第 27807 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-boundary；明示資料={"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[6,-6],"point":[7,1],"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-boundary；明示資料={"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[6,-6],"point":[7,1],"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-boundary]

derived = [[mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary]]

GD boundary ORIGINAL ORIGINAL gradient d*x coordinate0 := mul(1, 7)

GD boundary ORIGINAL ORIGINAL gradient d*x coordinate0 = 7

GD boundary ORIGINAL ORIGINAL gradient plus b coordinate0 := sum(7, 6)

GD boundary ORIGINAL ORIGINAL gradient plus b coordinate0 = 13

GD boundary ORIGINAL ORIGINAL gradient d*x coordinate1 := mul(2, 1)

GD boundary ORIGINAL ORIGINAL gradient d*x coordinate1 = 2

GD boundary ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(2, -6)

GD boundary ORIGINAL ORIGINAL gradient plus b coordinate1 = -4

GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(7, 7)

GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 49

GD boundary ORIGINAL ORIGINAL diagonal times square coordinate0 := mul(1, 49)

GD boundary ORIGINAL ORIGINAL diagonal times square coordinate0 = 49

GD boundary ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(49, 2)

GD boundary ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 49/2

GD boundary ORIGINAL ORIGINAL linear product coordinate0 := mul(6, 7)

GD boundary ORIGINAL ORIGINAL linear product coordinate0 = 42

GD boundary ORIGINAL ORIGINAL coordinate objective sum0 := sum(49/2, 42)

GD boundary ORIGINAL ORIGINAL coordinate objective sum0 = 133/2

GD boundary ORIGINAL ORIGINAL objective running sum0 := sum(0, 133/2)

GD boundary ORIGINAL ORIGINAL objective running sum0 = 133/2

GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 := mul(1, 1)

GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 = 1

GD boundary ORIGINAL ORIGINAL diagonal times square coordinate1 := mul(2, 1)

GD boundary ORIGINAL ORIGINAL diagonal times square coordinate1 = 2

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

GD boundary ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 1

GD boundary ORIGINAL ORIGINAL linear product coordinate1 := mul(-6, 1)

GD boundary ORIGINAL ORIGINAL linear product coordinate1 = -6

GD boundary ORIGINAL ORIGINAL coordinate objective sum1 := sum(1, -6)

GD boundary ORIGINAL ORIGINAL coordinate objective sum1 = -5

GD boundary ORIGINAL ORIGINAL objective running sum1 := sum(133/2, -5)

GD boundary ORIGINAL ORIGINAL objective running sum1 = 123/2

source roles = [GD boundary ORIGINAL, [1, 2], [6, -6], [7, 1]]

derived = [[1, 2], [6, -6], [7, 1], [13, -4], 123/2]

GD boundary step1 ORIGINAL alpha times gradient0 := mul(1/4, 13)

GD boundary step1 ORIGINAL alpha times gradient0 = 13/4

GD boundary step1 ORIGINAL next coordinate0 := sub(7, 13/4)

GD boundary step1 ORIGINAL next coordinate0 = 15/4

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

GD boundary step1 ORIGINAL alpha times gradient1 = -1

GD boundary step1 ORIGINAL next coordinate1 := sub(1, -1)

GD boundary step1 ORIGINAL next coordinate1 = 2

GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate0 := mul(1, 15/4)

GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate0 = 15/4

GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate0 := sum(15/4, 6)

GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate0 = 39/4

GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate1 := mul(2, 2)

GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate1 = 4

GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate1 := sum(4, -6)

GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate1 = -2

GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(15/4, 15/4)

GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 225/16

GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate0 := mul(1, 225/16)

GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate0 = 225/16

GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate0 := div(225/16, 2)

GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate0 = 225/32

GD boundary step1 next recomputation ORIGINAL linear product coordinate0 := mul(6, 15/4)

GD boundary step1 next recomputation ORIGINAL linear product coordinate0 = 45/2

GD boundary step1 next recomputation ORIGINAL coordinate objective sum0 := sum(225/32, 45/2)

GD boundary step1 next recomputation ORIGINAL coordinate objective sum0 = 945/32

GD boundary step1 next recomputation ORIGINAL objective running sum0 := sum(0, 945/32)

GD boundary step1 next recomputation ORIGINAL objective running sum0 = 945/32

GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 := mul(2, 2)

GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 = 4

GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate1 := mul(2, 4)

GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate1 = 8

GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate1 := div(8, 2)

GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate1 = 4

GD boundary step1 next recomputation ORIGINAL linear product coordinate1 := mul(-6, 2)

GD boundary step1 next recomputation ORIGINAL linear product coordinate1 = -12

GD boundary step1 next recomputation ORIGINAL coordinate objective sum1 := sum(4, -12)

GD boundary step1 next recomputation ORIGINAL coordinate objective sum1 = -8

GD boundary step1 next recomputation ORIGINAL objective running sum1 := sum(945/32, -8)

GD boundary step1 next recomputation ORIGINAL objective running sum1 = 689/32

source roles = [GD boundary step1 next recomputation, [[1, 1], [2, 1]], [[6, 1], [-6, 1]], [[15, 4], [2, 1]]]

derived = [[1, 2], [6, -6], [15/4, 2], [39/4, -2], 689/32]

source roles = [1/4, 1]

derived = [[[0, [7, 1], 123/2], [1, [15/4, 2], 689/32]], 否]

逐步反向核對

• 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 — 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。
• S038 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL objective running sum1，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL coordinate objective sum1，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL linear product coordinate1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate1，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate1，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL objective running sum0，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL coordinate objective sum0，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL linear product coordinate0，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate0，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate0，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate1，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate0，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate0，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 ORIGINAL next coordinate1，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 ORIGINAL alpha times gradient1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 ORIGINAL next coordinate0，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 GD boundary step1 ORIGINAL alpha times gradient0，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 boundary 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 boundary 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 boundary 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 boundary 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 boundary 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 boundary 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 boundary 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 boundary 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 boundary 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 boundary 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 boundary 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 boundary 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 boundary 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 boundary 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 boundary 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 boundary 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

(finite_trace_only, 1, [[0, [7, 1], 123/2], [1, [15/4, 2], 689/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)

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。