# Question 27246: 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-0071) 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. R0053-27246 · q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0071 ## Original question and givens Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=2, 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=2, 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 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore 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 dependenciesRule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S011 S013 · 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, S012 S014 · 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, S013 S015 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore source roles = [6, 1, 1/2, 1/2] After derived = [6, 1/2, 1/32, 3/32] 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, S020 S022 · 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, S021 S023 · 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, S022 S024 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · impliesBefore source roles = [6, 2, 1/2, 1/2] After derived = [15, 1/4, 1/16, 15/64] 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 dependenciesRule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S029 S031 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · impliesBefore source roles = [6, 3, 1/2, 1/2] After derived = [20, 1/8, 1/8, 5/16] 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 dependenciesRule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S038 S040 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · impliesBefore source roles = [6, 4, 1/2, 1/2] After derived = [15, 1/16, 1/4, 15/64] 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 dependenciesRule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S047 S049 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · impliesBefore source roles = [6, 5, 1/2, 1/2] After derived = [6, 1/32, 1/2, 3/32] 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, S056 S058 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S060 S062 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · impliesBefore 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 dependenciesRule: CLT_geometric_probability_power_zero_empty_product · Uses: G001, S063 S065 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · impliesBefore source roles = [6, 6, 1/2, 1/2] After derived = [1, 1/64, 1, 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 dependenciesRule: CLT_EACH_binomial_lattice_exact_count_and_probability · Uses: G001, S066 S068 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S073 S075 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S074 S076 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S075 S077 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · impliesBefore source roles = [6, 1/2, 2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]]] After derived = [1, 11/32, z_threshold=(2-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, S077 ## Reverse checks - S078 · 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. - S077 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum6 with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum5 with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Recompute CLT binomial ORIGINAL total mass sum4 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, 2, 11/32, Yes, z_threshold=(2-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: 553491c5a59c16eb474c7be4021518d85d705863d618dec8e6b6cbf239657f05 - candidate record SHA-256: bf4cfc1afa211db017c809773b489cd4e25354ee23f8639bcd05c47a9adc015d - 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-0071.json)