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R0066-33673 · q-20261014-advance-20261031-r06-finite-field-0098
Use the complete prime field Z_83, with prime p=83, 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=22, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
G001 — Use the complete prime field Z_83, with prime p=83, 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=22, 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 = [83, [2, 3, 4, 5, 6, 7, 8, 9]]
After
derived = [[1, 2, 3, 3, 5, 6, 3, 2], 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, 22)
After
Field ORIGINAL previous power times element k=1 = 22
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, 22, 83]
After
derived = [0, 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, S003
Before
Field ORIGINAL previous power times element k=2 := mul(22, 22)
After
Field ORIGINAL previous power times element k=2 = 484
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, 484, 83]
After
derived = [5, 69]
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(69, 22)
After
Field ORIGINAL previous power times element k=3 = 1518
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, 1518, 83]
After
derived = [18, 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, S007
Before
Field ORIGINAL previous power times element k=4 := mul(24, 22)
After
Field ORIGINAL previous power times element k=4 = 528
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, 528, 83]
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, S009
Before
Field ORIGINAL previous power times element k=5 := mul(30, 22)
After
Field ORIGINAL previous power times element k=5 = 660
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, 660, 83]
After
derived = [7, 79]
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(79, 22)
After
Field ORIGINAL previous power times element k=6 = 1738
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, 1738, 83]
After
derived = [20, 78]
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(78, 22)
After
Field ORIGINAL previous power times element k=7 = 1716
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, 1716, 83]
After
derived = [20, 56]
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(56, 22)
After
Field ORIGINAL previous power times element k=8 = 1232
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, 1232, 83]
After
derived = [14, 70]
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(70, 22)
After
Field ORIGINAL previous power times element k=9 = 1540
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, 1540, 83]
After
derived = [18, 46]
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(46, 22)
After
Field ORIGINAL previous power times element k=10 = 1012
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, 1012, 83]
After
derived = [12, 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, S021
Before
Field ORIGINAL previous power times element k=11 := mul(16, 22)
After
Field ORIGINAL previous power times element k=11 = 352
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, 352, 83]
After
derived = [4, 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, S023
Before
Field ORIGINAL previous power times element k=12 := mul(20, 22)
After
Field ORIGINAL previous power times element k=12 = 440
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, 440, 83]
After
derived = [5, 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, 22)
After
Field ORIGINAL previous power times element k=13 = 550
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, 550, 83]
After
derived = [6, 52]
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(52, 22)
After
Field ORIGINAL previous power times element k=14 = 1144
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, 1144, 83]
After
derived = [13, 65]
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(65, 22)
After
Field ORIGINAL previous power times element k=15 = 1430
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, 1430, 83]
After
derived = [17, 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, S031
Before
Field ORIGINAL previous power times element k=16 := mul(19, 22)
After
Field ORIGINAL previous power times element k=16 = 418
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, 418, 83]
After
derived = [5, 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, S033
Before
Field ORIGINAL previous power times element k=17 := mul(3, 22)
After
Field ORIGINAL previous power times element k=17 = 66
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, 66, 83]
After
derived = [0, 66]
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(66, 22)
After
Field ORIGINAL previous power times element k=18 = 1452
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, 1452, 83]
After
derived = [17, 41]
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(41, 22)
After
Field ORIGINAL previous power times element k=19 = 902
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, 902, 83]
After
derived = [10, 72]
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(72, 22)
After
Field ORIGINAL previous power times element k=20 = 1584
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, 1584, 83]
After
derived = [19, 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, S041
Before
Field ORIGINAL previous power times element k=21 := mul(7, 22)
After
Field ORIGINAL previous power times element k=21 = 154
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, 154, 83]
After
derived = [1, 71]
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(71, 22)
After
Field ORIGINAL previous power times element k=22 = 1562
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, 1562, 83]
After
derived = [18, 68]
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(68, 22)
After
Field ORIGINAL previous power times element k=23 = 1496
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, 1496, 83]
After
derived = [18, 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, S047
Before
Field ORIGINAL previous power times element k=24 := mul(2, 22)
After
Field ORIGINAL previous power times element k=24 = 44
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, 44, 83]
After
derived = [0, 44]
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(44, 22)
After
Field ORIGINAL previous power times element k=25 = 968
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, 968, 83]
After
derived = [11, 55]
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(55, 22)
After
