# Question 11723: A Whole-Edge Dirichlet Coefficient Identity [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-0048) 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. R0022-11723 · q-20261013-advance-20261025-r01-boundary-value-problem-0048 ## Original question and givens Use the quadratic candidate coefficients=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-24, 24, -23, 25]. 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. Selected edge=bottom; supplied coefficients=[56, 0, 3]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_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=[10, 0, -2, 3, 0, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-24, 24, -23, 25]. 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. Selected edge=bottom; supplied coefficients=[56, 0, 3]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_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=[10, 0, -2, 3, 0, 0];basis=[1,x,y,x^2,xy,y^2] After u(x,y)=(10)+(0)*x+(-2)*y+(3)*x^2+(0)*x*y+(0)*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 selected boundary fixed coordinate squared := mul(-23, -23) After selected boundary fixed coordinate squared = 529 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 selected boundary fixed linear term := mul(-2, -23) After selected boundary fixed linear term = 46 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 · calculationBefore selected boundary fixed square term := mul(0, 529) After selected boundary fixed square 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, S003 S005 · calculationBefore selected boundary constant coefficient := sum(10, 46, 0) After selected boundary constant coefficient = 56 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 selected boundary cross contribution := mul(0, -23) After selected boundary 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, S005 S007 · calculationBefore selected boundary linear coefficient := sum(0, 0) After selected boundary 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, S006 S008 · impliesBefore selected boundary:vary x;fix y=-23;original coefficients=[10, 0, -2, 3, 0, 0] After ascending x slice coefficients=[56, 0, 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, S007 S009 · calculationBefore selected boundary coefficient residual0 := sub(56, 56) After selected boundary 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, S008 S010 · calculationBefore selected boundary coefficient residual1 := sub(0, 0) After selected boundary 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, S009 S011 · calculationBefore selected boundary coefficient residual2 := sub(3, 3) After selected boundary 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, S010 S012 · impliesBefore selected boundary:edge=bottom;bounds=[-24, 24, -23, 25];remaining variable=x;restriction=[56, 0, 3];supplied=[56, 0, 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, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute selected boundary coefficient residual2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute selected boundary coefficient residual1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute selected boundary coefficient residual0 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 — Recompute selected boundary linear coefficient with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute selected boundary cross contribution with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute selected boundary constant coefficient with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute selected boundary fixed square term with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute selected boundary fixed linear term with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute selected boundary fixed coordinate squared 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 ([-24, 24, -23, 25], bottom, x, [56, 0, 3], [56, 0, 3], Yes, declared_edge_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: 3c471d11eec090eaf2e340cc0130329908a0e22a14773d1a8fb5402b1fd5e59d - candidate record SHA-256: cf5f8850431886637eda3058f98ccd25d8ac296cc01dda79e39541f7bdfeeee1 - 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-0048.json)