# Question 27800: All Nonzero Prime Residues Pair with Inverses, with a Special p=2 Case [Return to catalogue page 139](https://ocme.evemisslab.com/en/questions/page/139/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261118-r04-wilson-theorem-foundations/#q-20261015-advance-20261118-r04-wilson-theorem-0050) 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. R0054-27800 · q-20261015-advance-20261118-r04-wilson-theorem-0050 ## Original question and givens Use prime modulus p=97. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence). - G001 — Use prime modulus p=97. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence). ## 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-wilson-theorem](https://ocme.evemisslab.com/en/mko/mko-wilson-theorem/) ## Step-by-step derivation S001 · impliesBefore source roles = [wilson-prime-inverse-pairing] After derived = [[p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence]] Conditions: none List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [97, 1] After derived = [1, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S001 S003 · impliesBefore source roles = [97, 2] After derived = [49, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S002 S004 · impliesBefore source roles = [97, 3] After derived = [65, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S003 S005 · impliesBefore source roles = [97, 4] After derived = [73, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S004 S006 · impliesBefore source roles = [97, 5] After derived = [39, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S005 S007 · impliesBefore source roles = [97, 6] After derived = [81, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S006 S008 · impliesBefore source roles = [97, 7] After derived = [14, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S007 S009 · impliesBefore source roles = [97, 8] After derived = [85, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S008 S010 · impliesBefore source roles = [97, 9] After derived = [54, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S009 S011 · impliesBefore source roles = [97, 10] After derived = [68, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S010 S012 · impliesBefore source roles = [97, 11] After derived = [53, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S011 S013 · impliesBefore source roles = [97, 12] After derived = [89, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S012 S014 · impliesBefore source roles = [97, 13] After derived = [15, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S013 S015 · impliesBefore source roles = [97, 14] After derived = [7, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S014 S016 · impliesBefore source roles = [97, 15] After derived = [13, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S015 S017 · impliesBefore source roles = [97, 16] After derived = [91, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S016 S018 · impliesBefore source roles = [97, 17] After derived = [40, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S017 S019 · impliesBefore source roles = [97, 18] After derived = [27, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S018 S020 · impliesBefore source roles = [97, 19] After derived = [46, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S019 S021 · impliesBefore source roles = [97, 20] After derived = [34, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S020 S022 · impliesBefore source roles = [97, 21] After derived = [37, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S021 S023 · impliesBefore source roles = [97, 22] After derived = [75, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S022 S024 · impliesBefore source roles = [97, 23] After derived = [38, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S023 S025 · impliesBefore source roles = [97, 24] After derived = [93, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S024 S026 · impliesBefore source roles = [97, 25] After derived = [66, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S025 S027 · impliesBefore source roles = [97, 26] After derived = [56, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S026 S028 · impliesBefore source roles = [97, 27] After derived = [18, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S027 S029 · impliesBefore source roles = [97, 28] After derived = [52, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S028 S030 · impliesBefore source roles = [97, 29] After derived = [87, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S029 S031 · impliesBefore source roles = [97, 30] After derived = [55, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S030 S032 · impliesBefore source roles = [97, 31] After derived = [72, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S031 S033 · impliesBefore source roles = [97, 32] After derived = [94, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S032 S034 · impliesBefore source roles = [97, 33] After derived = [50, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S033 S035 · impliesBefore source roles = [97, 34] After derived = [20, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S034 S036 · impliesBefore source roles = [97, 35] After derived = [61, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S035 S037 · impliesBefore source roles = [97, 36] After derived = [62, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S036 S038 · impliesBefore source roles = [97, 37] After derived = [21, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S037 S039 · impliesBefore source roles = [97, 38] After derived = [23, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S038 S040 · impliesBefore source roles = [97, 39] After derived = [5, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S039 S041 · impliesBefore source roles = [97, 40] After derived = [17, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S040 S042 · impliesBefore source roles = [97, 41] After derived = [71, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S041 S043 · impliesBefore source roles = [97, 42] After derived = [67, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S042 S044 · impliesBefore source roles = [97, 43] After derived = [88, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S043 S045 · impliesBefore source roles = [97, 44] After derived = [86, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S044 S046 · impliesBefore source roles = [97, 45] After derived = [69, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S045 S047 · impliesBefore source roles = [97, 46] After derived = [19, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S046 S048 · impliesBefore source roles = [97, 47] After derived = [64, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S047 S049 · impliesBefore source roles = [97, 48] After derived = [95, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S048 S050 · impliesBefore source roles = [97, 49] After derived = [2, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S049 S051 · impliesBefore source roles = [97, 50] After derived = [33, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S050 S052 · impliesBefore source roles = [97, 51] After derived = [78, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S051 S053 · impliesBefore source roles = [97, 52] After derived = [28, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S052 S054 · impliesBefore source roles = [97, 53] After derived = [11, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S053 S055 · impliesBefore source roles = [97, 54] After derived = [9, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S054 S056 · impliesBefore source roles = [97, 55] After derived = [30, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S055 S057 · impliesBefore source roles = [97, 56] After derived = [26, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S056 S058 · impliesBefore source roles = [97, 57] After derived = [80, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S057 S059 · impliesBefore source roles = [97, 58] After derived = [92, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S058 S060 · impliesBefore source roles = [97, 59] After derived = [74, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S059 S061 · impliesBefore source roles = [97, 60] After derived = [76, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S060 S062 · impliesBefore source roles = [97, 61] After derived = [35, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S061 S063 · impliesBefore source roles = [97, 62] After derived = [36, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S062 S064 · impliesBefore source roles = [97, 63] After derived = [77, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S063 S065 · impliesBefore source roles = [97, 64] After derived = [47, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S064 S066 · impliesBefore source roles = [97, 65] After derived = [3, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S065 S067 · impliesBefore source roles = [97, 66] After derived = [25, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S066 S068 · impliesBefore source roles = [97, 67] After derived = [42, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S067 S069 · impliesBefore source roles = [97, 68] After derived = [10, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S068 S070 · impliesBefore source roles = [97, 69] After derived = [45, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S069 S071 · impliesBefore source roles = [97, 70] After derived = [79, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S070 S072 · impliesBefore source roles = [97, 71] After derived = [41, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S071 S073 · impliesBefore source roles = [97, 72] After derived = [31, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S072 S074 · impliesBefore source roles = [97, 73] After derived = [4, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S073 S075 · impliesBefore source roles = [97, 74] After derived = [59, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S074 S076 · impliesBefore source roles = [97, 75] After derived = [22, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S075 S077 · impliesBefore source roles = [97, 76] After derived = [60, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S076 S078 · impliesBefore source roles = [97, 77] After derived = [63, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S077 S079 · impliesBefore source roles = [97, 78] After derived = [51, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S078 S080 · impliesBefore source roles = [97, 79] After derived = [70, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S079 S081 · impliesBefore source roles = [97, 80] After derived = [57, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S080 S082 · impliesBefore source roles = [97, 81] After derived = [6, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S081 S083 · impliesBefore source roles = [97, 82] After derived = [84, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S082 S084 · impliesBefore source roles = [97, 83] After derived = [90, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S083 S085 · impliesBefore source roles = [97, 84] After derived = [82, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S084 S086 · impliesBefore source roles = [97, 85] After derived = [8, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S085 S087 · impliesBefore source roles = [97, 86] After derived = [44, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S086 S088 · impliesBefore source roles = [97, 87] After derived = [29, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S087 S089 · impliesBefore source roles = [97, 88] After derived = [43, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S088 S090 · impliesBefore source roles = [97, 89] After derived = [12, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S089 S091 · impliesBefore source roles = [97, 90] After derived = [83, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S090 S092 · impliesBefore source roles = [97, 91] After derived = [16, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S091 S093 · impliesBefore source roles = [97, 92] After derived = [58, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S092 S094 · impliesBefore source roles = [97, 93] After derived = [24, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S093 S095 · impliesBefore source roles = [97, 94] After derived = [32, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S094 S096 · impliesBefore source roles = [97, 95] After derived = [48, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S095 S097 · impliesBefore source roles = [97, 96] After derived = [96, 1] Conditions: none Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a. Step identifiers and dependenciesRule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S096 S098 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 := mul(1, 1) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 = 1 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S097 S099 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 1, 1, 1] After derived = [0, 1, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S098 S100 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 := mul(1, 2) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 = 2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S099 S101 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 2, 1, 2] After derived = [0, 2, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S100 S102 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 := mul(2, 3) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 = 6 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S101 S103 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 3, 2, 6] After derived = [0, 6, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S102 S104 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 := mul(6, 4) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 = 24 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S103 S105 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 4, 6, 24] After derived = [0, 24, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S104 S106 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 := mul(24, 5) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 = 120 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S105 S107 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 5, 24, 120] After derived = [1, 23, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S106 S108 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(23, 6) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 138 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S107 S109 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 6, 23, 138] After derived = [1, 41, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S108 S110 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(41, 7) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 287 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S109 S111 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 7, 41, 287] After derived = [2, 93, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S110 S112 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(93, 8) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 744 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S111 S113 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 8, 93, 744] After derived = [7, 65, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S112 S114 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(65, 9) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 585 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S113 S115 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 9, 65, 585] After derived = [6, 3, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S114 S116 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(3, 10) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 30 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S115 S117 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 10, 3, 30] After derived = [0, 30, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S116 S118 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(30, 11) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 330 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S117 S119 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 11, 30, 330] After derived = [3, 39, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S118 S120 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(39, 12) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 468 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S119 S121 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 12, 39, 468] After derived = [4, 80, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S120 S122 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(80, 13) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 1040 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S121 S123 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 13, 80, 1040] After derived = [10, 70, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S122 S124 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(70, 14) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 980 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S123 S125 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 14, 70, 980] After derived = [10, 10, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S124 S126 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(10, 15) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 150 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S125 S127 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 15, 10, 150] After derived = [1, 53, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S126 S128 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(53, 16) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 848 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S127 S129 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 16, 53, 848] After derived = [8, 72, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S128 S130 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(72, 17) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 1224 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S129 S131 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 17, 72, 1224] After derived = [12, 60, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S130 S132 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(60, 18) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 1080 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S131 S133 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 18, 60, 1080] After derived = [11, 13, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S132 S134 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(13, 19) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 247 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S133 S135 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 19, 13, 247] After derived = [2, 53, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S134 S136 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(53, 20) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 1060 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S135 S137 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 20, 53, 1060] After derived = [10, 90, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S136 S138 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(90, 21) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 1890 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S137 S139 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 21, 90, 1890] After derived = [19, 47, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S138 S140 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(47, 22) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 1034 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S139 S141 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 22, 47, 1034] After derived = [10, 64, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S140 S142 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(64, 23) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 1472 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S141 S143 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 23, 64, 1472] After derived = [15, 17, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S142 S144 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(17, 24) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 408 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S143 S145 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 24, 17, 408] After derived = [4, 20, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S144 S146 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(20, 25) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 500 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S145 S147 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 25, 20, 500] After derived = [5, 15, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S146 S148 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(15, 26) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 390 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S147 S149 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 26, 15, 390] After derived = [4, 2, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S148 S150 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(2, 27) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 54 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S149 S151 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 27, 2, 54] After derived = [0, 54, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S150 S152 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(54, 28) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 1512 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S151 S153 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 28, 54, 1512] After derived = [15, 57, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S152 S154 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(57, 29) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 1653 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S153 S155 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 29, 57, 1653] After derived = [17, 4, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S154 S156 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(4, 30) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 120 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S155 S157 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 30, 4, 120] After derived = [1, 23, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S156 S158 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(23, 31) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 713 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S157 S159 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 31, 23, 713] After derived = [7, 34, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S158 S160 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(34, 32) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 1088 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S159 S161 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 32, 34, 1088] After derived = [11, 21, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S160 S162 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(21, 33) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 693 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S161 S163 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 33, 21, 693] After derived = [7, 14, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S162 S164 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(14, 34) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 476 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S163 S165 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 34, 14, 476] After derived = [4, 88, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S164 S166 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(88, 35) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 3080 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S165 S167 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 35, 88, 3080] After derived = [31, 73, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S166 S168 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(73, 36) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 2628 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S167 S169 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 36, 73, 2628] After derived = [27, 9, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S168 S170 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 := mul(9, 37) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 = 333 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S169 S171 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 37, 9, 333] After derived = [3, 42, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S170 S172 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 := mul(42, 38) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 = 1596 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S171 S173 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 38, 42, 1596] After derived = [16, 44, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S172 S174 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 := mul(44, 39) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 = 1716 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S173 S175 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 39, 44, 1716] After derived = [17, 67, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S174 S176 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 := mul(67, 40) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 = 2680 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S175 S177 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 40, 67, 2680] After derived = [27, 61, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S176 S178 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 := mul(61, 41) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 = 2501 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S177 S179 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 41, 61, 2501] After derived = [25, 76, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S178 S180 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 := mul(76, 42) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 = 3192 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S179 S181 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 42, 76, 3192] After derived = [32, 88, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S180 S182 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 := mul(88, 43) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 = 3784 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S181 S183 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 43, 88, 3784] After derived = [39, 1, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S182 S184 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 := mul(1, 44) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 = 44 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S183 S185 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 44, 1, 44] After derived = [0, 44, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S184 S186 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 := mul(44, 45) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 = 1980 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S185 S187 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 45, 44, 1980] After derived = [20, 40, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S186 S188 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 := mul(40, 46) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 = 1840 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S187 S189 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 46, 40, 1840] After derived = [18, 94, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S188 S190 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 := mul(94, 47) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 = 4418 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S189 S191 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 47, 94, 4418] After derived = [45, 53, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S190 S192 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 := mul(53, 48) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 = 2544 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S191 S193 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 48, 53, 2544] After derived = [26, 22, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S192 S194 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 := mul(22, 49) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 = 1078 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S193 S195 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 49, 22, 1078] After derived = [11, 11, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S194 S196 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 := mul(11, 50) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 = 550 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S195 S197 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 50, 11, 550] After derived = [5, 65, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S196 S198 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 := mul(65, 51) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 = 3315 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S197 S199 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 51, 65, 3315] After derived = [34, 17, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S198 S200 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 := mul(17, 52) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 = 884 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S199 S201 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 52, 17, 884] After derived = [9, 11, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S200 S202 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 := mul(11, 53) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 = 583 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S201 S203 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 53, 11, 583] After derived = [6, 1, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S202 S204 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 := mul(1, 54) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 = 54 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S203 S205 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 54, 1, 54] After derived = [0, 54, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S204 S206 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 := mul(54, 55) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 = 2970 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S205 S207 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 55, 54, 2970] After derived = [30, 60, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S206 S208 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 := mul(60, 56) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 = 3360 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S207 S209 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 56, 60, 3360] After derived = [34, 62, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S208 S210 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 := mul(62, 57) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 = 3534 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S209 S211 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 57, 62, 3534] After derived = [36, 42, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S210 S212 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 := mul(42, 58) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 = 2436 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S211 S213 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 58, 42, 2436] After derived = [25, 11, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S212 S214 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 := mul(11, 59) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 = 649 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S213 S215 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 59, 11, 649] After derived = [6, 67, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S214 S216 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 := mul(67, 60) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 = 4020 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S215 S217 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 60, 67, 4020] After derived = [41, 43, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S216 S218 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 := mul(43, 61) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 = 2623 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S217 S219 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 61, 43, 2623] After derived = [27, 4, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S218 S220 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 := mul(4, 62) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 = 248 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S219 S221 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 62, 4, 248] After derived = [2, 54, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S220 S222 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 := mul(54, 63) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 = 3402 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S221 S223 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 63, 54, 3402] After derived = [35, 7, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S222 S224 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 := mul(7, 64) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 = 448 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S223 S225 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 64, 7, 448] After derived = [4, 60, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S224 S226 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 := mul(60, 65) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 = 3900 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S225 S227 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 65, 60, 3900] After derived = [40, 20, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S226 S228 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 := mul(20, 66) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 = 1320 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S227 S229 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 66, 20, 1320] After derived = [13, 59, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S228 S230 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product67 := mul(59, 67) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product67 = 3953 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S229 S231 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 67, 59, 3953] After derived = [40, 73, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S230 S232 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product68 := mul(73, 68) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product68 = 4964 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S231 S233 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 68, 73, 4964] After derived = [51, 17, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S232 S234 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product69 := mul(17, 69) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product69 = 1173 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S233 S235 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 69, 17, 1173] After derived = [12, 9, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S234 S236 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product70 := mul(9, 70) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product70 = 630 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S235 S237 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 70, 9, 630] After derived = [6, 48, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S236 S238 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product71 := mul(48, 71) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product71 = 3408 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S237 S239 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 71, 48, 3408] After derived = [35, 13, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S238 S240 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product72 := mul(13, 72) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product72 = 936 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S239 S241 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 72, 13, 936] After derived = [9, 63, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S240 S242 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product73 := mul(63, 73) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product73 = 4599 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S241 S243 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 73, 63, 4599] After derived = [47, 40, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S242 S244 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product74 := mul(40, 74) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product74 = 2960 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S243 S245 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 74, 40, 2960] After derived = [30, 50, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S244 S246 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product75 := mul(50, 75) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product75 = 3750 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S245 S247 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 75, 50, 3750] After derived = [38, 64, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S246 S248 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product76 := mul(64, 76) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product76 = 4864 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S247 S249 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 76, 64, 4864] After derived = [50, 14, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S248 S250 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product77 := mul(14, 77) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product77 = 1078 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S249 S251 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 77, 14, 1078] After derived = [11, 11, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S250 S252 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product78 := mul(11, 78) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product78 = 858 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S251 S253 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 78, 11, 858] After derived = [8, 82, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S252 S254 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product79 := mul(82, 79) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product79 = 6478 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S253 S255 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 79, 82, 6478] After derived = [66, 76, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S254 S256 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product80 := mul(76, 80) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product80 = 6080 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S255 S257 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 80, 76, 6080] After derived = [62, 66, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S256 S258 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product81 := mul(66, 81) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product81 = 5346 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S257 S259 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 81, 66, 5346] After derived = [55, 11, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S258 S260 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product82 := mul(11, 82) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product82 = 902 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S259 S261 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 82, 11, 902] After derived = [9, 29, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S260 S262 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product83 := mul(29, 83) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product83 = 2407 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S261 S263 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 83, 29, 2407] After derived = [24, 79, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S262 S264 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product84 := mul(79, 84) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product84 = 6636 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S263 S265 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 84, 79, 6636] After derived = [68, 40, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S264 S266 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product85 := mul(40, 85) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product85 = 3400 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S265 S267 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 85, 40, 3400] After derived = [35, 5, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S266 S268 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product86 := mul(5, 86) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product86 = 430 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S267 S269 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 86, 5, 430] After derived = [4, 42, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S268 S270 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product87 := mul(42, 87) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product87 = 3654 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S269 S271 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 87, 42, 3654] After derived = [37, 65, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S270 S272 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product88 := mul(65, 88) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product88 = 5720 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S271 S273 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 88, 65, 5720] After derived = [58, 94, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S272 S274 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product89 := mul(94, 89) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product89 = 8366 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S273 S275 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 89, 94, 8366] After derived = [86, 24, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S274 S276 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product90 := mul(24, 90) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product90 = 2160 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S275 S277 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 90, 24, 2160] After derived = [22, 26, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S276 S278 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product91 := mul(26, 91) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product91 = 2366 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S277 S279 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 91, 26, 2366] After derived = [24, 38, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S278 S280 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product92 := mul(38, 92) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product92 = 3496 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S279 S281 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 92, 38, 3496] After derived = [36, 4, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S280 S282 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product93 := mul(4, 93) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product93 = 372 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S281 S283 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 93, 4, 372] After derived = [3, 81, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S282 S284 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product94 := mul(81, 94) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product94 = 7614 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S283 S285 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 94, 81, 7614] After derived = [78, 48, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S284 S286 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product95 := mul(48, 95) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product95 = 4560 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S285 S287 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 95, 48, 4560] After derived = [47, 1, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S286 S288 · calculationBefore Wilson ORIGINAL prime factorial ORIGINAL running factorial product96 := mul(1, 96) After Wilson ORIGINAL prime factorial ORIGINAL running factorial product96 = 96 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S287 S289 · impliesBefore source roles = [Wilson ORIGINAL prime factorial, 97, 96, 1, 96] After derived = [0, 96, Yes, Yes] Conditions: none Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry. Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S288 S290 · impliesBefore source roles = [97, [[1, 1], [2, 49], [3, 65], [4, 73], [5, 39], [6, 81], [7, 14], [8, 85], [9, 54], [10, 68], [11, 53], [12, 89], [13, 15], [14, 7], [15, 13], [16, 91], [17, 40], [18, 27], [19, 46], [20, 34], [21, 37], [22, 75], [23, 38], [24, 93], [25, 66], [26, 56], [27, 18], [28, 52], [29, 87], [30, 55], [31, 72], [32, 94], [33, 50], [34, 20], [35, 61], [36, 62], [37, 21], [38, 23], [39, 5], [40, 17], [41, 71], [42, 67], [43, 88], [44, 86], [45, 69], [46, 19], [47, 64], [48, 95], [49, 2], [50, 33], [51, 78], [52, 28], [53, 11], [54, 9], [55, 30], [56, 26], [57, 80], [58, 92], [59, 74], [60, 76], [61, 35], [62, 36], [63, 77], [64, 47], [65, 3], [66, 25], [67, 42], [68, 10], [69, 45], [70, 79], [71, 41], [72, 31], [73, 4], [74, 59], [75, 22], [76, 60], [77, 63], [78, 51], [79, 70], [80, 57], [81, 6], [82, 84], [83, 90], [84, 82], [85, 8], [86, 44], [87, 29], [88, 43], [89, 12], [90, 83], [91, 16], [92, 58], [93, 24], [94, 32], [95, 48], [96, 96]]] After derived = [[1, 96], [[2, 49, 1], [3, 65, 1], [4, 73, 1], [5, 39, 1], [6, 81, 1], [7, 14, 1], [8, 85, 1], [9, 54, 1], [10, 68, 1], [11, 53, 1], [12, 89, 1], [13, 15, 1], [16, 91, 1], [17, 40, 1], [18, 27, 1], [19, 46, 1], [20, 34, 1], [21, 37, 1], [22, 75, 1], [23, 38, 1], [24, 93, 1], [25, 66, 1], [26, 56, 1], [28, 52, 1], [29, 87, 1], [30, 55, 1], [31, 72, 1], [32, 94, 1], [33, 50, 1], [35, 61, 1], [36, 62, 1], [41, 71, 1], [42, 67, 1], [43, 88, 1], [44, 86, 1], [45, 69, 1], [47, 64, 1], [48, 95, 1], [51, 78, 1], [57, 80, 1], [58, 92, 1], [59, 74, 1], [60, 76, 1], [63, 77, 1], [70, 79, 1], [82, 84, 1], [83, 90, 1]], 96, Yes] Conditions: none Nonself inverse pairs multiply to1.A self-inverse satisfies(r-1)(r+1)=0 modulo a prime,so is±1.For p>2 their product is-1;at p2 only the single1=-1 remains.This general teaching argument does not turn the finite table/program into a Lean proof. Step identifiers and dependenciesRule: Wilson_GENERAL_inverse_pairing_and_p2_exception · Uses: G001, S289 ## Reverse checks - S290 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S289 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S288 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product96 with exact rational arithmetic, not rounded decimals. - S287 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S286 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product95 with exact rational arithmetic, not rounded decimals. - S285 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S284 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product94 with exact rational arithmetic, not rounded decimals. - S283 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S282 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product93 with exact rational arithmetic, not rounded decimals. - S281 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S280 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product92 with exact rational arithmetic, not rounded decimals. - S279 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S278 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product91 with exact rational arithmetic, not rounded decimals. - S277 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S276 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product90 with exact rational arithmetic, not rounded decimals. - S275 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S274 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product89 with exact rational arithmetic, not rounded decimals. - S273 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S272 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product88 with exact rational arithmetic, not rounded decimals. - S271 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S270 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product87 with exact rational arithmetic, not rounded decimals. - S269 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S268 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product86 with exact rational arithmetic, not rounded decimals. - S267 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S266 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product85 with exact rational arithmetic, not rounded decimals. - S265 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S264 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product84 with exact rational arithmetic, not rounded decimals. - S263 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S262 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product83 with exact rational arithmetic, not rounded decimals. - S261 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S260 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product82 with exact rational arithmetic, not rounded decimals. - S259 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S258 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product81 with exact rational arithmetic, not rounded decimals. - S257 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S256 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product80 with exact rational arithmetic, not rounded decimals. - S255 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S254 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product79 with exact rational arithmetic, not rounded decimals. - S253 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S252 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product78 with exact rational arithmetic, not rounded decimals. - S251 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S250 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product77 with exact rational arithmetic, not rounded decimals. - S249 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S248 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product76 with exact rational arithmetic, not rounded decimals. - S247 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S246 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product75 with exact rational arithmetic, not rounded decimals. - S245 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S244 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product74 with exact rational arithmetic, not rounded decimals. - S243 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S242 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product73 with exact rational arithmetic, not rounded decimals. - S241 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S240 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product72 with exact rational arithmetic, not rounded decimals. - S239 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S238 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product71 with exact rational arithmetic, not rounded decimals. - S237 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S236 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product70 with exact rational arithmetic, not rounded decimals. - S235 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S234 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product69 with exact rational arithmetic, not rounded decimals. - S233 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S232 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product68 with exact rational arithmetic, not rounded decimals. - S231 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S230 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product67 with exact rational arithmetic, not rounded decimals. - S229 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S228 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 with exact rational arithmetic, not rounded decimals. - S227 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S226 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 with exact rational arithmetic, not rounded decimals. - S225 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S224 