Question 27878: 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=[3, 4], linear b=[1, 0], and point x=[0, -1] 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=[3, 4], linear b=[1, 0], and point x=[0, -1] 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(3, 0)

GD ORIGINAL ORIGINAL gradient d*x coordinate0 = 0

GD ORIGINAL ORIGINAL gradient plus b coordinate0 := sum(0, 1)

GD ORIGINAL ORIGINAL gradient plus b coordinate0 = 1

GD ORIGINAL ORIGINAL gradient d*x coordinate1 := mul(4, -1)

GD ORIGINAL ORIGINAL gradient d*x coordinate1 = -4

GD ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(-4, 0)

GD ORIGINAL ORIGINAL gradient plus b coordinate1 = -4

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(0, 0)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 0

GD ORIGINAL ORIGINAL diagonal times square coordinate0 := mul(3, 0)

GD ORIGINAL ORIGINAL diagonal times square coordinate0 = 0

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(0, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 0

GD ORIGINAL ORIGINAL linear product coordinate0 := mul(1, 0)

GD ORIGINAL ORIGINAL linear product coordinate0 = 0

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

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(-1, -1)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 = 1

GD ORIGINAL ORIGINAL diagonal times square coordinate1 := mul(4, 1)

GD ORIGINAL ORIGINAL diagonal times square coordinate1 = 4

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 := div(4, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 2

GD ORIGINAL ORIGINAL linear product coordinate1 := mul(0, -1)

GD ORIGINAL ORIGINAL linear product coordinate1 = 0

GD ORIGINAL ORIGINAL coordinate objective sum1 := sum(2, 0)

GD ORIGINAL ORIGINAL coordinate objective sum1 = 2

GD ORIGINAL ORIGINAL objective running sum1 := sum(0, 2)

GD ORIGINAL ORIGINAL objective running sum1 = 2

source roles = [GD ORIGINAL, [3, 4], [1, 0], [0, -1]]

derived = [[3, 4], [1, 0], [0, -1], [1, -4], 2]

GD safe ORIGINAL2 over largest diagonal := div(2, 4)

GD safe ORIGINAL2 over largest diagonal = 1/2

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

GD safe ORIGINAL step1 ORIGINAL alpha times gradient0 = 1/4

GD safe ORIGINAL step1 ORIGINAL next coordinate0 := sub(0, 1/4)

GD safe ORIGINAL step1 ORIGINAL next coordinate0 = -1/4

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

GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 = -1

GD safe ORIGINAL step1 ORIGINAL next coordinate1 := sub(-1, -1)

GD safe ORIGINAL step1 ORIGINAL next coordinate1 = 0

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 := mul(3, -1/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 = -3/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 := sum(-3/4, 1)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 = 1/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1 := mul(4, 0)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1 = 0

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1 := sum(0, 0)

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

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(-1/4, -1/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 1/16

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 := mul(3, 1/16)

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 = 3/16

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 := div(3/16, 2)

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 = 3/32

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

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 = -1/4

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 := sum(3/32, -1/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 = -5/32

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 := sum(0, -5/32)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 = -5/32

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 := mul(0, 0)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 = 0

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1 := mul(4, 0)

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1 = 0

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 := div(0, 2)

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 = 0

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 := mul(0, 0)

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 = 0

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 := sum(0, 0)

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 = 0

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 := sum(-5/32, 0)

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

source roles = [GD safe ORIGINAL step1 next recomputation, [[3, 1], [4, 1]], [[1, 1], [0, 1]], [[-1, 4], [0, 1]]]

derived = [[3, 4], [1, 0], [-1/4, 0], [1/4, 0], -5/32]

source roles = [[3, 4], 1/4, 4, 1/2, 1]

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

(4, 1/4, 1/2, 1, [[0, [0, -1], 2, [1, -4]], [1, [-1/4, 0], -5/32, [1/4, 0]]], 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.