# Question 27884: A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace [Return to catalogue page 140](https://ocme.evemisslab.com/en/questions/page/140/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261119-r03-gradient-descent-foundations/#q-20261015-pulse-20261119-r03-0059) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0054-27884 · q-20261015-pulse-20261119-r03-0059 ## 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, 2], linear b=[2, -1], and point x=[1, -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=[3, 2], linear b=[2, -1], and point x=[1, -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](https://ocme.evemisslab.com/en/mko/mko-gradient-descent/) ## Step-by-step derivation S001 · impliesBefore source roles = [gd-safe-quadratic-trace] After derived = [[largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope]] Conditions: none List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore GD ORIGINAL ORIGINAL gradient d*x coordinate0 := mul(3, 1) After GD ORIGINAL ORIGINAL gradient d*x coordinate0 = 3 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore GD ORIGINAL ORIGINAL gradient plus b coordinate0 := sum(3, 2) After GD ORIGINAL ORIGINAL gradient plus b coordinate0 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore GD ORIGINAL ORIGINAL gradient d*x coordinate1 := mul(2, -2) After GD ORIGINAL ORIGINAL gradient d*x coordinate1 = -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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore GD ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(-4, -1) After GD ORIGINAL ORIGINAL gradient plus b 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(1, 1) After GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore GD ORIGINAL ORIGINAL diagonal times square coordinate0 := mul(3, 1) After GD ORIGINAL ORIGINAL diagonal times square coordinate0 = 3 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(3, 2) After GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 3/2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore GD ORIGINAL ORIGINAL linear product coordinate0 := mul(2, 1) After GD ORIGINAL ORIGINAL linear product coordinate0 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore GD ORIGINAL ORIGINAL coordinate objective sum0 := sum(3/2, 2) After GD ORIGINAL ORIGINAL coordinate objective sum0 = 7/2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore GD ORIGINAL ORIGINAL objective running sum0 := sum(0, 7/2) After GD ORIGINAL ORIGINAL objective running sum0 = 7/2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 := mul(-2, -2) After GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore GD ORIGINAL ORIGINAL diagonal times square coordinate1 := mul(2, 4) After GD ORIGINAL ORIGINAL diagonal times square 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 := div(8, 2) After GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore GD ORIGINAL ORIGINAL linear product coordinate1 := mul(-1, -2) After GD ORIGINAL ORIGINAL linear product 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore GD ORIGINAL ORIGINAL coordinate objective sum1 := sum(4, 2) After GD ORIGINAL ORIGINAL coordinate objective sum1 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore GD ORIGINAL ORIGINAL objective running sum1 := sum(7/2, 6) After GD ORIGINAL ORIGINAL objective running sum1 = 19/2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore source roles = [GD ORIGINAL, [3, 2], [2, -1], [1, -2]] After derived = [[3, 2], [2, -1], [1, -2], [5, -5], 19/2] 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. Step identifiers and dependenciesRule: GD_WHOLE_positive_diagonal_quadratic_ordered_gradient_and_objective · Uses: G001, S017 S019 · calculationBefore GD safe ORIGINAL2 over largest diagonal := div(2, 3) After GD safe ORIGINAL2 over largest diagonal = 2/3 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore GD safe ORIGINAL step1 ORIGINAL alpha times gradient0 := mul(1/4, 5) After GD safe ORIGINAL step1 ORIGINAL alpha times gradient0 = 5/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore GD safe ORIGINAL step1 ORIGINAL next coordinate0 := sub(1, 5/4) After GD safe ORIGINAL step1 ORIGINAL next coordinate0 = -1/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 := mul(1/4, -5) After GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 = -5/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore GD safe ORIGINAL step1 ORIGINAL next coordinate1 := sub(-2, -5/4) After GD safe ORIGINAL step1 ORIGINAL next coordinate1 = -3/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 := mul(3, -1/4) After GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 = -3/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 := sum(-3/4, 2) After GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 = 5/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1 := mul(2, -3/4) After GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1 = -3/2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1 := sum(-3/2, -1) After GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1 = -5/2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(-1/4, -1/4) After GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 1/16 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 := mul(3, 1/16) After GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 = 3/16 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 := div(3/16, 2) After GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 = 3/32 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 := mul(2, -1/4) After GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 = -1/2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 := sum(3/32, -1/2) After GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 = -13/32 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 := sum(0, -13/32) After GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 = -13/32 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 := mul(-3/4, -3/4) After GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 = 9/16 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1 := mul(2, 9/16) After GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1 = 9/8 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 := div(9/8, 2) After GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 = 9/16 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 := mul(-1, -3/4) After GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 = 3/4 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 := sum(9/16, 3/4) After GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 = 21/16 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 := sum(-13/32, 21/16) After GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 = 29/32 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · impliesBefore source roles = [GD safe ORIGINAL step1 next recomputation, [[3, 1], [2, 1]], [[2, 1], [-1, 1]], [[-1, 4], [-3, 4]]] After derived = [[3, 2], [2, -1], [-1/4, -3/4], [5/4, -5/2], 29/32] 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. Step identifiers and dependenciesRule: GD_WHOLE_positive_diagonal_quadratic_ordered_gradient_and_objective · Uses: G001, S039 S041 · impliesBefore source roles = [[3, 2], 1/4, 3, 2/3, 1] After derived = [f(x-alpha*g)-f(x)=-alpha*sum_i(1-alpha*d_i/2)*g_i^2, [[0, [1, -2], 19/2, [5, -5]], [1, [-1/4, -3/4], 29/32, [5/4, -5/2]]], Yes, No] Conditions: none Expand the WHOLE quadratic to obtain the difference identity.With0