]
Conditions: none
Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together.
Rule: fixed_other_coordinate_before_derivative · Uses: G001, S014
Before
boundary record0 coefficient residual0 := sub(1090, 1090)
After
boundary record0 coefficient residual0 = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
boundary record0 coefficient residual1 := sub(20, 20)
After
boundary record0 coefficient residual1 = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
boundary record0 coefficient residual2 := sub(-1, -1)
After
boundary record0 coefficient residual2 = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
boundary record0:edge=left;bounds=[-23, 23, -22, 24];remaining variable=y;restriction=[1090, 20, -1];supplied=[1090, 20, -1]
After
all coefficient residuals=[0, 0, 0];polynomial identity=Yes
Conditions: none
Compare all three coefficients of the restriction polynomial,not sampled boundary values.This protocol audits a polynomial identity.If an edge collapses to a point,a coefficient mismatch rejects this stronger certificate but does not by itself refute equality at that single point.
Rule: whole_edge_polynomial_coefficient_certificate · Uses: G001, S018
Before
boundary record1 fixed coordinate squared := mul(23, 23)
After
boundary record1 fixed coordinate squared = 529
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
boundary record1 fixed linear term := mul(-1, 23)
After
boundary record1 fixed linear term = -23
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
boundary record1 fixed square term := mul(2, 529)
After
boundary record1 fixed square term = 1058
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
boundary record1 constant coefficient := sum(9, -23, 1058)
After
boundary record1 constant coefficient = 1044
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
boundary record1 cross contribution := mul(-1, 23)
After
boundary record1 cross contribution = -23
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
boundary record1 linear coefficient := sum(-3, -23)
After
boundary record1 linear coefficient = -26
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
boundary record1:vary y;fix x=23;original coefficients=[9, -1, -3, 2, -1, -1]
After
ascending y slice coefficients=[1044, -26, -1]
Conditions: none
Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together.
Rule: fixed_other_coordinate_before_derivative · Uses: G001, S025
Before
boundary record1 coefficient residual0 := sub(1044, 1044)
After
boundary record1 coefficient residual0 = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
boundary record1 coefficient residual1 := sub(-26, -26)
After
boundary record1 coefficient residual1 = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
boundary record1 coefficient residual2 := sub(-1, -1)
After
boundary record1 coefficient residual2 = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
boundary record1:edge=right;bounds=[-23, 23, -22, 24];remaining variable=y;restriction=[1044, -26, -1];supplied=[1044, -26, -1]
After
all coefficient residuals=[0, 0, 0];polynomial identity=Yes
Conditions: none
Compare all three coefficients of the restriction polynomial,not sampled boundary values.This protocol audits a polynomial identity.If an edge collapses to a point,a coefficient mismatch rejects this stronger certificate but does not by itself refute equality at that single point.
Rule: whole_edge_polynomial_coefficient_certificate · Uses: G001, S029
Before
boundary record2 fixed coordinate squared := mul(-22, -22)
After
boundary record2 fixed coordinate squared = 484
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
boundary record2 fixed linear term := mul(-3, -22)
After
boundary record2 fixed linear term = 66
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
boundary record2 fixed square term := mul(-1, 484)
After
boundary record2 fixed square term = -484
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
boundary record2 constant coefficient := sum(9, 66, -484)
After
boundary record2 constant coefficient = -409
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
boundary record2 cross contribution := mul(-1, -22)
After
boundary record2 cross contribution = 22
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
boundary record2 linear coefficient := sum(-1, 22)
After
boundary record2 linear coefficient = 21
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
boundary record2:vary x;fix y=-22;original coefficients=[9, -1, -3, 2, -1, -1]
After
ascending x slice coefficients=[-409, 21, 2]
Conditions: none
Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together.
Rule: fixed_other_coordinate_before_derivative · Uses: G001, S036
Before
boundary record2 coefficient residual0 := sub(-409, -409)
After
boundary record2 coefficient residual0 = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
boundary record2 coefficient residual1 := sub(21, 21)
After
boundary record2 coefficient residual1 = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
boundary record2 coefficient residual2 := sub(2, 2)
After
boundary record2 coefficient residual2 = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
boundary record2:edge=bottom;bounds=[-23, 23, -22, 24];remaining variable=x;restriction=[-409, 21, 2];supplied=[-409, 21, 2]
After
all coefficient residuals=[0, 0, 0];polynomial identity=Yes
Conditions: none
Compare all three coefficients of the restriction polynomial,not sampled boundary values.This protocol audits a polynomial identity.If an edge collapses to a point,a coefficient mismatch rejects this stronger certificate but does not by itself refute equality at that single point.
Rule: whole_edge_polynomial_coefficient_certificate · Uses: G001, S040
Before
boundary record3 fixed coordinate squared := mul(24, 24)
After
boundary record3 fixed coordinate squared = 576
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
boundary record3 fixed linear term := mul(-3, 24)
After
boundary record3 fixed linear term = -72
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
boundary record3 fixed square term := mul(-1, 576)
After
boundary record3 fixed square term = -576
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
boundary record3 constant coefficient := sum(9, -72, -576)
After
boundary record3 constant coefficient = -639
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
boundary record3 cross contribution := mul(-1, 24)
After
boundary record3 cross contribution = -24
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
boundary record3 linear coefficient := sum(-1, -24)
After
boundary record3 linear coefficient = -25
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
boundary record3:vary x;fix y=24;original coefficients=[9, -1, -3, 2, -1, -1]
After
ascending x slice coefficients=[-639, -25, 2]
Conditions: none
Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together.
Rule: fixed_other_coordinate_before_derivative · Uses: G001, S047
Before
boundary record3 coefficient residual0 := sub(-639, -639)
After
boundary record3 coefficient residual0 = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
boundary record3 coefficient residual1 := sub(-25, -25)
After
boundary record3 coefficient residual1 = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
boundary record3 coefficient residual2 := sub(2, 2)
After
boundary record3 coefficient residual2 = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
boundary record3:edge=top;bounds=[-23, 23, -22, 24];remaining variable=x;restriction=[-639, -25, 2];supplied=[-639, -25, 2]
After
all coefficient residuals=[0, 0, 0];polynomial identity=Yes
Conditions: none
Compare all three coefficients of the restriction polynomial,not sampled boundary values.This protocol audits a polynomial identity.If an edge collapses to a point,a coefficient mismatch rejects this stronger certificate but does not by itself refute equality at that single point.
Rule: whole_edge_polynomial_coefficient_certificate · Uses: G001, S051
Before
interior identity=No;boundary row flags=[Yes, Yes, Yes, Yes]
After
all boundary identities=Yes;joint certificate=No;state=boundary_pass_interior_fail
Conditions: none
Finish interior and all boundary certificates separately before conjunction.One passing group does not repair another.This audits the declared candidate,not general BVP existence,uniqueness,stability,regularity or well-posedness.
Rule: BVP_two_obligation_groups_conjunction · Uses: G001, S052
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 — Recompute boundary record3 coefficient residual2 with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Recompute boundary record3 coefficient residual1 with exact rational arithmetic, not rounded decimals.S049 · premise_and_result_audit — Recompute boundary record3 coefficient residual0 with exact rational arithmetic, not rounded decimals.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 — Recompute boundary record3 linear coefficient with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Recompute boundary record3 cross contribution with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Recompute boundary record3 constant coefficient with exact rational arithmetic, not rounded decimals.S044 · premise_and_result_audit — Recompute boundary record3 fixed square term with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute boundary record3 fixed linear term with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute boundary record3 fixed coordinate squared with exact rational arithmetic, not rounded decimals.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 — Recompute boundary record2 coefficient residual2 with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Recompute boundary record2 coefficient residual1 with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute boundary record2 coefficient residual0 with exact rational arithmetic, not rounded decimals.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 — Recompute boundary record2 linear coefficient with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute boundary record2 cross contribution with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute boundary record2 constant coefficient with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute boundary record2 fixed square term with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute boundary record2 fixed linear term with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute boundary record2 fixed coordinate squared with exact rational arithmetic, not rounded decimals.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 — Recompute boundary record1 coefficient residual2 with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute boundary record1 coefficient residual1 with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute boundary record1 coefficient residual0 with exact rational arithmetic, not rounded decimals.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 — Recompute boundary record1 linear coefficient with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute boundary record1 cross contribution with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute boundary record1 constant coefficient with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute boundary record1 fixed square term with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute boundary record1 fixed linear term with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute boundary record1 fixed coordinate squared 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 boundary record0 coefficient residual2 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute boundary record0 coefficient residual1 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute boundary record0 coefficient residual0 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Recompute boundary record0 linear coefficient with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute boundary record0 cross contribution with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute boundary record0 constant coefficient with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute boundary record0 fixed square term with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute boundary record0 fixed linear term with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute boundary record0 fixed coordinate squared with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Recompute interior constant signed residual with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute interior Laplacian 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 u_yy constant with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute u_xx constant with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([9, -1, -3, 2, -1, -1], 2, 3, No, [[left, y, [1090, 20, -1], [1090, 20, -1], Yes], [right, y, [1044, -26, -1], [1044, -26, -1], Yes], [bottom, x, [-409, 21, 2], [-409, 21, 2], Yes], [top, x, [-639, -25, 2], [-639, -25, 2], Yes]], Yes, No, candidate_satisfies_declared_BVP_only)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
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