Ordered Gradient Linearity, Negative Directions and Metric Identification
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batch-20261015-advance-20261119-r06-gradient-vector-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
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Field-order and source-binding data
1. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[1, 0], grad_g=[0, -1], alpha=1/2, beta=1/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([1, 0], [0, -1], 1/2, 1/3, [1/2, -1/3], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([1, 0], [0, -1], 1/2, 1/3, [1/2, -1/3], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
2. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[2, 1], grad_g=[1, -2], alpha=1/2, beta=2/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([2, 1], [1, -2], 1/2, 2/3, [5/3, -5/6], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([2, 1], [1, -2], 1/2, 2/3, [5/3, -5/6], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
3. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[3, 2], grad_g=[2, -3], alpha=1/2, beta=3/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([3, 2], [2, -3], 1/2, 1, [7/2, -2], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([3, 2], [2, -3], 1/2, 1, [7/2, -2], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
4. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[4, 3], grad_g=[3, -4], alpha=1/2, beta=4/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([4, 3], [3, -4], 1/2, 4/3, [6, -23/6], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([4, 3], [3, -4], 1/2, 4/3, [6, -23/6], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
5. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[5, 4], grad_g=[4, -5], alpha=1/2, beta=1/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([5, 4], [4, -5], 1/2, 1/3, [23/6, 1/3], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([5, 4], [4, -5], 1/2, 1/3, [23/6, 1/3], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
6. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[6, 0], grad_g=[5, -6], alpha=1/2, beta=2/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([6, 0], [5, -6], 1/2, 2/3, [19/3, -4], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([6, 0], [5, -6], 1/2, 2/3, [19/3, -4], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
7. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[7, 1], grad_g=[6, -7], alpha=1/2, beta=3/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([7, 1], [6, -7], 1/2, 1, [19/2, -13/2], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([7, 1], [6, -7], 1/2, 1, [19/2, -13/2], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
8. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[8, 2], grad_g=[0, -8], alpha=1/2, beta=4/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([8, 2], [0, -8], 1/2, 4/3, [4, -29/3], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([8, 2], [0, -8], 1/2, 4/3, [4, -29/3], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
9. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[9, 3], grad_g=[1, -9], alpha=1/2, beta=1/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([9, 3], [1, -9], 1/2, 1/3, [29/6, -3/2], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([9, 3], [1, -9], 1/2, 1/3, [29/6, -3/2], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
10. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[10, 4], grad_g=[2, -10], alpha=1/2, beta=2/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([10, 4], [2, -10], 1/2, 2/3, [19/3, -14/3], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([10, 4], [2, -10], 1/2, 2/3, [19/3, -14/3], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
11. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[11, 0], grad_g=[3, -11], alpha=1/2, beta=3/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([11, 0], [3, -11], 1/2, 1, [17/2, -11], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([11, 0], [3, -11], 1/2, 1, [17/2, -11], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
12. Gradient Linearity Preserves the Fixed Coordinate Order
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).
View answer and explanation
([12, 1], [4, -12], 1/2, 4/3, [34/3, -31/2], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([12, 1], [4, -12], 1/2, 4/3, [34/3, -31/2], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
13. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[13, 2], grad_g=[5, -13], alpha=1/2, beta=1/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([13, 2], [5, -13], 1/2, 1/3, [49/6, -10/3], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([13, 2], [5, -13], 1/2, 1/3, [49/6, -10/3], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
14. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[14, 3], grad_g=[6, -14], alpha=1/2, beta=2/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([14, 3], [6, -14], 1/2, 2/3, [11, -47/6], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([14, 3], [6, -14], 1/2, 2/3, [11, -47/6], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
15. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[15, 4], grad_g=[0, -15], alpha=1/2, beta=3/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([15, 4], [0, -15], 1/2, 1, [15/2, -13], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([15, 4], [0, -15], 1/2, 1, [15/2, -13], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
16. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[16, 0], grad_g=[1, -16], alpha=1/2, beta=4/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([16, 0], [1, -16], 1/2, 4/3, [28/3, -64/3], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([16, 0], [1, -16], 1/2, 4/3, [28/3, -64/3], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
17. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[17, 1], grad_g=[2, -17], alpha=1/2, beta=1/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([17, 1], [2, -17], 1/2, 1/3, [55/6, -31/6], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([17, 1], [2, -17], 1/2, 1/3, [55/6, -31/6], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
18. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[18, 2], grad_g=[3, -18], alpha=1/2, beta=2/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([18, 2], [3, -18], 1/2, 2/3, [11, -11], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([18, 2], [3, -18], 1/2, 2/3, [11, -11], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
19. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[19, 3], grad_g=[4, -19], alpha=1/2, beta=3/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([19, 3], [4, -19], 1/2, 1, [27/2, -35/2], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([19, 3], [4, -19], 1/2, 1, [27/2, -35/2], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
20. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[20, 4], grad_g=[5, -20], alpha=1/2, beta=4/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([20, 4], [5, -20], 1/2, 4/3, [50/3, -74/3], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([20, 4], [5, -20], 1/2, 4/3, [50/3, -74/3], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
21. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[21, 0], grad_g=[6, -21], alpha=1/2, beta=1/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([21, 0], [6, -21], 1/2, 1/3, [25/2, -7], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([21, 0], [6, -21], 1/2, 1/3, [25/2, -7], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
22. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[22, 1], grad_g=[0, -22], alpha=1/2, beta=2/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([22, 1], [0, -22], 1/2, 2/3, [11, -85/6], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([22, 1], [0, -22], 1/2, 2/3, [11, -85/6], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
23. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[23, 2], grad_g=[1, -23], alpha=1/2, beta=3/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([23, 2], [1, -23], 1/2, 1, [25/2, -22], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([23, 2], [1, -23], 1/2, 1, [25/2, -22], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
24. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[24, 3], grad_g=[2, -24], alpha=1/2, beta=4/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([24, 3], [2, -24], 1/2, 4/3, [44/3, -61/2], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([24, 3], [2, -24], 1/2, 4/3, [44/3, -61/2], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
25. Gradient Linearity Preserves the Fixed Coordinate Order
Use gradients of differentiable f and g at the same point: grad_f=[25, 4], grad_g=[3, -25], alpha=1/2, beta=1/3. Give (gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope).
View answer and explanation
([25, 4], [3, -25], 1/2, 1/3, [27/2, -19/3], pointwise_gradient_linearity_in_fixed_coordinates)
Compute alpha*grad_f+beta*grad_g component by component in the fixed Euclidean basis. The supplied derivative vectors and differentiability premise are not inferred from a finite difference. This identity is not an extremum or convergence certificate. The exact result, retaining the requested order, is ([25, 4], [3, -25], 1/2, 1/3, [27/2, -19/3], pointwise_gradient_linearity_in_fixed_coordinates).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-linearity · candidate
26. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[1, -1] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([1, -1], [-1, 1], -2, 2, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([1, -1], [-1, 1], -2, 2, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
27. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[2, -2] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([2, -2], [-2, 2], -8, 8, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([2, -2], [-2, 2], -8, 8, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
28. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[3, -3] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([3, -3], [-3, 3], -18, 18, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([3, -3], [-3, 3], -18, 18, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
29. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[4, -4] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([4, -4], [-4, 4], -32, 32, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([4, -4], [-4, 4], -32, 32, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
30. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[5, -5] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([5, -5], [-5, 5], -50, 50, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([5, -5], [-5, 5], -50, 50, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
31. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[6, -6] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([6, -6], [-6, 6], -72, 72, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([6, -6], [-6, 6], -72, 72, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
32. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[7, -7] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([7, -7], [-7, 7], -98, 98, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([7, -7], [-7, 7], -98, 98, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
33. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[8, -8] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([8, -8], [-8, 8], -128, 128, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([8, -8], [-8, 8], -128, 128, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
34. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[9, -9] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([9, -9], [-9, 9], -162, 162, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([9, -9], [-9, 9], -162, 162, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
35. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[10, -10] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([10, -10], [-10, 10], -200, 200, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([10, -10], [-10, 10], -200, 200, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
36. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[11, -11] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([11, -11], [-11, 11], -242, 242, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([11, -11], [-11, 11], -242, 242, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
37. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[12, -12] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([12, -12], [-12, 12], -288, 288, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([12, -12], [-12, 12], -288, 288, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
38. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[13, -13] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([13, -13], [-13, 13], -338, 338, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([13, -13], [-13, 13], -338, 338, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
39. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[14, -14] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([14, -14], [-14, 14], -392, 392, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([14, -14], [-14, 14], -392, 392, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
40. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[15, -15] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([15, -15], [-15, 15], -450, 450, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([15, -15], [-15, 15], -450, 450, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
41. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[16, -16] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([16, -16], [-16, 16], -512, 512, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([16, -16], [-16, 16], -512, 512, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
42. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[17, -17] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([17, -17], [-17, 17], -578, 578, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([17, -17], [-17, 17], -578, 578, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
43. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[18, -18] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([18, -18], [-18, 18], -648, 648, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([18, -18], [-18, 18], -648, 648, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
44. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[19, -19] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([19, -19], [-19, 19], -722, 722, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([19, -19], [-19, 19], -722, 722, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
45. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[20, -20] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([20, -20], [-20, 20], -800, 800, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([20, -20], [-20, 20], -800, 800, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
46. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[21, -21] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([21, -21], [-21, 21], -882, 882, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([21, -21], [-21, 21], -882, 882, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
47. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[22, -22] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([22, -22], [-22, 22], -968, 968, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([22, -22], [-22, 22], -968, 968, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
48. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[23, -23] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([23, -23], [-23, 23], -1058, 1058, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([23, -23], [-23, 23], -1058, 1058, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
49. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[24, -24] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([24, -24], [-24, 24], -1152, 1152, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([24, -24], [-24, 24], -1152, 1152, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
50. A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product
Use nonzero Euclidean gradient=[25, -25] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).
View answer and explanation
([25, -25], [-25, 25], -1250, 1250, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)
Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([25, -25], [-25, 25], -1250, 1250, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-negative-direction · candidate
51. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[1, 1], [1, 2]], x=[0, 1], y=[1, 0], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[1, 1], [1, 2]], [0, 1], [1, 0], [-1, 1], [0, 1], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[1, 1], [1, 2]], [0, 1], [1, 0], [-1, 1], [0, 1], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
52. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[2, 1], [1, 3]], x=[1, 1], y=[1, -1], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[2, 1], [1, 3]], [1, 1], [1, -1], [0, 2], [2, 6], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[2, 1], [1, 3]], [1, 1], [1, -1], [0, 2], [2, 6], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
53. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[3, 1], [1, 4]], x=[2, 1], y=[1, -2], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[3, 1], [1, 4]], [2, 1], [1, -2], [1, 3], [6, 13], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[3, 1], [1, 4]], [2, 1], [1, -2], [1, 3], [6, 13], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
54. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[4, 1], [1, 5]], x=[3, 1], y=[1, -3], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[4, 1], [1, 5]], [3, 1], [1, -3], [2, 4], [12, 22], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[4, 1], [1, 5]], [3, 1], [1, -3], [2, 4], [12, 22], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
55. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[5, 1], [1, 6]], x=[4, 1], y=[1, -4], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[5, 1], [1, 6]], [4, 1], [1, -4], [3, 5], [20, 33], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[5, 1], [1, 6]], [4, 1], [1, -4], [3, 5], [20, 33], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
56. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[6, 1], [1, 2]], x=[5, 1], y=[1, -5], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[6, 1], [1, 2]], [5, 1], [1, -5], [4, 6], [30, 16], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[6, 1], [1, 2]], [5, 1], [1, -5], [4, 6], [30, 16], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
57. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[7, 1], [1, 3]], x=[6, 1], y=[1, -6], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[7, 1], [1, 3]], [6, 1], [1, -6], [5, 7], [42, 26], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[7, 1], [1, 3]], [6, 1], [1, -6], [5, 7], [42, 26], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
58. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[8, 1], [1, 4]], x=[7, 1], y=[1, -7], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[8, 1], [1, 4]], [7, 1], [1, -7], [6, 8], [56, 38], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[8, 1], [1, 4]], [7, 1], [1, -7], [6, 8], [56, 38], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
59. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[9, 1], [1, 5]], x=[8, 1], y=[1, -8], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[9, 1], [1, 5]], [8, 1], [1, -8], [7, 9], [72, 52], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[9, 1], [1, 5]], [8, 1], [1, -8], [7, 9], [72, 52], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
60. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[10, 1], [1, 6]], x=[9, 1], y=[1, -9], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[10, 1], [1, 6]], [9, 1], [1, -9], [8, 10], [90, 68], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[10, 1], [1, 6]], [9, 1], [1, -9], [8, 10], [90, 68], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
61. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[11, 1], [1, 2]], x=[10, 1], y=[1, -10], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[11, 1], [1, 2]], [10, 1], [1, -10], [9, 11], [110, 31], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[11, 1], [1, 2]], [10, 1], [1, -10], [9, 11], [110, 31], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
62. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[12, 1], [1, 3]], x=[11, 1], y=[1, -11], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[12, 1], [1, 3]], [11, 1], [1, -11], [10, 12], [132, 46], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[12, 1], [1, 3]], [11, 1], [1, -11], [10, 12], [132, 46], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
63. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[13, 1], [1, 4]], x=[12, 1], y=[1, -12], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[13, 1], [1, 4]], [12, 1], [1, -12], [11, 13], [156, 63], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[13, 1], [1, 4]], [12, 1], [1, -12], [11, 13], [156, 63], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
64. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[14, 1], [1, 5]], x=[13, 1], y=[1, -13], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[14, 1], [1, 5]], [13, 1], [1, -13], [12, 14], [182, 82], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[14, 1], [1, 5]], [13, 1], [1, -13], [12, 14], [182, 82], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
65. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[15, 1], [1, 6]], x=[14, 1], y=[1, -14], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[15, 1], [1, 6]], [14, 1], [1, -14], [13, 15], [210, 103], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[15, 1], [1, 6]], [14, 1], [1, -14], [13, 15], [210, 103], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
66. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[16, 1], [1, 2]], x=[15, 1], y=[1, -15], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[16, 1], [1, 2]], [15, 1], [1, -15], [14, 16], [240, 46], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[16, 1], [1, 2]], [15, 1], [1, -15], [14, 16], [240, 46], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
67. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[17, 1], [1, 3]], x=[16, 1], y=[1, -16], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[17, 1], [1, 3]], [16, 1], [1, -16], [15, 17], [272, 66], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[17, 1], [1, 3]], [16, 1], [1, -16], [15, 17], [272, 66], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
68. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[18, 1], [1, 4]], x=[17, 1], y=[1, -17], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[18, 1], [1, 4]], [17, 1], [1, -17], [16, 18], [306, 88], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[18, 1], [1, 4]], [17, 1], [1, -17], [16, 18], [306, 88], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
69. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[19, 1], [1, 5]], x=[18, 1], y=[1, -18], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[19, 1], [1, 5]], [18, 1], [1, -18], [17, 19], [342, 112], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[19, 1], [1, 5]], [18, 1], [1, -18], [17, 19], [342, 112], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
70. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[20, 1], [1, 6]], x=[19, 1], y=[1, -19], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[20, 1], [1, 6]], [19, 1], [1, -19], [18, 20], [380, 138], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[20, 1], [1, 6]], [19, 1], [1, -19], [18, 20], [380, 138], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
71. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[21, 1], [1, 2]], x=[20, 1], y=[1, -20], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[21, 1], [1, 2]], [20, 1], [1, -20], [19, 21], [420, 61], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[21, 1], [1, 2]], [20, 1], [1, -20], [19, 21], [420, 61], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
72. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[22, 1], [1, 3]], x=[21, 1], y=[1, -21], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[22, 1], [1, 3]], [21, 1], [1, -21], [20, 22], [462, 86], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[22, 1], [1, 3]], [21, 1], [1, -21], [20, 22], [462, 86], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
73. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[23, 1], [1, 4]], x=[22, 1], y=[1, -22], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[23, 1], [1, 4]], [22, 1], [1, -22], [21, 23], [506, 113], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[23, 1], [1, 4]], [22, 1], [1, -22], [21, 23], [506, 113], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
74. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[24, 1], [1, 5]], x=[23, 1], y=[1, -23], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[24, 1], [1, 5]], [23, 1], [1, -23], [22, 24], [552, 142], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[24, 1], [1, 5]], [23, 1], [1, -23], [22, 24], [552, 142], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
75. A Symmetric Quadratic Has Gradient Difference Q Times Point Difference
Use symmetric square Q=[[25, 1], [1, 6]], x=[24, 1], y=[1, -24], for f(u)=1/2*u^T*Q*u. Give (symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity).
View answer and explanation
([[25, 1], [1, 6]], [24, 1], [1, -24], [23, 25], [600, 173], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)
Retain x−y and compute Q*(x−y) in exact coordinate order. Symmetry supplies gradient Q*u. The identity does not assert positive definiteness, convexity, a minimum or convergence. The exact result, retaining the requested order, is ([[25, 1], [1, 6]], [24, 1], [1, -24], [23, 25], [600, 173], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-quadratic-difference · candidate
76. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=zero_gradient, complete model={"dimension":1}. Give (mode, dimension, zero_gradient, stationary, minimum_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(zero_gradient, 1, [0], Yes, No, zero_gradient_only_certifies_stationarity)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (zero_gradient, 1, [0], Yes, No, zero_gradient_only_certifies_stationarity).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
77. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=positive_gradient_direction, complete model={"gradient":[2,-1]}. Give (mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(positive_gradient_direction, [2, -1], [2, -1], 5, Yes, No, positive_gradient_is_first_order_ascent_not_descent)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (positive_gradient_direction, [2, -1], [2, -1], 5, Yes, No, positive_gradient_is_first_order_ascent_not_descent).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
78. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=non_euclidean_metric, complete model={"coordinate_gradient":[3,1],"metric":[[2,0],[0,3]]}. Give (mode, coordinate_partials_covector, positive_definite_metric, metric_gradient, metric_independence_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(non_euclidean_metric, [3, 1], [[2, 0], [0, 3]], [3/2, 1/3], No, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (non_euclidean_metric, [3, 1], [[2, 0], [0, 3]], [3/2, 1/3], No, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
79. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=finite_difference_confusion, complete model={"difference_quotient":4,"step":[1,5]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(finite_difference_confusion, 4, 1/5, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 4, 1/5, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
80. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=finite_difference_confusion, complete model={"difference_quotient":5,"step":[1,6]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(finite_difference_confusion, 5, 1/6, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 5, 1/6, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
81. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=positive_gradient_direction, complete model={"gradient":[6,-1]}. Give (mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(positive_gradient_direction, [6, -1], [6, -1], 37, Yes, No, positive_gradient_is_first_order_ascent_not_descent)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (positive_gradient_direction, [6, -1], [6, -1], 37, Yes, No, positive_gradient_is_first_order_ascent_not_descent).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
82. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=non_euclidean_metric, complete model={"coordinate_gradient":[7,1],"metric":[[2,0],[0,3]]}. Give (mode, coordinate_partials_covector, positive_definite_metric, metric_gradient, metric_independence_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(non_euclidean_metric, [7, 1], [[2, 0], [0, 3]], [7/2, 1/3], No, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (non_euclidean_metric, [7, 1], [[2, 0], [0, 3]], [7/2, 1/3], No, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
83. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=finite_difference_confusion, complete model={"difference_quotient":8,"step":[1,9]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(finite_difference_confusion, 8, 1/9, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 8, 1/9, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
84. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=finite_difference_confusion, complete model={"difference_quotient":9,"step":[1,10]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(finite_difference_confusion, 9, 1/10, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 9, 1/10, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
85. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=positive_gradient_direction, complete model={"gradient":[10,-1]}. Give (mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(positive_gradient_direction, [10, -1], [10, -1], 101, Yes, No, positive_gradient_is_first_order_ascent_not_descent)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (positive_gradient_direction, [10, -1], [10, -1], 101, Yes, No, positive_gradient_is_first_order_ascent_not_descent).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
86. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=non_euclidean_metric, complete model={"coordinate_gradient":[11,1],"metric":[[2,0],[0,3]]}. Give (mode, coordinate_partials_covector, positive_definite_metric, metric_gradient, metric_independence_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(non_euclidean_metric, [11, 1], [[2, 0], [0, 3]], [11/2, 1/3], No, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (non_euclidean_metric, [11, 1], [[2, 0], [0, 3]], [11/2, 1/3], No, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
87. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=finite_difference_confusion, complete model={"difference_quotient":12,"step":[1,13]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(finite_difference_confusion, 12, 1/13, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 12, 1/13, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
88. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=finite_difference_confusion, complete model={"difference_quotient":13,"step":[1,14]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(finite_difference_confusion, 13, 1/14, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 13, 1/14, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
89. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=positive_gradient_direction, complete model={"gradient":[14,-1]}. Give (mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(positive_gradient_direction, [14, -1], [14, -1], 197, Yes, No, positive_gradient_is_first_order_ascent_not_descent)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (positive_gradient_direction, [14, -1], [14, -1], 197, Yes, No, positive_gradient_is_first_order_ascent_not_descent).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
90. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
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).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(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)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (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).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
91. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=finite_difference_confusion, complete model={"difference_quotient":16,"step":[1,17]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(finite_difference_confusion, 16, 1/17, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 16, 1/17, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
92. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=finite_difference_confusion, complete model={"difference_quotient":17,"step":[1,18]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(finite_difference_confusion, 17, 1/18, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 17, 1/18, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
93. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=positive_gradient_direction, complete model={"gradient":[18,-1]}. Give (mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(positive_gradient_direction, [18, -1], [18, -1], 325, Yes, No, positive_gradient_is_first_order_ascent_not_descent)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (positive_gradient_direction, [18, -1], [18, -1], 325, Yes, No, positive_gradient_is_first_order_ascent_not_descent).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
94. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
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).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(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)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (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).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
95. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=finite_difference_confusion, complete model={"difference_quotient":20,"step":[1,21]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(finite_difference_confusion, 20, 1/21, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 20, 1/21, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
96. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=finite_difference_confusion, complete model={"difference_quotient":21,"step":[1,22]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(finite_difference_confusion, 21, 1/22, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 21, 1/22, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
97. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=positive_gradient_direction, complete model={"gradient":[22,-1]}. Give (mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(positive_gradient_direction, [22, -1], [22, -1], 485, Yes, No, positive_gradient_is_first_order_ascent_not_descent)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (positive_gradient_direction, [22, -1], [22, -1], 485, Yes, No, positive_gradient_is_first_order_ascent_not_descent).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
98. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=non_euclidean_metric, complete model={"coordinate_gradient":[23,1],"metric":[[2,0],[0,3]]}. Give (mode, coordinate_partials_covector, positive_definite_metric, metric_gradient, metric_independence_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(non_euclidean_metric, [23, 1], [[2, 0], [0, 3]], [23/2, 1/3], No, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (non_euclidean_metric, [23, 1], [[2, 0], [0, 3]], [23/2, 1/3], No, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
99. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=finite_difference_confusion, complete model={"difference_quotient":24,"step":[1,25]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(finite_difference_confusion, 24, 1/25, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 24, 1/25, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
100. Stationarity, Metric Identification and Finite Differences Have Different Responsibilities
Use mode=finite_difference_confusion, complete model={"difference_quotient":25,"step":[1,26]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
The four modes have6/7/6/5 fields. A non-Euclidean gradient uses the metric identification and is not automatically the coordinate-partial covector. A finite difference is not a derivative certificate.
View answer and explanation
(finite_difference_confusion, 25, 1/26, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
A zero gradient certifies stationarity only. The positive gradient has positive directional inner product and is first-order ascent. For a declared positive-definite metric M, identify the coordinate-partial covector by solving M*grad=partials rather than assuming the Euclidean vector. A single nonzero-step finite difference is not a derivative or gradient certificate. The exact result, retaining the requested order, is (finite_difference_confusion, 25, 1/26, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate).
Reviewed step-by-step derivation
Gradient vector · qf-mod50-graddeep-boundary · candidate
Canonical question JSONL · English JSONL · Validation report