# Question 33831: Stationarity, Metric Identification and Finite Differences Have Different Responsibilities [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-r06-0081) 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. R0066-33831 · q-20261015-pulse-20261119-r06-0081 ## Original question and givens Use mode=positive_gradient_direction, complete model={"gradient":[6,-1]}. Give (mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary). - G001 — Use mode=positive_gradient_direction, complete model={"gradient":[6,-1]}. Give (mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary). ## 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-boundary, [domain, mode, model, operation]] After derived = [graddeep-boundary, 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 = [positive direction ORIGINAL gradient raw coordinate0, 100000000, 6, 1] After derived = [[6, 1], 6] 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 positive direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator := div(6, 1) After positive direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator = 6 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 = [positive direction ORIGINAL gradient 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S003 S005 · calculationBefore positive direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator := div(-1, 1) After positive direction ORIGINAL gradient 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [positive direction ORIGINAL gradient, [6, -1]] After derived = [[6, -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 dependenciesRule: R66_VECTOR_ALL_original_order_typed_nonempty · Uses: G001, S005 S007 · impliesBefore source roles = [positive gradient inner product product0 SOURCE naive mul, 100000000, 36, 1] After derived = [[36, 1], 36] 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 positive gradient inner product product0 SOURCE naive mul ORIGINAL numerator over denominator := div(36, 1) After positive gradient inner product product0 SOURCE naive mul ORIGINAL numerator over denominator = 36 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 positive gradient inner product product0 ORIGINAL rational mul := mul(6, 6) After positive gradient inner product product0 ORIGINAL rational mul = 36 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 = [positive gradient inner product prefix0 SOURCE naive sum, 100000000, 36, 1] After derived = [[36, 1], 36] 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 positive gradient inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(36, 1) After positive gradient inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 36 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 positive gradient inner product prefix0 ORIGINAL rational sum := sum(0, 36) After positive gradient inner product prefix0 ORIGINAL rational sum = 36 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 = [positive gradient inner product product1 SOURCE naive mul, 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 dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S012 S014 · calculationBefore positive gradient inner product product1 SOURCE naive mul ORIGINAL numerator over denominator := div(1, 1) After positive gradient inner product product1 SOURCE naive mul 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore positive gradient inner product product1 ORIGINAL rational mul := mul(-1, -1) After positive gradient inner product product1 ORIGINAL rational mul = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore source roles = [positive gradient inner product prefix1 SOURCE naive sum, 100000000, 37, 1] After derived = [[37, 1], 37] 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 positive gradient inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(37, 1) After positive gradient inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 37 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 positive gradient inner product prefix1 ORIGINAL rational sum := sum(36, 1) After positive gradient inner product prefix1 ORIGINAL rational sum = 37 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 = [positive gradient inner product, [6, -1], [6, -1]] After derived = [37] 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, S018 S020 · impliesBefore source roles = [[6, -1]] After derived = [37, Yes, No] Conditions: none For nonzero g,the directional derivative along+g is g·g>0:first-order ASCENT,not descent.This is not an arbitrary finite-distance value guarantee for an unspecified function. Step identifiers and dependenciesRule: Gradient_POSITIVE_gradient_first_order_ASCENT_not_descent · Uses: G001, S019 ## Reverse checks - S020 · 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. - 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 positive gradient inner product prefix1 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute positive gradient inner product prefix1 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 positive gradient inner product product1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute positive gradient inner product product1 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 positive gradient inner product prefix0 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute positive gradient inner product prefix0 SOURCE naive sum 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 positive gradient inner product product0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute positive gradient inner product product0 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 positive 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 positive 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 (positive_gradient_direction, [6, -1], [6, -1], 37, Yes, No, positive_gradient_is_first_order_ascent_not_descent) 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: 0d2d72927c779cb10fdfaa5f8f9553bce6aaf23b2578112fffc25068c7008dc2 - candidate record SHA-256: 4337125409149b7a6f6888124e427dff67c413d787cf0f6028de174fd99f6855 - 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-r06-0081.json)