第 11776 題：邊界值問題

原題與已知條件

內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={"coefficients":[-12,-2,-3,1,-1,-2],"bounds":[-2,2,-1,3],"source":-1,"boundary_data":[["left",[-4,-1,-2]],["right",[-12,-5,-2]],["bottom",[-11,-1,1]],["top",[-39,-5,1]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。

• G001 — 內部PDE責任與矩形邊界資料分開。操作=bvp-obligation-separation；資料={"coefficients":[-12,-2,-3,1,-1,-2],"bounds":[-2,2,-1,3],"source":-1,"boundary_data":[["left",[-4,-1,-2]],["right",[-12,-5,-2]],["bottom",[-11,-1,1]],["top",[-39,-5,1]]]}。請依 [laplacian,source,interior_identity,boundary_rows,boundary_identity,state] 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-boundary-value-problem

逐步推導

original coefficients=[-12, -2, -3, 1, -1, -2];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(-12)+(-2)*x+(-3)*y+(1)*x^2+(-1)*x*y+(-2)*y^2

u_xx constant := mul(2, 1)

u_xx constant = 2

u_yy constant := mul(2, -2)

u_yy constant = -4

u coefficients=[-12, -2, -3, 1, -1, -2];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y

u_xx=2;u_xy=-1;u_yy=-4

interior Laplacian := sum(2, -4)

interior Laplacian = -2

interior constant signed residual := sub(-2, -1)

interior constant signed residual = -1

u coefficients=[-12, -2, -3, 1, -1, -2];constant Laplacian=-2;source=-1

constant residual=-1;interior polynomial identity=否

ordered bounds=[-2, 2, -1, 3];supplied edge order=[left, right, bottom, top]

all four distinct edges present;preserve original record order

boundary record0 fixed coordinate squared := mul(-2, -2)

boundary record0 fixed coordinate squared = 4

boundary record0 fixed linear term := mul(-2, -2)

boundary record0 fixed linear term = 4

boundary record0 fixed square term := mul(1, 4)

boundary record0 fixed square term = 4

boundary record0 constant coefficient := sum(-12, 4, 4)

boundary record0 constant coefficient = -4

boundary record0 cross contribution := mul(-1, -2)

boundary record0 cross contribution = 2

boundary record0 linear coefficient := sum(-3, 2)

boundary record0 linear coefficient = -1

boundary record0:vary y;fix x=-2;original coefficients=[-12, -2, -3, 1, -1, -2]

ascending y slice coefficients=[-4, -1, -2]

boundary record0 coefficient residual0 := sub(-4, -4)

boundary record0 coefficient residual0 = 0

boundary record0 coefficient residual1 := sub(-1, -1)

boundary record0 coefficient residual1 = 0

boundary record0 coefficient residual2 := sub(-2, -2)

boundary record0 coefficient residual2 = 0

boundary record0:edge=left;bounds=[-2, 2, -1, 3];remaining variable=y;restriction=[-4, -1, -2];supplied=[-4, -1, -2]

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

boundary record1 fixed coordinate squared := mul(2, 2)

boundary record1 fixed coordinate squared = 4

boundary record1 fixed linear term := mul(-2, 2)

boundary record1 fixed linear term = -4

boundary record1 fixed square term := mul(1, 4)

boundary record1 fixed square term = 4

boundary record1 constant coefficient := sum(-12, -4, 4)

boundary record1 constant coefficient = -12

boundary record1 cross contribution := mul(-1, 2)

boundary record1 cross contribution = -2

boundary record1 linear coefficient := sum(-3, -2)

boundary record1 linear coefficient = -5

boundary record1:vary y;fix x=2;original coefficients=[-12, -2, -3, 1, -1, -2]

ascending y slice coefficients=[-12, -5, -2]

boundary record1 coefficient residual0 := sub(-12, -12)

boundary record1 coefficient residual0 = 0

boundary record1 coefficient residual1 := sub(-5, -5)

boundary record1 coefficient residual1 = 0

boundary record1 coefficient residual2 := sub(-2, -2)

boundary record1 coefficient residual2 = 0

boundary record1:edge=right;bounds=[-2, 2, -1, 3];remaining variable=y;restriction=[-12, -5, -2];supplied=[-12, -5, -2]

