Question 27247: An Exact Binomial Lattice Is Not a Normal Approximation

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0053-27247 · q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0072

Original question and givens

Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=3, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).

G001 — Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=3, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).

Definition and 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.

mko-central-limit-theorem-finite-variance-iid

Step-by-step derivation

S001 · implies

Before

source roles = [clt-binomial-lattice]

After

derived = [[n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope]]

Conditions: none

List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

CLT binomial ORIGINAL one minus probability := sub(1, 1/2)

After

CLT binomial ORIGINAL one minus probability = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

source roles = [CLT binomial k0 success, 1/2, 0]

After

derived = [1]

Conditions: none

A zeroth power is the empty product1,not0 because n or the base is0.

Step identifiers and dependencies

Rule: CLT_geometric_probability_power_zero_empty_product · Uses: G001, S002

S004 · calculation

Before

CLT binomial k0 failure ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k0 failure ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

CLT binomial k0 failure ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k0 failure ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

CLT binomial k0 failure ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k0 failure ORIGINAL power multiply3 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

CLT binomial k0 failure ORIGINAL power multiply4 := mul(1/8, 1/2)

After

CLT binomial k0 failure ORIGINAL power multiply4 = 1/16

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

CLT binomial k0 failure ORIGINAL power multiply5 := mul(1/16, 1/2)

After

CLT binomial k0 failure ORIGINAL power multiply5 = 1/32

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

CLT binomial k0 failure ORIGINAL power multiply6 := mul(1/32, 1/2)

After

CLT binomial k0 failure ORIGINAL power multiply6 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

CLT binomial ORIGINAL power product k0 := mul(1, 1/64)

After

CLT binomial ORIGINAL power product k0 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

CLT binomial ORIGINAL choose times joint k0 := mul(1, 1/64)

After

CLT binomial ORIGINAL choose times joint k0 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [6, 0, 1/2, 1/2]

After

derived = [1, 1, 1/64, 1/64]

Conditions: none

Every choice of k success positions has probability p^k(1-p)^(n-k),with C(n,k) choices.This complete finite-independence calculation gives an exact lattice mass,not a normal approximation.Later sums reuse the SAME established pure term recipe without changing any source profile.

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S011

S013 · calculation

Before

CLT binomial k1 success ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k1 success ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

CLT binomial k1 failure ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k1 failure ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

CLT binomial k1 failure ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k1 failure ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

CLT binomial k1 failure ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k1 failure ORIGINAL power multiply3 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

CLT binomial k1 failure ORIGINAL power multiply4 := mul(1/8, 1/2)

After

CLT binomial k1 failure ORIGINAL power multiply4 = 1/16

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

CLT binomial k1 failure ORIGINAL power multiply5 := mul(1/16, 1/2)

After

CLT binomial k1 failure ORIGINAL power multiply5 = 1/32

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

CLT binomial ORIGINAL power product k1 := mul(1/2, 1/32)

After

CLT binomial ORIGINAL power product k1 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

CLT binomial ORIGINAL choose times joint k1 := mul(6, 1/64)

After

CLT binomial ORIGINAL choose times joint k1 = 3/32

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

source roles = [6, 1, 1/2, 1/2]

After

derived = [6, 1/2, 1/32, 3/32]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S020

S022 · calculation

Before

CLT binomial k2 success ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k2 success ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

CLT binomial k2 success ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k2 success ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

CLT binomial k2 failure ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k2 failure ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

CLT binomial k2 failure ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k2 failure ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

CLT binomial k2 failure ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k2 failure ORIGINAL power multiply3 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

CLT binomial k2 failure ORIGINAL power multiply4 := mul(1/8, 1/2)

After

CLT binomial k2 failure ORIGINAL power multiply4 = 1/16

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

CLT binomial ORIGINAL power product k2 := mul(1/4, 1/16)

After

CLT binomial ORIGINAL power product k2 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

CLT binomial ORIGINAL choose times joint k2 := mul(15, 1/64)

