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R0049-25291 · q-20261015-advance-20261114-r04-standard-normal-distribution-0016
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).
G001 — 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).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [normaldist-symbolic-density]
After
derived = [[x, symbolic_density, density_law, normalization_scope]]
Conditions: none
List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
Gaussian ORIGINAL x squared := mul(-16/17, -16/17)
After
Gaussian ORIGINAL x squared = 256/289
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Gaussian square times exact half := mul(256/289, 1/2)
After
Gaussian square times exact half = 128/289
Conditions: none
Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [-16/17, -128/289]
After
derived = [(2*pi)^(-1/2)*exp(-128/289), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization is a NAMED Gaussian integral premise]
Conditions: none
Exact rational arithmetic computes only the exponent−x^2/2;retain exp and its normalization factor symbolically.The whole Gaussian law and normalization are declared premises,not proved by one density value.
Rule: Gaussian_WHOLE_declared_density_symbolic_value_named_normalization · Uses: G001, S003
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute Gaussian square times exact half with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Gaussian ORIGINAL x squared with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(-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)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 5fb922bf2aeed6383490232dc02d646df7fe76288dd58c0fe523842a9e97269ccandidate record SHA-256: ee146e474824bc660ad333987dffee59f585ab5662363353820db8f80e069b99semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3Lean: NOT_RUN · human review: NOT_PERFORMED