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R0022-11771 · q-20261013-advance-20261025-r01-boundary-value-problem-0071
Use the quadratic candidate coefficients=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-22, 22, -21, 23]. 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=-2; supplied [edge,coefficients] records=[[left, [536, -19, -2]], [right, [448, 25, -2]], [bottom, [-937, -23, 1]], [top, [-981, 21, 1]]] 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=[8, -2, 3, 1, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-22, 22, -21, 23]. 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=-2; supplied [edge,coefficients] records=[[left, [536, -19, -2]], [right, [448, 25, -2]], [bottom, [-937, -23, 1]], [top, [-981, 21, 1]]] 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=[8, -2, 3, 1, 1, -2];basis=[1,x,y,x^2,xy,y^2]
After
u(x,y)=(8)+(-2)*x+(3)*y+(1)*x^2+(1)*x*y+(-2)*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, 1)
After
u_xx 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, S001
Before
u_yy constant := mul(2, -2)
After
u_yy 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, S002
Before
u coefficients=[8, -2, 3, 1, 1, -2];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y
After
u_xx=2;u_xy=1;u_yy=-4
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(2, -4)
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, -2)
After
interior constant signed residual = 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, S005
Before
u coefficients=[8, -2, 3, 1, 1, -2];constant Laplacian=-2;source=-2
After
constant residual=0;interior polynomial identity=Yes
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=[-22, 22, -21, 23];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(-22, -22)
After
boundary record0 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, S008
Before
boundary record0 fixed linear term := mul(-2, -22)
After
boundary record0 fixed linear term = 44
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(1, 484)
After
boundary record0 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, S010
Before
boundary record0 constant coefficient := sum(8, 44, 484)
After
boundary record0 constant coefficient = 536
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, -22)
After
boundary record0 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, S012
Before
boundary record0 linear coefficient := sum(3, -22)
After
boundary record0 linear coefficient = -19
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=-22;original coefficients=[8, -2, 3, 1, 1, -2]
After
ascending y slice coefficients=[536, -19, -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, S014
Before
boundary record0 coefficient residual0 := sub(536, 536)
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(-19, -19)
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(-2, -2)
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=[-22, 22, -21, 23];remaining variable=y;restriction=[536, -19, -2];supplied=[536, -19, -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, S018
Before
boundary record1 fixed coordinate squared := mul(22, 22)
After
boundary record1 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, S019
Before
boundary record1 fixed linear term := mul(-2, 22)
After
boundary record1 fixed linear term = -44
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(1, 484)
After
boundary record1 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, S021
Before
boundary record1 constant coefficient := sum(8, -44, 484)
After
boundary record1 constant coefficient = 448
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, 22)
After
boundary record1 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, S023
Before
boundary record1 linear coefficient := sum(3, 22)
After
boundary record1 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, S024
Before
boundary record1:vary y;fix x=22;original coefficients=[8, -2, 3, 1, 1, -2]
After
ascending y slice coefficients=[448, 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, S025
Before
boundary record1 coefficient residual0 := sub(448, 448)
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(25, 25)
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(-2, -2)
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=[-22, 22, -21, 23];remaining variable=y;restriction=[448, 25, -2];supplied=[448, 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, S029
Before
boundary record2 fixed coordinate squared := mul(-21, -21)
After
boundary record2 fixed coordinate squared = 441
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, -21)
After
boundary record2 fixed linear term = -63
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(-2, 441)
After
boundary record2 fixed square term = -882
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(8, -63, -882)
After
boundary record2 constant coefficient = -937
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, -21)
After
boundary record2 cross contribution = -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, S034
Before
boundary record2 linear coefficient := sum(-2, -21)
After
boundary record2 linear coefficient = -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, S035
Before
boundary record2:vary x;fix y=-21;original coefficients=[8, -2, 3, 1, 1, -2]
After
ascending x slice coefficients=[-937, -23, 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, S036
Before
boundary record2 coefficient residual0 := sub(-937, -937)
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(-23, -23)
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(1, 1)
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=[-22, 22, -21, 23];remaining variable=x;restriction=[-937, -23, 1];supplied=[-937, -23, 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, S040
Before
boundary record3 fixed coordinate squared := mul(23, 23)
After
boundary record3 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, S041
Before
boundary record3 fixed linear term := mul(3, 23)
After
boundary record3 fixed linear term = 69
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(-2, 529)
After
boundary record3 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, S043
Before
boundary record3 constant coefficient := sum(8, 69, -1058)
After
boundary record3 constant coefficient = -981
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, 23)
After
boundary record3 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, S045
Before
boundary record3 linear coefficient := sum(-2, 23)
After
boundary record3 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, S046
Before
boundary record3:vary x;fix y=23;original coefficients=[8, -2, 3, 1, 1, -2]
After
ascending x slice coefficients=[-981, 21, 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, S047
Before
boundary record3 coefficient residual0 := sub(-981, -981)
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(21, 21)
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(1, 1)
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=[-22, 22, -21, 23];remaining variable=x;restriction=[-981, 21, 1];supplied=[-981, 21, 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, S051
Before
interior identity=Yes;boundary row flags=[Yes, Yes, Yes, Yes]
After
all boundary identities=Yes;joint certificate=Yes;state=both_pass
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.([8, -2, 3, 1, 1, -2], -2, -2, Yes, [[left, y, [536, -19, -2], [536, -19, -2], Yes], [right, y, [448, 25, -2], [448, 25, -2], Yes], [bottom, x, [-937, -23, 1], [-937, -23, 1], Yes], [top, x, [-981, 21, 1], [-981, 21, 1], Yes]], Yes, Yes, 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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: c7849076523a7ae1757432b109128b0375bae876cebc7a87eaaadc1636378121candidate record SHA-256: f7be55193c37157a9b48703f4260ea1dff4142af75edfeee39ab1992d2f61e6asemantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8facquestion-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01bLean: NOT_RUN · human review: NOT_PERFORMED