# Question 11786: Interior Pass and Boundary Pass Must Not Be Conflated [Return to catalogue page 59](https://ocme.evemisslab.com/en/questions/page/59/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261025-r01-boundary-value-problem-foundations/#q-20261013-advance-20261025-r01-boundary-value-problem-0086) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0022-11786 · q-20261013-advance-20261025-r01-boundary-value-problem-0086 ## Original question and givens Use the quadratic candidate coefficients=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-12, 12, -11, 13]. 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, boundary data=[[left, [454, 0, -2]], [right, [406, 0, -2]], [bottom, [-244, -2, 3]], [top, [-340, -2, 3]]] 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=[-2, -2, 0, 3, 0, -2] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-12, 12, -11, 13]. 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, boundary data=[[left, [454, 0, -2]], [right, [406, 0, -2]], [bottom, [-244, -2, 3]], [top, [-340, -2, 3]]] 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. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-boundary-value-problem](https://ocme.evemisslab.com/en/mko/mko-boundary-value-problem/) ## Step-by-step derivation S001 · impliesBefore original coefficients=[-2, -2, 0, 3, 0, -2];basis=[1,x,y,x^2,xy,y^2] After u(x,y)=(-2)+(-2)*x+(0)*y+(3)*x^2+(0)*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. Step identifiers and dependenciesRule: bivariate_scalar_polynomial_basis_roles · Uses: G001 S002 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore u coefficients=[-2, -2, 0, 3, 0, -2];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y After u_xx=6;u_xy=0;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. Step identifiers and dependenciesRule: polynomial_second_partials_explicit_monomials · Uses: G001, S003 S005 · calculationBefore interior Laplacian := sum(6, -4) After interior Laplacian = 2 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore u coefficients=[-2, -2, 0, 3, 0, -2];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. Step identifiers and dependenciesRule: interior_identity_separate_boundary_obligation · Uses: G001, S006 S008 · impliesBefore ordered bounds=[-12, 12, -11, 13];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. Step identifiers and dependenciesRule: four_boundary_records_complete_not_sorted_away · Uses: G001, S007 S009 · calculationBefore boundary record0 fixed coordinate squared := mul(-12, -12) After boundary record0 fixed coordinate squared = 144 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore boundary record0 fixed linear term := mul(-2, -12) After boundary record0 fixed linear term = 24 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore boundary record0 fixed square term := mul(3, 144) After boundary record0 fixed square term = 432 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore boundary record0 constant coefficient := sum(-2, 24, 432) After boundary record0 constant coefficient = 454 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore boundary record0 cross contribution := mul(0, -12) After boundary record0 cross contribution = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore boundary record0 linear coefficient := sum(0, 0) After boundary record0 linear coefficient = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore boundary record0:vary y;fix x=-12;original coefficients=[-2, -2, 0, 3, 0, -2] After ascending y slice coefficients=[454, 0, -2] Conditions: none Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together. Step identifiers and dependenciesRule: fixed_other_coordinate_before_derivative · Uses: G001, S014 S016 · calculationBefore boundary record0 coefficient residual0 := sub(454, 454) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore boundary record0 coefficient residual1 := sub(0, 0) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore boundary record0:edge=left;bounds=[-12, 12, -11, 13];remaining variable=y;restriction=[454, 0, -2];supplied=[454, 0, -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. Step identifiers and dependenciesRule: whole_edge_polynomial_coefficient_certificate · Uses: G001, S018 S020 · calculationBefore boundary record1 fixed coordinate squared := mul(12, 12) After boundary record1 fixed coordinate squared = 144 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore boundary record1 fixed linear term := mul(-2, 12) After boundary record1 fixed linear term = -24 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore boundary record1 fixed square term := mul(3, 144) After boundary record1 fixed square term = 432 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore boundary record1 constant coefficient := sum(-2, -24, 432) After boundary record1 constant coefficient = 406 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore boundary record1 cross contribution := mul(0, 12) After boundary record1 cross contribution = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore boundary record1 linear coefficient := sum(0, 0) After boundary record1 linear coefficient = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore boundary record1:vary y;fix x=12;original coefficients=[-2, -2, 0, 3, 0, -2] After ascending y slice coefficients=[406, 0, -2] Conditions: none Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together. Step identifiers and dependenciesRule: fixed_other_coordinate_before_derivative · Uses: G001, S025 S027 · calculationBefore boundary record1 coefficient residual0 := sub(406, 406) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore boundary record1 coefficient residual1 := sub(0, 0) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · impliesBefore boundary record1:edge=right;bounds=[-12, 12, -11, 13];remaining variable=y;restriction=[406, 0, -2];supplied=[406, 0, -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. Step identifiers and dependenciesRule: whole_edge_polynomial_coefficient_certificate · Uses: G001, S029 S031 · calculationBefore boundary record2 fixed coordinate squared := mul(-11, -11) After boundary record2 fixed coordinate squared = 121 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore boundary record2 fixed linear term := mul(0, -11) After boundary record2 fixed linear term = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore boundary record2 fixed square term := mul(-2, 121) After boundary record2 fixed square term = -242 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore boundary record2 constant coefficient := sum(-2, 0, -242) After boundary record2 constant coefficient = -244 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore boundary record2 cross contribution := mul(0, -11) After boundary record2 cross contribution = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore boundary record2 linear coefficient := sum(-2, 0) After boundary record2 linear coefficient = -2 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · impliesBefore boundary record2:vary x;fix y=-11;original coefficients=[-2, -2, 0, 3, 0, -2] After ascending x slice coefficients=[-244, -2, 3] Conditions: none Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together. Step identifiers and dependenciesRule: fixed_other_coordinate_before_derivative · Uses: G001, S036 S038 · calculationBefore boundary record2 coefficient residual0 := sub(-244, -244) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore boundary record2 coefficient residual1 := sub(-2, -2) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · impliesBefore boundary record2:edge=bottom;bounds=[-12, 12, -11, 13];remaining variable=x;restriction=[-244, -2, 3];supplied=[-244, -2, 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. Step identifiers and dependenciesRule: whole_edge_polynomial_coefficient_certificate · Uses: G001, S040 S042 · calculationBefore boundary record3 fixed coordinate squared := mul(13, 13) After boundary record3 fixed coordinate squared = 169 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore boundary record3 fixed linear term := mul(0, 13) After boundary record3 fixed linear term = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore boundary record3 fixed square term := mul(-2, 169) After boundary record3 fixed square term = -338 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore boundary record3 constant coefficient := sum(-2, 0, -338) After boundary record3 constant coefficient = -340 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore boundary record3 cross contribution := mul(0, 13) After boundary record3 cross contribution = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore boundary record3 linear coefficient := sum(-2, 0) After boundary record3 linear coefficient = -2 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · impliesBefore boundary record3:vary x;fix y=13;original coefficients=[-2, -2, 0, 3, 0, -2] After ascending x slice coefficients=[-340, -2, 3] Conditions: none Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together. Step identifiers and dependenciesRule: fixed_other_coordinate_before_derivative · Uses: G001, S047 S049 · calculationBefore boundary record3 coefficient residual0 := sub(-340, -340) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore boundary record3 coefficient residual1 := sub(-2, -2) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · impliesBefore boundary record3:edge=top;bounds=[-12, 12, -11, 13];remaining variable=x;restriction=[-340, -2, 3];supplied=[-340, -2, 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. Step identifiers and dependenciesRule: whole_edge_polynomial_coefficient_certificate · Uses: G001, S051 S053 · impliesBefore 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. Step identifiers and dependenciesRule: BVP_two_obligation_groups_conjunction · Uses: G001, S052 ## Reverse checks - 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. ## Result (2, 3, No, [[left, y, [454, 0, -2], [454, 0, -2], Yes], [right, y, [406, 0, -2], [406, 0, -2], Yes], [bottom, x, [-244, -2, 3], [-244, -2, 3], Yes], [top, x, [-340, -2, 3], [-340, -2, 3], Yes]], Yes, boundary_pass_interior_fail) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 0056c878a9d248f2810dd7b04a36439ff0254a7829971699415d01936523fc80 - candidate record SHA-256: 2e34252a54dd995996671360d81f9acce413d20cc61d68544647089335f29c17 - semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac - question-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261025-r01-boundary-value-problem-0086.json)