OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 27803: Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 140

← Question derivation index · Original question batch

R0054-27803 · q-20261015-pulse-20261119-r03-0078

Original question and givens

Use mode=finite_trace_only, complete model={"diagonal":[1,2],"linear":[2,-2],"point":[3,1],"step_size":[1,4],"steps":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).

  • G001 — Use mode=finite_trace_only, complete model={"diagonal":[1,2],"linear":[2,-2],"point":[3,1],"step_size":[1,4],"steps":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).

Definition and scope

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.

Step-by-step derivation

S001 · implies

Before

source roles = [gd-boundary]

After

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

Conditions: none

List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

GD boundary ORIGINAL ORIGINAL gradient d*x coordinate0 := mul(1, 3)

After

GD boundary ORIGINAL ORIGINAL gradient d*x coordinate0 = 3

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL gradient plus b coordinate0 = 5

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL gradient d*x coordinate1 = 2

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL gradient plus b coordinate1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 9

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL diagonal times square coordinate0 = 9

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 9/2

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL linear product coordinate0 = 6

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL coordinate objective sum0 = 21/2

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL objective running sum0 = 21/2

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 = 1

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL diagonal times square coordinate1 = 2

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 1

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL linear product coordinate1 = -2

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL coordinate objective sum1 = -1

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

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

After

GD boundary ORIGINAL ORIGINAL objective running sum1 = 19/2

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

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

After

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

Conditions: none

For the WHOLE specified f(x)=1/2 sum d_i x_i²+sum b_i x_i,differentiate each coordinate to d_i x_i+b_i,not2d_i x_i.Compute ALL gradient entries first,then the objective in original order,squaring before multiplying d_i.Qualify positive diagonals,dimensions and coordinate order.

Step identifiers and dependencies

Rule: GD_WHOLE_positive_diagonal_quadratic_ordered_gradient_and_objective · Uses: G001, S017

S019 · calculation

Before

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

After

GD boundary step1 ORIGINAL alpha times gradient0 = 5/4

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

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

After

GD boundary step1 ORIGINAL next coordinate0 = 7/4

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

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

After

GD boundary step1 ORIGINAL alpha times gradient1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

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

After

GD boundary step1 ORIGINAL next coordinate1 = 1

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

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

After

GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate1 = 1

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

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

After

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

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · implies

Before

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

After

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

Conditions: none

For the WHOLE specified f(x)=1/2 sum d_i x_i²+sum b_i x_i,differentiate each coordinate to d_i x_i+b_i,not2d_i x_i.Compute ALL gradient entries first,then the objective in original order,squaring before multiplying d_i.Qualify positive diagonals,dimensions and coordinate order.

Step identifiers and dependencies

Rule: GD_WHOLE_positive_diagonal_quadratic_ordered_gradient_and_objective · Uses: G001, S038

S040 · implies

Before

source roles = [1/4, 1]

After

derived = [[[0, [3, 1], 19/2], [1, [7/4, 1], 129/32]], No]

Conditions: none

This mode requires only alpha>0,not the safe range.Retain the initial point and every later objective.A finite trace,whether decreasing or increasing,is not a general convergence/global-optimality certificate.

Step identifiers and dependencies

Rule: GD_finite_trace_ONLY_positive_not_safe_no_limit_claim · Uses: G001, S039

Reverse checks

  • S040 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S039 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S038 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL objective running sum1 with exact rational arithmetic, not rounded decimals.
  • S037 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL coordinate objective sum1 with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL linear product coordinate1 with exact rational arithmetic, not rounded decimals.
  • S035 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate1 with exact rational arithmetic, not rounded decimals.
  • S034 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate1 with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL objective running sum0 with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL coordinate objective sum0 with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL linear product coordinate0 with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate0 with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate0 with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate1 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate1 with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate0 with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate0 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute GD boundary step1 ORIGINAL next coordinate1 with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute GD boundary step1 ORIGINAL alpha times gradient1 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute GD boundary step1 ORIGINAL next coordinate0 with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute GD boundary step1 ORIGINAL alpha times gradient0 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S017 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL objective running sum1 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL coordinate objective sum1 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL linear product coordinate1 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL quadratic divide2 coordinate1 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL diagonal times square coordinate1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL objective running sum0 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL coordinate objective sum0 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL linear product coordinate0 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL quadratic divide2 coordinate0 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL diagonal times square coordinate0 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL gradient plus b coordinate1 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL gradient d*x coordinate1 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL gradient plus b coordinate0 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute GD boundary ORIGINAL ORIGINAL gradient d*x coordinate0 with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.

Result

(finite_trace_only, 1, [[0, [3, 1], 19/2], [1, [7/4, 1], 129/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: e2bc07e63595e2be41a4c5bebd8c14b3c2417130eda805e57f4881d0485d9040
  • candidate record SHA-256: abfcd09cec82140a06fbd7e48e971b69daf59130ef03585b482a71fe0636193f
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON