Question 33700: The Nonzero Multiplicative Group Gives a p−1 Power Identity

Original question and givens

Use the complete prime field Z_97, with prime p=97, 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. Nonzero element=25. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.

• G001 — Use the complete prime field Z_97, with prime p=97, 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. Nonzero element=25. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-finite-field

Step-by-step derivation

source roles = [qf-mod31-fielddeep-power-cycle, [domain, element, operation, p]]

derived = [fielddeep-power-cycle, Exact_prime_field_Zp]

source roles = [97, [2, 3, 4, 5, 6, 7, 8, 9]]

derived = [[1, 1, 1, 2, 1, 6, 1, 7], Yes]

Field ORIGINAL previous power times element k=1 := mul(1, 25)

Field ORIGINAL previous power times element k=1 = 25

source roles = [power exponent1, 25, 97]

derived = [0, 25]

Field ORIGINAL previous power times element k=2 := mul(25, 25)

Field ORIGINAL previous power times element k=2 = 625

source roles = [power exponent2, 625, 97]

derived = [6, 43]

Field ORIGINAL previous power times element k=3 := mul(43, 25)

Field ORIGINAL previous power times element k=3 = 1075

source roles = [power exponent3, 1075, 97]

derived = [11, 8]

Field ORIGINAL previous power times element k=4 := mul(8, 25)

Field ORIGINAL previous power times element k=4 = 200

source roles = [power exponent4, 200, 97]

derived = [2, 6]

Field ORIGINAL previous power times element k=5 := mul(6, 25)

Field ORIGINAL previous power times element k=5 = 150

source roles = [power exponent5, 150, 97]

derived = [1, 53]

Field ORIGINAL previous power times element k=6 := mul(53, 25)

Field ORIGINAL previous power times element k=6 = 1325

source roles = [power exponent6, 1325, 97]

derived = [13, 64]

Field ORIGINAL previous power times element k=7 := mul(64, 25)

Field ORIGINAL previous power times element k=7 = 1600

source roles = [power exponent7, 1600, 97]

derived = [16, 48]

Field ORIGINAL previous power times element k=8 := mul(48, 25)

Field ORIGINAL previous power times element k=8 = 1200

source roles = [power exponent8, 1200, 97]

derived = [12, 36]

Field ORIGINAL previous power times element k=9 := mul(36, 25)

Field ORIGINAL previous power times element k=9 = 900

source roles = [power exponent9, 900, 97]

derived = [9, 27]

Field ORIGINAL previous power times element k=10 := mul(27, 25)

Field ORIGINAL previous power times element k=10 = 675

source roles = [power exponent10, 675, 97]

derived = [6, 93]

Field ORIGINAL previous power times element k=11 := mul(93, 25)

Field ORIGINAL previous power times element k=11 = 2325

source roles = [power exponent11, 2325, 97]

derived = [23, 94]

Field ORIGINAL previous power times element k=12 := mul(94, 25)

Field ORIGINAL previous power times element k=12 = 2350

source roles = [power exponent12, 2350, 97]

derived = [24, 22]

Field ORIGINAL previous power times element k=13 := mul(22, 25)

Field ORIGINAL previous power times element k=13 = 550

source roles = [power exponent13, 550, 97]

derived = [5, 65]

Field ORIGINAL previous power times element k=14 := mul(65, 25)

Field ORIGINAL previous power times element k=14 = 1625

source roles = [power exponent14, 1625, 97]

derived = [16, 73]

Field ORIGINAL previous power times element k=15 := mul(73, 25)

Field ORIGINAL previous power times element k=15 = 1825

source roles = [power exponent15, 1825, 97]

derived = [18, 79]

Field ORIGINAL previous power times element k=16 := mul(79, 25)

Field ORIGINAL previous power times element k=16 = 1975

source roles = [power exponent16, 1975, 97]

derived = [20, 35]

Field ORIGINAL previous power times element k=17 := mul(35, 25)

Field ORIGINAL previous power times element k=17 = 875

source roles = [power exponent17, 875, 97]

derived = [9, 2]

Field ORIGINAL previous power times element k=18 := mul(2, 25)

Field ORIGINAL previous power times element k=18 = 50

source roles = [power exponent18, 50, 97]

derived = [0, 50]

Field ORIGINAL previous power times element k=19 := mul(50, 25)

Field ORIGINAL previous power times element k=19 = 1250

source roles = [power exponent19, 1250, 97]

derived = [12, 86]

Field ORIGINAL previous power times element k=20 := mul(86, 25)

Field ORIGINAL previous power times element k=20 = 2150

source roles = [power exponent20, 2150, 97]

derived = [22, 16]

Field ORIGINAL previous power times element k=21 := mul(16, 25)

Field ORIGINAL previous power times element k=21 = 400

source roles = [power exponent21, 400, 97]

derived = [4, 12]

Field ORIGINAL previous power times element k=22 := mul(12, 25)

Field ORIGINAL previous power times element k=22 = 300

source roles = [power exponent22, 300, 97]

derived = [3, 9]

Field ORIGINAL previous power times element k=23 := mul(9, 25)

Field ORIGINAL previous power times element k=23 = 225

source roles = [power exponent23, 225, 97]

derived = [2, 31]

Field ORIGINAL previous power times element k=24 := mul(31, 25)

Field ORIGINAL previous power times element k=24 = 775

source roles = [power exponent24, 775, 97]

derived = [7, 96]

Field ORIGINAL previous power times element k=25 := mul(96, 25)

Field ORIGINAL previous power times element k=25 = 2400

source roles = [power exponent25, 2400, 97]

derived = [24, 72]

Field ORIGINAL previous power times element k=26 := mul(72, 25)

Field ORIGINAL previous power times element k=26 = 1800

source roles = [power exponent26, 1800, 97]

derived = [18, 54]

Field ORIGINAL previous power times element k=27 := mul(54, 25)

Field ORIGINAL previous power times element k=27 = 1350

source roles = [power exponent27, 1350, 97]

derived = [13, 89]

Field ORIGINAL previous power times element k=28 := mul(89, 25)

Field ORIGINAL previous power times element k=28 = 2225

source roles = [power exponent28, 2225, 97]

derived = [22, 91]

Field ORIGINAL previous power times element k=29 := mul(91, 25)

Field ORIGINAL previous power times element k=29 = 2275

source roles = [power exponent29, 2275, 97]

derived = [23, 44]

Field ORIGINAL previous power times element k=30 := mul(44, 25)

Field ORIGINAL previous power times element k=30 = 1100

source roles = [power exponent30, 1100, 97]

derived = [11, 33]

Field ORIGINAL previous power times element k=31 := mul(33, 25)

Field ORIGINAL previous power times element k=31 = 825

source roles = [power exponent31, 825, 97]

derived = [8, 49]

Field ORIGINAL previous power times element k=32 := mul(49, 25)

Field ORIGINAL previous power times element k=32 = 1225

source roles = [power exponent32, 1225, 97]

derived = [12, 61]

Field ORIGINAL previous power times element k=33 := mul(61, 25)

Field ORIGINAL previous power times element k=33 = 1525

source roles = [power exponent33, 1525, 97]

derived = [15, 70]

Field ORIGINAL previous power times element k=34 := mul(70, 25)

Field ORIGINAL previous power times element k=34 = 1750

source roles = [power exponent34, 1750, 97]

derived = [18, 4]

Field ORIGINAL previous power times element k=35 := mul(4, 25)

Field ORIGINAL previous power times element k=35 = 100

source roles = [power exponent35, 100, 97]

derived = [1, 3]

Field ORIGINAL previous power times element k=36 := mul(3, 25)

Field ORIGINAL previous power times element k=36 = 75

source roles = [power exponent36, 75, 97]

derived = [0, 75]

Field ORIGINAL previous power times element k=37 := mul(75, 25)

Field ORIGINAL previous power times element k=37 = 1875

source roles = [power exponent37, 1875, 97]

derived = [19, 32]

Field ORIGINAL previous power times element k=38 := mul(32, 25)

Field ORIGINAL previous power times element k=38 = 800

source roles = [power exponent38, 800, 97]

derived = [8, 24]

Field ORIGINAL previous power times element k=39 := mul(24, 25)

Field ORIGINAL previous power times element k=39 = 600

source roles = [power exponent39, 600, 97]

derived = [6, 18]

Field ORIGINAL previous power times element k=40 := mul(18, 25)

