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R0049-25205 · q-20261015-advance-20261114-r04-standard-normal-distribution-0055
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).
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 x=-8. Give (x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary).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-cdf-symmetry]
After
derived = [[x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary]]
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(-8, -8)
After
Gaussian ORIGINAL x squared = 64
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(64, 1/2)
After
Gaussian square times exact half = 32
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 = [-8, -32]
After
derived = [(2*pi)^(-1/2)*exp(-32), 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
Before
source roles = [-8, 8, even continuous density and declared integral1]
After
derived = [Phi(8)=1-Phi(-8), Phi(0)=1/2, Yes]
Conditions: none
Substitute u=−t to turn the lower tail to−x into the upper tail from x;evenness keeps the density.A continuous law has no point atom and the declared total integral1 gives1−Phi(x).At0 the equal halves give1/2,not a general nonzero elementary CDF value.
Rule: Gaussian_CDF_symmetry_integral_substitution_USES_named_total1 · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(-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)
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: f794700d737906affed44594bc576551faa343dc0996e9699ece9b5c2a5f5b17candidate record SHA-256: 9a1a17839208a21eefeddbb4f183d82b7475362ea3484b7fed758e48b19f1906semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3Lean: NOT_RUN · human review: NOT_PERFORMED