Field ORIGINAL previous power times element k=26 = 1210
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, 1210, 83]
After
derived = [14, 48]
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(48, 22)
After
Field ORIGINAL previous power times element k=27 = 1056
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, 1056, 83]
After
derived = [12, 60]
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(60, 22)
After
Field ORIGINAL previous power times element k=28 = 1320
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, 1320, 83]
After
derived = [15, 75]
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(75, 22)
After
Field ORIGINAL previous power times element k=29 = 1650
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, 1650, 83]
After
derived = [19, 73]
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(73, 22)
After
Field ORIGINAL previous power times element k=30 = 1606
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, 1606, 83]
After
derived = [19, 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, S061
Before
Field ORIGINAL previous power times element k=31 := mul(29, 22)
After
Field ORIGINAL previous power times element k=31 = 638
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, 638, 83]
After
derived = [7, 57]
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(57, 22)
After
Field ORIGINAL previous power times element k=32 = 1254
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, 1254, 83]
After
derived = [15, 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, S065
Before
Field ORIGINAL previous power times element k=33 := mul(9, 22)
After
Field ORIGINAL previous power times element k=33 = 198
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, 198, 83]
After
derived = [2, 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, S067
Before
Field ORIGINAL previous power times element k=34 := mul(32, 22)
After
Field ORIGINAL previous power times element k=34 = 704
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, 704, 83]
After
derived = [8, 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, S069
Before
Field ORIGINAL previous power times element k=35 := mul(40, 22)
After
Field ORIGINAL previous power times element k=35 = 880
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, 880, 83]
After
derived = [10, 50]
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(50, 22)
After
Field ORIGINAL previous power times element k=36 = 1100
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, 1100, 83]
After
derived = [13, 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, S073
Before
Field ORIGINAL previous power times element k=37 := mul(21, 22)
After
Field ORIGINAL previous power times element k=37 = 462
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, 462, 83]
After
derived = [5, 47]
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(47, 22)
After
Field ORIGINAL previous power times element k=38 = 1034
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, 1034, 83]
After
derived = [12, 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, S077
Before
Field ORIGINAL previous power times element k=39 := mul(38, 22)
After
Field ORIGINAL previous power times element k=39 = 836
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, 836, 83]
After
derived = [10, 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, S079
Before
Field ORIGINAL previous power times element k=40 := mul(6, 22)
After
Field ORIGINAL previous power times element k=40 = 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, S080
Before
source roles = [power exponent40, 132, 83]
After
derived = [1, 49]
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(49, 22)
After
Field ORIGINAL previous power times element k=41 = 1078
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, 1078, 83]
After
derived = [12, 82]
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
Field ORIGINAL previous power times element k=42 := mul(82, 22)
After
Field ORIGINAL previous power times element k=42 = 1804
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, S084
Before
source roles = [power exponent42, 1804, 83]
After
derived = [21, 61]
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, S085
Before
Field ORIGINAL previous power times element k=43 := mul(61, 22)
After
Field ORIGINAL previous power times element k=43 = 1342
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, S086
Before
source roles = [power exponent43, 1342, 83]
After
derived = [16, 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, S087
Before
Field ORIGINAL previous power times element k=44 := mul(14, 22)
After
Field ORIGINAL previous power times element k=44 = 308
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, S088
Before
source roles = [power exponent44, 308, 83]
After
derived = [3, 59]
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, S089
Before
Field ORIGINAL previous power times element k=45 := mul(59, 22)
After
Field ORIGINAL previous power times element k=45 = 1298
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, S090
Before
source roles = [power exponent45, 1298, 83]
After
derived = [15, 53]
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, S091
Before
Field ORIGINAL previous power times element k=46 := mul(53, 22)
After
Field ORIGINAL previous power times element k=46 = 1166
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, S092
Before
source roles = [power exponent46, 1166, 83]
After
derived = [14, 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, S093
Before
Field ORIGINAL previous power times element k=47 := mul(4, 22)
After
Field ORIGINAL previous power times element k=47 = 88
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, S094
Before
source roles = [power exponent47, 88, 83]
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.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S095
Before
Field ORIGINAL previous power times element k=48 := mul(5, 22)
After
Field ORIGINAL previous power times element k=48 = 110
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, S096
Before
source roles = [power exponent48, 110, 83]
After
derived = [1, 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, S097
Before
Field ORIGINAL previous power times element k=49 := mul(27, 22)
After
Field ORIGINAL previous power times element k=49 = 594
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, S098
Before
source roles = [power exponent49, 594, 83]
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, S099
Before
Field ORIGINAL previous power times element k=50 := mul(13, 22)
After
Field ORIGINAL previous power times element k=50 = 286
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, S100
Before
source roles = [power exponent50, 286, 83]
After
derived = [3, 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, S101
Before
Field ORIGINAL previous power times element k=51 := mul(37, 22)
After
Field ORIGINAL previous power times element k=51 = 814
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, S102
Before
source roles = [power exponent51, 814, 83]
After
derived = [9, 67]
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, S103
Before
Field ORIGINAL previous power times element k=52 := mul(67, 22)
After
Field ORIGINAL previous power times element k=52 = 1474
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, S104
Before
source roles = [power exponent52, 1474, 83]
After
derived = [17, 63]
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, S105
Before
Field ORIGINAL previous power times element k=53 := mul(63, 22)
After
Field ORIGINAL previous power times element k=53 = 1386
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, S106
Before
source roles = [power exponent53, 1386, 83]
After
derived = [16, 58]
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, S107
Before
Field ORIGINAL previous power times element k=54 := mul(58, 22)
After
Field ORIGINAL previous power times element k=54 = 1276
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, S108
Before
source roles = [power exponent54, 1276, 83]
After
derived = [15, 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, S109
Before
Field ORIGINAL previous power times element k=55 := mul(31, 22)
After
Field ORIGINAL previous power times element k=55 = 682
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, S110
Before
source roles = [power exponent55, 682, 83]
After
derived = [8, 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, S111
Before
Field ORIGINAL previous power times element k=56 := mul(18, 22)
After
Field ORIGINAL previous power times element k=56 = 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, S112
Before
source roles = [power exponent56, 396, 83]
After
derived = [4, 64]
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, S113
Before
Field ORIGINAL previous power times element k=57 := mul(64, 22)
After
Field ORIGINAL previous power times element k=57 = 1408
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, S114
Before
source roles = [power exponent57, 1408, 83]
After
derived = [16, 80]
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, S115
Before
Field ORIGINAL previous power times element k=58 := mul(80, 22)
After
Field ORIGINAL previous power times element k=58 = 1760
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, S116
Before
source roles = [power exponent58, 1760, 83]
After
derived = [21, 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, S117
Before
Field ORIGINAL previous power times element k=59 := mul(17, 22)
After
Field ORIGINAL previous power times element k=59 = 374
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, S118
Before
source roles = [power exponent59, 374, 83]
After
derived = [4, 42]
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, S119
Before
Field ORIGINAL previous power times element k=60 := mul(42, 22)
After
Field ORIGINAL previous power times element k=60 = 924
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, S120
Before
source roles = [power exponent60, 924, 83]
After
derived = [11, 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, S121
Before
Field ORIGINAL previous power times element k=61 := mul(11, 22)
After
Field ORIGINAL previous power times element k=61 = 242
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, S122
Before
source roles = [power exponent61, 242, 83]
After
derived = [2, 76]
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, S123
Before
Field ORIGINAL previous power times element k=62 := mul(76, 22)
After
Field ORIGINAL previous power times element k=62 = 1672
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, S124
Before
source roles = [power exponent62, 1672, 83]
After
derived = [20, 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, S125
Before
Field ORIGINAL previous power times element k=63 := mul(12, 22)
After
Field ORIGINAL previous power times element k=63 = 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, S126
Before
source roles = [power exponent63, 264, 83]
After
derived = [3, 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, S127
Before
Field ORIGINAL previous power times element k=64 := mul(15, 22)
After
Field ORIGINAL previous power times element k=64 = 330
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, S128
Before
source roles = [power exponent64, 330, 83]
After
derived = [3, 81]
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, S129
Before
Field ORIGINAL previous power times element k=65 := mul(81, 22)
After
Field ORIGINAL previous power times element k=65 = 1782
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, S130
Before
source roles = [power exponent65, 1782, 83]
After
derived = [21, 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, S131
Before
Field ORIGINAL previous power times element k=66 := mul(39, 22)
After
Field ORIGINAL previous power times element k=66 = 858
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, S132
Before
source roles = [power exponent66, 858, 83]
After
derived = [10, 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, S133
Before
Field ORIGINAL previous power times element k=67 := mul(28, 22)
After
Field ORIGINAL previous power times element k=67 = 616
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, S134
Before
source roles = [power exponent67, 616, 83]
After
derived = [7, 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, S135
Before
Field ORIGINAL previous power times element k=68 := mul(35, 22)
After
Field ORIGINAL previous power times element k=68 = 770
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, S136
Before
source roles = [power exponent68, 770, 83]
After
derived = [9, 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, S137
Before
Field ORIGINAL previous power times element k=69 := mul(23, 22)
After
Field ORIGINAL previous power times element k=69 = 506
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, S138
Before
source roles = [power exponent69, 506, 83]
After
derived = [6, 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, S139
Before
Field ORIGINAL previous power times element k=70 := mul(8, 22)
After
Field ORIGINAL previous power times element k=70 = 176
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, S140
Before
source roles = [power exponent70, 176, 83]
After
derived = [2, 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, S141
Before
Field ORIGINAL previous power times element k=71 := mul(10, 22)
After
Field ORIGINAL previous power times element k=71 = 220
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, S142
Before
source roles = [power exponent71, 220, 83]
After
derived = [2, 54]
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, S143
Before
Field ORIGINAL previous power times element k=72 := mul(54, 22)
After
Field ORIGINAL previous power times element k=72 = 1188
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, S144
Before
source roles = [power exponent72, 1188, 83]
After
derived = [14, 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, S145
Before
Field ORIGINAL previous power times element k=73 := mul(26, 22)
After
Field ORIGINAL previous power times element k=73 = 572
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, S146
Before
source roles = [power exponent73, 572, 83]
After
derived = [6, 74]
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, S147
Before
Field ORIGINAL previous power times element k=74 := mul(74, 22)
After
Field ORIGINAL previous power times element k=74 = 1628
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, S148
Before
source roles = [power exponent74, 1628, 83]
After
derived = [19, 51]
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, S149
Before
Field ORIGINAL previous power times element k=75 := mul(51, 22)
After
Field ORIGINAL previous power times element k=75 = 1122
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, S150
Before
source roles = [power exponent75, 1122, 83]
After
derived = [13, 43]
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, S151
Before
Field ORIGINAL previous power times element k=76 := mul(43, 22)
After
Field ORIGINAL previous power times element k=76 = 946
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, S152
Before
source roles = [power exponent76, 946, 83]
After
derived = [11, 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, S153
Before
Field ORIGINAL previous power times element k=77 := mul(33, 22)
After
Field ORIGINAL previous power times element k=77 = 726
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, S154
Before
source roles = [power exponent77, 726, 83]
After
derived = [8, 62]
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, S155
Before
Field ORIGINAL previous power times element k=78 := mul(62, 22)
After
Field ORIGINAL previous power times element k=78 = 1364
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, S156
Before
source roles = [power exponent78, 1364, 83]
After
derived = [16, 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, S157
Before
Field ORIGINAL previous power times element k=79 := mul(36, 22)
After
Field ORIGINAL previous power times element k=79 = 792
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, S158
Before
source roles = [power exponent79, 792, 83]
After
derived = [9, 45]
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, S159
Before
Field ORIGINAL previous power times element k=80 := mul(45, 22)
After
Field ORIGINAL previous power times element k=80 = 990
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, S160
Before
source roles = [power exponent80, 990, 83]
After
derived = [11, 77]
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, S161
Before
Field ORIGINAL previous power times element k=81 := mul(77, 22)
After
Field ORIGINAL previous power times element k=81 = 1694
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, S162
Before
source roles = [power exponent81, 1694, 83]
After
derived = [20, 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, S163
Before
Field ORIGINAL previous power times element k=82 := mul(34, 22)
After
Field ORIGINAL previous power times element k=82 = 748
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, S164
Before
source roles = [power exponent82, 748, 83]
After
derived = [9, 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, S165
Before
Field ORIGINAL previous power times element k=83 := mul(1, 22)
After
Field ORIGINAL previous power times element k=83 = 22
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, S166
Before
source roles = [power exponent83, 22, 83]
After
derived = [0, 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, S167
Before
source roles = [83, 22, [22, 69, 24, 30, 79, 78, 56, 70, 46, 16, 20, 25, 52, 65, 19, 3, 66, 41, 72, 7, 71, 68, 2, 44, 55, 48, 60, 75, 73, 29, 57, 9, 32, 40, 50, 21, 47, 38, 6, 49, 82, 61, 14, 59, 53, 4, 5, 27, 13, 37, 67, 63, 58, 31, 18, 64, 80, 17, 42, 11, 76, 12, 15, 81, 39, 28, 35, 23, 8, 10, 54, 26, 74, 51, 43, 33, 62, 36, 45, 77, 34, 1, 22]]
After
derived = [22, 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, S168
S169 · 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.S168 · 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.S167 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=83 with exact rational arithmetic, not rounded decimals.S166 · 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.S165 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=82 with exact rational arithmetic, not rounded decimals.S164 · 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.S163 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=81 with exact rational arithmetic, not rounded decimals.S162 · 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.S161 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=80 with exact rational arithmetic, not rounded decimals.S160 · 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.S159 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=79 with exact rational arithmetic, not rounded decimals.S158 · 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.S157 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=78 with exact rational arithmetic, not rounded decimals.S156 · 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.S155 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=77 with exact rational arithmetic, not rounded decimals.S154 · 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.S153 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=76 with exact rational arithmetic, not rounded decimals.S152 · 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.S151 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=75 with exact rational arithmetic, not rounded decimals.S150 · 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.S149 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=74 with exact rational arithmetic, not rounded decimals.S148 · 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.S147 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=73 with exact rational arithmetic, not rounded decimals.S146 · 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.S145 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=72 with exact rational arithmetic, not rounded decimals.S144 · 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.S143 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=71 with exact rational arithmetic, not rounded decimals.S142 · 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.S141 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=70 with exact rational arithmetic, not rounded decimals.S140 · 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.S139 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=69 with exact rational arithmetic, not rounded decimals.S138 · 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.S137 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=68 with exact rational arithmetic, not rounded decimals.S136 · 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.S135 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=67 with exact rational arithmetic, not rounded decimals.S134 · 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.S133 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=66 with exact rational arithmetic, not rounded decimals.S132 · 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.S131 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=65 with exact rational arithmetic, not rounded decimals.S130 · 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.S129 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=64 with exact rational arithmetic, not rounded decimals.S128 · 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.S127 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=63 with exact rational arithmetic, not rounded decimals.S126 · 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.S125 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=62 with exact rational arithmetic, not rounded decimals.S124 · 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.S123 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=61 with exact rational arithmetic, not rounded decimals.S122 · 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.S121 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=60 with exact rational arithmetic, not rounded decimals.S120 · 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.S119 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=59 with exact rational arithmetic, not rounded decimals.S118 · 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.S117 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=58 with exact rational arithmetic, not rounded decimals.S116 · 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.S115 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=57 with exact rational arithmetic, not rounded decimals.S114 · 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.S113 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=56 with exact rational arithmetic, not rounded decimals.S112 · 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.S111 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=55 with exact rational arithmetic, not rounded decimals.S110 · 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.S109 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=54 with exact rational arithmetic, not rounded decimals.S108 · 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.S107 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=53 with exact rational arithmetic, not rounded decimals.S106 · 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.S105 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=52 with exact rational arithmetic, not rounded decimals.S104 · 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.S103 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=51 with exact rational arithmetic, not rounded decimals.S102 · 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.S101 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=50 with exact rational arithmetic, not rounded decimals.S100 · 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.S099 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=49 with exact rational arithmetic, not rounded decimals.S098 · 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.S097 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=48 with exact rational arithmetic, not rounded decimals.S096 · 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.S095 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=47 with exact rational arithmetic, not rounded decimals.S094 · 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.S093 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=46 with exact rational arithmetic, not rounded decimals.S092 · 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.S091 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=45 with exact rational arithmetic, not rounded decimals.S090 · 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.S089 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=44 with exact rational arithmetic, not rounded decimals.S088 · 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.S087 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=43 with exact rational arithmetic, not rounded decimals.S086 · 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.S085 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=42 with exact rational arithmetic, not rounded decimals.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.(22, 83, 22, 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: cb4c73506b71f478984b0196c970a3ffe72dee7188690088c12051f6a2276667candidate record SHA-256: 0e6b8d817675ef4c10f93a19eb81c5b369abc55220f17d5f51514a88de378f9asemantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710ccLean: NOT_RUN · human review: NOT_PERFORMED