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R0054-27819 · q-20261015-pulse-20261119-r03-0094
Use mode=finite_trace_only, complete model={"diagonal":[1,2],"linear":[18,-18],"point":[19,1],"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":[18,-18],"point":[19,1],"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, 19)
After
GD boundary ORIGINAL ORIGINAL gradient d*x coordinate0 = 19
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(19, 18)
After
GD boundary ORIGINAL ORIGINAL gradient plus b coordinate0 = 37
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, -18)
After
GD boundary ORIGINAL ORIGINAL gradient plus b coordinate1 = -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, S004
Before
GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(19, 19)
After
GD boundary ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 361
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, 361)
After
GD boundary ORIGINAL ORIGINAL diagonal times square coordinate0 = 361
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(361, 2)
After
GD boundary ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 361/2
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(18, 19)
After
GD boundary ORIGINAL ORIGINAL linear product coordinate0 = 342
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(361/2, 342)
After
GD boundary ORIGINAL ORIGINAL coordinate objective sum0 = 1045/2
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, 1045/2)
After
GD boundary ORIGINAL ORIGINAL objective running sum0 = 1045/2
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(-18, 1)
After
GD boundary ORIGINAL ORIGINAL linear product coordinate1 = -18
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, -18)
After
GD boundary ORIGINAL ORIGINAL coordinate objective sum1 = -17
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(1045/2, -17)
After
GD boundary ORIGINAL ORIGINAL objective running sum1 = 1011/2
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], [18, -18], [19, 1]]
After
derived = [[1, 2], [18, -18], [19, 1], [37, -16], 1011/2]
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, 37)
After
GD boundary step1 ORIGINAL alpha times gradient0 = 37/4
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(19, 37/4)
After
GD boundary step1 ORIGINAL next coordinate0 = 39/4
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, -16)
After
GD boundary step1 ORIGINAL alpha times gradient1 = -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, S020
Before
GD boundary step1 ORIGINAL next coordinate1 := sub(1, -4)
After
GD boundary step1 ORIGINAL next coordinate1 = 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, S021
Before
GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate0 := mul(1, 39/4)
After
GD boundary step1 next recomputation ORIGINAL gradient d*x coordinate0 = 39/4
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(39/4, 18)
After
GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate0 = 111/4
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, 5)
After
GD boundary step1 next recomputation ORIGINAL gradient d*x 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, S024
Before
GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate1 := sum(10, -18)
After
GD boundary step1 next recomputation ORIGINAL gradient plus b coordinate1 = -8
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(39/4, 39/4)
After
GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 1521/16
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, 1521/16)
After
GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate0 = 1521/16
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(1521/16, 2)
After
GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate0 = 1521/32
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(18, 39/4)
After
GD boundary step1 next recomputation ORIGINAL linear product coordinate0 = 351/2
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(1521/32, 351/2)
After
GD boundary step1 next recomputation ORIGINAL coordinate objective sum0 = 7137/32
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, 7137/32)
After
GD boundary step1 next recomputation ORIGINAL objective running sum0 = 7137/32
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(5, 5)
After
GD boundary step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 = 25
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, 25)
After
GD boundary step1 next recomputation ORIGINAL diagonal times square coordinate1 = 50
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(50, 2)
After
GD boundary step1 next recomputation ORIGINAL quadratic divide2 coordinate1 = 25
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(-18, 5)
After
GD boundary step1 next recomputation ORIGINAL linear product coordinate1 = -90
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(25, -90)
After
GD boundary step1 next recomputation ORIGINAL coordinate objective sum1 = -65
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(7137/32, -65)
After
GD boundary step1 next recomputation ORIGINAL objective running sum1 = 5057/32
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]], [[18, 1], [-18, 1]], [[39, 4], [5, 1]]]
After
derived = [[1, 2], [18, -18], [39/4, 5], [111/4, -8], 5057/32]
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, [19, 1], 1011/2], [1, [39/4, 5], 5057/32]], 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, [19, 1], 1011/2], [1, [39/4, 5], 5057/32]], 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: 3e58d2e96b579ade8cf75dddc921e9cf2f04344f7cd748bab9756d4942a81873candidate record SHA-256: 66b5a1fa3467f31d638ea1def4edca2f2d839bad7e3ac9a943d8cf86aa3ea581semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED