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Question 33909: A Symmetric Quadratic Has Gradient Difference Q Times Point Difference

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0066-33909 · q-20261015-pulse-20261119-r06-0059

Original question and givens

Use symmetric square Q=[[9, 1], [1, 5]], x=[8, 1], y=[1, -8], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).

  • G001 — Use symmetric square Q=[[9, 1], [1, 5]], x=[8, 1], y=[1, -8], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-mod50-graddeep-quadratic-difference, [domain, operation, symmetric_matrix, x, y]]

After

derived = [graddeep-quadratic-difference, Exact_gradient_vector_density]

Conditions: none

THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged.

Step identifiers and dependencies

Rule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001

S002 · implies

Before

source roles = [quadratic Q ORIGINAL row0 raw coordinate0, 100000000, 9, 1]

After

derived = [[9, 1], 9]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S001

S003 · calculation

Before

quadratic Q ORIGINAL row0 raw coordinate0 ORIGINAL numerator over denominator := div(9, 1)

After

quadratic Q ORIGINAL row0 raw coordinate0 ORIGINAL numerator over denominator = 9

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [quadratic Q ORIGINAL row0 raw coordinate1, 100000000, 1, 1]

After

derived = [[1, 1], 1]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S003

S005 · calculation

Before

quadratic Q ORIGINAL row0 raw coordinate1 ORIGINAL numerator over denominator := div(1, 1)

After

quadratic Q ORIGINAL row0 raw coordinate1 ORIGINAL numerator over denominator = 1

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [quadratic Q ORIGINAL row0, [9, 1]]

After

derived = [[9, 1]]

Conditions: none

Validate EVERY raw vector component,nonemptiness and required length BEFORE use.Keep ORIGINAL coordinate order,including cells not ultimately selected.

Step identifiers and dependencies

Rule: R66_VECTOR_ALL_original_order_typed_nonempty · Uses: G001, S005

S007 · implies

Before

source roles = [quadratic Q ORIGINAL row1 raw coordinate0, 100000000, 1, 1]

After

derived = [[1, 1], 1]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S006

S008 · calculation

Before

quadratic Q ORIGINAL row1 raw coordinate0 ORIGINAL numerator over denominator := div(1, 1)

After

quadratic Q ORIGINAL row1 raw coordinate0 ORIGINAL numerator over denominator = 1

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

source roles = [quadratic Q ORIGINAL row1 raw coordinate1, 100000000, 5, 1]

After

derived = [[5, 1], 5]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S008

S010 · calculation

Before

quadratic Q ORIGINAL row1 raw coordinate1 ORIGINAL numerator over denominator := div(5, 1)

After

quadratic Q ORIGINAL row1 raw coordinate1 ORIGINAL numerator over denominator = 5

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

source roles = [quadratic Q ORIGINAL row1, [1, 5]]

After

derived = [[1, 5]]

Conditions: none

Validate EVERY raw vector component,nonemptiness and required length BEFORE use.Keep ORIGINAL coordinate order,including cells not ultimately selected.

Step identifiers and dependencies

Rule: R66_VECTOR_ALL_original_order_typed_nonempty · Uses: G001, S010

S012 · implies

Before

source roles = [quadratic x ORIGINAL raw coordinate0, 100000000, 8, 1]

After

derived = [[8, 1], 8]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S011

S013 · calculation

Before

quadratic x ORIGINAL raw coordinate0 ORIGINAL numerator over denominator := div(8, 1)

After

quadratic x ORIGINAL raw coordinate0 ORIGINAL numerator over denominator = 8

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [quadratic x ORIGINAL raw coordinate1, 100000000, 1, 1]

After

derived = [[1, 1], 1]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S013

S015 · calculation

Before

quadratic x ORIGINAL raw coordinate1 ORIGINAL numerator over denominator := div(1, 1)

After

quadratic x ORIGINAL raw coordinate1 ORIGINAL numerator over denominator = 1

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

source roles = [quadratic x ORIGINAL, [8, 1]]

After

derived = [[8, 1]]

Conditions: none

Validate EVERY raw vector component,nonemptiness and required length BEFORE use.Keep ORIGINAL coordinate order,including cells not ultimately selected.

Step identifiers and dependencies

Rule: R66_VECTOR_ALL_original_order_typed_nonempty · Uses: G001, S015

S017 · implies

Before

source roles = [quadratic y ORIGINAL raw coordinate0, 100000000, 1, 1]

After

derived = [[1, 1], 1]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S016

S018 · calculation

Before

quadratic y ORIGINAL raw coordinate0 ORIGINAL numerator over denominator := div(1, 1)

After

quadratic y ORIGINAL raw coordinate0 ORIGINAL numerator over denominator = 1

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

source roles = [quadratic y ORIGINAL raw coordinate1, 100000000, -8, 1]

After

derived = [[-8, 1], -8]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S018

S020 · calculation

Before

quadratic y ORIGINAL raw coordinate1 ORIGINAL numerator over denominator := div(-8, 1)

After

quadratic y ORIGINAL raw coordinate1 ORIGINAL numerator over denominator = -8

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

source roles = [quadratic y ORIGINAL, [1, -8]]

After

derived = [[1, -8]]

Conditions: none

Validate EVERY raw vector component,nonemptiness and required length BEFORE use.Keep ORIGINAL coordinate order,including cells not ultimately selected.

Step identifiers and dependencies

Rule: R66_VECTOR_ALL_original_order_typed_nonempty · Uses: G001, S020

S022 · implies

Before

source roles = [Quadratic point difference coordinate0 SOURCE naive sub, 100000000, 7, 1]

After

derived = [[7, 1], 7]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S021

S023 · calculation

Before

Quadratic point difference coordinate0 SOURCE naive sub ORIGINAL numerator over denominator := div(7, 1)

After

Quadratic point difference coordinate0 SOURCE naive sub ORIGINAL numerator over denominator = 7

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

Quadratic point difference coordinate0 ORIGINAL rational sub := sub(8, 1)

After

Quadratic point difference coordinate0 ORIGINAL rational sub = 7

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · implies

Before

source roles = [Quadratic point difference coordinate1 SOURCE naive sub, 100000000, 9, 1]

After

derived = [[9, 1], 9]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S024

S026 · calculation

Before

Quadratic point difference coordinate1 SOURCE naive sub ORIGINAL numerator over denominator := div(9, 1)

After

Quadratic point difference coordinate1 SOURCE naive sub ORIGINAL numerator over denominator = 9

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

Quadratic point difference coordinate1 ORIGINAL rational sub := sub(1, -8)

After

Quadratic point difference coordinate1 ORIGINAL rational sub = 9

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

source roles = [Quadratic Q times point difference row0 product0 SOURCE naive mul, 100000000, 63, 1]

After

derived = [[63, 1], 63]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S027

S029 · calculation

Before

Quadratic Q times point difference row0 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(63, 1)

After

Quadratic Q times point difference row0 product0 SOURCE naive mul ORIGINAL numerator over denominator = 63

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

Quadratic Q times point difference row0 product0 ORIGINAL rational mul := mul(9, 7)

After

Quadratic Q times point difference row0 product0 ORIGINAL rational mul = 63

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · implies

Before

source roles = [Quadratic Q times point difference row0 prefix0 SOURCE naive sum, 100000000, 63, 1]

After

derived = [[63, 1], 63]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S030

S032 · calculation

Before

Quadratic Q times point difference row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(63, 1)

After

Quadratic Q times point difference row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 63

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Quadratic Q times point difference row0 prefix0 ORIGINAL rational sum := sum(0, 63)

After

Quadratic Q times point difference row0 prefix0 ORIGINAL rational sum = 63

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · implies

Before

source roles = [Quadratic Q times point difference row0 product1 SOURCE naive mul, 100000000, 9, 1]

After

derived = [[9, 1], 9]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S033

S035 · calculation

Before

Quadratic Q times point difference row0 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(9, 1)

After

Quadratic Q times point difference row0 product1 SOURCE naive mul ORIGINAL numerator over denominator = 9

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

Quadratic Q times point difference row0 product1 ORIGINAL rational mul := mul(1, 9)

After

Quadratic Q times point difference row0 product1 ORIGINAL rational mul = 9

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · implies

Before

source roles = [Quadratic Q times point difference row0 prefix1 SOURCE naive sum, 100000000, 72, 1]

After

derived = [[72, 1], 72]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S036

S038 · calculation

Before

Quadratic Q times point difference row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(72, 1)

After

Quadratic Q times point difference row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 72

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

Quadratic Q times point difference row0 prefix1 ORIGINAL rational sum := sum(63, 9)

After

Quadratic Q times point difference row0 prefix1 ORIGINAL rational sum = 72

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · implies

Before

source roles = [Quadratic Q times point difference row0, [9, 1], [7, 9]]

After

derived = [72]

Conditions: none

Multiply EVERY aligned pair,then accumulate from0 in SOURCE order.Check every pre-reduction intermediate;the final value alone or a reordered sum cannot bypass a failing prefix.

Step identifiers and dependencies

Rule: R66_DOT_complete_ordered_terms_and_source_prefixes · Uses: G001, S039

S041 · implies

Before

source roles = [Quadratic Q times point difference row1 product0 SOURCE naive mul, 100000000, 7, 1]

After

derived = [[7, 1], 7]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S040

S042 · calculation

Before

Quadratic Q times point difference row1 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(7, 1)

After

Quadratic Q times point difference row1 product0 SOURCE naive mul ORIGINAL numerator over denominator = 7

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

Quadratic Q times point difference row1 product0 ORIGINAL rational mul := mul(1, 7)

After

Quadratic Q times point difference row1 product0 ORIGINAL rational mul = 7

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · implies

Before

source roles = [Quadratic Q times point difference row1 prefix0 SOURCE naive sum, 100000000, 7, 1]

After

derived = [[7, 1], 7]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S043

S045 · calculation

Before

Quadratic Q times point difference row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(7, 1)

After

Quadratic Q times point difference row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 7

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

Quadratic Q times point difference row1 prefix0 ORIGINAL rational sum := sum(0, 7)

After

Quadratic Q times point difference row1 prefix0 ORIGINAL rational sum = 7

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · implies

Before

source roles = [Quadratic Q times point difference row1 product1 SOURCE naive mul, 100000000, 45, 1]

After

derived = [[45, 1], 45]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S046

S048 · calculation

Before

Quadratic Q times point difference row1 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(45, 1)

After

Quadratic Q times point difference row1 product1 SOURCE naive mul ORIGINAL numerator over denominator = 45

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

Quadratic Q times point difference row1 product1 ORIGINAL rational mul := mul(5, 9)

After

Quadratic Q times point difference row1 product1 ORIGINAL rational mul = 45

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · implies

Before

source roles = [Quadratic Q times point difference row1 prefix1 SOURCE naive sum, 100000000, 52, 1]

After

derived = [[52, 1], 52]

Conditions: none

Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S049

S051 · calculation

Before

Quadratic Q times point difference row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(52, 1)

After

Quadratic Q times point difference row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 52

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

Quadratic Q times point difference row1 prefix1 ORIGINAL rational sum := sum(7, 45)

After

Quadratic Q times point difference row1 prefix1 ORIGINAL rational sum = 52

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · implies

Before

source roles = [Quadratic Q times point difference row1, [1, 5], [7, 9]]

After

derived = [52]

Conditions: none

Multiply EVERY aligned pair,then accumulate from0 in SOURCE order.Check every pre-reduction intermediate;the final value alone or a reordered sum cannot bypass a failing prefix.

Step identifiers and dependencies

Rule: R66_DOT_complete_ordered_terms_and_source_prefixes · Uses: G001, S052

S054 · implies

Before

source roles = [[[9, 1], [1, 5]], [8, 1], [1, -8]]

After

derived = [[7, 9], [72, 52]]

Conditions: none

For f(u)=1/2*u^T Q u,differentiation gives1/2*(Q+Q^T)u;SYMMETRY makes it Qu,so the gradient difference is Q(x−y).Qualify the COMPLETE symmetric square matrix,without inventing positive definiteness,convexity,minimality or convergence.

Step identifiers and dependencies

Rule: Gradient_SYMMETRIC_quadratic_full_differential_identity · Uses: G001, S053

Reverse checks

  • S054 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S053 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S052 · premise_and_result_audit — Recompute Quadratic Q times point difference row1 prefix1 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals.
  • S051 · premise_and_result_audit — Recompute Quadratic Q times point difference row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S050 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S049 · premise_and_result_audit — Recompute Quadratic Q times point difference row1 product1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S048 · premise_and_result_audit — Recompute Quadratic Q times point difference row1 product1 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S047 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S046 · premise_and_result_audit — Recompute Quadratic Q times point difference row1 prefix0 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals.
  • S045 · premise_and_result_audit — Recompute Quadratic Q times point difference row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S044 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S043 · premise_and_result_audit — Recompute Quadratic Q times point difference row1 product0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S042 · premise_and_result_audit — Recompute Quadratic Q times point difference row1 product0 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S041 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S040 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S039 · premise_and_result_audit — Recompute Quadratic Q times point difference row0 prefix1 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals.
  • S038 · premise_and_result_audit — Recompute Quadratic Q times point difference row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S037 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S036 · premise_and_result_audit — Recompute Quadratic Q times point difference row0 product1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S035 · premise_and_result_audit — Recompute Quadratic Q times point difference row0 product1 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S034 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S033 · premise_and_result_audit — Recompute Quadratic Q times point difference row0 prefix0 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute Quadratic Q times point difference row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S030 · premise_and_result_audit — Recompute Quadratic Q times point difference row0 product0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute Quadratic Q times point difference row0 product0 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S027 · premise_and_result_audit — Recompute Quadratic point difference coordinate1 ORIGINAL rational sub with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute Quadratic point difference coordinate1 SOURCE naive sub ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S024 · premise_and_result_audit — Recompute Quadratic point difference coordinate0 ORIGINAL rational sub with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute Quadratic point difference coordinate0 SOURCE naive sub ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S021 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S020 · premise_and_result_audit — Recompute quadratic y ORIGINAL raw coordinate1 ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S018 · premise_and_result_audit — Recompute quadratic y ORIGINAL raw coordinate0 ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S016 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S015 · premise_and_result_audit — Recompute quadratic x ORIGINAL raw coordinate1 ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S013 · premise_and_result_audit — Recompute quadratic x ORIGINAL raw coordinate0 ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S011 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S010 · premise_and_result_audit — Recompute quadratic Q ORIGINAL row1 raw coordinate1 ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S008 · premise_and_result_audit — Recompute quadratic Q ORIGINAL row1 raw coordinate0 ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S006 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S005 · premise_and_result_audit — Recompute quadratic Q ORIGINAL row0 raw coordinate1 ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S003 · premise_and_result_audit — Recompute quadratic Q ORIGINAL row0 raw coordinate0 ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S001 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.

Result

([[9, 1], [1, 5]], [8, 1], [1, -8], [7, 9], [72, 52], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: ec3d27a98293010e8a10902b5222adef4c6caddf5284fcabc4efaffcdae5aaba
  • candidate record SHA-256: 17e3b45302dc48234d8d8c887452ea72c9c5ed3d97c015cefb56144511eb47ec
  • semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
  • question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON