Question 27876: 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=[1, 2], linear b=[-1, -2], and point x=[-2, -3] 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=[1, 2], linear b=[-1, -2], and point x=[-2, -3] 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(1, -2)

GD ORIGINAL ORIGINAL gradient d*x coordinate0 = -2

GD ORIGINAL ORIGINAL gradient plus b coordinate0 := sum(-2, -1)

GD ORIGINAL ORIGINAL gradient plus b coordinate0 = -3

GD ORIGINAL ORIGINAL gradient d*x coordinate1 := mul(2, -3)

GD ORIGINAL ORIGINAL gradient d*x coordinate1 = -6

GD ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(-6, -2)

GD ORIGINAL ORIGINAL gradient plus b coordinate1 = -8

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(1, 4)

GD ORIGINAL ORIGINAL diagonal times square coordinate0 = 4

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(4, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 2

GD ORIGINAL ORIGINAL linear product coordinate0 := mul(-1, -2)

GD ORIGINAL ORIGINAL linear product coordinate0 = 2

GD ORIGINAL ORIGINAL coordinate objective sum0 := sum(2, 2)

GD ORIGINAL ORIGINAL coordinate objective sum0 = 4

GD ORIGINAL ORIGINAL objective running sum0 := sum(0, 4)

GD ORIGINAL ORIGINAL objective running sum0 = 4

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 := mul(-3, -3)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 = 9

GD ORIGINAL ORIGINAL diagonal times square coordinate1 := mul(2, 9)

GD ORIGINAL ORIGINAL diagonal times square coordinate1 = 18

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 := div(18, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 9

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

GD ORIGINAL ORIGINAL linear product coordinate1 = 6

GD ORIGINAL ORIGINAL coordinate objective sum1 := sum(9, 6)

GD ORIGINAL ORIGINAL coordinate objective sum1 = 15

GD ORIGINAL ORIGINAL objective running sum1 := sum(4, 15)

GD ORIGINAL ORIGINAL objective running sum1 = 19

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

derived = [[1, 2], [-1, -2], [-2, -3], [-3, -8], 19]

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

GD safe ORIGINAL2 over largest diagonal = 1

GD safe ORIGINAL step1 ORIGINAL alpha times gradient0 := mul(1/4, -3)

GD safe ORIGINAL step1 ORIGINAL alpha times gradient0 = -3/4

GD safe ORIGINAL step1 ORIGINAL next coordinate0 := sub(-2, -3/4)

GD safe ORIGINAL step1 ORIGINAL next coordinate0 = -5/4

GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 := mul(1/4, -8)

GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 = -2

GD safe ORIGINAL step1 ORIGINAL next coordinate1 := sub(-3, -2)

GD safe ORIGINAL step1 ORIGINAL next coordinate1 = -1

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 := mul(1, -5/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 = -5/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 := sum(-5/4, -1)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 = -9/4

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

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

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

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

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(-5/4, -5/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 25/16

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 := mul(1, 25/16)

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 = 25/16

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 := div(25/16, 2)

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 = 25/32

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 := mul(-1, -5/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 = 5/4

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 := sum(25/32, 5/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 = 65/32

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 := sum(0, 65/32)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 = 65/32

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

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

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

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

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

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 = 1

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

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 = 2

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

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 = 3

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 := sum(65/32, 3)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 = 161/32

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

derived = [[1, 2], [-1, -2], [-5/4, -1], [-9/4, -4], 161/32]

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

derived = [f(x-alpha*g)-f(x)=-alpha*sum_i(1-alpha*d_i/2)*g_i^2, [[0, [-2, -3], 19, [-3, -8]], [1, [-5/4, -1], 161/32, [-9/4, -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

(2, 1/4, 1, 1, [[0, [-2, -3], 19, [-3, -8]], [1, [-5/4, -1], 161/32, [-9/4, -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.