Step-by-step derivation
S001 · implies
Before
source roles = [qf-mod47-morder-qualification, [a, domain, n, operation]]
After
derived = [morder-qualification, Exact_multiplicative_order_mod_n]
Conditions: none
THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged.
Step identifiers and dependencies
Rule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001
S002 · implies
Before
source roles = [ORIGINAL representative, 3, 22]
After
derived = [0, 3]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S001
S003 · implies
Before
source roles = [22, 3, 3, 1]
After
derived = [[1, 3, 5, 7, 9, 13, 15, 17, 19, 21], 10]
Conditions: none
FIRST require n≥2,normalize into0..n−1 and qualify gcd1.Enumerate ALL unit residues to obtain phi(n);do not replace general phi with n−1 or assign a numeric order to a nonunit.
Step identifiers and dependencies
Rule: Order_ALL_residues_UNIT_gcd_and_complete_phi_carrier · Uses: G001, S002
S004 · calculation
Before
Order ORIGINAL previous residue times unit exponent1 := mul(1, 3)
After
Order ORIGINAL previous residue times unit exponent1 = 3
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
source roles = [first-return exponent1, 3, 22]
After
derived = [0, 3]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S004
S006 · calculation
Before
Order ORIGINAL previous residue times unit exponent2 := mul(3, 3)
After
Order ORIGINAL previous residue times unit exponent2 = 9
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · implies
Before
source roles = [first-return exponent2, 9, 22]
After
derived = [0, 9]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S006
S008 · calculation
Before
Order ORIGINAL previous residue times unit exponent3 := mul(9, 3)
After
Order ORIGINAL previous residue times unit exponent3 = 27
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · implies
Before
source roles = [first-return exponent3, 27, 22]
After
derived = [1, 5]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S008
S010 · calculation
Before
Order ORIGINAL previous residue times unit exponent4 := mul(5, 3)
After
Order ORIGINAL previous residue times unit exponent4 = 15
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · implies
Before
source roles = [first-return exponent4, 15, 22]
After
derived = [0, 15]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S010
S012 · calculation
Before
Order ORIGINAL previous residue times unit exponent5 := mul(15, 3)
After
Order ORIGINAL previous residue times unit exponent5 = 45
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · implies
Before
source roles = [first-return exponent5, 45, 22]
After
derived = [2, 1]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Step identifiers and dependencies
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S012
S014 · implies
Before
source roles = [22, 3, 10]
After
derived = [[[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], 5]
Conditions: none
The SOURCE computes the complete first-return prefix EVEN when qualification outputs only gcd/phi.Retain this UNUSED computation obligation.Start at POSITIVE exponent1 with no earlier return.Do not claim order=phi,a primitive root or additive order.
Step identifiers and dependencies
Rule: Order_SOURCE_UNUSED_qualification_return_table_still_checked · Uses: G001, S013