Symbolic Gaussian Densities, CDF Symmetry and Named-Theorem Boundaries
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batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
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Field-order and source-binding data
1. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=1/2. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(1/2, (2*pi)^(-1/2)*exp(-1/8), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (1/2, (2*pi)^(-1/2)*exp(-1/8), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
2. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=-2/3. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(-2/3, (2*pi)^(-1/2)*exp(-2/9), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (-2/3, (2*pi)^(-1/2)*exp(-2/9), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
3. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=3/4. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(3/4, (2*pi)^(-1/2)*exp(-9/32), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (3/4, (2*pi)^(-1/2)*exp(-9/32), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
4. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=-4/5. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(-4/5, (2*pi)^(-1/2)*exp(-8/25), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (-4/5, (2*pi)^(-1/2)*exp(-8/25), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
5. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=5/6. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(5/6, (2*pi)^(-1/2)*exp(-25/72), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (5/6, (2*pi)^(-1/2)*exp(-25/72), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
6. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=-6/7. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(-6/7, (2*pi)^(-1/2)*exp(-18/49), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (-6/7, (2*pi)^(-1/2)*exp(-18/49), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
7. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=7/8. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(7/8, (2*pi)^(-1/2)*exp(-49/128), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (7/8, (2*pi)^(-1/2)*exp(-49/128), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
8. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=-8/9. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(-8/9, (2*pi)^(-1/2)*exp(-32/81), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (-8/9, (2*pi)^(-1/2)*exp(-32/81), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
9. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=9/10. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(9/10, (2*pi)^(-1/2)*exp(-81/200), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (9/10, (2*pi)^(-1/2)*exp(-81/200), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
10. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=-10/11. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(-10/11, (2*pi)^(-1/2)*exp(-50/121), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (-10/11, (2*pi)^(-1/2)*exp(-50/121), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
11. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=11/12. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(11/12, (2*pi)^(-1/2)*exp(-121/288), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (11/12, (2*pi)^(-1/2)*exp(-121/288), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
12. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=-12/13. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(-12/13, (2*pi)^(-1/2)*exp(-72/169), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (-12/13, (2*pi)^(-1/2)*exp(-72/169), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
13. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=13/14. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(13/14, (2*pi)^(-1/2)*exp(-169/392), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (13/14, (2*pi)^(-1/2)*exp(-169/392), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
14. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=-14/15. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(-14/15, (2*pi)^(-1/2)*exp(-98/225), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (-14/15, (2*pi)^(-1/2)*exp(-98/225), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
15. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=15/16. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(15/16, (2*pi)^(-1/2)*exp(-225/512), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (15/16, (2*pi)^(-1/2)*exp(-225/512), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
16. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=-16/17. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(-16/17, (2*pi)^(-1/2)*exp(-128/289), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (-16/17, (2*pi)^(-1/2)*exp(-128/289), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
17. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=17/18. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(17/18, (2*pi)^(-1/2)*exp(-289/648), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (17/18, (2*pi)^(-1/2)*exp(-289/648), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
18. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=-18/19. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(-18/19, (2*pi)^(-1/2)*exp(-162/361), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (-18/19, (2*pi)^(-1/2)*exp(-162/361), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
19. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=19/20. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(19/20, (2*pi)^(-1/2)*exp(-361/800), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (19/20, (2*pi)^(-1/2)*exp(-361/800), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
20. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=-20/21. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(-20/21, (2*pi)^(-1/2)*exp(-200/441), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (-20/21, (2*pi)^(-1/2)*exp(-200/441), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
21. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=21/22. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(21/22, (2*pi)^(-1/2)*exp(-441/968), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (21/22, (2*pi)^(-1/2)*exp(-441/968), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
22. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=-22/23. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(-22/23, (2*pi)^(-1/2)*exp(-242/529), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (-22/23, (2*pi)^(-1/2)*exp(-242/529), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
23. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=23/24. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(23/24, (2*pi)^(-1/2)*exp(-529/1152), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (23/24, (2*pi)^(-1/2)*exp(-529/1152), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
24. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=-24/25. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(-24/25, (2*pi)^(-1/2)*exp(-288/625), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (-24/25, (2*pi)^(-1/2)*exp(-288/625), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
25. The Standard-Normal Density Value Remains an Exact Symbolic Expression
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use exact x=25/26. Give (x, symbolic_density, density_law, normalization_scope).
View answer and explanation
(25/26, (2*pi)^(-1/2)*exp(-625/1352), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)
Compute the rational exponent −x^2/2 and retain the symbolic normalization factor and exponential. A symbolic finite argument is not a proof of Gaussian normalization, a point mass or an elementary CDF evaluation. The exact result, retaining the requested order, is (25/26, (2*pi)^(-1/2)*exp(-625/1352), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-symbolic-density · candidate
26. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=1/2 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(1/2, -1/2, (2*pi)^(-1/2)*exp(-1/8), (2*pi)^(-1/2)*exp(-1/8), Yes, phi(x)/phi(0)=exp(-1/8), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (1/2, -1/2, (2*pi)^(-1/2)*exp(-1/8), (2*pi)^(-1/2)*exp(-1/8), Yes, phi(x)/phi(0)=exp(-1/8), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
27. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-2/3 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(-2/3, 2/3, (2*pi)^(-1/2)*exp(-2/9), (2*pi)^(-1/2)*exp(-2/9), Yes, phi(x)/phi(0)=exp(-2/9), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (-2/3, 2/3, (2*pi)^(-1/2)*exp(-2/9), (2*pi)^(-1/2)*exp(-2/9), Yes, phi(x)/phi(0)=exp(-2/9), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
28. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=3/4 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(3/4, -3/4, (2*pi)^(-1/2)*exp(-9/32), (2*pi)^(-1/2)*exp(-9/32), Yes, phi(x)/phi(0)=exp(-9/32), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (3/4, -3/4, (2*pi)^(-1/2)*exp(-9/32), (2*pi)^(-1/2)*exp(-9/32), Yes, phi(x)/phi(0)=exp(-9/32), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
29. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-4/5 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(-4/5, 4/5, (2*pi)^(-1/2)*exp(-8/25), (2*pi)^(-1/2)*exp(-8/25), Yes, phi(x)/phi(0)=exp(-8/25), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (-4/5, 4/5, (2*pi)^(-1/2)*exp(-8/25), (2*pi)^(-1/2)*exp(-8/25), Yes, phi(x)/phi(0)=exp(-8/25), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
30. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=5/6 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(5/6, -5/6, (2*pi)^(-1/2)*exp(-25/72), (2*pi)^(-1/2)*exp(-25/72), Yes, phi(x)/phi(0)=exp(-25/72), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (5/6, -5/6, (2*pi)^(-1/2)*exp(-25/72), (2*pi)^(-1/2)*exp(-25/72), Yes, phi(x)/phi(0)=exp(-25/72), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
31. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-6/7 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(-6/7, 6/7, (2*pi)^(-1/2)*exp(-18/49), (2*pi)^(-1/2)*exp(-18/49), Yes, phi(x)/phi(0)=exp(-18/49), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (-6/7, 6/7, (2*pi)^(-1/2)*exp(-18/49), (2*pi)^(-1/2)*exp(-18/49), Yes, phi(x)/phi(0)=exp(-18/49), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
32. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=7/8 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(7/8, -7/8, (2*pi)^(-1/2)*exp(-49/128), (2*pi)^(-1/2)*exp(-49/128), Yes, phi(x)/phi(0)=exp(-49/128), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (7/8, -7/8, (2*pi)^(-1/2)*exp(-49/128), (2*pi)^(-1/2)*exp(-49/128), Yes, phi(x)/phi(0)=exp(-49/128), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
33. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-8/9 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(-8/9, 8/9, (2*pi)^(-1/2)*exp(-32/81), (2*pi)^(-1/2)*exp(-32/81), Yes, phi(x)/phi(0)=exp(-32/81), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (-8/9, 8/9, (2*pi)^(-1/2)*exp(-32/81), (2*pi)^(-1/2)*exp(-32/81), Yes, phi(x)/phi(0)=exp(-32/81), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
34. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=9/10 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(9/10, -9/10, (2*pi)^(-1/2)*exp(-81/200), (2*pi)^(-1/2)*exp(-81/200), Yes, phi(x)/phi(0)=exp(-81/200), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (9/10, -9/10, (2*pi)^(-1/2)*exp(-81/200), (2*pi)^(-1/2)*exp(-81/200), Yes, phi(x)/phi(0)=exp(-81/200), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
35. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-10/11 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(-10/11, 10/11, (2*pi)^(-1/2)*exp(-50/121), (2*pi)^(-1/2)*exp(-50/121), Yes, phi(x)/phi(0)=exp(-50/121), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (-10/11, 10/11, (2*pi)^(-1/2)*exp(-50/121), (2*pi)^(-1/2)*exp(-50/121), Yes, phi(x)/phi(0)=exp(-50/121), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
36. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=11/12 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(11/12, -11/12, (2*pi)^(-1/2)*exp(-121/288), (2*pi)^(-1/2)*exp(-121/288), Yes, phi(x)/phi(0)=exp(-121/288), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (11/12, -11/12, (2*pi)^(-1/2)*exp(-121/288), (2*pi)^(-1/2)*exp(-121/288), Yes, phi(x)/phi(0)=exp(-121/288), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
37. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-12/13 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(-12/13, 12/13, (2*pi)^(-1/2)*exp(-72/169), (2*pi)^(-1/2)*exp(-72/169), Yes, phi(x)/phi(0)=exp(-72/169), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (-12/13, 12/13, (2*pi)^(-1/2)*exp(-72/169), (2*pi)^(-1/2)*exp(-72/169), Yes, phi(x)/phi(0)=exp(-72/169), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
38. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=13/14 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(13/14, -13/14, (2*pi)^(-1/2)*exp(-169/392), (2*pi)^(-1/2)*exp(-169/392), Yes, phi(x)/phi(0)=exp(-169/392), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (13/14, -13/14, (2*pi)^(-1/2)*exp(-169/392), (2*pi)^(-1/2)*exp(-169/392), Yes, phi(x)/phi(0)=exp(-169/392), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
39. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-14/15 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(-14/15, 14/15, (2*pi)^(-1/2)*exp(-98/225), (2*pi)^(-1/2)*exp(-98/225), Yes, phi(x)/phi(0)=exp(-98/225), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (-14/15, 14/15, (2*pi)^(-1/2)*exp(-98/225), (2*pi)^(-1/2)*exp(-98/225), Yes, phi(x)/phi(0)=exp(-98/225), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
40. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=15/16 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(15/16, -15/16, (2*pi)^(-1/2)*exp(-225/512), (2*pi)^(-1/2)*exp(-225/512), Yes, phi(x)/phi(0)=exp(-225/512), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (15/16, -15/16, (2*pi)^(-1/2)*exp(-225/512), (2*pi)^(-1/2)*exp(-225/512), Yes, phi(x)/phi(0)=exp(-225/512), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
41. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-16/17 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(-16/17, 16/17, (2*pi)^(-1/2)*exp(-128/289), (2*pi)^(-1/2)*exp(-128/289), Yes, phi(x)/phi(0)=exp(-128/289), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (-16/17, 16/17, (2*pi)^(-1/2)*exp(-128/289), (2*pi)^(-1/2)*exp(-128/289), Yes, phi(x)/phi(0)=exp(-128/289), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
42. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=17/18 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(17/18, -17/18, (2*pi)^(-1/2)*exp(-289/648), (2*pi)^(-1/2)*exp(-289/648), Yes, phi(x)/phi(0)=exp(-289/648), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (17/18, -17/18, (2*pi)^(-1/2)*exp(-289/648), (2*pi)^(-1/2)*exp(-289/648), Yes, phi(x)/phi(0)=exp(-289/648), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
43. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-18/19 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(-18/19, 18/19, (2*pi)^(-1/2)*exp(-162/361), (2*pi)^(-1/2)*exp(-162/361), Yes, phi(x)/phi(0)=exp(-162/361), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (-18/19, 18/19, (2*pi)^(-1/2)*exp(-162/361), (2*pi)^(-1/2)*exp(-162/361), Yes, phi(x)/phi(0)=exp(-162/361), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
44. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=19/20 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(19/20, -19/20, (2*pi)^(-1/2)*exp(-361/800), (2*pi)^(-1/2)*exp(-361/800), Yes, phi(x)/phi(0)=exp(-361/800), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (19/20, -19/20, (2*pi)^(-1/2)*exp(-361/800), (2*pi)^(-1/2)*exp(-361/800), Yes, phi(x)/phi(0)=exp(-361/800), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
45. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-20/21 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(-20/21, 20/21, (2*pi)^(-1/2)*exp(-200/441), (2*pi)^(-1/2)*exp(-200/441), Yes, phi(x)/phi(0)=exp(-200/441), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (-20/21, 20/21, (2*pi)^(-1/2)*exp(-200/441), (2*pi)^(-1/2)*exp(-200/441), Yes, phi(x)/phi(0)=exp(-200/441), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
46. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=21/22 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(21/22, -21/22, (2*pi)^(-1/2)*exp(-441/968), (2*pi)^(-1/2)*exp(-441/968), Yes, phi(x)/phi(0)=exp(-441/968), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (21/22, -21/22, (2*pi)^(-1/2)*exp(-441/968), (2*pi)^(-1/2)*exp(-441/968), Yes, phi(x)/phi(0)=exp(-441/968), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
47. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-22/23 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(-22/23, 22/23, (2*pi)^(-1/2)*exp(-242/529), (2*pi)^(-1/2)*exp(-242/529), Yes, phi(x)/phi(0)=exp(-242/529), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (-22/23, 22/23, (2*pi)^(-1/2)*exp(-242/529), (2*pi)^(-1/2)*exp(-242/529), Yes, phi(x)/phi(0)=exp(-242/529), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
48. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=23/24 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(23/24, -23/24, (2*pi)^(-1/2)*exp(-529/1152), (2*pi)^(-1/2)*exp(-529/1152), Yes, phi(x)/phi(0)=exp(-529/1152), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (23/24, -23/24, (2*pi)^(-1/2)*exp(-529/1152), (2*pi)^(-1/2)*exp(-529/1152), Yes, phi(x)/phi(0)=exp(-529/1152), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
49. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-24/25 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(-24/25, 24/25, (2*pi)^(-1/2)*exp(-288/625), (2*pi)^(-1/2)*exp(-288/625), Yes, phi(x)/phi(0)=exp(-288/625), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (-24/25, 24/25, (2*pi)^(-1/2)*exp(-288/625), (2*pi)^(-1/2)*exp(-288/625), Yes, phi(x)/phi(0)=exp(-288/625), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
50. Evenness and the Density Ratio Follow from the Squared Argument
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=25/26 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).
View answer and explanation
(25/26, -25/26, (2*pi)^(-1/2)*exp(-625/1352), (2*pi)^(-1/2)*exp(-625/1352), Yes, phi(x)/phi(0)=exp(-625/1352), evenness_follows_from_x_squared)
The density expressions agree because the exponent uses x^2. The normalization factor cancels in phi(x)/phi(0)=exp(−x^2/2). Keep these symbolic density identities separate from normalization and CDF values. The exact result, retaining the requested order, is (25/26, -25/26, (2*pi)^(-1/2)*exp(-625/1352), (2*pi)^(-1/2)*exp(-625/1352), Yes, phi(x)/phi(0)=exp(-625/1352), evenness_follows_from_x_squared).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-evenness-ratio · candidate
51. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-12. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(-12, Phi(12)=1-Phi(-12), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (-12, Phi(12)=1-Phi(-12), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
52. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-11. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(-11, Phi(11)=1-Phi(-11), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (-11, Phi(11)=1-Phi(-11), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
53. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-10. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(-10, Phi(10)=1-Phi(-10), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (-10, Phi(10)=1-Phi(-10), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
54. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-9. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(-9, Phi(9)=1-Phi(-9), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (-9, Phi(9)=1-Phi(-9), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
55. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-8. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(-8, Phi(8)=1-Phi(-8), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (-8, Phi(8)=1-Phi(-8), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
56. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-7. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(-7, Phi(7)=1-Phi(-7), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (-7, Phi(7)=1-Phi(-7), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
57. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-6. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(-6, Phi(6)=1-Phi(-6), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (-6, Phi(6)=1-Phi(-6), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
58. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-5. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(-5, Phi(5)=1-Phi(-5), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (-5, Phi(5)=1-Phi(-5), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
59. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-4. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(-4, Phi(4)=1-Phi(-4), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (-4, Phi(4)=1-Phi(-4), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
60. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-3. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(-3, Phi(3)=1-Phi(-3), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (-3, Phi(3)=1-Phi(-3), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
61. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-2. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(-2, Phi(2)=1-Phi(-2), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (-2, Phi(2)=1-Phi(-2), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
62. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=-1. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(-1, Phi(1)=1-Phi(-1), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (-1, Phi(1)=1-Phi(-1), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
63. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=0. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(0, Phi(0)=1-Phi(0), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (0, Phi(0)=1-Phi(0), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
64. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=1. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(1, Phi(-1)=1-Phi(1), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (1, Phi(-1)=1-Phi(1), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
65. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=2. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(2, Phi(-2)=1-Phi(2), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (2, Phi(-2)=1-Phi(2), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
66. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=3. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(3, Phi(-3)=1-Phi(3), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (3, Phi(-3)=1-Phi(3), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
67. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=4. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(4, Phi(-4)=1-Phi(4), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (4, Phi(-4)=1-Phi(4), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
68. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=5. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(5, Phi(-5)=1-Phi(5), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (5, Phi(-5)=1-Phi(5), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
69. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=6. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(6, Phi(-6)=1-Phi(6), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (6, Phi(-6)=1-Phi(6), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
70. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=7. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(7, Phi(-7)=1-Phi(7), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (7, Phi(-7)=1-Phi(7), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
71. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=8. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(8, Phi(-8)=1-Phi(8), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (8, Phi(-8)=1-Phi(8), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
72. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=9. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(9, Phi(-9)=1-Phi(9), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (9, Phi(-9)=1-Phi(9), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
73. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=10. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(10, Phi(-10)=1-Phi(10), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (10, Phi(-10)=1-Phi(10), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
74. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=11. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(11, Phi(-11)=1-Phi(11), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (11, Phi(-11)=1-Phi(11), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
75. Declared Normalization and Even Density Give CDF Symmetry, Not Arbitrary Exact Values
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Use x=12. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).
View answer and explanation
(12, Phi(-12)=1-Phi(12), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)
With the declared normalized even law, Phi(−x)=1−Phi(x) and Phi(0)=1/2. Do not interpret this as computing a general nonzero elementary CDF value or as proving normalization from finitely many evaluations. The exact result, retaining the requested order, is (12, Phi(-12)=1-Phi(12), Phi(0)=1/2, Yes, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-cdf-symmetry · candidate
76. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=1>0, panel count=10. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(1, 10, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (1, 10, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
77. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=2>0, panel count=11. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(2, 11, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (2, 11, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
78. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=3>0, panel count=12. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(3, 12, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (3, 12, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
79. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=4>0, panel count=13. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(4, 13, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (4, 13, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
80. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=5>0, panel count=14. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(5, 14, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (5, 14, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
81. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=6>0, panel count=15. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(6, 15, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (6, 15, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
82. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=7>0, panel count=16. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(7, 16, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (7, 16, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
83. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=8>0, panel count=17. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(8, 17, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (8, 17, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
84. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=9>0, panel count=18. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(9, 18, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (9, 18, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
85. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=10>0, panel count=19. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(10, 19, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (10, 19, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
86. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=11>0, panel count=20. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(11, 20, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (11, 20, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
87. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=12>0, panel count=21. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(12, 21, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (12, 21, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
88. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=13>0, panel count=22. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(13, 22, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (13, 22, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
89. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=14>0, panel count=23. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(14, 23, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (14, 23, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
90. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=15>0, panel count=24. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(15, 24, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (15, 24, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
91. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=16>0, panel count=25. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(16, 25, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (16, 25, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
92. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=17>0, panel count=26. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(17, 26, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (17, 26, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
93. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=18>0, panel count=27. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(18, 27, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (18, 27, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
94. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=19>0, panel count=28. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(19, 28, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (19, 28, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
95. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=20>0, panel count=29. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(20, 29, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (20, 29, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
96. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=21>0, panel count=30. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(21, 30, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (21, 30, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
97. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=22>0, panel count=31. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(22, 31, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (22, 31, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
98. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=23>0, panel count=32. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(23, 32, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (23, 32, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
99. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=24>0, panel count=33. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(24, 33, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (24, 33, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
100. Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments
Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=25>0, panel count=34. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).
This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag.
View answer and explanation
(25, 34, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)
This boundary records finite-method scope only; it does not even supply a quadrature result. Retain the No proof claim separately from named normalization/mean/variance facts. Those facts need their own theorem-level derivations, not a larger finite window or panel count. The exact result, retaining the requested order, is (25, 34, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments).
Reviewed step-by-step derivation
Standard normal distribution · qf-mod45-normaldist-normalization-moment-boundary · candidate
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