Question 11703: A Whole-Edge Dirichlet Coefficient Identity

Original question and givens

Use the quadratic candidate coefficients=[-10, 0, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-4, 4, -3, 5]. 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=[-7, -3, 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, -1, 3, 1, 0] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-4, 4, -3, 5]. 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=[-7, -3, 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

Step-by-step derivation

original coefficients=[-10, 0, -1, 3, 1, 0];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(-10)+(0)*x+(-1)*y+(3)*x^2+(1)*x*y+(0)*y^2

selected boundary fixed coordinate squared := mul(-3, -3)

selected boundary fixed coordinate squared = 9

selected boundary fixed linear term := mul(-1, -3)

selected boundary fixed linear term = 3

selected boundary fixed square term := mul(0, 9)

selected boundary fixed square term = 0

selected boundary constant coefficient := sum(-10, 3, 0)

selected boundary constant coefficient = -7

selected boundary cross contribution := mul(1, -3)

selected boundary cross contribution = -3

selected boundary linear coefficient := sum(0, -3)

selected boundary linear coefficient = -3

selected boundary:vary x;fix y=-3;original coefficients=[-10, 0, -1, 3, 1, 0]

ascending x slice coefficients=[-7, -3, 3]

selected boundary coefficient residual0 := sub(-7, -7)

selected boundary coefficient residual0 = 0

selected boundary coefficient residual1 := sub(-3, -3)

selected boundary coefficient residual1 = 0

selected boundary coefficient residual2 := sub(3, 3)

selected boundary coefficient residual2 = 0

selected boundary:edge=bottom;bounds=[-4, 4, -3, 5];remaining variable=x;restriction=[-7, -3, 3];supplied=[-7, -3, 3]

all coefficient residuals=[0, 0, 0];polynomial identity=Yes

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

([-4, 4, -3, 5], bottom, x, [-7, -3, 3], [-7, -3, 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.