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R0054-27823 · q-20261015-pulse-20261119-r03-0098
Use mode=finite_trace_only, complete model={"diagonal":[1,2],"linear":[22,-22],"point":[23,1],"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":[22,-22],"point":[23,1],"step_size":[1,4],"steps":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).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.
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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
GD boundary ORIGINAL ORIGINAL gradient d*x coordinate0 := mul(1, 23)
After
GD boundary ORIGINAL ORIGINAL gradient d*x coordinate0 = 23
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
GD boundary ORIGINAL ORIGINAL gradient plus b coordinate0 := sum(23, 22)
After
GD boundary ORIGINAL ORIGINAL gradient plus b coordinate0 = 45
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S002
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
GD boundary ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(2, -22)
After
GD boundary ORIGINAL ORIGINAL gradient plus b coordinate1 = -20
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(23, 23)
After
GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 529
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
GD boundary ORIGINAL ORIGINAL diagonal times square coordinate0 := mul(1, 529)
After
GD boundary ORIGINAL ORIGINAL diagonal times square coordinate0 = 529
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
GD boundary ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(529, 2)
After
GD boundary ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 529/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
GD boundary ORIGINAL ORIGINAL linear product coordinate0 := mul(22, 23)
After
GD boundary ORIGINAL ORIGINAL linear product coordinate0 = 506
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
GD boundary ORIGINAL ORIGINAL coordinate objective sum0 := sum(529/2, 506)
After
GD boundary ORIGINAL ORIGINAL coordinate objective sum0 = 1541/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
GD boundary ORIGINAL ORIGINAL objective running sum0 := sum(0, 1541/2)
After
GD boundary ORIGINAL ORIGINAL objective running sum0 = 1541/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S010
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.
Rule: exact_rational_arithmetic · Uses: G001, S011
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.
Rule: exact_rational_arithmetic · Uses: G001, S012
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.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
GD boundary ORIGINAL ORIGINAL linear product coordinate1 := mul(-22, 1)
After
GD boundary ORIGINAL ORIGINAL linear product coordinate1 = -22
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
GD boundary ORIGINAL ORIGINAL coordinate objective sum1 := sum(1, -22)
After
GD boundary ORIGINAL ORIGINAL coordinate objective sum1 = -21
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
GD boundary ORIGINAL ORIGINAL objective running sum1 := sum(1541/2, -21)
After
GD boundary ORIGINAL ORIGINAL objective running sum1 = 1499/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
source roles = [GD boundary ORIGINAL, [1, 2], [22, -22], [23, 1]]
After
derived = [[1, 2], [22, -22], [23, 1], [45, -20], 1499/2]
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.
Rule: GD_WHOLE_positive_diagonal_quadratic_ordered_gradient_and_objective · Uses: G001, S017
Before
GD boundary step1 ORIGINAL alpha times gradient0 := mul(1/4, 45)
After
GD boundary step1 ORIGINAL alpha times gradient0 = 45/4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
GD boundary step1 ORIGINAL next coordinate0 := sub(23, 45/4)
After
GD boundary step1 ORIGINAL next coordinate0 = 47/4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
GD boundary step1 ORIGINAL alpha times gradient1 := mul(1/4, -20)
After
GD boundary step1 ORIGINAL alpha times gradient1 = -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.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
GD boundary step1 ORIGINAL next coordinate1 := sub(1, -5)
After
GD boundary step1 ORIGINAL next coordinate1 = 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.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate0 := mul(1, 47/4)
After
GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate0 = 47/4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate0 := sum(47/4, 22)
After
GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate0 = 135/4
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate1 := mul(2, 6)
After
GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate1 = 12
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate1 := sum(12, -22)
After
GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate1 = -10
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(47/4, 47/4)
After
GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 2209/16
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate0 := mul(1, 2209/16)
After
GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate0 = 2209/16
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate0 := div(2209/16, 2)
After
GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate0 = 2209/32
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
GD boundary step1 next recomputation ORIGINAL linear product coordinate0 := mul(22, 47/4)
After
GD boundary step1 next recomputation ORIGINAL linear product coordinate0 = 517/2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
GD boundary step1 next recomputation ORIGINAL coordinate objective sum0 := sum(2209/32, 517/2)
After
GD boundary step1 next recomputation ORIGINAL coordinate objective sum0 = 10481/32
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
GD boundary step1 next recomputation ORIGINAL objective running sum0 := sum(0, 10481/32)
After
GD boundary step1 next recomputation ORIGINAL objective running sum0 = 10481/32
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 := mul(6, 6)
After
GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 = 36
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate1 := mul(2, 36)
After
GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate1 = 72
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate1 := div(72, 2)
After
GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate1 = 36
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
GD boundary step1 next recomputation ORIGINAL linear product coordinate1 := mul(-22, 6)
After
GD boundary step1 next recomputation ORIGINAL linear product coordinate1 = -132
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
GD boundary step1 next recomputation ORIGINAL coordinate objective sum1 := sum(36, -132)
After
GD boundary step1 next recomputation ORIGINAL coordinate objective sum1 = -96
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
GD boundary step1 next recomputation ORIGINAL objective running sum1 := sum(10481/32, -96)
After
GD boundary step1 next recomputation ORIGINAL objective running sum1 = 7409/32
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
source roles = [GD boundary step1 next recomputation, [[1, 1], [2, 1]], [[22, 1], [-22, 1]], [[47, 4], [6, 1]]]
After
derived = [[1, 2], [22, -22], [47/4, 6], [135/4, -10], 7409/32]
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.
Rule: GD_WHOLE_positive_diagonal_quadratic_ordered_gradient_and_objective · Uses: G001, S038
Before
source roles = [1/4, 1]
After
derived = [[[0, [23, 1], 1499/2], [1, [47/4, 6], 7409/32]], 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.
Rule: GD_finite_trace_ONLY_positive_not_safe_no_limit_claim · Uses: G001, S039
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.(finite_trace_only, 1, [[0, [23, 1], 1499/2], [1, [47/4, 6], 7409/32]], 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.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 0ec680728ed6d19ab07389a8f2af8ed385204f544eb663cc252af876da963f22candidate record SHA-256: 5f790ef6f08bdae3313025aacd3747945bc98acda46fcd95092f3afa3040239fsemantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED