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Question 33896: A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product

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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← Question derivation index · Original question batch

R0066-33896 · q-20261015-pulse-20261119-r06-0046

Original question and givens

Use nonzero Euclidean gradient=[21, -21] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).

  • G001 — Use nonzero Euclidean gradient=[21, -21] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-mod50-graddeep-negative-direction, [domain, gradient, operation]]

After

derived = [graddeep-negative-direction, 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 = [negative direction ORIGINAL gradient raw coordinate0, 100000000, 21, 1]

After

derived = [[21, 1], 21]

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

negative direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator := div(21, 1)

After

negative direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator = 21

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 = [negative direction ORIGINAL gradient raw coordinate1, 100000000, -21, 1]

After

derived = [[-21, 1], -21]

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

negative direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator := div(-21, 1)

After

negative direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator = -21

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 = [negative direction ORIGINAL gradient, [21, -21]]

After

derived = [[21, -21]]

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 = [Negative gradient coordinate0 SOURCE naive mul, 100000000, -21, 1]

After

derived = [[-21, 1], -21]

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

Negative gradient coordinate0 SOURCE naive mul ORIGINAL numerator over denominator := div(-21, 1)

After

Negative gradient coordinate0 SOURCE naive mul ORIGINAL numerator over denominator = -21

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 · calculation

Before

Negative gradient coordinate0 ORIGINAL rational mul := mul(-1, 21)

After

Negative gradient coordinate0 ORIGINAL rational mul = -21

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, S008

S010 · implies

Before

source roles = [Negative gradient coordinate1 SOURCE naive mul, 100000000, 21, 1]

After

derived = [[21, 1], 21]

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, S009

S011 · calculation

Before

Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator := div(21, 1)

After

Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator = 21

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, S010

S012 · calculation

Before

Negative gradient coordinate1 ORIGINAL rational mul := mul(-1, -21)

After

Negative gradient coordinate1 ORIGINAL rational mul = 21

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, S011

S013 · implies

Before

source roles = [negative direction inner product product0 SOURCE naive mul, 100000000, -441, 1]

After

derived = [[-441, 1], -441]

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, S012

S014 · calculation

Before

negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator := div(-441, 1)

After

negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator = -441

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, S013

S015 · calculation

Before

negative direction inner product product0 ORIGINAL rational mul := mul(21, -21)

After

negative direction inner product product0 ORIGINAL rational mul = -441

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 = [negative direction inner product prefix0 SOURCE naive sum, 100000000, -441, 1]

After

derived = [[-441, 1], -441]

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, S015

S017 · calculation

Before

negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(-441, 1)

After

negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator = -441

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, S016

S018 · calculation

Before

negative direction inner product prefix0 ORIGINAL rational sum := sum(0, -441)

After

negative direction inner product prefix0 ORIGINAL rational sum = -441

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 = [negative direction inner product product1 SOURCE naive mul, 100000000, -441, 1]

After

derived = [[-441, 1], -441]

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

negative direction inner product product1 SOURCE naive mul ORIGINAL numerator over denominator := div(-441, 1)

After

negative direction inner product product1 SOURCE naive mul ORIGINAL numerator over denominator = -441

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 · calculation

Before

negative direction inner product product1 ORIGINAL rational mul := mul(-21, 21)

After

negative direction inner product product1 ORIGINAL rational mul = -441

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, S020

S022 · implies

Before

source roles = [negative direction inner product prefix1 SOURCE naive sum, 100000000, -882, 1]

After

derived = [[-882, 1], -882]

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

negative direction inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(-882, 1)

After

negative direction inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator = -882

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

negative direction inner product prefix1 ORIGINAL rational sum := sum(-441, -441)

After

negative direction inner product prefix1 ORIGINAL rational sum = -882

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 = [negative direction inner product, [21, -21], [-21, 21]]

After

derived = [-882]

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, S024

S026 · implies

Before

source roles = [negative direction norm squared product0 SOURCE naive mul, 100000000, 441, 1]

After

derived = [[441, 1], 441]

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, S025

S027 · calculation

Before

negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator := div(441, 1)

After

negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator = 441

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 · calculation

Before

negative direction norm squared product0 ORIGINAL rational mul := mul(21, 21)

After

negative direction norm squared product0 ORIGINAL rational mul = 441

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, S027

S029 · implies

Before

source roles = [negative direction norm squared prefix0 SOURCE naive sum, 100000000, 441, 1]

After

derived = [[441, 1], 441]

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, S028

S030 · calculation

Before

negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(441, 1)

After

negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 441

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 · calculation

Before

negative direction norm squared prefix0 ORIGINAL rational sum := sum(0, 441)

After

negative direction norm squared prefix0 ORIGINAL rational sum = 441

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, S030

S032 · implies

Before

source roles = [negative direction norm squared product1 SOURCE naive mul, 100000000, 441, 1]

After

derived = [[441, 1], 441]

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, S031

S033 · calculation

Before

negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator := div(441, 1)

After

negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator = 441

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 · calculation

Before

negative direction norm squared product1 ORIGINAL rational mul := mul(-21, -21)

After

negative direction norm squared product1 ORIGINAL rational mul = 441

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, S033

S035 · implies

Before

source roles = [negative direction norm squared prefix1 SOURCE naive sum, 100000000, 882, 1]

After

derived = [[882, 1], 882]

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, S034

S036 · calculation

Before

negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(882, 1)

After

negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 882

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 · calculation

Before

negative direction norm squared prefix1 ORIGINAL rational sum := sum(441, 441)

After

negative direction norm squared prefix1 ORIGINAL rational sum = 882

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, S036

S038 · implies

Before

source roles = [negative direction norm squared, [21, -21], [21, -21]]

After

derived = [882]

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, S037

S039 · implies

Before

source roles = [[21, -21]]

After

derived = [[-21, 21], -882, 882, Yes]

Conditions: none

A NONZERO gradient has positive squared norm.The UNNORMALIZED direction−gradient gives inner product−norm²<0.Differentiability supports a first-order descent direction,not descent for every positive finite step.

Step identifiers and dependencies

Rule: Gradient_NEGATIVE_direction_NONZERO_first_order_NOT_finite_step · Uses: G001, S038

Reverse checks

  • S039 · 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.
  • S038 · 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.
  • S037 · premise_and_result_audit — Recompute negative direction norm squared prefix1 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Recompute negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S035 · 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.
  • S034 · premise_and_result_audit — Recompute negative direction norm squared product1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Recompute negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S032 · 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.
  • S031 · premise_and_result_audit — Recompute negative direction norm squared prefix0 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S029 · 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.
  • S028 · premise_and_result_audit — Recompute negative direction norm squared product0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S026 · 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.
  • 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 negative direction inner product prefix1 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute negative direction inner product prefix1 SOURCE naive sum 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 — Recompute negative direction inner product product1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute negative direction inner product product1 SOURCE naive mul 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 negative direction inner product prefix0 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • 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 negative direction inner product product0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S013 · 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.
  • S012 · premise_and_result_audit — Recompute Negative gradient coordinate1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S010 · 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.
  • S009 · premise_and_result_audit — Recompute Negative gradient coordinate0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Negative gradient coordinate0 SOURCE naive mul 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 negative direction ORIGINAL gradient 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 negative direction ORIGINAL gradient 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

([21, -21], [-21, 21], -882, 882, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

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: 6160a91e1c78c2bf29b768e8816e3c61660325534090d195be9dca1519117a79
  • candidate record SHA-256: 20a99786cd778ba9a8171a3f25bdf2bfd107d4f64d8933ea2361c1158080061d
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  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON