第 27891 題：梯度下降

原題與已知條件

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[1,5],"linear":[-1,-1],"point":[-2,-2],"step_size":[1,4],"steps":1}；請依 exact_ordered_tuple_declared_by_interface 回答。

• G001 — standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={"diagonal":[1,5],"linear":[-1,-1],"point":[-2,-2],"step_size":[1,4],"steps":1}；請依 exact_ordered_tuple_declared_by_interface 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-gradient-descent

逐步推導

source roles = [gd-safe-quadratic-trace]

derived = [[largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope]]

GD ORIGINAL ORIGINAL gradient d*x coordinate0 := mul(1, -2)

GD ORIGINAL ORIGINAL gradient d*x coordinate0 = -2

GD ORIGINAL ORIGINAL gradient plus b coordinate0 := sum(-2, -1)

GD ORIGINAL ORIGINAL gradient plus b coordinate0 = -3

GD ORIGINAL ORIGINAL gradient d*x coordinate1 := mul(5, -2)

GD ORIGINAL ORIGINAL gradient d*x coordinate1 = -10

GD ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(-10, -1)

GD ORIGINAL ORIGINAL gradient plus b coordinate1 = -11

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(-2, -2)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 4

GD ORIGINAL ORIGINAL diagonal times square coordinate0 := mul(1, 4)

GD ORIGINAL ORIGINAL diagonal times square coordinate0 = 4

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(4, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 2

GD ORIGINAL ORIGINAL linear product coordinate0 := mul(-1, -2)

GD ORIGINAL ORIGINAL linear product coordinate0 = 2

GD ORIGINAL ORIGINAL coordinate objective sum0 := sum(2, 2)

GD ORIGINAL ORIGINAL coordinate objective sum0 = 4

GD ORIGINAL ORIGINAL objective running sum0 := sum(0, 4)

GD ORIGINAL ORIGINAL objective running sum0 = 4

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 := mul(-2, -2)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 = 4

GD ORIGINAL ORIGINAL diagonal times square coordinate1 := mul(5, 4)

GD ORIGINAL ORIGINAL diagonal times square coordinate1 = 20

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 := div(20, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 10

GD ORIGINAL ORIGINAL linear product coordinate1 := mul(-1, -2)

GD ORIGINAL ORIGINAL linear product coordinate1 = 2

GD ORIGINAL ORIGINAL coordinate objective sum1 := sum(10, 2)

GD ORIGINAL ORIGINAL coordinate objective sum1 = 12

GD ORIGINAL ORIGINAL objective running sum1 := sum(4, 12)

GD ORIGINAL ORIGINAL objective running sum1 = 16

source roles = [GD ORIGINAL, [1, 5], [-1, -1], [-2, -2]]

derived = [[1, 5], [-1, -1], [-2, -2], [-3, -11], 16]

GD safe ORIGINAL2 over largest diagonal := div(2, 5)

GD safe ORIGINAL2 over largest diagonal = 2/5

GD safe ORIGINAL step1 ORIGINAL alpha times gradient0 := mul(1/4, -3)

GD safe ORIGINAL step1 ORIGINAL alpha times gradient0 = -3/4

GD safe ORIGINAL step1 ORIGINAL next coordinate0 := sub(-2, -3/4)

GD safe ORIGINAL step1 ORIGINAL next coordinate0 = -5/4

GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 := mul(1/4, -11)

GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 = -11/4

GD safe ORIGINAL step1 ORIGINAL next coordinate1 := sub(-2, -11/4)

GD safe ORIGINAL step1 ORIGINAL next coordinate1 = 3/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 := mul(1, -5/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 = -5/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 := sum(-5/4, -1)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 = -9/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1 := mul(5, 3/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1 = 15/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1 := sum(15/4, -1)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1 = 11/4

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(-5/4, -5/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 25/16

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 := mul(1, 25/16)

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 = 25/16

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 := div(25/16, 2)

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 = 25/32

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 := mul(-1, -5/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 = 5/4

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 := sum(25/32, 5/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 = 65/32

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 := sum(0, 65/32)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 = 65/32

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 := mul(3/4, 3/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 = 9/16

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1 := mul(5, 9/16)

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1 = 45/16

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 := div(45/16, 2)

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 = 45/32

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 := mul(-1, 3/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 = -3/4

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 := sum(45/32, -3/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 = 21/32

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 := sum(65/32, 21/32)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 = 43/16

source roles = [GD safe ORIGINAL step1 next recomputation, [[1, 1], [5, 1]], [[-1, 1], [-1, 1]], [[-5, 4], [3, 4]]]

derived = [[1, 5], [-1, -1], [-5/4, 3/4], [-9/4, 11/4], 43/16]

source roles = [[1, 5], 1/4, 5, 2/5, 1]

derived = [f(x-alpha*g)-f(x)=-alpha*sum_i(1-alpha*d_i/2)*g_i^2, [[0, [-2, -2], 16, [-3, -11]], [1, [-5/4, 3/4], 43/16, [-9/4, 11/4]]], 是, 否]

逐步反向核對

• S041 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S040 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S039 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 ORIGINAL next coordinate1，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 ORIGINAL alpha times gradient1，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 ORIGINAL next coordinate0，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 ORIGINAL alpha times gradient0，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL2 over largest diagonal，不以小數近似替代。
• S018 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S017 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL objective running sum1，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL coordinate objective sum1，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL linear product coordinate1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL quadratic divide2 coordinate1，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL diagonal times square coordinate1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL objective running sum0，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL coordinate objective sum0，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL linear product coordinate0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL quadratic divide2 coordinate0，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL diagonal times square coordinate0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL gradient plus b coordinate1，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL gradient d*x coordinate1，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL gradient plus b coordinate0，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL gradient d*x coordinate0，不以小數近似替代。
• S001 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。

結論

(5, 1/4, 2/5, 1, [[0, [-2, -2], 16, [-3, -11]], [1, [-5/4, 3/4], 43/16, [-9/4, 11/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

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。