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Question 33862: Gradient Linearity Preserves the Fixed Coordinate Order

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

R0066-33862 · q-20261015-pulse-20261119-r06-0012

Original question and givens

Use gradients of differentiable f and g at the same point: grad_f=[12, 1], grad_g=[4, -12], alpha=1/2, beta=4/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).

  • G001 — Use gradients of differentiable f and g at the same point: grad_f=[12, 1], grad_g=[4, -12], alpha=1/2, beta=4/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-mod50-graddeep-linearity, [alpha, beta, domain, gradient_f, gradient_g, operation]]

After

derived = [graddeep-linearity, 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 = [gradient f ORIGINAL raw coordinate0, 100000000, 12, 1]

After

derived = [[12, 1], 12]

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

gradient f ORIGINAL raw coordinate0 ORIGINAL numerator over denominator := div(12, 1)

After

gradient f ORIGINAL raw coordinate0 ORIGINAL numerator over denominator = 12

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 = [gradient f ORIGINAL 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

gradient f ORIGINAL raw coordinate1 ORIGINAL numerator over denominator := div(1, 1)

After

gradient f ORIGINAL 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 = [gradient f ORIGINAL, [12, 1]]

After

derived = [[12, 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 = [gradient g ORIGINAL raw coordinate0, 100000000, 4, 1]

After

derived = [[4, 1], 4]

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

gradient g ORIGINAL raw coordinate0 ORIGINAL numerator over denominator := div(4, 1)

After

gradient g ORIGINAL raw coordinate0 ORIGINAL numerator over denominator = 4

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 = [gradient g ORIGINAL raw coordinate1, 100000000, -12, 1]

After

derived = [[-12, 1], -12]

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

gradient g ORIGINAL raw coordinate1 ORIGINAL numerator over denominator := div(-12, 1)

After

gradient g ORIGINAL raw coordinate1 ORIGINAL numerator over denominator = -12

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 = [gradient g ORIGINAL, [4, -12]]

After

derived = [[4, -12]]

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 = [alpha ORIGINAL, 100000000, 1, 2]

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, S011

S013 · calculation

Before

alpha ORIGINAL ORIGINAL numerator over denominator := div(1, 2)

After

alpha ORIGINAL 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, S012

S014 · implies

Before

source roles = [beta ORIGINAL, 100000000, 4, 3]

After

derived = [[4, 3], 4/3]

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

beta ORIGINAL ORIGINAL numerator over denominator := div(4, 3)

After

beta ORIGINAL ORIGINAL numerator over denominator = 4/3

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 = [Gradient linearity alpha f coordinate0 SOURCE naive mul, 100000000, 12, 2]

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, S015

S017 · calculation

Before

Gradient linearity alpha f coordinate0 SOURCE naive mul ORIGINAL numerator over denominator := div(12, 2)

After

Gradient linearity alpha f coordinate0 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, S016

S018 · calculation

Before

Gradient linearity alpha f coordinate0 ORIGINAL rational mul := mul(1/2, 12)

After

Gradient linearity alpha f coordinate0 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, S017

S019 · implies

Before

source roles = [Gradient linearity beta g coordinate0 SOURCE naive mul, 100000000, 16, 3]

After

derived = [[16, 3], 16/3]

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, S018

S020 · calculation

Before

Gradient linearity beta g coordinate0 SOURCE naive mul ORIGINAL numerator over denominator := div(16, 3)

After

Gradient linearity beta g coordinate0 SOURCE naive mul ORIGINAL numerator over denominator = 16/3

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, S019

S021 · calculation

Before

Gradient linearity beta g coordinate0 ORIGINAL rational mul := mul(4/3, 4)

After

Gradient linearity beta g coordinate0 ORIGINAL rational mul = 16/3

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, S020

S022 · implies

Before

source roles = [Gradient linearity sum coordinate0 SOURCE naive sum, 100000000, 34, 3]

After

derived = [[34, 3], 34/3]

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, S021

S023 · calculation

Before

Gradient linearity sum coordinate0 SOURCE naive sum ORIGINAL numerator over denominator := div(34, 3)

After

Gradient linearity sum coordinate0 SOURCE naive sum ORIGINAL numerator over denominator = 34/3

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, S022

S024 · calculation

Before

Gradient linearity sum coordinate0 ORIGINAL rational sum := sum(6, 16/3)

After

Gradient linearity sum coordinate0 ORIGINAL rational sum = 34/3

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, S023

S025 · implies

Before

source roles = [Gradient linearity alpha f coordinate1 SOURCE naive mul, 100000000, 1, 2]

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, S024

S026 · calculation

Before

Gradient linearity alpha f coordinate1 SOURCE naive mul ORIGINAL numerator over denominator := div(1, 2)

After

Gradient linearity alpha f coordinate1 SOURCE naive mul 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, S025

S027 · calculation

Before

Gradient linearity alpha f coordinate1 ORIGINAL rational mul := mul(1/2, 1)

After

Gradient linearity alpha f coordinate1 ORIGINAL rational mul = 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 · implies

Before

source roles = [Gradient linearity beta g coordinate1 SOURCE naive mul, 100000000, -48, 3]

After

derived = [[-16, 1], -16]

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, S027

S029 · calculation

Before

Gradient linearity beta g coordinate1 SOURCE naive mul ORIGINAL numerator over denominator := div(-48, 3)

After

Gradient linearity beta g coordinate1 SOURCE naive mul ORIGINAL numerator over denominator = -16

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, S028

S030 · calculation

Before

Gradient linearity beta g coordinate1 ORIGINAL rational mul := mul(4/3, -12)

After

Gradient linearity beta g coordinate1 ORIGINAL rational mul = -16

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 · implies

Before

source roles = [Gradient linearity sum coordinate1 SOURCE naive sum, 100000000, -31, 2]

After

derived = [[-31, 2], -31/2]

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, S030

S032 · calculation

Before

Gradient linearity sum coordinate1 SOURCE naive sum ORIGINAL numerator over denominator := div(-31, 2)

After

Gradient linearity sum coordinate1 SOURCE naive sum ORIGINAL numerator over denominator = -31/2

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, S031

S033 · calculation

Before

Gradient linearity sum coordinate1 ORIGINAL rational sum := sum(1/2, -16)

After

Gradient linearity sum coordinate1 ORIGINAL rational sum = -31/2

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, S032

S034 · implies

Before

source roles = [[12, 1], [4, -12], 1/2, 4/3]

After

derived = [[34/3, -31/2]]

Conditions: none

Premises give f/g gradients at the SAME differentiable point in a fixed Euclidean basis.Combine EACH coordinate linearly;the supplied derivatives are not certified from finite differences.The identity does not establish extrema or convergence.

Step identifiers and dependencies

Rule: Gradient_LINEARITY_same_point_differentiability_fixed_coordinates · Uses: G001, S033

Reverse checks

  • S034 · 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.
  • S033 · premise_and_result_audit — Recompute Gradient linearity sum coordinate1 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute Gradient linearity sum coordinate1 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S031 · 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.
  • S030 · premise_and_result_audit — Recompute Gradient linearity beta g coordinate1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute Gradient linearity beta g coordinate1 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S028 · 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.
  • S027 · premise_and_result_audit — Recompute Gradient linearity alpha f coordinate1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute Gradient linearity alpha f coordinate1 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S024 · premise_and_result_audit — Recompute Gradient linearity sum coordinate0 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute Gradient linearity sum coordinate0 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S021 · premise_and_result_audit — Recompute Gradient linearity beta g coordinate0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute Gradient linearity beta g coordinate0 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S018 · premise_and_result_audit — Recompute Gradient linearity alpha f coordinate0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Gradient linearity alpha f coordinate0 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
  • S015 · premise_and_result_audit — Recompute beta ORIGINAL 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 alpha ORIGINAL 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 gradient g ORIGINAL 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 gradient g ORIGINAL 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 gradient f ORIGINAL 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 gradient f ORIGINAL 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

([12, 1], [4, -12], 1/2, 4/3, [34/3, -31/2], pointwise_gradient_linearity_in_fixed_coordinates)

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: 892f4c11b116078b7b92d9c65d94f288ff97d86059916ca0520620f014561d6a
  • candidate record SHA-256: 8048a8fe6c3a330216e42329b11e2950629809cce6ed2cbbab7834bfd848a0e0
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  • Lean: NOT_RUN · human review: NOT_PERFORMED
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