Field ORIGINAL previous power times element k=40 = 450

source roles = [power exponent40, 450, 97]

derived = [4, 62]

Field ORIGINAL previous power times element k=41 := mul(62, 25)

Field ORIGINAL previous power times element k=41 = 1550

source roles = [power exponent41, 1550, 97]

derived = [15, 95]

Field ORIGINAL previous power times element k=42 := mul(95, 25)

Field ORIGINAL previous power times element k=42 = 2375

source roles = [power exponent42, 2375, 97]

derived = [24, 47]

Field ORIGINAL previous power times element k=43 := mul(47, 25)

Field ORIGINAL previous power times element k=43 = 1175

source roles = [power exponent43, 1175, 97]

derived = [12, 11]

Field ORIGINAL previous power times element k=44 := mul(11, 25)

Field ORIGINAL previous power times element k=44 = 275

source roles = [power exponent44, 275, 97]

derived = [2, 81]

Field ORIGINAL previous power times element k=45 := mul(81, 25)

Field ORIGINAL previous power times element k=45 = 2025

source roles = [power exponent45, 2025, 97]

derived = [20, 85]

Field ORIGINAL previous power times element k=46 := mul(85, 25)

Field ORIGINAL previous power times element k=46 = 2125

source roles = [power exponent46, 2125, 97]

derived = [21, 88]

Field ORIGINAL previous power times element k=47 := mul(88, 25)

Field ORIGINAL previous power times element k=47 = 2200

source roles = [power exponent47, 2200, 97]

derived = [22, 66]

Field ORIGINAL previous power times element k=48 := mul(66, 25)

Field ORIGINAL previous power times element k=48 = 1650

source roles = [power exponent48, 1650, 97]

derived = [17, 1]

Field ORIGINAL previous power times element k=49 := mul(1, 25)

Field ORIGINAL previous power times element k=49 = 25

source roles = [power exponent49, 25, 97]

derived = [0, 25]

Field ORIGINAL previous power times element k=50 := mul(25, 25)

Field ORIGINAL previous power times element k=50 = 625

source roles = [power exponent50, 625, 97]

derived = [6, 43]

Field ORIGINAL previous power times element k=51 := mul(43, 25)

Field ORIGINAL previous power times element k=51 = 1075

source roles = [power exponent51, 1075, 97]

derived = [11, 8]

Field ORIGINAL previous power times element k=52 := mul(8, 25)

Field ORIGINAL previous power times element k=52 = 200

source roles = [power exponent52, 200, 97]

derived = [2, 6]

Field ORIGINAL previous power times element k=53 := mul(6, 25)

Field ORIGINAL previous power times element k=53 = 150

source roles = [power exponent53, 150, 97]

derived = [1, 53]

Field ORIGINAL previous power times element k=54 := mul(53, 25)

Field ORIGINAL previous power times element k=54 = 1325

source roles = [power exponent54, 1325, 97]

derived = [13, 64]

Field ORIGINAL previous power times element k=55 := mul(64, 25)

Field ORIGINAL previous power times element k=55 = 1600

source roles = [power exponent55, 1600, 97]

derived = [16, 48]

Field ORIGINAL previous power times element k=56 := mul(48, 25)

Field ORIGINAL previous power times element k=56 = 1200

source roles = [power exponent56, 1200, 97]

derived = [12, 36]

Field ORIGINAL previous power times element k=57 := mul(36, 25)

Field ORIGINAL previous power times element k=57 = 900

source roles = [power exponent57, 900, 97]

derived = [9, 27]

Field ORIGINAL previous power times element k=58 := mul(27, 25)

Field ORIGINAL previous power times element k=58 = 675

source roles = [power exponent58, 675, 97]

derived = [6, 93]

Field ORIGINAL previous power times element k=59 := mul(93, 25)

Field ORIGINAL previous power times element k=59 = 2325

source roles = [power exponent59, 2325, 97]

derived = [23, 94]

Field ORIGINAL previous power times element k=60 := mul(94, 25)

Field ORIGINAL previous power times element k=60 = 2350

source roles = [power exponent60, 2350, 97]

derived = [24, 22]

Field ORIGINAL previous power times element k=61 := mul(22, 25)

Field ORIGINAL previous power times element k=61 = 550

source roles = [power exponent61, 550, 97]

derived = [5, 65]

