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R0022-11765 · q-20261013-advance-20261025-r01-boundary-value-problem-0065
Use the quadratic candidate coefficients=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-16, 16, -15, 17]. 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=11; supplied [edge,coefficients] records=[[left, [738, -19, 2]], [right, [802, 13, 2]], [bottom, [497, -13, 3]], [top, [529, 19, 3]]] 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=[2, 2, -3, 3, 1, 2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-16, 16, -15, 17]. 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=11; supplied [edge,coefficients] records=[[left, [738, -19, 2]], [right, [802, 13, 2]], [bottom, [497, -13, 3]], [top, [529, 19, 3]]] 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=[2, 2, -3, 3, 1, 2];basis=[1,x,y,x^2,xy,y^2]
After
u(x,y)=(2)+(2)*x+(-3)*y+(3)*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, 3)
After
u_xx constant = 6
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=[2, 2, -3, 3, 1, 2];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y
After
u_xx=6;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(6, 4)
After
interior Laplacian = 10
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(10, 11)
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=[2, 2, -3, 3, 1, 2];constant Laplacian=10;source=11
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=[-16, 16, -15, 17];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(-16, -16)
After
boundary record0 fixed coordinate squared = 256
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, -16)
After
boundary record0 fixed linear term = -32
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(3, 256)
After
boundary record0 fixed square term = 768
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(2, -32, 768)
After
boundary record0 constant coefficient = 738
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, -16)
After
boundary record0 cross contribution = -16
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, -16)
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=-16;original coefficients=[2, 2, -3, 3, 1, 2]
After
ascending y slice coefficients=[738, -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(738, 738)
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=[-16, 16, -15, 17];remaining variable=y;restriction=[738, -19, 2];supplied=[738, -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(16, 16)
After
boundary record1 fixed coordinate squared = 256
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, 16)
After
boundary record1 fixed linear term = 32
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(3, 256)
After
boundary record1 fixed square term = 768
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(2, 32, 768)
After
boundary record1 constant coefficient = 802
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, 16)
After
boundary record1 cross contribution = 16
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, 16)
After
boundary record1 linear coefficient = 13
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=16;original coefficients=[2, 2, -3, 3, 1, 2]
After
ascending y slice coefficients=[802, 13, 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(802, 802)
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(13, 13)
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=[-16, 16, -15, 17];remaining variable=y;restriction=[802, 13, 2];supplied=[802, 13, 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(-15, -15)
After
boundary record2 fixed coordinate squared = 225
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, -15)
After
boundary record2 fixed linear term = 45
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, 225)
After
boundary record2 fixed square term = 450
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(2, 45, 450)
After
boundary record2 constant coefficient = 497
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, -15)
After
boundary record2 cross contribution = -15
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, -15)
After
boundary record2 linear coefficient = -13
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=-15;original coefficients=[2, 2, -3, 3, 1, 2]
After
ascending x slice coefficients=[497, -13, 3]
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(497, 497)
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(-13, -13)
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(3, 3)
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=[-16, 16, -15, 17];remaining variable=x;restriction=[497, -13, 3];supplied=[497, -13, 3]
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(17, 17)
After
boundary record3 fixed coordinate squared = 289
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, 17)
After
boundary record3 fixed linear term = -51
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, 289)
After
boundary record3 fixed square term = 578
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(2, -51, 578)
After
boundary record3 constant coefficient = 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, S044
Before
boundary record3 cross contribution := mul(1, 17)
After
boundary record3 cross contribution = 17
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, 17)
After
boundary record3 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, S046
Before
boundary record3:vary x;fix y=17;original coefficients=[2, 2, -3, 3, 1, 2]
After
ascending x slice coefficients=[529, 19, 3]
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(529, 529)
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(19, 19)
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(3, 3)
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=[-16, 16, -15, 17];remaining variable=x;restriction=[529, 19, 3];supplied=[529, 19, 3]
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.([2, 2, -3, 3, 1, 2], 10, 11, No, [[left, y, [738, -19, 2], [738, -19, 2], Yes], [right, y, [802, 13, 2], [802, 13, 2], Yes], [bottom, x, [497, -13, 3], [497, -13, 3], Yes], [top, x, [529, 19, 3], [529, 19, 3], 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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 97d502f201dd81e1701c39c2b4e6e357a4e5bc066c07c6e0ff50fddf67c47dc9candidate record SHA-256: 0ca6d108f4cddf18092fdc210498e5bd6027789bfda26d1089ee77e24857a385semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8facquestion-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01bLean: NOT_RUN · human review: NOT_PERFORMED