# Question 11741: An Interior Poisson Identity without a Boundary Claim [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-0016) 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. R0022-11741 · q-20261013-advance-20261025-r01-boundary-value-problem-0016 ## Original question and givens Use the quadratic candidate coefficients=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=4. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_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=[3, -2, -2, 4, -1, -2] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=4. Give (coefficients,u_xx+u_yy=source,constant Laplacian,source,signed residual,interior identity?,interior_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. ## 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=[3, -2, -2, 4, -1, -2];basis=[1,x,y,x^2,xy,y^2] After u(x,y)=(3)+(-2)*x+(-2)*y+(4)*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. Step identifiers and dependenciesRule: bivariate_scalar_polynomial_basis_roles · Uses: G001 S002 · calculationBefore u_xx constant := mul(2, 4) After u_xx constant = 8 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=[3, -2, -2, 4, -1, -2];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y After u_xx=8;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. Step identifiers and dependenciesRule: polynomial_second_partials_explicit_monomials · Uses: G001, S003 S005 · calculationBefore interior Laplacian := sum(8, -4) After interior Laplacian = 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, S004 S006 · calculationBefore interior constant signed residual := sub(4, 4) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore u coefficients=[3, -2, -2, 4, -1, -2];constant Laplacian=4;source=4 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. Step identifiers and dependenciesRule: interior_identity_separate_boundary_obligation · Uses: G001, S006 ## Reverse checks - 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 ([3, -2, -2, 4, -1, -2], u_xx+u_yy=source, 4, 4, 0, Yes, interior_only) 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: 57908ee32aedc15fd9b2a30cc71b3f0e5e029aba6492243961cec154a3efde32 - candidate record SHA-256: 475b187092976d271fb7012c1643a91aca595fb6c0b0f19cc977ba7f9fe0863b - 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-0016.json)