Field ORIGINAL previous power times element k=62 := mul(65, 25)

Field ORIGINAL previous power times element k=62 = 1625

source roles = [power exponent62, 1625, 97]

derived = [16, 73]

Field ORIGINAL previous power times element k=63 := mul(73, 25)

Field ORIGINAL previous power times element k=63 = 1825

source roles = [power exponent63, 1825, 97]

derived = [18, 79]

Field ORIGINAL previous power times element k=64 := mul(79, 25)

Field ORIGINAL previous power times element k=64 = 1975

source roles = [power exponent64, 1975, 97]

derived = [20, 35]

Field ORIGINAL previous power times element k=65 := mul(35, 25)

Field ORIGINAL previous power times element k=65 = 875

source roles = [power exponent65, 875, 97]

derived = [9, 2]

Field ORIGINAL previous power times element k=66 := mul(2, 25)

Field ORIGINAL previous power times element k=66 = 50

source roles = [power exponent66, 50, 97]

derived = [0, 50]

Field ORIGINAL previous power times element k=67 := mul(50, 25)

Field ORIGINAL previous power times element k=67 = 1250

source roles = [power exponent67, 1250, 97]

derived = [12, 86]

Field ORIGINAL previous power times element k=68 := mul(86, 25)

Field ORIGINAL previous power times element k=68 = 2150

source roles = [power exponent68, 2150, 97]

derived = [22, 16]

Field ORIGINAL previous power times element k=69 := mul(16, 25)

Field ORIGINAL previous power times element k=69 = 400

source roles = [power exponent69, 400, 97]

derived = [4, 12]

Field ORIGINAL previous power times element k=70 := mul(12, 25)

Field ORIGINAL previous power times element k=70 = 300

source roles = [power exponent70, 300, 97]

derived = [3, 9]

Field ORIGINAL previous power times element k=71 := mul(9, 25)

Field ORIGINAL previous power times element k=71 = 225

source roles = [power exponent71, 225, 97]

derived = [2, 31]

Field ORIGINAL previous power times element k=72 := mul(31, 25)

Field ORIGINAL previous power times element k=72 = 775

source roles = [power exponent72, 775, 97]

derived = [7, 96]

Field ORIGINAL previous power times element k=73 := mul(96, 25)

Field ORIGINAL previous power times element k=73 = 2400

source roles = [power exponent73, 2400, 97]

derived = [24, 72]

Field ORIGINAL previous power times element k=74 := mul(72, 25)

Field ORIGINAL previous power times element k=74 = 1800

source roles = [power exponent74, 1800, 97]

derived = [18, 54]

Field ORIGINAL previous power times element k=75 := mul(54, 25)

Field ORIGINAL previous power times element k=75 = 1350

source roles = [power exponent75, 1350, 97]

derived = [13, 89]

Field ORIGINAL previous power times element k=76 := mul(89, 25)

Field ORIGINAL previous power times element k=76 = 2225

source roles = [power exponent76, 2225, 97]

derived = [22, 91]

Field ORIGINAL previous power times element k=77 := mul(91, 25)

Field ORIGINAL previous power times element k=77 = 2275

source roles = [power exponent77, 2275, 97]

derived = [23, 44]

Field ORIGINAL previous power times element k=78 := mul(44, 25)

Field ORIGINAL previous power times element k=78 = 1100

source roles = [power exponent78, 1100, 97]

derived = [11, 33]

Field ORIGINAL previous power times element k=79 := mul(33, 25)

Field ORIGINAL previous power times element k=79 = 825

source roles = [power exponent79, 825, 97]

derived = [8, 49]

Field ORIGINAL previous power times element k=80 := mul(49, 25)

Field ORIGINAL previous power times element k=80 = 1225

source roles = [power exponent80, 1225, 97]

derived = [12, 61]

Field ORIGINAL previous power times element k=81 := mul(61, 25)

Field ORIGINAL previous power times element k=81 = 1525

source roles = [power exponent81, 1525, 97]

derived = [15, 70]

Field ORIGINAL previous power times element k=82 := mul(70, 25)

Field ORIGINAL previous power times element k=82 = 1750

source roles = [power exponent82, 1750, 97]

derived = [18, 4]

