Question 27895: A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace

Original question and givens

Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[-2, 3], and point x=[2, 2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).

• G001 — Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[-2, 3], and point x=[2, 2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).

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.

• mko-gradient-descent

Step-by-step derivation

source roles = [gd-safe-quadratic-trace]

derived = [[largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope]]

GD ORIGINAL ORIGINAL gradient d*x coordinate0 := mul(2, 2)

GD ORIGINAL ORIGINAL gradient d*x coordinate0 = 4

GD ORIGINAL ORIGINAL gradient plus b coordinate0 := sum(4, -2)

GD ORIGINAL ORIGINAL gradient plus b coordinate0 = 2

GD ORIGINAL ORIGINAL gradient d*x coordinate1 := mul(5, 2)

GD ORIGINAL ORIGINAL gradient d*x coordinate1 = 10

GD ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(10, 3)

GD ORIGINAL ORIGINAL gradient plus b coordinate1 = 13

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(2, 2)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 4

GD ORIGINAL ORIGINAL diagonal times square coordinate0 := mul(2, 4)

GD ORIGINAL ORIGINAL diagonal times square coordinate0 = 8

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(8, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 4

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

GD ORIGINAL ORIGINAL linear product coordinate0 = -4

GD ORIGINAL ORIGINAL coordinate objective sum0 := sum(4, -4)

GD ORIGINAL ORIGINAL coordinate objective sum0 = 0

GD ORIGINAL ORIGINAL objective running sum0 := sum(0, 0)

GD ORIGINAL ORIGINAL objective running sum0 = 0

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 := mul(2, 2)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 = 4

GD ORIGINAL ORIGINAL diagonal times square coordinate1 := mul(5, 4)

GD ORIGINAL ORIGINAL diagonal times square coordinate1 = 20

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 := div(20, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 10

GD ORIGINAL ORIGINAL linear product coordinate1 := mul(3, 2)

GD ORIGINAL ORIGINAL linear product coordinate1 = 6

GD ORIGINAL ORIGINAL coordinate objective sum1 := sum(10, 6)

GD ORIGINAL ORIGINAL coordinate objective sum1 = 16

GD ORIGINAL ORIGINAL objective running sum1 := sum(0, 16)

GD ORIGINAL ORIGINAL objective running sum1 = 16

source roles = [GD ORIGINAL, [2, 5], [-2, 3], [2, 2]]

derived = [[2, 5], [-2, 3], [2, 2], [2, 13], 16]

GD safe ORIGINAL2 over largest diagonal := div(2, 5)

GD safe ORIGINAL2 over largest diagonal = 2/5

GD safe ORIGINAL step1 ORIGINAL alpha times gradient0 := mul(1/4, 2)

GD safe ORIGINAL step1 ORIGINAL alpha times gradient0 = 1/2

GD safe ORIGINAL step1 ORIGINAL next coordinate0 := sub(2, 1/2)

GD safe ORIGINAL step1 ORIGINAL next coordinate0 = 3/2

GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 := mul(1/4, 13)

GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 = 13/4

GD safe ORIGINAL step1 ORIGINAL next coordinate1 := sub(2, 13/4)

GD safe ORIGINAL step1 ORIGINAL next coordinate1 = -5/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 := mul(2, 3/2)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 = 3

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 := sum(3, -2)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 = 1

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1 := mul(5, -5/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1 = -25/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1 := sum(-25/4, 3)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1 = -13/4

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(3/2, 3/2)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 9/4

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 := mul(2, 9/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 = 9/2

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 := div(9/2, 2)

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 = 9/4

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 := mul(-2, 3/2)

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 = -3

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 := sum(9/4, -3)

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 = -3/4

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 := sum(0, -3/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 = -3/4

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 := mul(-5/4, -5/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 = 25/16

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1 := mul(5, 25/16)

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1 = 125/16

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 := div(125/16, 2)

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 = 125/32

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 := mul(3, -5/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 = -15/4

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 := sum(125/32, -15/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 = 5/32

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 := sum(-3/4, 5/32)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 = -19/32

source roles = [GD safe ORIGINAL step1 next recomputation, [[2, 1], [5, 1]], [[-2, 1], [3, 1]], [[3, 2], [-5, 4]]]

derived = [[2, 5], [-2, 3], [3/2, -5/4], [1, -13/4], -19/32]

source roles = [[2, 5], 1/4, 5, 2/5, 1]

derived = [f(x-alpha*g)-f(x)=-alpha*sum_i(1-alpha*d_i/2)*g_i^2, [[0, [2, 2], 16, [2, 13]], [1, [3/2, -5/4], -19/32, [1, -13/4]]], Yes, No]

Reverse checks

• S041 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 ORIGINAL next coordinate1 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 ORIGINAL next coordinate0 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute GD safe ORIGINAL step1 ORIGINAL alpha times gradient0 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute GD safe ORIGINAL2 over largest diagonal 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 ORIGINAL ORIGINAL objective running sum1 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL coordinate objective sum1 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL linear product coordinate1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL diagonal times square coordinate1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL objective running sum0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL coordinate objective sum0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL linear product coordinate0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL diagonal times square coordinate0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL gradient plus b coordinate1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL gradient d*x coordinate1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL gradient plus b coordinate0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute GD 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

(5, 1/4, 2/5, 1, [[0, [2, 2], 16, [2, 13]], [1, [3/2, -5/4], -19/32, [1, -13/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

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.