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Question 23382: The Complete Nonzero Square Table Has Half as Many Distinct Residues

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0045-23382 · q-20261014-advance-20261111-r06-quadratic-residue-0007

Original question and givens

Use bounded odd prime p=19 and the declared nonzero quadratic-residue convention. Enumerate every x=1..p−1 and x^2 modulo p; normalize input classes into0..p−1. Zero is excluded from nonzero residues. Do not assume an Euler-criterion or Legendre-symbol result. Give (p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal).

  • G001 — Use bounded odd prime p=19 and the declared nonzero quadratic-residue convention. Enumerate every x=1..p−1 and x^2 modulo p; normalize input classes into0..p−1. Zero is excluded from nonzero residues. Do not assume an Euler-criterion or Legendre-symbol result. Give (p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal).

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.

Step-by-step derivation

S001 · implies

Before

source operation=quadratic-square-table;original snapshot and mathematics retained

After

explicit ordered source output fields=[p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal]

Conditions: none

Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

Quadratic ORIGINAL modulus:ORIGINAL integer=19;complete possible trial divisors=[2, 3, 4]

After

ALL [divisor,remainder] rows=[[2, 1], [3, 1], [4, 3]];prime=Yes;a composite>1 has a factor at most its square root

Conditions: none

Check the entire square-root divisor range.The empty range qualifies prime2,but not0 or1.Samples,square tables or an Euler criterion do not replace primality qualification.

Step identifiers and dependencies

Rule: finite_prime_ALL_trial_divisors_through_square_root · Uses: G001, S001

S003 · calculation

Before

Quadratic ORIGINAL nonzero root1 square := mul(1, 1)

After

Quadratic ORIGINAL nonzero root1 square = 1

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

ORIGINAL nonzero root x=1 in1..18;actual square=1

After

Euclidean square=0*19+1;nonzero square class=1;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S003

S005 · calculation

Before

Quadratic ORIGINAL nonzero root2 square := mul(2, 2)

After

Quadratic ORIGINAL nonzero root2 square = 4

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

ORIGINAL nonzero root x=2 in1..18;actual square=4

After

Euclidean square=0*19+4;nonzero square class=4;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S005

S007 · calculation

Before

Quadratic ORIGINAL nonzero root3 square := mul(3, 3)

After

Quadratic ORIGINAL nonzero root3 square = 9

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

ORIGINAL nonzero root x=3 in1..18;actual square=9

After

Euclidean square=0*19+9;nonzero square class=9;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S007

S009 · calculation

Before

Quadratic ORIGINAL nonzero root4 square := mul(4, 4)

After

Quadratic ORIGINAL nonzero root4 square = 16

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

ORIGINAL nonzero root x=4 in1..18;actual square=16

After

Euclidean square=0*19+16;nonzero square class=16;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S009

S011 · calculation

Before

Quadratic ORIGINAL nonzero root5 square := mul(5, 5)

After

Quadratic ORIGINAL nonzero root5 square = 25

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

ORIGINAL nonzero root x=5 in1..18;actual square=25

After

Euclidean square=1*19+6;nonzero square class=6;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S011

S013 · calculation

Before

Quadratic ORIGINAL nonzero root6 square := mul(6, 6)

After

Quadratic ORIGINAL nonzero root6 square = 36

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

ORIGINAL nonzero root x=6 in1..18;actual square=36

After

Euclidean square=1*19+17;nonzero square class=17;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S013

S015 · calculation

Before

Quadratic ORIGINAL nonzero root7 square := mul(7, 7)

After

Quadratic ORIGINAL nonzero root7 square = 49

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

ORIGINAL nonzero root x=7 in1..18;actual square=49

After

Euclidean square=2*19+11;nonzero square class=11;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S015

S017 · calculation

Before

Quadratic ORIGINAL nonzero root8 square := mul(8, 8)

After

Quadratic ORIGINAL nonzero root8 square = 64

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

ORIGINAL nonzero root x=8 in1..18;actual square=64

After

Euclidean square=3*19+7;nonzero square class=7;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S017

S019 · calculation

Before

Quadratic ORIGINAL nonzero root9 square := mul(9, 9)

After

Quadratic ORIGINAL nonzero root9 square = 81

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

ORIGINAL nonzero root x=9 in1..18;actual square=81

After

Euclidean square=4*19+5;nonzero square class=5;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S019

S021 · calculation

Before

Quadratic ORIGINAL nonzero root10 square := mul(10, 10)

After

Quadratic ORIGINAL nonzero root10 square = 100

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

ORIGINAL nonzero root x=10 in1..18;actual square=100

After

Euclidean square=5*19+5;nonzero square class=5;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S021

S023 · calculation

Before

Quadratic ORIGINAL nonzero root11 square := mul(11, 11)

After

Quadratic ORIGINAL nonzero root11 square = 121

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

ORIGINAL nonzero root x=11 in1..18;actual square=121

After

Euclidean square=6*19+7;nonzero square class=7;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S023