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 with exact rational arithmetic, not rounded decimals. - S223 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S222 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 with exact rational arithmetic, not rounded decimals. - S221 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S220 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 with exact rational arithmetic, not rounded decimals. - S219 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S218 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 with exact rational arithmetic, not rounded decimals. - S217 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S216 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 with exact rational arithmetic, not rounded decimals. - S215 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S214 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 with exact rational arithmetic, not rounded decimals. - S213 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S212 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 with exact rational arithmetic, not rounded decimals. - S211 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S210 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 with exact rational arithmetic, not rounded decimals. - S209 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S208 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 with exact rational arithmetic, not rounded decimals. - S207 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S206 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 with exact rational arithmetic, not rounded decimals. - S205 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S204 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 with exact rational arithmetic, not rounded decimals. - S203 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S202 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 with exact rational arithmetic, not rounded decimals. - S201 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S200 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 with exact rational arithmetic, not rounded decimals. - S199 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S198 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 with exact rational arithmetic, not rounded decimals. - S197 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S196 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 with exact rational arithmetic, not rounded decimals. - S195 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S194 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 with exact rational arithmetic, not rounded decimals. - S193 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S192 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 with exact rational arithmetic, not rounded decimals. - S191 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S190 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 with exact rational arithmetic, not rounded decimals. - S189 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S188 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 with exact rational arithmetic, not rounded decimals. - S187 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S186 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 with exact rational arithmetic, not rounded decimals. - S185 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S184 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 with exact rational arithmetic, not rounded decimals. - S183 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S182 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 with exact rational arithmetic, not rounded decimals. - S181 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S180 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 with exact rational arithmetic, not rounded decimals. - S179 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S178 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 with exact rational arithmetic, not rounded decimals. - S177 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S176 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 with exact rational arithmetic, not rounded decimals. - S175 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S174 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 with exact rational arithmetic, not rounded decimals. - S173 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S172 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 with exact rational arithmetic, not rounded decimals. - S171 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S170 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 with exact rational arithmetic, not rounded decimals. - S169 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S168 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 with exact rational arithmetic, not rounded decimals. - S167 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S166 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 with exact rational arithmetic, not rounded decimals. - S165 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S164 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 with exact rational arithmetic, not rounded decimals. - S163 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S162 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 with exact rational arithmetic, not rounded decimals. - S161 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S160 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 with exact rational arithmetic, not rounded decimals. - S159 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S158 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 with exact rational arithmetic, not rounded decimals. - S157 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S156 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 with exact rational arithmetic, not rounded decimals. - S155 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S154 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 with exact rational arithmetic, not rounded decimals. - S153 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S152 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 with exact rational arithmetic, not rounded decimals. - S151 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S150 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 with exact rational arithmetic, not rounded decimals. - S149 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S148 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 with exact rational arithmetic, not rounded decimals. - S147 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S146 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 with exact rational arithmetic, not rounded decimals. - S145 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S144 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 with exact rational arithmetic, not rounded decimals. - S143 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S142 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 with exact rational arithmetic, not rounded decimals. - S141 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S140 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 with exact rational arithmetic, not rounded decimals. - S139 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S138 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 with exact rational arithmetic, not rounded decimals. - S137 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S136 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 with exact rational arithmetic, not rounded decimals. - S135 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S134 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 with exact rational arithmetic, not rounded decimals. - S133 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S132 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 with exact rational arithmetic, not rounded decimals. - S131 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S130 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 with exact rational arithmetic, not rounded decimals. - S129 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S128 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 with exact rational arithmetic, not rounded decimals. - S127 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S126 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 with exact rational arithmetic, not rounded decimals. - S125 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S124 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 with exact rational arithmetic, not rounded decimals. - S123 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S122 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 with exact rational arithmetic, not rounded decimals. - S121 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S120 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 with