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Question 33840: Stationarity, Metric Identification and Finite Differences Have Different Responsibilities

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

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← Question derivation index · Original question batch

R0066-33840 · q-20261015-pulse-20261119-r06-0090

Original question and givens

Use mode=non_euclidean_metric, complete model={"coordinate_gradient":[15,1],"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":[15,1],"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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001

S002 · implies

Before

source roles = [metric ORIGINAL covector raw coordinate0, 100000000, 15, 1]

After

derived = [[15, 1], 15]

Conditions: none

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

Step identifiers and dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S001

S003 · calculation

Before

metric ORIGINAL covector raw coordinate0 ORIGINAL numerator over denominator := div(15, 1)

After

metric ORIGINAL covector raw coordinate0 ORIGINAL numerator over denominator = 15

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S003

S005 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [metric ORIGINAL covector, [15, 1]]

After

derived = [[15, 1]]

Conditions: none

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

Step identifiers and dependencies

Rule: R66_VECTOR_ALL_original_order_typed_nonempty · Uses: G001, S005

S007 · implies

Before

source roles = [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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S006

S008 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S008

S010 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

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 dependencies

Rule: R66_VECTOR_ALL_original_order_typed_nonempty · Uses: G001, S010

S012 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S011

S013 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S013

S015 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

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 dependencies

Rule: R66_VECTOR_ALL_original_order_typed_nonempty · Uses: G001, S015

S017 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S016

S018 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S019

S021 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S022

S024 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S025

S027 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S028

S030 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S031

S033 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S034

S036 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

source roles = [Metric inverse times covector row0 product0 SOURCE naive mul, 100000000, 15, 2]

After

derived = [[15, 2], 15/2]

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S037

S039 · calculation

Before

Metric inverse times covector row0 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(15, 2)

After

Metric inverse times covector row0 product0 SOURCE naive mul ORIGINAL numerator over denominator = 15/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

Metric inverse times covector row0 product0 ORIGINAL rational mul := mul(1/2, 15)

After

Metric inverse times covector row0 product0 ORIGINAL rational mul = 15/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · implies

Before

source roles = [Metric inverse times covector row0 prefix0 SOURCE naive sum, 100000000, 15, 2]

After

derived = [[15, 2], 15/2]

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S040

S042 · calculation

Before

Metric inverse times covector row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(15, 2)

After

Metric inverse times covector row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 15/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

Metric inverse times covector row0 prefix0 ORIGINAL rational sum := sum(0, 15/2)

After

Metric inverse times covector row0 prefix0 ORIGINAL rational sum = 15/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S043

S045 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · implies

Before

source roles = [Metric inverse times covector row0 prefix1 SOURCE naive sum, 100000000, 15, 2]

After

derived = [[15, 2], 15/2]

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S046

S048 · calculation

Before

Metric inverse times covector row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(15, 2)

After

Metric inverse times covector row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 15/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

Metric inverse times covector row0 prefix1 ORIGINAL rational sum := sum(15/2, 0)

After

Metric inverse times covector row0 prefix1 ORIGINAL rational sum = 15/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · implies

Before

source roles = [Metric inverse times covector row0, [1/2, 0], [15, 1]]

After

derived = [15/2]

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 dependencies

Rule: R66_DOT_complete_ordered_terms_and_source_prefixes · Uses: G001, S049

S051 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S050

S052 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

Metric inverse times covector row1 product0 ORIGINAL rational mul := mul(0, 15)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S053

S055 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S056

S058 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · implies

Before

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 dependencies

Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S059

S061 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · implies

Before

source roles = [Metric inverse times covector row1, [0, 1/3], [15, 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 dependencies

Rule: R66_DOT_complete_ordered_terms_and_source_prefixes · Uses: G001, S062

S064 · implies

Before

source roles = [[15, 1], [[2, 0], [0, 3]], 6]

After

derived = [[[1/2, 0], [0, 1/3]], [15/2, 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 dependencies

Rule: 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, [15, 1], [[2, 0], [0, 3]], [15/2, 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: ea75c0e8c41692866298505a17ffc8e5273ca83cdcc742d939896adf0e4acf64
  • candidate record SHA-256: c20a0dafb4c848b0a1ac668790543bd4961ad531b30041f88a569473caf7737c
  • semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
  • question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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