第 27803 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":[2,-2],"point":[3,1],"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":[2,-2],"point":[3,1],"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, 3)

GD boundary ORIGINAL ORIGINAL gradient d*x coordinate0 = 3

GD boundary ORIGINAL ORIGINAL gradient plus b coordinate0 := sum(3, 2)

GD boundary ORIGINAL ORIGINAL gradient plus b coordinate0 = 5

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, -2)

GD boundary ORIGINAL ORIGINAL gradient plus b coordinate1 = 0

GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(3, 3)

GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 9

GD boundary ORIGINAL ORIGINAL diagonal times square coordinate0 := mul(1, 9)

GD boundary ORIGINAL ORIGINAL diagonal times square coordinate0 = 9

GD boundary ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(9, 2)

GD boundary ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 9/2

GD boundary ORIGINAL ORIGINAL linear product coordinate0 := mul(2, 3)

GD boundary ORIGINAL ORIGINAL linear product coordinate0 = 6

GD boundary ORIGINAL ORIGINAL coordinate objective sum0 := sum(9/2, 6)

GD boundary ORIGINAL ORIGINAL coordinate objective sum0 = 21/2

GD boundary ORIGINAL ORIGINAL objective running sum0 := sum(0, 21/2)

GD boundary ORIGINAL ORIGINAL objective running sum0 = 21/2

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(-2, 1)

GD boundary ORIGINAL ORIGINAL linear product coordinate1 = -2

GD boundary ORIGINAL ORIGINAL coordinate objective sum1 := sum(1, -2)

GD boundary ORIGINAL ORIGINAL coordinate objective sum1 = -1

GD boundary ORIGINAL ORIGINAL objective running sum1 := sum(21/2, -1)

GD boundary ORIGINAL ORIGINAL objective running sum1 = 19/2

source roles = [GD boundary ORIGINAL, [1, 2], [2, -2], [3, 1]]

derived = [[1, 2], [2, -2], [3, 1], [5, 0], 19/2]

GD boundary step1 ORIGINAL alpha times gradient0 := mul(1/4, 5)

GD boundary step1 ORIGINAL alpha times gradient0 = 5/4

GD boundary step1 ORIGINAL next coordinate0 := sub(3, 5/4)

GD boundary step1 ORIGINAL next coordinate0 = 7/4

GD boundary step1 ORIGINAL alpha times gradient1 := mul(1/4, 0)

GD boundary step1 ORIGINAL alpha times gradient1 = 0

GD boundary step1 ORIGINAL next coordinate1 := sub(1, 0)

GD boundary step1 ORIGINAL next coordinate1 = 1

GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate0 := mul(1, 7/4)

GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate0 = 7/4

GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate0 := sum(7/4, 2)

GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate0 = 15/4

GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate1 := mul(2, 1)

GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate1 = 2

GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate1 := sum(2, -2)

GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate1 = 0

GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(7/4, 7/4)

GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 49/16

GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate0 := mul(1, 49/16)

GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate0 = 49/16

GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate0 := div(49/16, 2)

GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate0 = 49/32

GD boundary step1 next recomputation ORIGINAL linear product coordinate0 := mul(2, 7/4)

GD boundary step1 next recomputation ORIGINAL linear product coordinate0 = 7/2

GD boundary step1 next recomputation ORIGINAL coordinate objective sum0 := sum(49/32, 7/2)

GD boundary step1 next recomputation ORIGINAL coordinate objective sum0 = 161/32

GD boundary step1 next recomputation ORIGINAL objective running sum0 := sum(0, 161/32)

GD boundary step1 next recomputation ORIGINAL objective running sum0 = 161/32

GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 := mul(1, 1)

GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 = 1

GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate1 := mul(2, 1)

GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate1 = 2

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

GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate1 = 1

GD boundary step1 next recomputation ORIGINAL linear product coordinate1 := mul(-2, 1)

GD boundary step1 next recomputation ORIGINAL linear product coordinate1 = -2

GD boundary step1 next recomputation ORIGINAL coordinate objective sum1 := sum(1, -2)

GD boundary step1 next recomputation ORIGINAL coordinate objective sum1 = -1

GD boundary step1 next recomputation ORIGINAL objective running sum1 := sum(161/32, -1)

GD boundary step1 next recomputation ORIGINAL objective running sum1 = 129/32

source roles = [GD boundary step1 next recomputation, [[1, 1], [2, 1]], [[2, 1], [-2, 1]], [[7, 4], [1, 1]]]

derived = [[1, 2], [2, -2], [7/4, 1], [15/4, 0], 129/32]

source roles = [1/4, 1]

derived = [[[0, [3, 1], 19/2], [1, [7/4, 1], 129/32]], 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, [3, 1], 19/2], [1, [7/4, 1], 129/32]], 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。