Step-by-step derivation
S001 · implies
Before
source operation=legendre-zero-class;original snapshot and mathematics retained
After
explicit ordered source output fields=[prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate]
Conditions: none
Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
Legendre ORIGINAL odd prime:ORIGINAL modulus=17;ALL possible trial divisors through square root=[2, 3, 4]
After
ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 1]];prime=Yes
Conditions: none
Qualify primality over the full trial range,not by guessing from Legendre/Euler classification.This is a modulus premise,not the square status of a.
Step identifiers and dependencies
Rule: legendre_complete_prime_trial_not_Euler_classification · Uses: G001, S001
S003 · calculation
Before
Legendre ORIGINAL nonzero root1 exact square := mul(1, 1)
After
Legendre ORIGINAL nonzero root1 exact square = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
ORIGINAL root x=1 in1..16;exact square=1
After
square=0*17+1;canonical square class=1;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S003
S005 · calculation
Before
Legendre ORIGINAL nonzero root2 exact square := mul(2, 2)
After
Legendre ORIGINAL nonzero root2 exact square = 4
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
ORIGINAL root x=2 in1..16;exact square=4
After
square=0*17+4;canonical square class=4;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S005
S007 · calculation
Before
Legendre ORIGINAL nonzero root3 exact square := mul(3, 3)
After
Legendre ORIGINAL nonzero root3 exact square = 9
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
ORIGINAL root x=3 in1..16;exact square=9
After
square=0*17+9;canonical square class=9;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S007
S009 · calculation
Before
Legendre ORIGINAL nonzero root4 exact square := mul(4, 4)
After
Legendre ORIGINAL nonzero root4 exact square = 16
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
ORIGINAL root x=4 in1..16;exact square=16
After
square=0*17+16;canonical square class=16;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S009
S011 · calculation
Before
Legendre ORIGINAL nonzero root5 exact square := mul(5, 5)
After
Legendre ORIGINAL nonzero root5 exact square = 25
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
ORIGINAL root x=5 in1..16;exact square=25
After
square=1*17+8;canonical square class=8;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S011
S013 · calculation
Before
Legendre ORIGINAL nonzero root6 exact square := mul(6, 6)
After
Legendre ORIGINAL nonzero root6 exact square = 36
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · implies
Before
ORIGINAL root x=6 in1..16;exact square=36
After
square=2*17+2;canonical square class=2;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S013
S015 · calculation
Before
Legendre ORIGINAL nonzero root7 exact square := mul(7, 7)
After
Legendre ORIGINAL nonzero root7 exact square = 49
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · implies
Before
ORIGINAL root x=7 in1..16;exact square=49
After
square=2*17+15;canonical square class=15;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S015
S017 · calculation
Before
Legendre ORIGINAL nonzero root8 exact square := mul(8, 8)
After
Legendre ORIGINAL nonzero root8 exact square = 64
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · implies
Before
ORIGINAL root x=8 in1..16;exact square=64
After
square=3*17+13;canonical square class=13;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S017
S019 · calculation
Before
Legendre ORIGINAL nonzero root9 exact square := mul(9, 9)
After
Legendre ORIGINAL nonzero root9 exact square = 81
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · implies
Before
ORIGINAL root x=9 in1..16;exact square=81
After
square=4*17+13;canonical square class=13;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S019
S021 · calculation
Before
Legendre ORIGINAL nonzero root10 exact square := mul(10, 10)
After
Legendre ORIGINAL nonzero root10 exact square = 100
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S020
S022 · implies
Before
ORIGINAL root x=10 in1..16;exact square=100
After
square=5*17+15;canonical square class=15;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S021
S023 · calculation
Before
Legendre ORIGINAL nonzero root11 exact square := mul(11, 11)
After
Legendre ORIGINAL nonzero root11 exact square = 121
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S022
S024 · implies
Before
ORIGINAL root x=11 in1..16;exact square=121
After
square=7*17+2;canonical square class=2;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S023
S025 · calculation
Before
Legendre ORIGINAL nonzero root12 exact square := mul(12, 12)
After
Legendre ORIGINAL nonzero root12 exact square = 144
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S024
S026 · implies
Before
ORIGINAL root x=12 in1..16;exact square=144
After
square=8*17+8;canonical square class=8;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S025
S027 · calculation
Before
Legendre ORIGINAL nonzero root13 exact square := mul(13, 13)
After
Legendre ORIGINAL nonzero root13 exact square = 169
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S026
S028 · implies
Before
ORIGINAL root x=13 in1..16;exact square=169
After
square=9*17+16;canonical square class=16;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S027
S029 · calculation
Before
Legendre ORIGINAL nonzero root14 exact square := mul(14, 14)
After
Legendre ORIGINAL nonzero root14 exact square = 196
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S028
S030 · implies
Before
ORIGINAL root x=14 in1..16;exact square=196
After
square=11*17+9;canonical square class=9;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S029
S031 · calculation
Before
Legendre ORIGINAL nonzero root15 exact square := mul(15, 15)
After
Legendre ORIGINAL nonzero root15 exact square = 225
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S030
S032 · implies
Before
ORIGINAL root x=15 in1..16;exact square=225
After
square=13*17+4;canonical square class=4;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S031
S033 · calculation
Before
Legendre ORIGINAL nonzero root16 exact square := mul(16, 16)
After
Legendre ORIGINAL nonzero root16 exact square = 256
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S032
S034 · implies
Before
ORIGINAL root x=16 in1..16;exact square=256
After
square=15*17+1;canonical square class=1;ALL root indices retained
Conditions: none
Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.
Step identifiers and dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S033
S035 · implies
Before
ORIGINAL prime=17;COMPLETE nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 8], [6, 2], [7, 15], [8, 13], [9, 13], [10, 15], [11, 2], [12, 8], [13, 16], [14, 9], [15, 4], [16, 1]];nonzero square classes=[1, 2, 4, 8, 9, 13, 15, 16]
After
for EACH normalized class r,first r0 ->symbol0;else square-table member ->symbol1;otherwise ->symbol-1;these cases partition ALL0..p-1
Conditions: none
Separate zero first;the fact that0 is a square does not give symbol+1.Nonzero classes are classified by the complete square table.This is definition plus exhaustive evidence,not a later multiplicativity or Euler formula.
Step identifiers and dependencies
Rule: legendre_ZERO_then_residue_nonresidue_three_way_definition · Uses: G001, S034
S036 · calculation
Before
Legendre ORIGINAL candidate class quotient times modulus := mul(6, 17)
After
Legendre ORIGINAL candidate class quotient times modulus = 102
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S035
S037 · calculation
Before
Legendre ORIGINAL candidate class ORIGINAL signed value minus multiple := sub(102, 102)
After
Legendre ORIGINAL candidate class ORIGINAL signed value minus multiple = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S036
S038 · implies
Before
Legendre ORIGINAL candidate class:ORIGINAL a=102,positive modulus=17;actual quotient=6
After
normalized class=0;a=quotient*modulus+class;ORIGINAL a retained in output where declared
Conditions: none
Retain the same class for negative a and normalize into0..p−1,rather than using a negative language-specific remainder.
Step identifiers and dependencies
Rule: legendre_ORIGINAL_signed_class_Euclidean_normalization · Uses: G001, S037