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R0054-27751 · q-20261015-advance-20261118-r04-wilson-theorem-0001
Use exact integer n=2>1. Give (n, prime, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, instance_equivalence, instance_scope).
G001 — Use exact integer n=2>1. Give (n, prime, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, instance_equivalence, instance_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-domain-factorial]
After
derived = [[n, prime, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, instance_equivalence, instance_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, 2, 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
source roles = [2, [[1, 1]]]
After
derived = [Yes, 1, 1, Yes, 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, S003
S004 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S003 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S002 · premise_and_result_audit — 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.(2, Yes, [[1, 1]], 1, 1, Yes, Yes, Wilson_equivalence_instance)
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: 9a77eb15eeda6655eb214643f43bd3b26c4491fd452d168e391e9968f06f018ccandidate record SHA-256: c1bc7363b6a72e31817d14c3b7aac7bcdb17c3d849dd80a63b050ac236c7973asemantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED