# Question 27719: Wilson Domain, Finite Testing, Fermat Comparison and the p=2 Exception Stay Separate

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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.

R0054-27719 · q-20261015-advance-20261118-r04-wilson-theorem-0094



## Original question and givens

Use mode=fermat_confusion and exact integer n=67. Give (mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary).


- G001 — Use mode=fermat_confusion and exact integer n=67. Give (mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary).





## 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-boundary]

After

derived = [[mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary]]

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 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product1 := mul(1, 1)

After

Wilson boundary ORIGINAL prime 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, S001





S003 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 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, S002





S004 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product2 := mul(1, 2)

After

Wilson boundary ORIGINAL prime 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, S003





S005 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 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, S004





S006 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product3 := mul(2, 3)

After

Wilson boundary ORIGINAL prime 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, S005





S007 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 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, S006





S008 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product4 := mul(6, 4)

After

Wilson boundary ORIGINAL prime 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, S007





S009 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 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, S008





S010 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product5 := mul(24, 5)

After

Wilson boundary ORIGINAL prime 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, S009





S011 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 5, 24, 120]

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, S010





S012 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 := mul(53, 6)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 = 318

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, S011





S013 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 6, 53, 318]

After

derived = [4, 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, S012





S014 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 := mul(50, 7)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 = 350

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, S013





S015 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 7, 50, 350]

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, S014





S016 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 := mul(15, 8)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 = 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, S015





S017 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 8, 15, 120]

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, S016





S018 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 := mul(53, 9)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 = 477

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, S017





S019 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 9, 53, 477]

After

derived = [7, 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.

Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S018





S020 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 := mul(8, 10)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 = 80

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, S019





S021 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 10, 8, 80]

After

derived = [1, 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, S020





S022 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 := mul(13, 11)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 = 143

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, S021





S023 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 11, 13, 143]

After

derived = [2, 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, S022





S024 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 := mul(9, 12)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 = 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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 12, 9, 108]

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, S024





S026 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 := mul(41, 13)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 = 533

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, S025





S027 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 13, 41, 533]

After

derived = [7, 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, S026





S028 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 := mul(64, 14)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 = 896

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, S027





S029 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 14, 64, 896]

After

derived = [13, 25, 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, S028





S030 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 := mul(25, 15)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 = 375

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, S029





S031 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 15, 25, 375]

After

derived = [5, 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, S030





S032 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 := mul(40, 16)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 = 640

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, S031





S033 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 16, 40, 640]

After

derived = [9, 37, 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, S032





S034 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 := mul(37, 17)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 = 629

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, S033





S035 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 17, 37, 629]

After

derived = [9, 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, S034





S036 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 := mul(26, 18)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 = 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, S035





S037 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 18, 26, 468]

After

derived = [6, 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, S036





S038 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 := mul(66, 19)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 = 1254

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, S037





S039 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 19, 66, 1254]

After

derived = [18, 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, S038





S040 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 := mul(48, 20)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 = 960

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, S039





S041 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 20, 48, 960]

After

derived = [14, 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, S040





S042 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 := mul(22, 21)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 = 462

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, S041





S043 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 21, 22, 462]

After

derived = [6, 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, S042





S044 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 := mul(60, 22)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 = 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, S043





S045 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 22, 60, 1320]

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, S044





S046 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 := mul(47, 23)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 = 1081

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, S045





S047 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 23, 47, 1081]

After

derived = [16, 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, S046





S048 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 := mul(9, 24)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 = 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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047





S049 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 24, 9, 216]

After

derived = [3, 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, S048





S050 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 := mul(15, 25)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 = 375

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, S049





S051 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 25, 15, 375]

After

derived = [5, 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, S050





S052 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 := mul(40, 26)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 = 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, S051





S053 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 26, 40, 1040]

After

derived = [15, 35, 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, S052





S054 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 := mul(35, 27)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 = 945

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, S053





S055 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 27, 35, 945]

After

derived = [14, 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, S054





S056 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 := mul(7, 28)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 = 196

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, S055





S057 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 28, 7, 196]

After

derived = [2, 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, S056





S058 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 := mul(62, 29)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 = 1798

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, S057





S059 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 29, 62, 1798]

After

derived = [26, 56, 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, S058





S060 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 := mul(56, 30)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 = 1680

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, S059





S061 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 30, 56, 1680]

After

derived = [25, 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, S060





S062 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 := mul(5, 31)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 = 155

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, S061





S063 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 31, 5, 155]

After

derived = [2, 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, S062





S064 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 := mul(21, 32)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 = 672

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, S063





S065 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 32, 21, 672]

After

derived = [10, 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, S064





S066 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 := mul(2, 33)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 = 66

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, S065





S067 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 33, 2, 66]

After

derived = [0, 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, S066





S068 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 := mul(66, 34)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 = 2244

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, S067





S069 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 34, 66, 2244]

After

derived = [33, 33, 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, S068





S070 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 := mul(33, 35)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 = 1155

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, S069





S071 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 35, 33, 1155]

After

derived = [17, 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.

Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S070





S072 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 := mul(16, 36)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 = 576

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, S071





S073 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 36, 16, 576]

After

derived = [8, 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, S072





S074 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 := mul(40, 37)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 = 1480

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, S073





S075 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 37, 40, 1480]

After

derived = [22, 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, S074





S076 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 := mul(6, 38)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 = 228

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, S075





S077 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 38, 6, 228]

After

derived = [3, 27, 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, S076





S078 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 := mul(27, 39)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 = 1053

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, S077





S079 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 39, 27, 1053]

After

derived = [15, 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, S078





S080 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 := mul(48, 40)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 = 1920

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, S079





S081 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 40, 48, 1920]

After

derived = [28, 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, S080





S082 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 := mul(44, 41)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 = 1804

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, S081





S083 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 41, 44, 1804]

After

derived = [26, 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, S082





S084 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 := mul(62, 42)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 = 2604

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, S083





S085 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 42, 62, 2604]

After

derived = [38, 58, 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, S084





S086 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 := mul(58, 43)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 = 2494

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, S085





S087 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 43, 58, 2494]

After

derived = [37, 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, S086





S088 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 := mul(15, 44)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 = 660

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, S087





S089 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 44, 15, 660]

After

derived = [9, 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, S088





S090 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 := mul(57, 45)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 = 2565

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, S089





S091 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 45, 57, 2565]

After

derived = [38, 19, 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, S090





S092 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 := mul(19, 46)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 = 874

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, S091





S093 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 46, 19, 874]

After

derived = [13, 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, S092





S094 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 := mul(3, 47)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 = 141

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, S093





S095 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 47, 3, 141]

After

derived = [2, 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, S094





S096 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 := mul(7, 48)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 = 336

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, S095





S097 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 48, 7, 336]

After

derived = [5, 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, S096





S098 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 := mul(1, 49)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 = 49

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 boundary ORIGINAL prime, 67, 49, 1, 49]

After

derived = [0, 49, 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 boundary ORIGINAL prime ORIGINAL running factorial product50 := mul(49, 50)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product50 = 2450

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 boundary ORIGINAL prime, 67, 50, 49, 2450]

After

derived = [36, 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, S100





S102 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 := mul(38, 51)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 = 1938

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 boundary ORIGINAL prime, 67, 51, 38, 1938]

After

derived = [28, 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, S102





S104 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 := mul(62, 52)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 = 3224

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 boundary ORIGINAL prime, 67, 52, 62, 3224]

After

derived = [48, 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.

Step identifiers and dependenciesRule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S104





S106 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product53 := mul(8, 53)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product53 = 424

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 boundary ORIGINAL prime, 67, 53, 8, 424]

After

derived = [6, 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, S106





S108 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product54 := mul(22, 54)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product54 = 1188

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 boundary ORIGINAL prime, 67, 54, 22, 1188]

After

derived = [17, 49, 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 boundary ORIGINAL prime ORIGINAL running factorial product55 := mul(49, 55)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product55 = 2695

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 boundary ORIGINAL prime, 67, 55, 49, 2695]

After

derived = [40, 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, S110





S112 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product56 := mul(15, 56)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product56 = 840

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 boundary ORIGINAL prime, 67, 56, 15, 840]

After

derived = [12, 36, 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 boundary ORIGINAL prime ORIGINAL running factorial product57 := mul(36, 57)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product57 = 2052

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 boundary ORIGINAL prime, 67, 57, 36, 2052]

After

derived = [30, 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, S114





S116 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product58 := mul(42, 58)

After

Wilson boundary ORIGINAL prime 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, S115





S117 · impliesBefore

source roles = [Wilson boundary ORIGINAL prime, 67, 58, 42, 2436]

After

derived = [36, 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, S116





S118 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product59 := mul(24, 59)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product59 = 1416

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 boundary ORIGINAL prime, 67, 59, 24, 1416]

After

derived = [21, 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, S118





S120 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product60 := mul(9, 60)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product60 = 540

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 boundary ORIGINAL prime, 67, 60, 9, 540]

After

derived = [8, 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, S120





S122 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product61 := mul(4, 61)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product61 = 244

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 boundary ORIGINAL prime, 67, 61, 4, 244]

After

derived = [3, 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, S122





S124 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product62 := mul(43, 62)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product62 = 2666

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 boundary ORIGINAL prime, 67, 62, 43, 2666]

After

derived = [39, 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, S124





S126 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product63 := mul(53, 63)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product63 = 3339

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 boundary ORIGINAL prime, 67, 63, 53, 3339]

After

derived = [49, 56, 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 boundary ORIGINAL prime ORIGINAL running factorial product64 := mul(56, 64)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product64 = 3584

