# Question 33897: A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product [Return to catalogue page 170](https://ocme.evemisslab.com/en/questions/page/170/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261119-r06-gradient-vector-foundations/#q-20261015-pulse-20261119-r06-0047) 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. R0066-33897 · q-20261015-pulse-20261119-r06-0047 ## Original question and givens Use nonzero Euclidean gradient=[22, -22] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope). - G001 — Use nonzero Euclidean gradient=[22, -22] 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. - [mko-gradient-vector](https://ocme.evemisslab.com/en/mko/mko-gradient-vector/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001 S002 · impliesBefore source roles = [negative direction ORIGINAL gradient raw coordinate0, 100000000, 22, 1] After derived = [[22, 1], 22] 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S001 S003 · calculationBefore negative direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator := div(22, 1) After negative direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator = 22 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [negative direction ORIGINAL gradient raw coordinate1, 100000000, -22, 1] After derived = [[-22, 1], -22] 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S003 S005 · calculationBefore negative direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator := div(-22, 1) After negative direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator = -22 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [negative direction ORIGINAL gradient, [22, -22]] After derived = [[22, -22]] 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 dependenciesRule: R66_VECTOR_ALL_original_order_typed_nonempty · Uses: G001, S005 S007 · impliesBefore source roles = [Negative gradient coordinate0 SOURCE naive mul, 100000000, -22, 1] After derived = [[-22, 1], -22] 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S006 S008 · calculationBefore Negative gradient coordinate0 SOURCE naive mul ORIGINAL numerator over denominator := div(-22, 1) After Negative gradient coordinate0 SOURCE naive mul ORIGINAL numerator over denominator = -22 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Negative gradient coordinate0 ORIGINAL rational mul := mul(-1, 22) After Negative gradient coordinate0 ORIGINAL rational mul = -22 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [Negative gradient coordinate1 SOURCE naive mul, 100000000, 22, 1] After derived = [[22, 1], 22] 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S009 S011 · calculationBefore Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator := div(22, 1) After Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator = 22 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Negative gradient coordinate1 ORIGINAL rational mul := mul(-1, -22) After Negative gradient coordinate1 ORIGINAL rational mul = 22 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore source roles = [negative direction inner product product0 SOURCE naive mul, 100000000, -484, 1] After derived = [[-484, 1], -484] 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S012 S014 · calculationBefore negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator := div(-484, 1) After negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator = -484 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore negative direction inner product product0 ORIGINAL rational mul := mul(22, -22) After negative direction inner product product0 ORIGINAL rational mul = -484 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore source roles = [negative direction inner product prefix0 SOURCE naive sum, 100000000, -484, 1] After derived = [[-484, 1], -484] 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S015 S017 · calculationBefore negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(-484, 1) After negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator = -484 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore negative direction inner product prefix0 ORIGINAL rational sum := sum(0, -484) After negative direction inner product prefix0 ORIGINAL rational sum = -484 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore source roles = [negative direction inner product product1 SOURCE naive mul, 100000000, -484, 1] After derived = [[-484, 1], -484] 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S018 S020 · calculationBefore negative direction inner product product1 SOURCE naive mul ORIGINAL numerator over denominator := div(-484, 1) After negative direction inner product product1 SOURCE naive mul ORIGINAL numerator over denominator = -484 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore negative direction inner product product1 ORIGINAL rational mul := mul(-22, 22) After negative direction inner product product1 ORIGINAL rational mul = -484 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · impliesBefore source roles = [negative direction inner product prefix1 SOURCE naive sum, 100000000, -968, 1] After derived = [[-968, 1], -968] 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S021 S023 · calculationBefore negative direction inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(-968, 1) After negative direction inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator = -968 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore negative direction inner product prefix1 ORIGINAL rational sum := sum(-484, -484) After negative direction inner product prefix1 ORIGINAL rational sum = -968 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore source roles = [negative direction inner product, [22, -22], [-22, 22]] After derived = [-968] 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 dependenciesRule: R66_DOT_complete_ordered_terms_and_source_prefixes · Uses: G001, S024 S026 · impliesBefore source roles = [negative direction norm squared product0 SOURCE naive mul, 100000000, 484, 1] After derived = [[484, 1], 484] 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S025 S027 · calculationBefore negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator := div(484, 1) After negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator = 484 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore negative direction norm squared product0 ORIGINAL rational mul := mul(22, 22) After negative direction norm squared product0 ORIGINAL rational mul = 484 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore source roles = [negative direction norm squared prefix0 SOURCE naive sum, 100000000, 484, 1] After derived = [[484, 1], 484] 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S028 S030 · calculationBefore negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(484, 1) After negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 484 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore negative direction norm squared prefix0 ORIGINAL rational sum := sum(0, 484) After negative direction norm squared prefix0 ORIGINAL rational sum = 484 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · impliesBefore source roles = [negative direction norm squared product1 SOURCE naive mul, 100000000, 484, 1] After derived = [[484, 1], 484] 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S031 S033 · calculationBefore negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator := div(484, 1) After negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator = 484 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore negative direction norm squared product1 ORIGINAL rational mul := mul(-22, -22) After negative direction norm squared product1 ORIGINAL rational mul = 484 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · impliesBefore source roles = [negative direction norm squared prefix1 SOURCE naive sum, 100000000, 968, 1] After derived = [[968, 1], 968] 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S034 S036 · calculationBefore negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(968, 1) After negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 968 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore negative direction norm squared prefix1 ORIGINAL rational sum := sum(484, 484) After negative direction norm squared prefix1 ORIGINAL rational sum = 968 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · impliesBefore source roles = [negative direction norm squared, [22, -22], [22, -22]] After derived = [968] 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 dependenciesRule: R66_DOT_complete_ordered_terms_and_source_prefixes · Uses: G001, S037 S039 · impliesBefore source roles = [[22, -22]] After derived = [[-22, 22], -968, 968, 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 dependenciesRule: 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 ([22, -22], [-22, 22], -968, 968, 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: 24703b1f4c699de63e84418905817c0ba9d6fe71efeebce61d9c979db03a7c3c - candidate record SHA-256: 204e3697bdfcc6bac38273feb8e22e9a3a4b0aa408722d344958dbf351316cc8 - semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267 - question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-pulse-20261119-r06-0047.json)