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R0022-11742 · q-20261013-advance-20261025-r01-boundary-value-problem-0017
Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=1. 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=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Interior equation u_xx+u_yy=source has source=1. 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.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=[4, -1, -1, 1, 0, -1];basis=[1,x,y,x^2,xy,y^2]
After
u(x,y)=(4)+(-1)*x+(-1)*y+(1)*x^2+(0)*x*y+(-1)*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, 1)
After
u_xx constant = 2
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, -1)
After
u_yy constant = -2
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=[4, -1, -1, 1, 0, -1];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y
After
u_xx=2;u_xy=0;u_yy=-2
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(2, -2)
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, 1)
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
u coefficients=[4, -1, -1, 1, 0, -1];constant Laplacian=0;source=1
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.
Rule: interior_identity_separate_boundary_obligation · Uses: G001, S006
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.([4, -1, -1, 1, 0, -1], u_xx+u_yy=source, 0, 1, -1, No, interior_only)
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 6f7b40eb11cfade82940ca786c06708f5f4d50d0cce09679dcefd7e997f2a658candidate record SHA-256: c658c21603cc2b46ce9642aa5f7ba4e8834d3e4ec73a6879c45b63574949dfcasemantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8facquestion-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01bLean: NOT_RUN · human review: NOT_PERFORMED