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R0066-33663 · q-20261014-advance-20261031-r06-finite-field-0088
Use the complete prime field Z_41, with prime p=41, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=12, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
G001 — Use the complete prime field Z_41, with prime p=41, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=12, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.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 = [qf-mod31-fielddeep-frobenius, [domain, element, operation, p]]
After
derived = [fielddeep-frobenius, Exact_prime_field_Zp]
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.
Rule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001
Before
source roles = [41, [2, 3, 4, 5, 6]]
After
derived = [[1, 2, 1, 1, 5], Yes]
Conditions: none
Trial EVERY divisor2 through floor(sqrt(p));a composite would have a factor there.The empty test for p=2 is valid.This qualifies THIS prime field Z_p,not every finite field’s order or every finite ring as a field.
Rule: Field_PRIME_complete_trial_divisor_qualification · Uses: G001, S001
Before
Field ORIGINAL previous power times element k=1 := mul(1, 12)
After
Field ORIGINAL previous power times element k=1 = 12
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [power exponent1, 12, 41]
After
derived = [0, 12]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S003
Before
Field ORIGINAL previous power times element k=2 := mul(12, 12)
After
Field ORIGINAL previous power times element k=2 = 144
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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
source roles = [power exponent2, 144, 41]
After
derived = [3, 21]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S005
Before
Field ORIGINAL previous power times element k=3 := mul(21, 12)
After
Field ORIGINAL previous power times element k=3 = 252
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.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
source roles = [power exponent3, 252, 41]
After
derived = [6, 6]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S007
Before
Field ORIGINAL previous power times element k=4 := mul(6, 12)
After
Field ORIGINAL previous power times element k=4 = 72
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.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
source roles = [power exponent4, 72, 41]
After
derived = [1, 31]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S009
Before
Field ORIGINAL previous power times element k=5 := mul(31, 12)
After
Field ORIGINAL previous power times element k=5 = 372
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.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
source roles = [power exponent5, 372, 41]
After
derived = [9, 3]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S011
Before
Field ORIGINAL previous power times element k=6 := mul(3, 12)
After
Field ORIGINAL previous power times element k=6 = 36
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.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
source roles = [power exponent6, 36, 41]
After
derived = [0, 36]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S013
Before
Field ORIGINAL previous power times element k=7 := mul(36, 12)
After
Field ORIGINAL previous power times element k=7 = 432
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.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
source roles = [power exponent7, 432, 41]
After
derived = [10, 22]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S015
Before
Field ORIGINAL previous power times element k=8 := mul(22, 12)
After
Field ORIGINAL previous power times element k=8 = 264
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.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
source roles = [power exponent8, 264, 41]
After
derived = [6, 18]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S017
Before
Field ORIGINAL previous power times element k=9 := mul(18, 12)
After
Field ORIGINAL previous power times element k=9 = 216
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.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
source roles = [power exponent9, 216, 41]
After
derived = [5, 11]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S019
Before
Field ORIGINAL previous power times element k=10 := mul(11, 12)
After
Field ORIGINAL previous power times element k=10 = 132
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.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
source roles = [power exponent10, 132, 41]
After
derived = [3, 9]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S021
Before
Field ORIGINAL previous power times element k=11 := mul(9, 12)
After
Field ORIGINAL previous power times element k=11 = 108
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.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
source roles = [power exponent11, 108, 41]
After
derived = [2, 26]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S023
Before
Field ORIGINAL previous power times element k=12 := mul(26, 12)
After
Field ORIGINAL previous power times element k=12 = 312
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.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
source roles = [power exponent12, 312, 41]
After
derived = [7, 25]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S025
Before
Field ORIGINAL previous power times element k=13 := mul(25, 12)
After
Field ORIGINAL previous power times element k=13 = 300
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.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
source roles = [power exponent13, 300, 41]
After
derived = [7, 13]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S027
Before
Field ORIGINAL previous power times element k=14 := mul(13, 12)
After
Field ORIGINAL previous power times element k=14 = 156
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.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
source roles = [power exponent14, 156, 41]
After
derived = [3, 33]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S029
Before
Field ORIGINAL previous power times element k=15 := mul(33, 12)
After
Field ORIGINAL previous power times element k=15 = 396
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.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
source roles = [power exponent15, 396, 41]
After
derived = [9, 27]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S031
Before
Field ORIGINAL previous power times element k=16 := mul(27, 12)
After
Field ORIGINAL previous power times element k=16 = 324
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.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
source roles = [power exponent16, 324, 41]
After
derived = [7, 37]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S033
Before
Field ORIGINAL previous power times element k=17 := mul(37, 12)
After
Field ORIGINAL previous power times element k=17 = 444
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.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
source roles = [power exponent17, 444, 41]
After
derived = [10, 34]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S035
Before
Field ORIGINAL previous power times element k=18 := mul(34, 12)
After
Field ORIGINAL previous power times element k=18 = 408
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.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
source roles = [power exponent18, 408, 41]
After
derived = [9, 39]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S037
Before
Field ORIGINAL previous power times element k=19 := mul(39, 12)
After
Field ORIGINAL previous power times element k=19 = 468
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.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
source roles = [power exponent19, 468, 41]
After
derived = [11, 17]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S039
Before
Field ORIGINAL previous power times element k=20 := mul(17, 12)
After
Field ORIGINAL previous power times element k=20 = 204
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.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
source roles = [power exponent20, 204, 41]
After
derived = [4, 40]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S041
Before
Field ORIGINAL previous power times element k=21 := mul(40, 12)
After
Field ORIGINAL previous power times element k=21 = 480
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.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
source roles = [power exponent21, 480, 41]
After
derived = [11, 29]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S043
Before
Field ORIGINAL previous power times element k=22 := mul(29, 12)
After
Field ORIGINAL previous power times element k=22 = 348
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.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
source roles = [power exponent22, 348, 41]
After
derived = [8, 20]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S045
Before
Field ORIGINAL previous power times element k=23 := mul(20, 12)
After
Field ORIGINAL previous power times element k=23 = 240
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.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
source roles = [power exponent23, 240, 41]
After
derived = [5, 35]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S047
Before
Field ORIGINAL previous power times element k=24 := mul(35, 12)
After
Field ORIGINAL previous power times element k=24 = 420
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.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
source roles = [power exponent24, 420, 41]
After
derived = [10, 10]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S049
Before
Field ORIGINAL previous power times element k=25 := mul(10, 12)
After
Field ORIGINAL previous power times element k=25 = 120
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.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
source roles = [power exponent25, 120, 41]
After
derived = [2, 38]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S051
Before
Field ORIGINAL previous power times element k=26 := mul(38, 12)
After
Field ORIGINAL previous power times element k=26 = 456
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.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
source roles = [power exponent26, 456, 41]
After
derived = [11, 5]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S053
Before
Field ORIGINAL previous power times element k=27 := mul(5, 12)
After
Field ORIGINAL previous power times element k=27 = 60
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.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
source roles = [power exponent27, 60, 41]
After
derived = [1, 19]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S055
Before
Field ORIGINAL previous power times element k=28 := mul(19, 12)
After
Field ORIGINAL previous power times element k=28 = 228
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.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
source roles = [power exponent28, 228, 41]
After
derived = [5, 23]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S057
Before
Field ORIGINAL previous power times element k=29 := mul(23, 12)
After
Field ORIGINAL previous power times element k=29 = 276
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.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
source roles = [power exponent29, 276, 41]
After
derived = [6, 30]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S059
Before
Field ORIGINAL previous power times element k=30 := mul(30, 12)
After
Field ORIGINAL previous power times element k=30 = 360
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.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
source roles = [power exponent30, 360, 41]
After
derived = [8, 32]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S061
Before
Field ORIGINAL previous power times element k=31 := mul(32, 12)
After
Field ORIGINAL previous power times element k=31 = 384
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.
Rule: exact_rational_arithmetic · Uses: G001, S062
Before
source roles = [power exponent31, 384, 41]
After
derived = [9, 15]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S063
Before
Field ORIGINAL previous power times element k=32 := mul(15, 12)
After
Field ORIGINAL previous power times element k=32 = 180
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.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
source roles = [power exponent32, 180, 41]
After
derived = [4, 16]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S065
Before
Field ORIGINAL previous power times element k=33 := mul(16, 12)
After
Field ORIGINAL previous power times element k=33 = 192
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.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
source roles = [power exponent33, 192, 41]
After
derived = [4, 28]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S067
Before
Field ORIGINAL previous power times element k=34 := mul(28, 12)
After
Field ORIGINAL previous power times element k=34 = 336
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.
Rule: exact_rational_arithmetic · Uses: G001, S068
Before
source roles = [power exponent34, 336, 41]
After
derived = [8, 8]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S069
Before
Field ORIGINAL previous power times element k=35 := mul(8, 12)
After
Field ORIGINAL previous power times element k=35 = 96
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.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
source roles = [power exponent35, 96, 41]
After
derived = [2, 14]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S071
Before
Field ORIGINAL previous power times element k=36 := mul(14, 12)
After
Field ORIGINAL previous power times element k=36 = 168
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.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
source roles = [power exponent36, 168, 41]
After
derived = [4, 4]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S073
Before
Field ORIGINAL previous power times element k=37 := mul(4, 12)
After
Field ORIGINAL previous power times element k=37 = 48
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.
Rule: exact_rational_arithmetic · Uses: G001, S074
Before
source roles = [power exponent37, 48, 41]
After
derived = [1, 7]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S075
Before
Field ORIGINAL previous power times element k=38 := mul(7, 12)
After
Field ORIGINAL previous power times element k=38 = 84
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.
Rule: exact_rational_arithmetic · Uses: G001, S076
Before
source roles = [power exponent38, 84, 41]
After
derived = [2, 2]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S077
Before
Field ORIGINAL previous power times element k=39 := mul(2, 12)
After
Field ORIGINAL previous power times element k=39 = 24
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.
Rule: exact_rational_arithmetic · Uses: G001, S078
Before
source roles = [power exponent39, 24, 41]
After
derived = [0, 24]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S079
Before
Field ORIGINAL previous power times element k=40 := mul(24, 12)
After
Field ORIGINAL previous power times element k=40 = 288
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.
Rule: exact_rational_arithmetic · Uses: G001, S080
Before
source roles = [power exponent40, 288, 41]
After
derived = [7, 1]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S081
Before
Field ORIGINAL previous power times element k=41 := mul(1, 12)
After
Field ORIGINAL previous power times element k=41 = 12
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.
Rule: exact_rational_arithmetic · Uses: G001, S082
Before
source roles = [power exponent41, 12, 41]
After
derived = [0, 12]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S083
Before
source roles = [41, 12, [12, 21, 6, 31, 3, 36, 22, 18, 11, 9, 26, 25, 13, 33, 27, 37, 34, 39, 17, 40, 29, 20, 35, 10, 38, 5, 19, 23, 30, 32, 15, 16, 28, 8, 14, 4, 7, 2, 24, 1, 12]]
After
derived = [12, Yes]
Conditions: none
Repeated multiplication checks element^p=element in THIS prime field,including zero.Frobenius need not fix every element of an arbitrary extension field;do not widen the conclusion.
Rule: Field_FROBENIUS_prime_field_INCLUDES_zero · Uses: G001, S084
S085 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S084 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S083 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=41 with exact rational arithmetic, not rounded decimals.S082 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S081 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=40 with exact rational arithmetic, not rounded decimals.S080 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S079 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=39 with exact rational arithmetic, not rounded decimals.S078 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S077 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=38 with exact rational arithmetic, not rounded decimals.S076 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S075 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=37 with exact rational arithmetic, not rounded decimals.S074 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S073 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=36 with exact rational arithmetic, not rounded decimals.S072 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S071 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=35 with exact rational arithmetic, not rounded decimals.S070 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S069 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=34 with exact rational arithmetic, not rounded decimals.S068 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S067 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=33 with exact rational arithmetic, not rounded decimals.S066 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S065 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=32 with exact rational arithmetic, not rounded decimals.S064 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S063 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=31 with exact rational arithmetic, not rounded decimals.S062 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S061 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=30 with exact rational arithmetic, not rounded decimals.S060 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S059 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=29 with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S057 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=28 with exact rational arithmetic, not rounded decimals.S056 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S055 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=27 with exact rational arithmetic, not rounded decimals.S054 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S053 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=26 with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S051 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=25 with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S049 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=24 with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S047 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=23 with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S045 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=22 with exact rational arithmetic, not rounded decimals.S044 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S043 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=21 with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S041 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=20 with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S039 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=19 with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S037 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=18 with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S035 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=17 with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S033 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=16 with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S031 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=15 with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S029 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=14 with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S027 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=13 with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S025 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=12 with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S023 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=11 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S021 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=10 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S019 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=9 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S017 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=8 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S015 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=7 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S013 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=6 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S011 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=5 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S009 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=4 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S007 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=3 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S005 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=2 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S003 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=1 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.S001 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.(12, 41, 12, Yes, Frobenius_on_prime_field)
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: c02555b3e060779bedfd2db0bc328c27b80e4089813b6bb7f1d7f158bf0a2610candidate record SHA-256: 59226f5e51c8ebbb575495a1111029576719b5e15a97088ef498bc767982c7basemantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710ccLean: NOT_RUN · human review: NOT_PERFORMED