exact rational arithmetic, not rounded decimals. - S119 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S118 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S116 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 with exact rational arithmetic, not rounded decimals. - S115 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S114 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S112 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 with exact rational arithmetic, not rounded decimals. - S111 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S110 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 with exact rational arithmetic, not rounded decimals. - S109 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S108 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 with exact rational arithmetic, not rounded decimals. - S107 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S106 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 with exact rational arithmetic, not rounded decimals. - S105 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S104 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S102 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 with exact rational arithmetic, not rounded decimals. - S101 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S100 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 with exact rational arithmetic, not rounded decimals. - S099 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S098 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S096 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S095 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S094 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S093 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S092 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S091 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S090 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S089 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S088 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S087 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S086 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S085 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S084 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S083 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S082 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S081 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S080 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S079 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S078 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S077 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S076 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S075 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S074 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S073 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S072 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S071 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S070 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S069 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S068 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S067 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S066 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S065 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S064 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S063 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S062 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S061 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S060 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S059 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S058 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S057 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S056 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S055 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S054 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S053 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S052 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S051 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S050 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S049 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S048 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S047 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S046 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S045 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S044 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S043 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S042 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S041 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S040 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S039 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S038 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S037 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S036 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S035 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S034 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S033 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S032 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S031 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S030 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S029 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S028 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S027 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S026 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S025 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S024 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S023 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S022 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S021 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S020 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S019 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S018 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S017 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S016 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S015 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S014 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S013 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S012 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S011 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S010 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S009 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S008 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S007 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S006 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S005 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S004 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S003 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S002 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. ## Result (97, [[1, 1], [2, 49], [3, 65], [4, 73], [5, 39], [6, 81], [7, 14], [8, 85], [9, 54], [10, 68], [11, 53], [12, 89], [13, 15], [14, 7], [15, 13], [16, 91], [17, 40], [18, 27], [19, 46], [20, 34], [21, 37], [22, 75], [23, 38], [24, 93], [25, 66], [26, 56], [27, 18], [28, 52], [29, 87], [30, 55], [31, 72], [32, 94], [33, 50], [34, 20], [35, 61], [36, 62], [37, 21], [38, 23], [39, 5], [40, 17], [41, 71], [42, 67], [43, 88], [44, 86], [45, 69], [46, 19], [47, 64], [48, 95], [49, 2], [50, 33], [51, 78], [52, 28], [53, 11], [54, 9], [55, 30], [56, 26], [57, 80], [58, 92], [59, 74], [60, 76], [61, 35], [62, 36], [63, 77], [64, 47], [65, 3], [66, 25], [67, 42], [68, 10], [69, 45], [70, 79], [71, 41], [72, 31], [73, 4], [74, 59], [75, 22], [76, 60], [77, 63], [78, 51], [79, 70], [80, 57], [81, 6], [82, 84], [83, 90], [84, 82], [85, 8], [86, 44], [87, 29], [88, 43], [89, 12], [90, 83], [91, 16], [92, 58], [93, 24], [94, 32], [95, 48], [96, 96]], [1, 96], [[2, 49, 1], [3, 65, 1], [4, 73, 1], [5, 39, 1], [6, 81, 1], [7, 14, 1], [8, 85, 1], [9, 54, 1], [10, 68, 1], [11, 53, 1], [12, 89, 1], [13, 15, 1], [16, 91, 1], [17, 40, 1], [18, 27, 1], [19, 46, 1], [20, 34, 1], [21, 37, 1], [22, 75, 1], [23, 38, 1], [24, 93, 1], [25, 66, 1], [26, 56, 1], [28, 52, 1], [29, 87, 1], [30, 55, 1], [31, 72, 1], [32, 94, 1], [33, 50, 1], [35, 61, 1], [36, 62, 1], [41, 71, 1], [42, 67, 1], [43, 88, 1], [44, 86, 1], [45, 69, 1], [47, 64, 1], [48, 95, 1], [51, 78, 1], [57, 80, 1], [58, 92, 1], [59, 74, 1], [60, 76, 1], [63, 77, 1], [70, 79, 1], [82, 84, 1], [83, 90, 1]], self_inverse_residues_1_and_p_minus_1, 96, Yes) 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: 21f8aa4e77f342131da77155a4304123637fd59f4175367d6b10129b4d669a8c - candidate record SHA-256: 9f561576b6635347c834aa5313009e4cb9148d38a678c3b91e8fef48d3a38606 - semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54 - question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261118-r04-wilson-theorem-0050.json)