S025 · calculation

Before

Quadratic ORIGINAL nonzero root12 square := mul(12, 12)

After

Quadratic ORIGINAL nonzero root12 square = 144

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

ORIGINAL nonzero root x=12 in1..18;actual square=144

After

Euclidean square=7*19+11;nonzero square class=11;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S025

S027 · calculation

Before

Quadratic ORIGINAL nonzero root13 square := mul(13, 13)

After

Quadratic ORIGINAL nonzero root13 square = 169

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

ORIGINAL nonzero root x=13 in1..18;actual square=169

After

Euclidean square=8*19+17;nonzero square class=17;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S027

S029 · calculation

Before

Quadratic ORIGINAL nonzero root14 square := mul(14, 14)

After

Quadratic ORIGINAL nonzero root14 square = 196

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

ORIGINAL nonzero root x=14 in1..18;actual square=196

After

Euclidean square=10*19+6;nonzero square class=6;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S029

S031 · calculation

Before

Quadratic ORIGINAL nonzero root15 square := mul(15, 15)

After

Quadratic ORIGINAL nonzero root15 square = 225

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

ORIGINAL nonzero root x=15 in1..18;actual square=225

After

Euclidean square=11*19+16;nonzero square class=16;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S031

S033 · calculation

Before

Quadratic ORIGINAL nonzero root16 square := mul(16, 16)

After

Quadratic ORIGINAL nonzero root16 square = 256

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · implies

Before

ORIGINAL nonzero root x=16 in1..18;actual square=256

After

Euclidean square=13*19+9;nonzero square class=9;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S033

S035 · calculation

Before

Quadratic ORIGINAL nonzero root17 square := mul(17, 17)

After

Quadratic ORIGINAL nonzero root17 square = 289

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

ORIGINAL nonzero root x=17 in1..18;actual square=289

After

Euclidean square=15*19+4;nonzero square class=4;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S035

S037 · calculation

Before

Quadratic ORIGINAL nonzero root18 square := mul(18, 18)

After

Quadratic ORIGINAL nonzero root18 square = 324

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

ORIGINAL nonzero root x=18 in1..18;actual square=324

After

Euclidean square=17*19+1;nonzero square class=1;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S037

S039 · calculation

Before

Quadratic distinct nonzero residue count := sum(1, 1, 1, 1, 1, 1, 1, 1, 1)

After

Quadratic distinct nonzero residue count = 9

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

Quadratic ORIGINAL modulus minus1 := sub(19, 1)

After

Quadratic ORIGINAL modulus minus1 = 18

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

Quadratic half of all nonzero roots := div(18, 2)

After

Quadratic half of all nonzero roots = 9

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · implies

Before

ORIGINAL ODD PRIME p=19;complete nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 6], [6, 17], [7, 11], [8, 7], [9, 5], [10, 5], [11, 7], [12, 11], [13, 17], [14, 6], [15, 16], [16, 9], [17, 4], [18, 1]];distinct classes=[1, 4, 5, 6, 7, 9, 11, 16, 17]

After

x^2=y^2 implies p divides(x-y)(x+y),so x=y or x=-y modp;odd p and nonzero x make opposite roots DISTINCT;each image has2 roots,so residue count=9=(19-1)/2=9

Conditions: none

The prime-field no-zero-divisor property gives x=±y;oddness makes2x nonzero.The opposite pairs partition all p−1 nonzero roots.This proves both counting and two-root statements,not their extension to zero,2 or composite moduli.

Step identifiers and dependencies

Rule: quadratic_prime_field_opposite_root_pairs_exactly_two · Uses: G001, S041

Reverse checks

  • S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S041 · premise_and_result_audit — Recompute Quadratic half of all nonzero roots with exact rational arithmetic, not rounded decimals.
  • S040 · premise_and_result_audit — Recompute Quadratic ORIGINAL modulus minus1 with exact rational arithmetic, not rounded decimals.
  • S039 · premise_and_result_audit — Recompute Quadratic distinct nonzero residue count with exact rational arithmetic, not rounded decimals.
  • S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S037 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root18 square with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root17 square with exact rational arithmetic, not rounded decimals.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root16 square with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root15 square with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root14 square with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root13 square with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root12 square with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root11 square with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root10 square with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root9 square with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root8 square with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root7 square with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root6 square with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root5 square with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root4 square with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root3 square with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root2 square with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root1 square with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(19, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 6], [6, 17], [7, 11], [8, 7], [9, 5], [10, 5], [11, 7], [12, 11], [13, 17], [14, 6], [15, 16], [16, 9], [17, 4], [18, 1]], [1, 4, 5, 6, 7, 9, 11, 16, 17], 9, 9, Yes)

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: a41a80d7eeb73a3f2f956bef802339d3bdccd078e063f8b55cbf26a3bdcfab6b
  • candidate record SHA-256: b6e2a77845630088bbc77798c331fef611cb26ee2c92307270660e7e4e539b4c
  • semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
  • question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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