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R0022-11781 · q-20261013-advance-20261025-r01-boundary-value-problem-0081
Use the quadratic candidate coefficients=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-7, 7, -6, 8]. 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=0, boundary data=[[left, [106, -5, -2]], [right, [77, 9, -2]], [bottom, [-91, -8, 2]], [top, [-119, 6, 2]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). 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=[-7, -2, 2, 2, 1, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-7, 7, -6, 8]. 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=0, boundary data=[[left, [106, -5, -2]], [right, [77, 9, -2]], [bottom, [-91, -8, 2]], [top, [-119, 6, 2]]] with all four distinct edges. Give (Laplacian,source,interior identity?,boundary rows in supplied order,boundary identities?,one of both_pass / interior_pass_boundary_fail / boundary_pass_interior_fail / both_fail). 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=[-7, -2, 2, 2, 1, -2];basis=[1,x,y,x^2,xy,y^2]
After
u(x,y)=(-7)+(-2)*x+(2)*y+(2)*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, 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, -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=[-7, -2, 2, 2, 1, -2];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=-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(4, -4)
After
interior Laplacian = 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, S004
Before
interior constant signed residual := sub(0, 0)
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=[-7, -2, 2, 2, 1, -2];constant Laplacian=0;source=0
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=[-7, 7, -6, 8];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(-7, -7)
After
boundary record0 fixed coordinate squared = 49
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, -7)
After
boundary record0 fixed linear term = 14
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, 49)
After
boundary record0 fixed square term = 98
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(-7, 14, 98)
After
boundary record0 constant coefficient = 105
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, -7)
After
boundary record0 cross contribution = -7
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(2, -7)
After
boundary record0 linear coefficient = -5
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=-7;original coefficients=[-7, -2, 2, 2, 1, -2]
After
ascending y slice coefficients=[105, -5, -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(105, 106)
After
boundary record0 coefficient residual0 = -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, S015
Before
boundary record0 coefficient residual1 := sub(-5, -5)
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=[-7, 7, -6, 8];remaining variable=y;restriction=[105, -5, -2];supplied=[106, -5, -2]
After
all coefficient residuals=[-1, 0, 0];polynomial identity=No
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(7, 7)
After
boundary record1 fixed coordinate squared = 49
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, 7)
After
boundary record1 fixed linear term = -14
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, 49)
After
boundary record1 fixed square term = 98
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(-7, -14, 98)
After
boundary record1 constant coefficient = 77
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, 7)
After
boundary record1 cross contribution = 7
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(2, 7)
After
boundary record1 linear coefficient = 9
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=7;original coefficients=[-7, -2, 2, 2, 1, -2]
After
ascending y slice coefficients=[77, 9, -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(77, 77)
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(9, 9)
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=[-7, 7, -6, 8];remaining variable=y;restriction=[77, 9, -2];supplied=[77, 9, -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(-6, -6)
After
boundary record2 fixed coordinate squared = 36
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(2, -6)
After
boundary record2 fixed linear term = -12
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, 36)
After
boundary record2 fixed square 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, S032
Before
boundary record2 constant coefficient := sum(-7, -12, -72)
After
boundary record2 constant coefficient = -91
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, -6)
After
boundary record2 cross contribution = -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, S034
Before
boundary record2 linear coefficient := sum(-2, -6)
After
boundary record2 linear coefficient = -8
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=-6;original coefficients=[-7, -2, 2, 2, 1, -2]
After
ascending x slice coefficients=[-91, -8, 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(-91, -91)
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(-8, -8)
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=[-7, 7, -6, 8];remaining variable=x;restriction=[-91, -8, 2];supplied=[-91, -8, 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(8, 8)
After
boundary record3 fixed coordinate squared = 64
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(2, 8)
After
boundary record3 fixed linear term = 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, S042
Before
boundary record3 fixed square term := mul(-2, 64)
After
boundary record3 fixed square term = -128
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(-7, 16, -128)
After
boundary record3 constant coefficient = -119
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, 8)
After
boundary record3 cross contribution = 8
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, 8)
After
boundary record3 linear coefficient = 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, S046
Before
boundary record3:vary x;fix y=8;original coefficients=[-7, -2, 2, 2, 1, -2]
After
ascending x slice coefficients=[-119, 6, 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(-119, -119)
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(6, 6)
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=[-7, 7, -6, 8];remaining variable=x;restriction=[-119, 6, 2];supplied=[-119, 6, 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=Yes;boundary row flags=[No, Yes, Yes, Yes]
After
all boundary identities=No;joint certificate=No;state=interior_pass_boundary_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.(0, 0, Yes, [[left, y, [105, -5, -2], [106, -5, -2], No], [right, y, [77, 9, -2], [77, 9, -2], Yes], [bottom, x, [-91, -8, 2], [-91, -8, 2], Yes], [top, x, [-119, 6, 2], [-119, 6, 2], Yes]], No, interior_pass_boundary_fail)
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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