Field ORIGINAL previous power times element k=83 := mul(4, 25)

Field ORIGINAL previous power times element k=83 = 100

source roles = [power exponent83, 100, 97]

derived = [1, 3]

Field ORIGINAL previous power times element k=84 := mul(3, 25)

Field ORIGINAL previous power times element k=84 = 75

source roles = [power exponent84, 75, 97]

derived = [0, 75]

Field ORIGINAL previous power times element k=85 := mul(75, 25)

Field ORIGINAL previous power times element k=85 = 1875

source roles = [power exponent85, 1875, 97]

derived = [19, 32]

Field ORIGINAL previous power times element k=86 := mul(32, 25)

Field ORIGINAL previous power times element k=86 = 800

source roles = [power exponent86, 800, 97]

derived = [8, 24]

Field ORIGINAL previous power times element k=87 := mul(24, 25)

Field ORIGINAL previous power times element k=87 = 600

source roles = [power exponent87, 600, 97]

derived = [6, 18]

Field ORIGINAL previous power times element k=88 := mul(18, 25)

Field ORIGINAL previous power times element k=88 = 450

source roles = [power exponent88, 450, 97]

derived = [4, 62]

Field ORIGINAL previous power times element k=89 := mul(62, 25)

Field ORIGINAL previous power times element k=89 = 1550

source roles = [power exponent89, 1550, 97]

derived = [15, 95]

Field ORIGINAL previous power times element k=90 := mul(95, 25)

Field ORIGINAL previous power times element k=90 = 2375

source roles = [power exponent90, 2375, 97]

derived = [24, 47]

Field ORIGINAL previous power times element k=91 := mul(47, 25)

Field ORIGINAL previous power times element k=91 = 1175

source roles = [power exponent91, 1175, 97]

derived = [12, 11]

Field ORIGINAL previous power times element k=92 := mul(11, 25)

Field ORIGINAL previous power times element k=92 = 275

source roles = [power exponent92, 275, 97]

derived = [2, 81]

Field ORIGINAL previous power times element k=93 := mul(81, 25)

Field ORIGINAL previous power times element k=93 = 2025

source roles = [power exponent93, 2025, 97]

derived = [20, 85]

Field ORIGINAL previous power times element k=94 := mul(85, 25)

Field ORIGINAL previous power times element k=94 = 2125

source roles = [power exponent94, 2125, 97]

derived = [21, 88]

Field ORIGINAL previous power times element k=95 := mul(88, 25)

Field ORIGINAL previous power times element k=95 = 2200

source roles = [power exponent95, 2200, 97]

derived = [22, 66]

Field ORIGINAL previous power times element k=96 := mul(66, 25)

Field ORIGINAL previous power times element k=96 = 1650

source roles = [power exponent96, 1650, 97]

derived = [17, 1]

source roles = [97, 25, [25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1, 25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1]]

derived = [1, Yes]

Reverse checks

• S195 · 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.
• S194 · 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.
• S193 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=96 with exact rational arithmetic, not rounded decimals.
• S192 · 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.
• S191 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=95 with exact rational arithmetic, not rounded decimals.
• S190 · 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.
• S189 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=94 with exact rational arithmetic, not rounded decimals.
• S188 · 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.
• S187 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=93 with exact rational arithmetic, not rounded decimals.
• S186 · 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.
• S185 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=92 with exact rational arithmetic, not rounded decimals.
• S184 · 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.
• S183 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=91 with exact rational arithmetic, not rounded decimals.
• S182 · 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.
• S181 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=90 with exact rational arithmetic, not rounded decimals.
• S180 · 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.
• S179 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=89 with exact rational arithmetic, not rounded decimals.
• S178 · 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.
• S177 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=88 with exact rational arithmetic, not rounded decimals.
• S176 · 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.
• S175 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=87 with exact rational arithmetic, not rounded decimals.
• S174 · 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.
• S173 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=86 with exact rational arithmetic, not rounded decimals.
• S172 · 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.
• S171 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=85 with exact rational arithmetic, not rounded decimals.
• S170 · 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.
• S169 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=84 with exact rational arithmetic, not rounded decimals.
• 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.

Result

(25, [25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1, 25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1], 1, Yes, nonzero_multiplicative_group_cycle)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

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.