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 boundary ORIGINAL prime, 67, 64, 56, 3584]

After

derived = [53, 33, 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 boundary ORIGINAL prime ORIGINAL running factorial product65 := mul(33, 65)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product65 = 2145

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 boundary ORIGINAL prime, 67, 65, 33, 2145]

After

derived = [32, 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, S130





S132 · calculationBefore

Wilson boundary ORIGINAL prime ORIGINAL running factorial product66 := mul(1, 66)

After

Wilson boundary ORIGINAL prime ORIGINAL running factorial product66 = 66

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 boundary ORIGINAL prime, 67, 66, 1, 66]

After

derived = [0, 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, S132





S134 · impliesBefore

source roles = [67, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 53], [6, 50], [7, 15], [8, 53], [9, 8], [10, 13], [11, 9], [12, 41], [13, 64], [14, 25], [15, 40], [16, 37], [17, 26], [18, 66], [19, 48], [20, 22], [21, 60], [22, 47], [23, 9], [24, 15], [25, 40], [26, 35], [27, 7], [28, 62], [29, 56], [30, 5], [31, 21], [32, 2], [33, 66], [34, 33], [35, 16], [36, 40], [37, 6], [38, 27], [39, 48], [40, 44], [41, 62], [42, 58], [43, 15], [44, 57], [45, 19], [46, 3], [47, 7], [48, 1], [49, 49], [50, 38], [51, 62], [52, 8], [53, 22], [54, 49], [55, 15], [56, 36], [57, 42], [58, 24], [59, 9], [60, 4], [61, 43], [62, 53], [63, 56], [64, 33], [65, 1], [66, 66]], 66]

After

derived = [66, Yes, No]

Conditions: none

Wilson concerns(n-1)!;Fermat concerns a^(n-1) when n does not divide a.This branch supplies no a or power computation.The named comparison neither identifies the theorems nor adds an executed power test.

Step identifiers and dependenciesRule: Wilson_and_Fermat_DISTINCT_named_statements_NO_unprovided_a_power · Uses: G001, S133









## Reverse checks


- S134 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
- 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 boundary ORIGINAL prime ORIGINAL running factorial product66 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 boundary ORIGINAL prime ORIGINAL running factorial product65 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 boundary ORIGINAL prime ORIGINAL running factorial product64 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 boundary ORIGINAL prime ORIGINAL running factorial product63 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 boundary ORIGINAL prime ORIGINAL running factorial product62 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 boundary ORIGINAL prime ORIGINAL running factorial product61 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 boundary ORIGINAL prime ORIGINAL running factorial product60 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 boundary ORIGINAL prime ORIGINAL running factorial product59 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 boundary ORIGINAL prime ORIGINAL running factorial product58 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 boundary ORIGINAL prime ORIGINAL running factorial product57 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 boundary ORIGINAL prime ORIGINAL running factorial product56 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 boundary ORIGINAL prime ORIGINAL running factorial product55 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 boundary ORIGINAL prime ORIGINAL running factorial product54 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 boundary ORIGINAL prime ORIGINAL running factorial product53 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 boundary ORIGINAL prime ORIGINAL running factorial product52 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 boundary ORIGINAL prime ORIGINAL running factorial product51 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 boundary ORIGINAL prime ORIGINAL running factorial product50 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 boundary ORIGINAL prime ORIGINAL running factorial product49 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 with exact rational arithmetic, not rounded decimals.
- 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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.





## Result

(fermat_confusion, 67, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 53], [6, 50], [7, 15], [8, 53], [9, 8], [10, 13], [11, 9], [12, 41], [13, 64], [14, 25], [15, 40], [16, 37], [17, 26], [18, 66], [19, 48], [20, 22], [21, 60], [22, 47], [23, 9], [24, 15], [25, 40], [26, 35], [27, 7], [28, 62], [29, 56], [30, 5], [31, 21], [32, 2], [33, 66], [34, 33], [35, 16], [36, 40], [37, 6], [38, 27], [39, 48], [40, 44], [41, 62], [42, 58], [43, 15], [44, 57], [45, 19], [46, 3], [47, 7], [48, 1], [49, 49], [50, 38], [51, 62], [52, 8], [53, 22], [54, 49], [55, 15], [56, 36], [57, 42], [58, 24], [59, 9], [60, 4], [61, 43], [62, 53], [63, 56], [64, 33], [65, 1], [66, 66]], 66, 66, Yes, Wilson:(n-1)!==-1_mod_67, Fermat:a^(67-1)==1_mod_67_when_n_does_not_divide_a, No, distinct_theorems_no_Fermat_power_claim_recomputed)

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: 88c99669ed29f8922e02d80ec8c9e9cd40fc74d723a1675c75e3db0b148a2279
- candidate record SHA-256: 1abb11f945c24f8db354d16f527ee6b1e9fa30a0af074cf0022e29e59ad113d9
- 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-0094.json)