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R0066-33664 · q-20261014-advance-20261031-r06-finite-field-0089
Use the complete prime field Z_43, with prime p=43, 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=13, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.
G001 — Use the complete prime field Z_43, with prime p=43, 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=13, 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 = [43, [2, 3, 4, 5, 6]]
After
derived = [[1, 1, 3, 3, 1], 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, 13)
After
Field ORIGINAL previous power times element k=1 = 13
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, 13, 43]
After
derived = [0, 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, S003
Before
Field ORIGINAL previous power times element k=2 := mul(13, 13)
After
Field ORIGINAL previous power times element k=2 = 169
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, 169, 43]
After
derived = [3, 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, S005
Before
Field ORIGINAL previous power times element k=3 := mul(40, 13)
After
Field ORIGINAL previous power times element k=3 = 520
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, 520, 43]
After
derived = [12, 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, S007
Before
Field ORIGINAL previous power times element k=4 := mul(4, 13)
After
Field ORIGINAL previous power times element k=4 = 52
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, 52, 43]
After
derived = [1, 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, S009
Before
Field ORIGINAL previous power times element k=5 := mul(9, 13)
After
Field ORIGINAL previous power times element k=5 = 117
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, 117, 43]
After
derived = [2, 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, S011
Before
Field ORIGINAL previous power times element k=6 := mul(31, 13)
After
Field ORIGINAL previous power times element k=6 = 403
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, 403, 43]
After
derived = [9, 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, S013
Before
Field ORIGINAL previous power times element k=7 := mul(16, 13)
After
Field ORIGINAL previous power times element k=7 = 208
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, 208, 43]
After
derived = [4, 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, S015
Before
Field ORIGINAL previous power times element k=8 := mul(36, 13)
After
Field ORIGINAL previous power times element k=8 = 468
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
source roles = [power exponent8, 468, 43]
After
derived = [10, 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, S017
Before
Field ORIGINAL previous power times element k=9 := mul(38, 13)
After
Field ORIGINAL previous power times element k=9 = 494
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, 494, 43]
After
derived = [11, 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, S019
Before
Field ORIGINAL previous power times element k=10 := mul(21, 13)
After
Field ORIGINAL previous power times element k=10 = 273
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, 273, 43]
After
derived = [6, 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, S021
Before
Field ORIGINAL previous power times element k=11 := mul(15, 13)
After
Field ORIGINAL previous power times element k=11 = 195
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, 195, 43]
After
derived = [4, 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, S023
Before
Field ORIGINAL previous power times element k=12 := mul(23, 13)
After
Field ORIGINAL previous power times element k=12 = 299
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, 299, 43]
After
derived = [6, 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, S025
Before
Field ORIGINAL previous power times element k=13 := mul(41, 13)
After
Field ORIGINAL previous power times element k=13 = 533
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, 533, 43]
After
derived = [12, 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, S027
Before
Field ORIGINAL previous power times element k=14 := mul(17, 13)
After
Field ORIGINAL previous power times element k=14 = 221
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, 221, 43]
After
derived = [5, 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, S029
Before
Field ORIGINAL previous power times element k=15 := mul(6, 13)
After
Field ORIGINAL previous power times element k=15 = 78
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, 78, 43]
After
derived = [1, 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, S031
Before
Field ORIGINAL previous power times element k=16 := mul(35, 13)
After
Field ORIGINAL previous power times element k=16 = 455
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, 455, 43]
After
derived = [10, 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, S033
Before
Field ORIGINAL previous power times element k=17 := mul(25, 13)
After
Field ORIGINAL previous power times element k=17 = 325
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, 325, 43]
After
derived = [7, 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, S035
Before
Field ORIGINAL previous power times element k=18 := mul(24, 13)
After
Field ORIGINAL previous power times element k=18 = 312
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
source roles = [power exponent18, 312, 43]
After
derived = [7, 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, S037
Before
Field ORIGINAL previous power times element k=19 := mul(11, 13)
After
Field ORIGINAL previous power times element k=19 = 143
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, 143, 43]
After
derived = [3, 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, S039
Before
Field ORIGINAL previous power times element k=20 := mul(14, 13)
After
Field ORIGINAL previous power times element k=20 = 182
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, 182, 43]
After
derived = [4, 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, S041
Before
Field ORIGINAL previous power times element k=21 := mul(10, 13)
After
Field ORIGINAL previous power times element k=21 = 130
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, 130, 43]
After
derived = [3, 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, S043
Before
Field ORIGINAL previous power times element k=22 := mul(1, 13)
After
Field ORIGINAL previous power times element k=22 = 13
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, 13, 43]
After
derived = [0, 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, S045
Before
Field ORIGINAL previous power times element k=23 := mul(13, 13)
After
Field ORIGINAL previous power times element k=23 = 169
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, 169, 43]
After
derived = [3, 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, S047
Before
Field ORIGINAL previous power times element k=24 := mul(40, 13)
After
Field ORIGINAL previous power times element k=24 = 520
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, 520, 43]
After
derived = [12, 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, S049
Before
Field ORIGINAL previous power times element k=25 := mul(4, 13)
After
Field ORIGINAL previous power times element k=25 = 52
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, 52, 43]
After
derived = [1, 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, S051
Before
Field ORIGINAL previous power times element k=26 := mul(9, 13)
After
Field ORIGINAL previous power times element k=26 = 117
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, 117, 43]
After
derived = [2, 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, S053
Before
Field ORIGINAL previous power times element k=27 := mul(31, 13)
After
Field ORIGINAL previous power times element k=27 = 403
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, 403, 43]
After
derived = [9, 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, S055
Before
Field ORIGINAL previous power times element k=28 := mul(16, 13)
After
Field ORIGINAL previous power times element k=28 = 208
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, 208, 43]
After
derived = [4, 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, S057
Before
Field ORIGINAL previous power times element k=29 := mul(36, 13)
After
Field ORIGINAL previous power times element k=29 = 468
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
source roles = [power exponent29, 468, 43]
After
derived = [10, 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, S059
Before
Field ORIGINAL previous power times element k=30 := mul(38, 13)
After
Field ORIGINAL previous power times element k=30 = 494
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, 494, 43]
After
derived = [11, 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, S061
Before
Field ORIGINAL previous power times element k=31 := mul(21, 13)
After
Field ORIGINAL previous power times element k=31 = 273
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, 273, 43]
After
derived = [6, 15]
Conditions: none
Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.
Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S063
Before
Field ORIGINAL previous power times element k=32 := mul(15, 13)
After
Field ORIGINAL previous power times element k=32 = 195
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, 195, 43]
After
derived = [4, 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, S065
Before
Field ORIGINAL previous power times element k=33 := mul(23, 13)
After
Field ORIGINAL previous power times element k=33 = 299
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, 299, 43]
After
derived = [6, 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, S067
Before
Field ORIGINAL previous power times element k=34 := mul(41, 13)
After
Field ORIGINAL previous power times element k=34 = 533
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, 533, 43]
After
derived = [12, 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, S069
Before
Field ORIGINAL previous power times element k=35 := mul(17, 13)
After
Field ORIGINAL previous power times element k=35 = 221
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, 221, 43]
After
derived = [5, 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, S071
Before
Field ORIGINAL previous power times element k=36 := mul(6, 13)
After
Field ORIGINAL previous power times element k=36 = 78
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, 78, 43]
After
derived = [1, 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, S073
Before
Field ORIGINAL previous power times element k=37 := mul(35, 13)
After
Field ORIGINAL previous power times element k=37 = 455
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, 455, 43]
After
derived = [10, 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, S075
Before
Field ORIGINAL previous power times element k=38 := mul(25, 13)
After
Field ORIGINAL previous power times element k=38 = 325
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, 325, 43]
After
derived = [7, 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, S077
Before
Field ORIGINAL previous power times element k=39 := mul(24, 13)
After
Field ORIGINAL previous power times element k=39 = 312
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Rule: exact_rational_arithmetic · Uses: G001, S078
Before
source roles = [power exponent39, 312, 43]
After
derived = [7, 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, S079
Before
Field ORIGINAL previous power times element k=40 := mul(11, 13)
After
Field ORIGINAL previous power times element k=40 = 143
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, 143, 43]
After
derived = [3, 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, S081
Before
Field ORIGINAL previous power times element k=41 := mul(14, 13)
After
Field ORIGINAL previous power times element k=41 = 182
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, 182, 43]
After
derived = [4, 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, S083
Before
Field ORIGINAL previous power times element k=42 := mul(10, 13)
After
Field ORIGINAL previous power times element k=42 = 130
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, 130, 43]
After
derived = [3, 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, S085
Before
Field ORIGINAL previous power times element k=43 := mul(1, 13)
After
Field ORIGINAL previous power times element k=43 = 13
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, 13, 43]
After
derived = [0, 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, S087
Before
source roles = [43, 13, [13, 40, 4, 9, 31, 16, 36, 38, 21, 15, 23, 41, 17, 6, 35, 25, 24, 11, 14, 10, 1, 13, 40, 4, 9, 31, 16, 36, 38, 21, 15, 23, 41, 17, 6, 35, 25, 24, 11, 14, 10, 1, 13]]
After
derived = [13, 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, S088
S089 · 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.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.(13, 43, 13, 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: 3b09735d23a180907a6063ebada55f8df0efebb81907e6cd7a396898e95379bccandidate record SHA-256: 4569851e9c935e5d5950b84c8465767ba99c4143f502cecfcf71aa0b7dcf3237semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710ccLean: NOT_RUN · human review: NOT_PERFORMED