# Question 33844: 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-r06-0094) 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. R0066-33844 · q-20261015-pulse-20261119-r06-0094 ## Original question and givens Use mode=non_euclidean_metric, complete model={"coordinate_gradient":[19,1],"metric":[[2,0],[0,3]]}. Give (mode, coordinate_partials_covector, positive_definite_metric, metric_gradient, metric_independence_claim, boundary). - G001 — Use mode=non_euclidean_metric, complete model={"coordinate_gradient":[19,1],"metric":[[2,0],[0,3]]}. Give (mode, coordinate_partials_covector, positive_definite_metric, metric_gradient, metric_independence_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 = [metric ORIGINAL covector raw coordinate0, 100000000, 19, 1] After derived = [[19, 1], 19] 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 metric ORIGINAL covector raw coordinate0 ORIGINAL numerator over denominator := div(19, 1) After metric ORIGINAL covector raw coordinate0 ORIGINAL numerator over denominator = 19 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 = [metric ORIGINAL covector 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 metric ORIGINAL covector raw coordinate1 ORIGINAL numerator over denominator := div(1, 1) After metric ORIGINAL covector 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 = [metric ORIGINAL covector, [19, 1]] After derived = [[19, 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 = [metric ORIGINAL matrix row0 raw coordinate0, 100000000, 2, 1] After derived = [[2, 1], 2] 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 metric ORIGINAL matrix row0 raw coordinate0 ORIGINAL numerator over denominator := div(2, 1) After metric ORIGINAL matrix row0 raw coordinate0 ORIGINAL numerator over denominator = 2 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 · impliesBefore source roles = [metric ORIGINAL matrix row0 raw coordinate1, 100000000, 0, 1] After derived = [[0, 1], 0] 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, S008 S010 · calculationBefore metric ORIGINAL matrix row0 raw coordinate1 ORIGINAL numerator over denominator := div(0, 1) After metric ORIGINAL matrix row0 raw coordinate1 ORIGINAL numerator over denominator = 0 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, S009 S011 · impliesBefore source roles = [metric ORIGINAL matrix row0, [2, 0]] After derived = [[2, 0]] 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, S010 S012 · impliesBefore source roles = [metric ORIGINAL matrix row1 raw coordinate0, 100000000, 0, 1] After derived = [[0, 1], 0] 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, S011 S013 · calculationBefore metric ORIGINAL matrix row1 raw coordinate0 ORIGINAL numerator over denominator := div(0, 1) After metric ORIGINAL matrix row1 raw coordinate0 ORIGINAL numerator over denominator = 0 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, S012 S014 · impliesBefore source roles = [metric ORIGINAL matrix row1 raw coordinate1, 100000000, 3, 1] After derived = [[3, 1], 3] 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, S013 S015 · calculationBefore metric ORIGINAL matrix row1 raw coordinate1 ORIGINAL numerator over denominator := div(3, 1) After metric ORIGINAL matrix row1 raw coordinate1 ORIGINAL numerator over denominator = 3 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 = [metric ORIGINAL matrix row1, [0, 3]] After derived = [[0, 3]] 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, S015 S017 · impliesBefore source roles = [Metric determinant main SOURCE naive mul, 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, S016 S018 · calculationBefore Metric determinant main SOURCE naive mul ORIGINAL numerator over denominator := div(6, 1) After Metric determinant main SOURCE naive mul 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, S017 S019 · calculationBefore Metric determinant main ORIGINAL rational mul := mul(2, 3) After Metric determinant main ORIGINAL rational mul = 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, S018 S020 · impliesBefore source roles = [Metric determinant cross SOURCE naive mul, 100000000, 0, 1] After derived = [[0, 1], 0] 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, S019 S021 · calculationBefore Metric determinant cross SOURCE naive mul ORIGINAL numerator over denominator := div(0, 1) After Metric determinant cross SOURCE naive mul ORIGINAL numerator over denominator = 0 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 · calculationBefore Metric determinant cross ORIGINAL rational mul := mul(0, 0) After Metric determinant cross ORIGINAL rational mul = 0 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, S021 S023 · impliesBefore source roles = [Metric determinant signed SOURCE naive sub, 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, S022 S024 · calculationBefore Metric determinant signed SOURCE naive sub ORIGINAL numerator over denominator := div(6, 1) After Metric determinant signed SOURCE naive sub 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, S023 S025 · calculationBefore Metric determinant signed ORIGINAL rational sub := sub(6, 0) After Metric determinant signed ORIGINAL rational sub = 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, S024 S026 · impliesBefore source roles = [Metric inverse00 SOURCE naive div, 100000000, 3, 6] After derived = [[1, 2], 1/2] 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 Metric inverse00 SOURCE naive div ORIGINAL numerator over denominator := div(3, 6) After Metric inverse00 SOURCE naive div ORIGINAL numerator over denominator = 1/2 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 Metric inverse00 ORIGINAL rational div := div(3, 6) After Metric inverse00 ORIGINAL rational div = 1/2 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 = [Metric inverse01 SOURCE naive div, 100000000, 0, 6] After derived = [[0, 1], 0] 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 Metric inverse01 SOURCE naive div ORIGINAL numerator over denominator := div(0, 6) After Metric inverse01 SOURCE naive div ORIGINAL numerator over denominator = 0 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 Metric inverse01 ORIGINAL rational div := div(0, 6) After Metric inverse01 ORIGINAL rational div = 0 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 = [Metric inverse10 SOURCE naive div, 100000000, 0, 6] After derived = [[0, 1], 0] 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 Metric inverse10 SOURCE naive div ORIGINAL numerator over denominator := div(0, 6) After Metric inverse10 SOURCE naive div ORIGINAL numerator over denominator = 0 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 Metric inverse10 ORIGINAL rational div := div(0, 6) After Metric inverse10 ORIGINAL rational div = 0 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 = [Metric inverse11 SOURCE naive div, 100000000, 2, 6] After derived = [[1, 3], 1/3] 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 Metric inverse11 SOURCE naive div ORIGINAL numerator over denominator := div(2, 6) After Metric inverse11 SOURCE naive div ORIGINAL numerator over denominator = 1/3 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 Metric inverse11 ORIGINAL rational div := div(2, 6) After Metric inverse11 ORIGINAL rational div = 1/3 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 = [Metric inverse times covector row0 product0 SOURCE naive mul, 100000000, 19, 2] After derived = [[19, 2], 19/2] 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, S037 S039 · calculationBefore Metric inverse times covector row0 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(19, 2) After Metric inverse times covector row0 product0 SOURCE naive mul ORIGINAL numerator over denominator = 19/2 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, S038 S040 · calculationBefore Metric inverse times covector row0 product0 ORIGINAL rational mul := mul(1/2, 19) After Metric inverse times covector row0 product0 ORIGINAL rational mul = 19/2 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, S039 S041 · impliesBefore source roles = [Metric inverse times covector row0 prefix0 SOURCE naive sum, 100000000, 19, 2] After derived = [[19, 2], 19/2] 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, S040 S042 · calculationBefore Metric inverse times covector row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(19, 2) After Metric inverse times covector row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 19/2 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, S041 S043 · calculationBefore Metric inverse times covector row0 prefix0 ORIGINAL rational sum := sum(0, 19/2) After Metric inverse times covector row0 prefix0 ORIGINAL rational sum = 19/2 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, S042 S044 · impliesBefore source roles = [Metric inverse times covector row0 product1 SOURCE naive mul, 100000000, 0, 1] After derived = [[0, 1], 0] 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, S043 S045 · calculationBefore Metric inverse times covector row0 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(0, 1) After Metric inverse times covector row0 product1 SOURCE naive mul ORIGINAL numerator over denominator = 0 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, S044 S046 · calculationBefore Metric inverse times covector row0 product1 ORIGINAL rational mul := mul(0, 1) After Metric inverse times covector row0 product1 ORIGINAL rational mul = 0 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, S045 S047 · impliesBefore source roles = [Metric inverse times covector row0 prefix1 SOURCE naive sum, 100000000, 19, 2] After derived = [[19, 2], 19/2] 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, S046 S048 · calculationBefore Metric inverse times covector row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(19, 2) After Metric inverse times covector row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 19/2 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, S047 S049 · calculationBefore Metric inverse times covector row0 prefix1 ORIGINAL rational sum := sum(19/2, 0) After Metric inverse times covector row0 prefix1 ORIGINAL rational sum = 19/2 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, S048 S050 · impliesBefore source roles = [Metric inverse times covector row0, [1/2, 0], [19, 1]] After derived = [19/2] 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, S049 S051 · impliesBefore source roles = [Metric inverse times covector row1 product0 SOURCE naive mul, 100000000, 0, 1] After derived = [[0, 1], 0] 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, S050 S052 · calculationBefore Metric inverse times covector row1 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(0, 1) After Metric inverse times covector row1 product0 SOURCE naive mul ORIGINAL numerator over denominator = 0 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, S051 S053 · calculationBefore Metric inverse times covector row1 product0 ORIGINAL rational mul := mul(0, 19) After Metric inverse times covector row1 product0 ORIGINAL rational mul = 0 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, S052 S054 · impliesBefore source roles = [Metric inverse times covector row1 prefix0 SOURCE naive sum, 100000000, 0, 1] After derived = [[0, 1], 0] 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, S053 S055 · calculationBefore Metric inverse times covector row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(0, 1) After Metric inverse times covector row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 0 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, S054 S056 · calculationBefore Metric inverse times covector row1 prefix0 ORIGINAL rational sum := sum(0, 0) After Metric inverse times covector row1 prefix0 ORIGINAL rational sum = 0 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, S055 S057 · impliesBefore source roles = [Metric inverse times covector row1 product1 SOURCE naive mul, 100000000, 1, 3] After derived = [[1, 3], 1/3] 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, S056 S058 · calculationBefore Metric inverse times covector row1 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(1, 3) After Metric inverse times covector row1 product1 SOURCE naive mul ORIGINAL numerator over denominator = 1/3 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, S057 S059 · calculationBefore Metric inverse times covector row1 product1 ORIGINAL rational mul := mul(1/3, 1) After Metric inverse times covector row1 product1 ORIGINAL rational mul = 1/3 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, S058 S060 · impliesBefore source roles = [Metric inverse times covector row1 prefix1 SOURCE naive sum, 100000000, 1, 3] After derived = [[1, 3], 1/3] 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, S059 S061 · calculationBefore Metric inverse times covector row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(1, 3) After Metric inverse times covector row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 1/3 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, S060 S062 · calculationBefore Metric inverse times covector row1 prefix1 ORIGINAL rational sum := sum(0, 1/3) After Metric inverse times covector row1 prefix1 ORIGINAL rational sum = 1/3 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, S061 S063 · impliesBefore source roles = [Metric inverse times covector row1, [0, 1/3], [19, 1]] After derived = [1/3] 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, S062 S064 · impliesBefore source roles = [[19, 1], [[2, 0], [0, 3]], 6] After derived = [[[1/2, 0], [0, 1/3]], [19/2, 1/3], No] Conditions: none Qualify symmetric2×2 M with M00>0 and determinant>0(Sylvester criterion),then identify the gradient by M*grad=coordinate-partial covector.EVERY inverse cell passes the source envelope,even when multiplied by0.This discriminating witness requires a different vector;Euclidean identification is not metric-independent. Step identifiers and dependenciesRule: Gradient_METRIC_positive_definite_identification_ALL_inverse_cells · Uses: G001, S063 ## Reverse checks - S064 · 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. - S063 · 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. - S062 · premise_and_result_audit — Recompute Metric inverse times covector row1 prefix1 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute Metric inverse times covector row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - S060 · 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. - S059 · premise_and_result_audit — Recompute Metric inverse times covector row1 product1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute Metric inverse times covector row1 product1 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - S057 · 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. - S056 · premise_and_result_audit — Recompute Metric inverse times covector row1 prefix0 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute Metric inverse times covector row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - 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 — Recompute Metric inverse times covector row1 product0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute Metric inverse times covector row1 product0 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - S051 · 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. - 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 Metric inverse times covector row0 prefix1 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Metric inverse times covector row0 prefix1 SOURCE naive sum 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 Metric inverse times covector row0 product1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Metric inverse times covector row0 product1 SOURCE naive mul 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 Metric inverse times covector row0 prefix0 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Metric inverse times covector row0 prefix0 SOURCE naive sum 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 — Recompute Metric inverse times covector row0 product0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Metric inverse times covector row0 product0 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - 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 Metric inverse11 ORIGINAL rational div with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Metric inverse11 SOURCE naive div 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 Metric inverse10 ORIGINAL rational div with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Metric inverse10 SOURCE naive div 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 Metric inverse01 ORIGINAL rational div with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Metric inverse01 SOURCE naive div 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 Metric inverse00 ORIGINAL rational div with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Metric inverse00 SOURCE naive div 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 — Recompute Metric determinant signed ORIGINAL rational sub with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Metric determinant signed SOURCE naive sub ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - S023 · 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. - S022 · premise_and_result_audit — Recompute Metric determinant cross ORIGINAL rational mul with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Metric determinant cross SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - 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 — Recompute Metric determinant main ORIGINAL rational mul with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Metric determinant main SOURCE naive mul 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 metric ORIGINAL matrix row1 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 metric ORIGINAL matrix row1 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 metric ORIGINAL matrix row0 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 metric ORIGINAL matrix row0 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 metric ORIGINAL covector 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 metric ORIGINAL covector 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 (non_euclidean_metric, [19, 1], [[2, 0], [0, 3]], [19/2, 1/3], No, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification) 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: 0e231fc1a87bf8ca7c6309416d15e933031956ac5c494858b6e3a1c08208c1b2 - candidate record SHA-256: 09b6874158d34a97c2ae91cca10df0b1312484e34c5c80978ee70e087d45d35d - 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-r06-0094.json)