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R0054-27747 · q-20261015-advance-20261118-r04-wilson-theorem-0072
Use composite integer n=34. Give (n, prime, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence_fails, finite_scope).
G001 — Use composite integer n=34. Give (n, prime, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence_fails, finite_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.
Before
source roles = [wilson-composite-instance]
After
derived = [[n, prime, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence_fails, finite_scope]]
Conditions: none
List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product1 := mul(1, 1)
After
Wilson ORIGINAL 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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
source roles = [Wilson ORIGINAL factorial, 34, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S002
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product2 := mul(1, 2)
After
Wilson ORIGINAL 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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
source roles = [Wilson ORIGINAL factorial, 34, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S004
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product3 := mul(2, 3)
After
Wilson ORIGINAL 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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
source roles = [Wilson ORIGINAL factorial, 34, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S006
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product4 := mul(6, 4)
After
Wilson ORIGINAL 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.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
source roles = [Wilson ORIGINAL factorial, 34, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S008
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product5 := mul(24, 5)
After
Wilson ORIGINAL 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.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
source roles = [Wilson ORIGINAL factorial, 34, 5, 24, 120]
After
derived = [3, 18, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S010
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product6 := mul(18, 6)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product6 = 108
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
source roles = [Wilson ORIGINAL factorial, 34, 6, 18, 108]
After
derived = [3, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S012
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product7 := mul(6, 7)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product7 = 42
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
source roles = [Wilson ORIGINAL factorial, 34, 7, 6, 42]
After
derived = [1, 8, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S014
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product8 := mul(8, 8)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product8 = 64
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
source roles = [Wilson ORIGINAL factorial, 34, 8, 8, 64]
After
derived = [1, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S016
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product9 := mul(30, 9)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product9 = 270
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
source roles = [Wilson ORIGINAL factorial, 34, 9, 30, 270]
After
derived = [7, 32, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S018
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product10 := mul(32, 10)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product10 = 320
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
source roles = [Wilson ORIGINAL factorial, 34, 10, 32, 320]
After
derived = [9, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S020
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product11 := mul(14, 11)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product11 = 154
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
source roles = [Wilson ORIGINAL factorial, 34, 11, 14, 154]
After
derived = [4, 18, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S022
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product12 := mul(18, 12)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product12 = 216
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
source roles = [Wilson ORIGINAL factorial, 34, 12, 18, 216]
After
derived = [6, 12, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S024
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product13 := mul(12, 13)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product13 = 156
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
source roles = [Wilson ORIGINAL factorial, 34, 13, 12, 156]
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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S026
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product14 := mul(20, 14)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product14 = 280
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
source roles = [Wilson ORIGINAL factorial, 34, 14, 20, 280]
After
derived = [8, 8, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S028
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product15 := mul(8, 15)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product15 = 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.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
source roles = [Wilson ORIGINAL factorial, 34, 15, 8, 120]
After
derived = [3, 18, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S030
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product16 := mul(18, 16)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product16 = 288
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
source roles = [Wilson ORIGINAL factorial, 34, 16, 18, 288]
After
derived = [8, 16, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S032
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product17 := mul(16, 17)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product17 = 272
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
source roles = [Wilson ORIGINAL factorial, 34, 17, 16, 272]
After
derived = [8, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S034
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product18 := mul(0, 18)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product18 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
source roles = [Wilson ORIGINAL factorial, 34, 18, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S036
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product19 := mul(0, 19)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product19 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
source roles = [Wilson ORIGINAL factorial, 34, 19, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S038
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product20 := mul(0, 20)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product20 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
source roles = [Wilson ORIGINAL factorial, 34, 20, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S040
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product21 := mul(0, 21)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product21 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
source roles = [Wilson ORIGINAL factorial, 34, 21, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S042
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product22 := mul(0, 22)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product22 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
source roles = [Wilson ORIGINAL factorial, 34, 22, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S044
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product23 := mul(0, 23)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product23 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
source roles = [Wilson ORIGINAL factorial, 34, 23, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S046
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product24 := mul(0, 24)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product24 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
source roles = [Wilson ORIGINAL factorial, 34, 24, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S048
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product25 := mul(0, 25)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product25 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
source roles = [Wilson ORIGINAL factorial, 34, 25, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S050
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product26 := mul(0, 26)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product26 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
source roles = [Wilson ORIGINAL factorial, 34, 26, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S052
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product27 := mul(0, 27)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product27 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
source roles = [Wilson ORIGINAL factorial, 34, 27, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S054
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product28 := mul(0, 28)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product28 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
source roles = [Wilson ORIGINAL factorial, 34, 28, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S056
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product29 := mul(0, 29)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product29 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
source roles = [Wilson ORIGINAL factorial, 34, 29, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S058
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product30 := mul(0, 30)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product30 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
source roles = [Wilson ORIGINAL factorial, 34, 30, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S060
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product31 := mul(0, 31)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product31 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
source roles = [Wilson ORIGINAL factorial, 34, 31, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S062
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product32 := mul(0, 32)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product32 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
source roles = [Wilson ORIGINAL factorial, 34, 32, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S064
Before
Wilson ORIGINAL factorial ORIGINAL running factorial product33 := mul(0, 33)
After
Wilson ORIGINAL factorial ORIGINAL running factorial product33 = 0
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
source roles = [Wilson ORIGINAL factorial, 34, 33, 0, 0]
After
derived = [0, 0, 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.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S066
Before
source roles = [34, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 18], [6, 6], [7, 8], [8, 30], [9, 32], [10, 14], [11, 18], [12, 12], [13, 20], [14, 8], [15, 18], [16, 16], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0]]]
After
derived = [No, 0, 33, No, Yes]
Conditions: none
Qualify primality and congruence separately.Both may be False at a composite while equivalence is True.The converse has a separate argument:a proper divisor d>=2 of composite n divides both n and(n-1)!;a congruence to-1 would force d to divide1,a contradiction.This reasoning is not a single instance or finite-cutoff test.
Rule: Wilson_prime_classification_instance_flags_AND_separate_converse_reason · Uses: G001, S067
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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product33 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product32 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product31 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product30 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product29 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product28 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product27 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product26 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product25 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product24 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product23 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product22 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product21 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product20 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product19 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product18 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product17 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product16 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product15 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product14 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product13 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product12 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product11 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product10 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product9 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product8 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product7 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product6 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product5 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product4 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product3 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product2 with exact rational arithmetic, not rounded decimals.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 — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product1 with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.(34, No, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 18], [6, 6], [7, 8], [8, 30], [9, 32], [10, 14], [11, 18], [12, 12], [13, 20], [14, 8], [15, 18], [16, 16], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0]], 0, 33, Yes, finite_composite_instance_not_converse_proof)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: f665d9ec85f2afc18444f85f48b9219462de8946fe66cf831e20350ca3cdededcandidate record SHA-256: 8a8b4266766c62c4ac17b5ca38860e4207689f42972a04ba08efe3d910f9ba08semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED