Named Norm-Induced Distances, Symmetry, Triangles and Evidence Modes
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batch-20261014-advance-20261027-r02-norm-induced-metric-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-12, -3, -5], right=[12, -6, -2]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [-12, -3, -5], [12, -6, -2], [-24, 3, -3], 30)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-12, -3, -5], [12, -6, -2], [-24, 3, -3], 30).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
2. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-11, -2, -2], right=[11, -1, -1]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(linfinity, [-11, -2, -2], [11, -1, -1], [-22, -1, -1], 22)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-11, -2, -2], [11, -1, -1], [-22, -1, -1], 22).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
3. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-10, -1, 1], right=[10, 4, 0]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [-10, -1, 1], [10, 4, 0], [-20, -5, 1], 26)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-10, -1, 1], [10, 4, 0], [-20, -5, 1], 26).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
4. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-9, 0, 4], right=[9, -4, 1]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(linfinity, [-9, 0, 4], [9, -4, 1], [-18, 4, 3], 18)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-9, 0, 4], [9, -4, 1], [-18, 4, 3], 18).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
5. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-8, 1, -4], right=[8, 1, -2]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [-8, 1, -4], [8, 1, -2], [-16, 0, -2], 18)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-8, 1, -4], [8, 1, -2], [-16, 0, -2], 18).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
6. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-7, 2, -1], right=[7, 6, -1]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(linfinity, [-7, 2, -1], [7, 6, -1], [-14, -4, 0], 14)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-7, 2, -1], [7, 6, -1], [-14, -4, 0], 14).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
7. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-6, 3, 2], right=[6, -2, 0]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [-6, 3, 2], [6, -2, 0], [-12, 5, 2], 19)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-6, 3, 2], [6, -2, 0], [-12, 5, 2], 19).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
8. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-5, -3, 5], right=[5, 3, 1]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(linfinity, [-5, -3, 5], [5, 3, 1], [-10, -6, 4], 10)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-5, -3, 5], [5, 3, 1], [-10, -6, 4], 10).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
9. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-4, -2, -3], right=[4, -5, -2]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [-4, -2, -3], [4, -5, -2], [-8, 3, -1], 12)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-4, -2, -3], [4, -5, -2], [-8, 3, -1], 12).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
10. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-3, -1, 0], right=[3, 0, -1]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(linfinity, [-3, -1, 0], [3, 0, -1], [-6, -1, 1], 6)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-3, -1, 0], [3, 0, -1], [-6, -1, 1], 6).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
11. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-2, 0, 3], right=[2, 5, 0]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [-2, 0, 3], [2, 5, 0], [-4, -5, 3], 12)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-2, 0, 3], [2, 5, 0], [-4, -5, 3], 12).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
12. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-1, 1, -5], right=[1, -3, 1]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(linfinity, [-1, 1, -5], [1, -3, 1], [-2, 4, -6], 6)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-1, 1, -5], [1, -3, 1], [-2, 4, -6], 6).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
13. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[0, 2, -2], right=[0, 2, -2]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [0, 2, -2], [0, 2, -2], [0, 0, 0], 0)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [0, 2, -2], [0, 2, -2], [0, 0, 0], 0).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
14. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[1, 3, 1], right=[-1, -6, -1]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(linfinity, [1, 3, 1], [-1, -6, -1], [2, 9, 2], 9)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [1, 3, 1], [-1, -6, -1], [2, 9, 2], 9).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
15. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[2, -3, 4], right=[-2, -1, 0]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [2, -3, 4], [-2, -1, 0], [4, -2, 4], 10)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [2, -3, 4], [-2, -1, 0], [4, -2, 4], 10).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
16. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[3, -2, -4], right=[-3, 4, 1]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(linfinity, [3, -2, -4], [-3, 4, 1], [6, -6, -5], 6)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [3, -2, -4], [-3, 4, 1], [6, -6, -5], 6).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
17. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[4, -1, -1], right=[-4, -4, -2]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [4, -1, -1], [-4, -4, -2], [8, 3, 1], 12)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [4, -1, -1], [-4, -4, -2], [8, 3, 1], 12).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
18. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[5, 0, 2], right=[-5, 1, -1]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 10)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 10).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
19. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[6, 1, 5], right=[-6, 6, 0]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [6, 1, 5], [-6, 6, 0], [12, -5, 5], 22)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [6, 1, 5], [-6, 6, 0], [12, -5, 5], 22).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
20. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[7, 2, -3], right=[-7, -2, 1]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(linfinity, [7, 2, -3], [-7, -2, 1], [14, 4, -4], 14)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [7, 2, -3], [-7, -2, 1], [14, 4, -4], 14).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
21. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[8, 3, 0], right=[-8, 3, -2]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [8, 3, 0], [-8, 3, -2], [16, 0, 2], 18)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [8, 3, 0], [-8, 3, -2], [16, 0, 2], 18).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
22. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[9, -3, 3], right=[-9, -5, -1]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(linfinity, [9, -3, 3], [-9, -5, -1], [18, 2, 4], 18)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [9, -3, 3], [-9, -5, -1], [18, 2, 4], 18).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
23. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[10, -2, -5], right=[-10, 0, 0]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [10, -2, -5], [-10, 0, 0], [20, -2, -5], 27)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [10, -2, -5], [-10, 0, 0], [20, -2, -5], 27).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
24. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[11, -1, -2], right=[-11, 5, 1]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
25. The Metric Induced by a Fixed Named Norm
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[12, 0, 1], right=[-12, -3, -2]. Give (norm_name,left,right,left−right,distance).
View answer and explanation
(l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30)
Subtract in the specified order and apply the same named norm. Keep the signed difference even though the resulting distance is symmetric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-distance · candidate
26. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-12, -3, -5], right=[12, -6, -2]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 30, 30, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 30, 30, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
27. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-11, -2, -2], right=[11, -1, -1]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(linfinity, 22, 22, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 22, 22, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
28. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-10, -1, 1], right=[10, 4, 0]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 26, 26, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 26, 26, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
29. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-9, 0, 4], right=[9, -4, 1]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(linfinity, 18, 18, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 18, 18, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
30. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-8, 1, -4], right=[8, 1, -2]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 18, 18, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 18, 18, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
31. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-7, 2, -1], right=[7, 6, -1]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(linfinity, 14, 14, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 14, 14, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
32. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-6, 3, 2], right=[6, -2, 0]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 19, 19, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 19, 19, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
33. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-5, -3, 5], right=[5, 3, 1]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(linfinity, 10, 10, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 10, 10, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
34. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-4, -2, -3], right=[4, -5, -2]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 12, 12, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 12, 12, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
35. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-3, -1, 0], right=[3, 0, -1]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(linfinity, 6, 6, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 6, 6, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
36. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[-2, 0, 3], right=[2, 5, 0]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 12, 12, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 12, 12, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
37. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[-1, 1, -5], right=[1, -3, 1]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(linfinity, 6, 6, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 6, 6, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
38. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[0, 2, -2], right=[0, 2, -2]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 0, 0, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 0, 0, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
39. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[1, 3, 1], right=[-1, -6, -1]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(linfinity, 9, 9, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 9, 9, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
40. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[2, -3, 4], right=[-2, -1, 0]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 10, 10, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 10, 10, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
41. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[3, -2, -4], right=[-3, 4, 1]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(linfinity, 6, 6, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 6, 6, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
42. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[4, -1, -1], right=[-4, -4, -2]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 12, 12, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 12, 12, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
43. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[5, 0, 2], right=[-5, 1, -1]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(linfinity, 10, 10, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 10, 10, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
44. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[6, 1, 5], right=[-6, 6, 0]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 22, 22, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 22, 22, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
45. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[7, 2, -3], right=[-7, -2, 1]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(linfinity, 14, 14, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 14, 14, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
46. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[8, 3, 0], right=[-8, 3, -2]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 18, 18, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 18, 18, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
47. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[9, -3, 3], right=[-9, -5, -1]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(linfinity, 18, 18, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 18, 18, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
48. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[10, -2, -5], right=[-10, 0, 0]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 27, 27, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 27, 27, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
49. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, left=[11, -1, -2], right=[-11, 5, 1]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(linfinity, 22, 22, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 22, 22, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
50. Reverse the Difference, Not the Norm Definition
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, left=[12, 0, 1], right=[-12, -3, -2]. Give (norm_name,d_xy,d_yx,equal?).
View answer and explanation
(l1, 30, 30, Yes)
Absolute homogeneity at scalar −1 gives equal norms for opposite differences. Compute both exact distances for this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 30, 30, Yes).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-symmetry · candidate
51. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[-12, -3], y=[-4, 12], z=[-6, -2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 7, 23, 16, 39, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 7, 23, 16, 39, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
52. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[-11, -2], y=[-3, 11], z=[-1, -1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(linfinity, 10, 13, 12, 25, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 10, 13, 12, 25, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
53. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[-10, -1], y=[-2, 10], z=[4, 0]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 15, 19, 16, 35, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 15, 19, 16, 35, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
54. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[-9, 0], y=[-1, 9], z=[-4, 1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(linfinity, 5, 9, 8, 17, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 5, 9, 8, 17, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
55. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[-8, 1], y=[0, 8], z=[1, 2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 10, 15, 7, 22, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 10, 15, 7, 22, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
56. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[-7, 2], y=[1, 7], z=[6, -2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(linfinity, 13, 8, 9, 17, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 13, 8, 9, 17, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
57. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[-6, 3], y=[2, 6], z=[-2, -1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 8, 11, 11, 22, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 8, 11, 11, 22, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
58. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[-5, -3], y=[3, 5], z=[3, 0]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(linfinity, 8, 8, 5, 13, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 8, 8, 5, 13, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
59. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[-4, -2], y=[4, 4], z=[-5, 1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 4, 14, 12, 26, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 4, 14, 12, 26, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
60. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[-3, -1], y=[-4, 3], z=[0, 2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(linfinity, 3, 4, 4, 8, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 3, 4, 4, 8, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
61. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[-2, 0], y=[-3, 2], z=[5, -2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 9, 3, 12, 15, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 9, 3, 12, 15, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
62. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[-1, 1], y=[-2, 1], z=[-3, -1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(linfinity, 2, 1, 2, 3, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 2, 1, 2, 3, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
63. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[0, 2], y=[-1, 0], z=[2, 0]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 4, 3, 3, 6, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 4, 3, 3, 6, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
64. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[1, 3], y=[0, -1], z=[-6, 1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(linfinity, 7, 4, 6, 10, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 7, 4, 6, 10, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
65. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[2, -3], y=[1, -2], z=[-1, 2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 8, 2, 6, 8, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 8, 2, 6, 8, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
66. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[3, -2], y=[2, -3], z=[4, -2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(linfinity, 1, 1, 2, 3, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 1, 1, 2, 3, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
67. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[4, -1], y=[3, -4], z=[-4, -1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 8, 4, 10, 14, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 8, 4, 10, 14, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
68. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[5, 0], y=[4, -5], z=[1, 0]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(linfinity, 4, 5, 5, 10, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 4, 5, 5, 10, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
69. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[6, 1], y=[-4, -6], z=[6, 1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 0, 17, 17, 34, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 0, 17, 17, 34, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
70. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[7, 2], y=[-3, -7], z=[-2, 2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(linfinity, 9, 10, 9, 19, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 9, 10, 9, 19, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
71. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[8, 3], y=[-2, -8], z=[3, -2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 10, 21, 11, 32, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 10, 21, 11, 32, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
72. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[9, -3], y=[-1, -9], z=[-5, -1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(linfinity, 14, 10, 8, 18, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 14, 10, 8, 18, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
73. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[10, -2], y=[0, -10], z=[0, 0]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 12, 18, 10, 28, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 12, 18, 10, 28, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
74. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, x=[11, -1], y=[1, -11], z=[5, 1]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(linfinity, 6, 10, 12, 22, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 6, 10, 12, 22, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
75. The Norm Triangle Inequality Supplies the Metric Triangle Bound
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, x=[12, 0], y=[2, -12], z=[-3, 2]. Give (norm_name,d_xz,d_xy,d_yz,rhs=d_xy+d_yz,pass?,norm_triangle_inequality).
View answer and explanation
(l1, 17, 22, 19, 41, Yes, norm_triangle_inequality)
Use x−z=(x−y)+(y−z) and the universal norm triangle inequality, then evaluate the three named distances. This is not an empirical proof from one passing triple. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 17, 22, 19, 41, Yes, norm_triangle_inequality).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-triangle-certificate · candidate
76. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=1, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 1, finite_samples, No, samples_do_not_establish_induced_metric)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 1, finite_samples, No, samples_do_not_establish_induced_metric).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
77. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=2, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(linfinity, 2, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 2, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
78. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=3, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 3, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 3, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
79. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=4, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(linfinity, 4, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 4, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
80. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=5, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 5, finite_samples, No, samples_do_not_establish_induced_metric)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 5, finite_samples, No, samples_do_not_establish_induced_metric).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
81. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=6, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(linfinity, 6, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 6, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
82. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=7, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 7, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 7, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
83. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=8, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(linfinity, 8, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 8, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
84. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=9, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 9, finite_samples, No, samples_do_not_establish_induced_metric)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 9, finite_samples, No, samples_do_not_establish_induced_metric).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
85. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=10, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(linfinity, 10, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 10, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
86. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=11, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 11, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 11, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
87. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=12, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(linfinity, 12, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 12, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
88. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=13, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 13, finite_samples, No, samples_do_not_establish_induced_metric)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 13, finite_samples, No, samples_do_not_establish_induced_metric).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
89. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=14, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(linfinity, 14, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 14, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
90. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=15, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 15, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 15, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
91. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=16, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(linfinity, 16, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 16, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
92. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=17, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 17, finite_samples, No, samples_do_not_establish_induced_metric)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 17, finite_samples, No, samples_do_not_establish_induced_metric).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
93. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=18, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(linfinity, 18, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 18, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
94. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=19, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 19, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 19, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
95. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=20, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(linfinity, 20, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 20, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
96. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=21, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 21, finite_samples, No, samples_do_not_establish_induced_metric)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 21, finite_samples, No, samples_do_not_establish_induced_metric).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
97. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=22, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(linfinity, 22, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 22, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
98. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=23, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 23, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 23, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
99. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=linfinity, dimension=24, evidence_mode=named_formula_certificate. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(linfinity, 24, named_formula_certificate, Yes, universal_from_norm_axioms)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 24, named_formula_certificate, Yes, universal_from_norm_axioms).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
100. Distinguish a Universal Norm Certificate from Finite Samples
Use one named, certified norm on a fixed real coordinate space: l1(v)=sum_i |v_i| or linfinity(v)=max_i |v_i|. Set d(x,y)=norm(x−y) on that same space. Its universal norm-axiom certificate establishes the metric axioms; finite distance samples do not. Do not infer that every metric comes from a norm, or claim completeness, compactness or an inner product. norm_name=l1, dimension=25, evidence_mode=finite_samples. Give (norm_name,dimension,evidence_mode,sufficient?,boundary). named_formula_certificate yields universal_from_norm_axioms; finite_samples yields samples_do_not_establish_induced_metric.
View answer and explanation
(l1, 25, finite_samples, No, samples_do_not_establish_induced_metric)
The No flag in sample mode means the proposed evidence is insufficient, not that this named formula fails to define a metric. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 25, finite_samples, No, samples_do_not_establish_induced_metric).
Reviewed step-by-step derivation
Norm-induced metric · qf-mod27-normmetric-induced-qualification · candidate
Canonical question JSONL · English JSONL · Validation report