Question 27234: 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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R0053-27234 · q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0059

Original question and givens

Use Bernoulli success probability p=1/2, sample size n=4, integer threshold=1, 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=4, integer threshold=1, 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 ORIGINAL power product k0 := mul(1, 1/16)

After

CLT binomial ORIGINAL power product k0 = 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, S007

S009 · calculation

Before

CLT binomial ORIGINAL choose times joint k0 := mul(1, 1/16)

After

CLT binomial ORIGINAL choose times joint k0 = 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, S008

S010 · implies

Before

source roles = [4, 0, 1/2, 1/2]

After

derived = [1, 1, 1/16, 1/16]

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, S009

S011 · 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, S010

S012 · 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, S011

S013 · 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, S012

S014 · 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, S013

S015 · calculation

Before

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

After

CLT binomial ORIGINAL power product k1 = 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, S014

S016 · calculation

Before

CLT binomial ORIGINAL choose times joint k1 := mul(4, 1/16)

After

CLT binomial ORIGINAL choose times joint k1 = 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, S015

S017 · implies

Before

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

After

derived = [4, 1/2, 1/8, 1/4]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S016

S018 · 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, S017

S019 · 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, S018

S020 · 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, S019

S021 · 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, S020

S022 · calculation

Before

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

After

CLT binomial ORIGINAL power product k2 = 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, S021

S023 · calculation

Before

CLT binomial ORIGINAL choose times joint k2 := mul(6, 1/16)

After

CLT binomial ORIGINAL choose times joint k2 = 3/8

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, S022

S024 · implies

Before

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

After

derived = [6, 1/4, 1/4, 3/8]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S023

S025 · 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, S024

S026 · 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, S025

S027 · 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, S026

S028 · 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, S027

S029 · calculation

Before

CLT binomial ORIGINAL power product k3 := mul(1/8, 1/2)

After

CLT binomial ORIGINAL power product k3 = 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, S028

S030 · calculation

Before

CLT binomial ORIGINAL choose times joint k3 := mul(4, 1/16)

After

CLT binomial ORIGINAL choose times joint k3 = 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, S029

S031 · implies

Before

source roles = [4, 3, 1/2, 1/2]

After

derived = [4, 1/8, 1/2, 1/4]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S030

S032 · 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, S031

S033 · 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, S032

S034 · 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, S033

S035 · 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, S034

S036 · implies

Before

source roles = [CLT binomial k4 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, S035

S037 · calculation

Before

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

After

CLT binomial ORIGINAL power product k4 = 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, S036

S038 · calculation

Before

CLT binomial ORIGINAL choose times joint k4 := mul(1, 1/16)

After

CLT binomial ORIGINAL choose times joint k4 = 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, S037

S039 · implies

Before

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

After

derived = [1, 1/16, 1, 1/16]

Conditions: none

Step identifiers and dependencies

Rule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S038

S040 · calculation

Before

CLT binomial ORIGINAL total mass sum0 := sum(0, 1/16)

After

CLT binomial ORIGINAL total mass sum0 = 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, S039

S041 · calculation

Before

CLT binomial ORIGINAL INCLUSIVE CDF sum0 := sum(0, 1/16)

After

CLT binomial ORIGINAL INCLUSIVE CDF sum0 = 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, S040

S042 · calculation

Before

CLT binomial ORIGINAL total mass sum1 := sum(1/16, 1/4)

After

CLT binomial ORIGINAL total mass sum1 = 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, S041

S043 · calculation

Before

CLT binomial ORIGINAL INCLUSIVE CDF sum1 := sum(1/16, 1/4)

After

CLT binomial ORIGINAL INCLUSIVE CDF sum1 = 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, S042

S044 · calculation

Before

CLT binomial ORIGINAL total mass sum2 := sum(5/16, 3/8)

After

CLT binomial ORIGINAL total mass sum2 = 11/16

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, S043

S045 · calculation

Before

CLT binomial ORIGINAL total mass sum3 := sum(11/16, 1/4)

After

CLT binomial ORIGINAL total mass sum3 = 15/16

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, S044

S046 · calculation

Before

CLT binomial ORIGINAL total mass sum4 := sum(15/16, 1/16)

After

CLT binomial ORIGINAL total mass sum4 = 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, S045

S047 · implies

Before

source roles = [4, 1/2, 1, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]]]

After

derived = [1, 5/16, z_threshold=(1-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, S046

Reverse checks

S047 · 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.
S046 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum4 with exact rational arithmetic, not rounded decimals.
S045 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum3 with exact rational arithmetic, not rounded decimals.
S044 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum2 with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum1 with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum1 with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum0 with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum0 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 k4 with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k4 with exact rational arithmetic, not rounded decimals.
S036 · 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.
S035 · premise_and_result_audit — Recompute CLT binomial k4 success ORIGINAL power multiply4 with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute CLT binomial k4 success ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute CLT binomial k4 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute CLT binomial k4 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S031 · 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.
S030 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k3 with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k3 with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute CLT binomial k3 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute CLT binomial k3 success ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute CLT binomial k3 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute CLT binomial k3 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S024 · 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.
S023 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k2 with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k2 with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute CLT binomial k2 failure ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute CLT binomial k2 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute CLT binomial k2 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute CLT binomial k2 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S017 · 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.
S016 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k1 with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k1 with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply3 with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute CLT binomial k1 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals.
S010 · 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.
S009 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k0 with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k0 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

(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 1, 5/16, Yes, z_threshold=(1-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.

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Lean: NOT_RUN · human review: NOT_PERFORMED

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