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

boundary record2 fixed coordinate squared := mul(-1, -1)

boundary record2 fixed coordinate squared = 1

boundary record2 fixed linear term := mul(-3, -1)

boundary record2 fixed linear term = 3

boundary record2 fixed square term := mul(-2, 1)

boundary record2 fixed square term = -2

boundary record2 constant coefficient := sum(-12, 3, -2)

boundary record2 constant coefficient = -11

boundary record2 cross contribution := mul(-1, -1)

boundary record2 cross contribution = 1

boundary record2 linear coefficient := sum(-2, 1)

boundary record2 linear coefficient = -1

boundary record2:vary x;fix y=-1;original coefficients=[-12, -2, -3, 1, -1, -2]

ascending x slice coefficients=[-11, -1, 1]

boundary record2 coefficient residual0 := sub(-11, -11)

boundary record2 coefficient residual0 = 0

boundary record2 coefficient residual1 := sub(-1, -1)

boundary record2 coefficient residual1 = 0

boundary record2 coefficient residual2 := sub(1, 1)

boundary record2 coefficient residual2 = 0

boundary record2:edge=bottom;bounds=[-2, 2, -1, 3];remaining variable=x;restriction=[-11, -1, 1];supplied=[-11, -1, 1]

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

boundary record3 fixed coordinate squared := mul(3, 3)

boundary record3 fixed coordinate squared = 9

boundary record3 fixed linear term := mul(-3, 3)

boundary record3 fixed linear term = -9

boundary record3 fixed square term := mul(-2, 9)

boundary record3 fixed square term = -18

boundary record3 constant coefficient := sum(-12, -9, -18)

boundary record3 constant coefficient = -39

boundary record3 cross contribution := mul(-1, 3)

boundary record3 cross contribution = -3

boundary record3 linear coefficient := sum(-2, -3)

boundary record3 linear coefficient = -5

boundary record3:vary x;fix y=3;original coefficients=[-12, -2, -3, 1, -1, -2]

ascending x slice coefficients=[-39, -5, 1]

boundary record3 coefficient residual0 := sub(-39, -39)

boundary record3 coefficient residual0 = 0

boundary record3 coefficient residual1 := sub(-5, -5)

boundary record3 coefficient residual1 = 0

boundary record3 coefficient residual2 := sub(1, 1)

boundary record3 coefficient residual2 = 0

boundary record3:edge=top;bounds=[-2, 2, -1, 3];remaining variable=x;restriction=[-39, -5, 1];supplied=[-39, -5, 1]

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

interior identity=否;boundary row flags=[是, 是, 是, 是]

all boundary identities=是;joint certificate=否;state=boundary_pass_interior_fail

逐步反向核對

• S053 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S052 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S051 · premise_and_result_audit — 以精確有理數重算 boundary record3 coefficient residual2，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 boundary record3 coefficient residual1，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 boundary record3 coefficient residual0，不以小數近似替代。
• S048 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S047 · premise_and_result_audit — 以精確有理數重算 boundary record3 linear coefficient，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 boundary record3 cross contribution，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 boundary record3 constant coefficient，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 boundary record3 fixed square term，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 boundary record3 fixed linear term，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 boundary record3 fixed coordinate squared，不以小數近似替代。
• S041 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S040 · premise_and_result_audit — 以精確有理數重算 boundary record2 coefficient residual2，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 boundary record2 coefficient residual1，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 boundary record2 coefficient residual0，不以小數近似替代。
• S037 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S036 · premise_and_result_audit — 以精確有理數重算 boundary record2 linear coefficient，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 boundary record2 cross contribution，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 boundary record2 constant coefficient，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 boundary record2 fixed square term，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 boundary record2 fixed linear term，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 boundary record2 fixed coordinate squared，不以小數近似替代。
• S030 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · premise_and_result_audit — 以精確有理數重算 boundary record1 coefficient residual2，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 boundary record1 coefficient residual1，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 boundary record1 coefficient residual0，不以小數近似替代。
• S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S025 · premise_and_result_audit — 以精確有理數重算 boundary record1 linear coefficient，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 boundary record1 cross contribution，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 boundary record1 constant coefficient，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 boundary record1 fixed square term，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 boundary record1 fixed linear term，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 boundary record1 fixed coordinate squared，不以小數近似替代。
• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · premise_and_result_audit — 以精確有理數重算 boundary record0 coefficient residual2，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 boundary record0 coefficient residual1，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 boundary record0 coefficient residual0，不以小數近似替代。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 以精確有理數重算 boundary record0 linear coefficient，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 boundary record0 cross contribution，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 boundary record0 constant coefficient，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 boundary record0 fixed square term，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 boundary record0 fixed linear term，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 boundary record0 fixed coordinate squared，不以小數近似替代。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 以精確有理數重算 interior constant signed residual，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 interior Laplacian，不以小數近似替代。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 以精確有理數重算 u_yy constant，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 u_xx constant，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(-2, -1, 否, [[left, y, [-4, -1, -2], [-4, -1, -2], 是], [right, y, [-12, -5, -2], [-12, -5, -2], 是], [bottom, x, [-11, -1, 1], [-11, -1, 1], 是], [top, x, [-39, -5, 1], [-39, -5, 1], 是]], 是, boundary_pass_interior_fail)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。