# Question 27226: An Exact Binomial Lattice Is Not a Normal Approximation [Return to catalogue page 137](https://ocme.evemisslab.com/en/questions/page/137/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations/#q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0051) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0053-27226 · q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0051 ## Original question and givens Use Bernoulli success probability p=1/2, sample size n=2, integer threshold=0, 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=2, integer threshold=0, 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](https://ocme.evemisslab.com/en/mko/mko-central-limit-theorem-finite-variance-iid/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore 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 dependenciesRule: CLT_geometric_probability_power_zero_empty_product · Uses: G001, S002 S004 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore CLT binomial ORIGINAL power product k0 := mul(1, 1/4) After CLT binomial ORIGINAL power product k0 = 1/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore CLT binomial ORIGINAL choose times joint k0 := mul(1, 1/4) After CLT binomial ORIGINAL choose times joint k0 = 1/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [2, 0, 1/2, 1/2] After derived = [1, 1, 1/4, 1/4] 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 dependenciesRule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S007 S009 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore CLT binomial ORIGINAL power product k1 := mul(1/2, 1/2) After CLT binomial ORIGINAL power product k1 = 1/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore CLT binomial ORIGINAL choose times joint k1 := mul(2, 1/4) After CLT binomial ORIGINAL choose times joint k1 = 1/2 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore source roles = [2, 1, 1/2, 1/2] After derived = [2, 1/2, 1/2, 1/2] 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 dependenciesRule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S012 S014 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore source roles = [CLT binomial k2 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 dependenciesRule: CLT_geometric_probability_power_zero_empty_product · Uses: G001, S015 S017 · calculationBefore CLT binomial ORIGINAL power product k2 := mul(1/4, 1) After CLT binomial ORIGINAL power product k2 = 1/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore CLT binomial ORIGINAL choose times joint k2 := mul(1, 1/4) After CLT binomial ORIGINAL choose times joint k2 = 1/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore source roles = [2, 2, 1/2, 1/2] After derived = [1, 1/4, 1, 1/4] 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 dependenciesRule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S018 S020 · calculationBefore CLT binomial ORIGINAL total mass sum0 := sum(0, 1/4) After CLT binomial ORIGINAL total mass sum0 = 1/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore CLT binomial ORIGINAL INCLUSIVE CDF sum0 := sum(0, 1/4) After CLT binomial ORIGINAL INCLUSIVE CDF sum0 = 1/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore CLT binomial ORIGINAL total mass sum1 := sum(1/4, 1/2) After CLT binomial ORIGINAL total mass sum1 = 3/4 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore CLT binomial ORIGINAL total mass sum2 := sum(3/4, 1/4) After CLT binomial ORIGINAL total mass sum2 = 1 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore source roles = [2, 1/2, 0, [[0, 1/4], [1, 1/2], [2, 1/4]]] After derived = [1, 1/4, z_threshold=(0-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 dependenciesRule: CLT_complete_binomial_total_inclusive_CDF_NO_normal_approximation · Uses: G001, S023 ## Reverse checks - 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 total mass sum2 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum1 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute CLT binomial ORIGINAL INCLUSIVE CDF sum0 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum0 with exact rational arithmetic, not rounded decimals. - S019 · 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. - S018 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k2 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k2 with exact rational arithmetic, not rounded decimals. - S016 · 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. - S015 · premise_and_result_audit — Recompute CLT binomial k2 success ORIGINAL power multiply2 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute CLT binomial k2 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals. - S013 · 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. - S012 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute CLT binomial k1 failure ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute CLT binomial k1 success ORIGINAL power multiply1 with exact rational arithmetic, not rounded decimals. - S008 · 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. - S007 · premise_and_result_audit — Recompute CLT binomial ORIGINAL choose times joint k0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute CLT binomial ORIGINAL power product k0 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 (2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 0, 1/4, Yes, z_threshold=(0-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: 25fe9d8a6c587c7bf833e953b4d8628f704512a4306ce42921d80c6147192fe2 - candidate record SHA-256: 5776fd883acaf817f29a3fa6fff43f231d50e270535c31d3a6f7b9b48e9b6cbf - semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62 - question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0051.json)