After

CLT binomial ORIGINAL choose times joint k2 = 15/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

source roles = [6, 2, 1/2, 1/2]

After

derived = [15, 1/4, 1/16, 15/64]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S029

S031 · calculation

Before

CLT binomial k3 success ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k3 success ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

CLT binomial k3 success ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k3 success ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

CLT binomial k3 success ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k3 success ORIGINAL power multiply3 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

CLT binomial k3 failure ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k3 failure ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

CLT binomial k3 failure ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k3 failure ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

CLT binomial k3 failure ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k3 failure ORIGINAL power multiply3 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

CLT binomial ORIGINAL power product k3 := mul(1/8, 1/8)

After

CLT binomial ORIGINAL power product k3 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

CLT binomial ORIGINAL choose times joint k3 := mul(20, 1/64)

After

CLT binomial ORIGINAL choose times joint k3 = 5/16

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · implies

Before

source roles = [6, 3, 1/2, 1/2]

After

derived = [20, 1/8, 1/8, 5/16]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S038

S040 · calculation

Before

CLT binomial k4 success ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k4 success ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

CLT binomial k4 success ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k4 success ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

CLT binomial k4 success ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k4 success ORIGINAL power multiply3 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

CLT binomial k4 success ORIGINAL power multiply4 := mul(1/8, 1/2)

After

CLT binomial k4 success ORIGINAL power multiply4 = 1/16

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

CLT binomial k4 failure ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k4 failure ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

CLT binomial k4 failure ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k4 failure ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

CLT binomial ORIGINAL power product k4 := mul(1/16, 1/4)

After

CLT binomial ORIGINAL power product k4 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

CLT binomial ORIGINAL choose times joint k4 := mul(15, 1/64)

After

CLT binomial ORIGINAL choose times joint k4 = 15/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · implies

Before

source roles = [6, 4, 1/2, 1/2]

After

derived = [15, 1/16, 1/4, 15/64]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S047

S049 · calculation

Before

CLT binomial k5 success ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k5 success ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

CLT binomial k5 success ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k5 success ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

CLT binomial k5 success ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k5 success ORIGINAL power multiply3 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

CLT binomial k5 success ORIGINAL power multiply4 := mul(1/8, 1/2)

After

CLT binomial k5 success ORIGINAL power multiply4 = 1/16

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

CLT binomial k5 success ORIGINAL power multiply5 := mul(1/16, 1/2)

After

CLT binomial k5 success ORIGINAL power multiply5 = 1/32

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

CLT binomial k5 failure ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k5 failure ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · calculation

Before

CLT binomial ORIGINAL power product k5 := mul(1/32, 1/2)

After

CLT binomial ORIGINAL power product k5 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

CLT binomial ORIGINAL choose times joint k5 := mul(6, 1/64)

After

CLT binomial ORIGINAL choose times joint k5 = 3/32

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · implies

Before

source roles = [6, 5, 1/2, 1/2]

After

derived = [6, 1/32, 1/2, 3/32]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S056

S058 · calculation

Before

CLT binomial k6 success ORIGINAL power multiply1 := mul(1, 1/2)

After

CLT binomial k6 success ORIGINAL power multiply1 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · calculation

Before

CLT binomial k6 success ORIGINAL power multiply2 := mul(1/2, 1/2)

After

CLT binomial k6 success ORIGINAL power multiply2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · calculation

Before

CLT binomial k6 success ORIGINAL power multiply3 := mul(1/4, 1/2)

After

CLT binomial k6 success ORIGINAL power multiply3 = 1/8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · calculation

Before

CLT binomial k6 success ORIGINAL power multiply4 := mul(1/8, 1/2)

After

CLT binomial k6 success ORIGINAL power multiply4 = 1/16

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · calculation

Before

CLT binomial k6 success ORIGINAL power multiply5 := mul(1/16, 1/2)

After

CLT binomial k6 success ORIGINAL power multiply5 = 1/32

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · calculation

Before

CLT binomial k6 success ORIGINAL power multiply6 := mul(1/32, 1/2)

After

CLT binomial k6 success ORIGINAL power multiply6 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · implies

Before

source roles = [CLT binomial k6 failure, 1/2, 0]

After

derived = [1]

Conditions: none

A zeroth power is the empty product1,not0 because n or the base is0.

Step identifiers and dependencies

Rule: CLT_geometric_probability_power_zero_empty_product · Uses: G001, S063

S065 · calculation

Before

CLT binomial ORIGINAL power product k6 := mul(1/64, 1)

After

CLT binomial ORIGINAL power product k6 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · calculation

Before

CLT binomial ORIGINAL choose times joint k6 := mul(1, 1/64)

After

CLT binomial ORIGINAL choose times joint k6 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S065

S067 · implies

Before

source roles = [6, 6, 1/2, 1/2]

After

derived = [1, 1/64, 1, 1/64]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S066

S068 · calculation

Before

CLT binomial ORIGINAL total mass sum0 := sum(0, 1/64)

After

CLT binomial ORIGINAL total mass sum0 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

CLT binomial ORIGINAL INCLUSIVE CDF sum0 := sum(0, 1/64)

After

CLT binomial ORIGINAL INCLUSIVE CDF sum0 = 1/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · calculation

Before

CLT binomial ORIGINAL total mass sum1 := sum(1/64, 3/32)

After

CLT binomial ORIGINAL total mass sum1 = 7/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · calculation

Before

CLT binomial ORIGINAL INCLUSIVE CDF sum1 := sum(1/64, 3/32)

After

CLT binomial ORIGINAL INCLUSIVE CDF sum1 = 7/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · calculation

Before

CLT binomial ORIGINAL total mass sum2 := sum(7/64, 15/64)

After

CLT binomial ORIGINAL total mass sum2 = 11/32

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · calculation

Before

CLT binomial ORIGINAL INCLUSIVE CDF sum2 := sum(7/64, 15/64)

After

CLT binomial ORIGINAL INCLUSIVE CDF sum2 = 11/32

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · calculation

Before

CLT binomial ORIGINAL total mass sum3 := sum(11/32, 5/16)

After

CLT binomial ORIGINAL total mass sum3 = 21/32

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S073

S075 · calculation

Before

CLT binomial ORIGINAL INCLUSIVE CDF sum3 := sum(11/32, 5/16)

After

CLT binomial ORIGINAL INCLUSIVE CDF sum3 = 21/32

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · calculation

Before

CLT binomial ORIGINAL total mass sum4 := sum(21/32, 15/64)

After

CLT binomial ORIGINAL total mass sum4 = 57/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S075

S077 · calculation

Before

CLT binomial ORIGINAL total mass sum5 := sum(57/64, 3/32)

After

CLT binomial ORIGINAL total mass sum5 = 63/64

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S076

S078 · calculation

Before

CLT binomial ORIGINAL total mass sum6 := sum(63/64, 1/64)

After

CLT binomial ORIGINAL total mass sum6 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S077

S079 · implies

Before

source roles = [6, 1/2, 3, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]]]

After

derived = [1, 21/32, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), No]

Conditions: none

List every k=0..n;the total is(p+(1-p))^n=1.The CDF includes k<=threshold,equality included.The z expression is a symbolic standardized threshold,not a Gaussian approximation,error bound or general CLT proof.

Step identifiers and dependencies

Rule: CLT_complete_binomial_total_inclusive_CDF_NO_normal_approximation · Uses: G001, S078

Reverse checks

S079 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S078 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum6 with exact rational arithmetic, not rounded decimals.
S077 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum5 with exact rational arithmetic, not rounded decimals.
S076 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum4 with exact rational arithmetic, not rounded decimals.
S075 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum3 with exact rational arithmetic, not rounded decimals.
S074 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum3 with exact rational arithmetic, not rounded decimals.
S073 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum2 with exact rational arithmetic, not rounded decimals.
S072 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum2 with exact rational arithmetic, not rounded decimals.
S071 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum1 with exact rational arithmetic, not rounded decimals.
S070 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum1 with exact rational arithmetic, not rounded decimals.
S069 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum0 with exact rational arithmetic, not rounded decimals.
S068 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum0 with exact rational arithmetic, not rounded decimals.
S067 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S066 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k6 with exact rational arithmetic, not rounded decimals.
S065 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k6 with exact rational arithmetic, not rounded decimals.
S064 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S063 · premise_and_result_audit — Recompute CLT binomial k6 success ORIGINAL power multiply6 with exact rational arithmetic, not rounded decimals.
S062 · premise_and_result_audit — Recompute CLT binomial k6 success ORIGINAL power multiply5 with exact rational arithmetic, not rounded decimals.
S061 · premise_and_result_audit — Recompute CLT binomial k6 success ORIGINAL power multiply4 with exact rational arithmetic, not rounded decimals.
S060 · premise_and_result_audit — Recompute CLT binomial k6 success ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
S059 · premise_and_result_audit — Recompute CLT binomial k6 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S058 · premise_and_result_audit — Recompute CLT binomial k6 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S057 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S056 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k5 with exact rational arithmetic, not rounded decimals.
S055 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k5 with exact rational arithmetic, not rounded decimals.
S054 · premise_and_result_audit — Recompute CLT binomial k5 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S053 · premise_and_result_audit — Recompute CLT binomial k5 success ORIGINAL power multiply5 with exact rational arithmetic, not rounded decimals.
S052 · premise_and_result_audit — Recompute CLT binomial k5 success ORIGINAL power multiply4 with exact rational arithmetic, not rounded decimals.
S051 · premise_and_result_audit — Recompute CLT binomial k5 success ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
S050 · premise_and_result_audit — Recompute CLT binomial k5 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S049 · premise_and_result_audit — Recompute CLT binomial k5 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S048 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S047 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k4 with exact rational arithmetic, not rounded decimals.
S046 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k4 with exact rational arithmetic, not rounded decimals.
S045 · premise_and_result_audit — Recompute CLT binomial k4 failure ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S044 · premise_and_result_audit — Recompute CLT binomial k4 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute CLT binomial k4 success ORIGINAL power multiply4 with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute CLT binomial k4 success ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute CLT binomial k4 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute CLT binomial k4 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S038 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k3 with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k3 with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Recompute CLT binomial k3 failure ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute CLT binomial k3 failure ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute CLT binomial k3 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute CLT binomial k3 success ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute CLT binomial k3 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute CLT binomial k3 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S029 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k2 with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k2 with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute CLT binomial k2 failure ORIGINAL power multiply4 with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute CLT binomial k2 failure ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute CLT binomial k2 failure ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute CLT binomial k2 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute CLT binomial k2 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute CLT binomial k2 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S020 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k1 with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k1 with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply5 with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply4 with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute CLT binomial k1 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S011 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k0 with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k0 with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute CLT binomial k0 failure ORIGINAL power multiply6 with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute CLT binomial k0 failure ORIGINAL power multiply5 with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute CLT binomial k0 failure ORIGINAL power multiply4 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute CLT binomial k0 failure ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute CLT binomial k0 failure ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute CLT binomial k0 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
S002 · premise_and_result_audit — Recompute CLT binomial ORIGINAL one minus probability with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.

Result

(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 3, 21/32, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

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.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: e9932a2781c56a8fb5ee686a1310152fc1f3eb934bfbfc5131589833889ed8b3
candidate record SHA-256: 0f07ab1dce11d720fba7115efc6f4851a0dd56b55818d6fadada7f7dd3aac515
semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
Lean: NOT_RUN · human review: NOT_PERFORMED

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