Question 32783: Inverse Table of a Finite Bijection

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0064-32783 · q-20261004-advance-20261005-r04-inverse-function-0008

Original question and givens

The declared total function f:A→B has A={-9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17}, and table {(-9, 17), (-8, 1), (-7, 3), (-6, 5), (-5, 7), (-4, 9), (-3, 11), (-2, 13), (-1, 15)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

G001 — The declared total function f:A→B has A={-9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17}, and table {(-9, 17), (-8, 1), (-7, 3), (-6, 5), (-5, 7), (-4, 9), (-3, 11), (-2, 13), (-1, 15)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

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-inverse-function

Step-by-step derivation

S001 · implies

Before

source roles = [qf-inverse-function-table, inverse_table, no-required-domain, []]

After

derived = [[complete_inverse_pairset_STRING]]

Conditions: none

Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status.

Step identifiers and dependencies

Rule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001

S002 · implies

Before

source roles = [ORIGINAL f domain, [-9, -8, -7, -6, -5, -4, -3, -2, -1]]

After

derived = [[-9, -8, -7, -6, -5, -4, -3, -2, -1]]

Conditions: none

RAW sets are distinct,SAFE and≤100.Sort only a derived work table,retaining original lists/hashes.

Step identifiers and dependencies

Rule: FiniteMap_RAW_distinct_SAFE_set_bound100 · Uses: G001, S001

S003 · implies

Before

source roles = [ORIGINAL f codomain, [1, 3, 5, 7, 9, 11, 13, 15, 17]]

After

derived = [[1, 3, 5, 7, 9, 11, 13, 15, 17]]

Conditions: none

RAW sets are distinct,SAFE and≤100.Sort only a derived work table,retaining original lists/hashes.

Step identifiers and dependencies

Rule: FiniteMap_RAW_distinct_SAFE_set_bound100 · Uses: G001, S002

S004 · implies

Before

source roles = [ORIGINAL f, [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [[[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

Conditions: none

FIRST type EVERY raw pair and enforce raw length≤200,THEN normalize identical rows as one ordered pair.Do not merge different outputs of one input.Empty domains do not waive RAW checks.

Step identifiers and dependencies

Rule: FiniteMap_RAW200_bound_BEFORE_identical_row_ASSET_normalization · Uses: G001, S003

S005 · implies

Before

source roles = [ORIGINAL f, 0, -9, 17, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S004

S006 · implies

Before

source roles = [ORIGINAL f, 1, -8, 1, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S005

S007 · implies

Before

source roles = [ORIGINAL f, 2, -7, 3, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S006

S008 · implies

Before

source roles = [ORIGINAL f, 3, -6, 5, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S007

S009 · implies

Before

source roles = [ORIGINAL f, 4, -5, 7, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S008

S010 · implies

Before

source roles = [ORIGINAL f, 5, -4, 9, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S009

S011 · implies

Before

source roles = [ORIGINAL f, 6, -3, 11, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S010

S012 · implies

Before

source roles = [ORIGINAL f, 7, -2, 13, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S011

S013 · implies

Before

source roles = [ORIGINAL f, 8, -1, 15, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S012

S014 · implies

Before

source roles = [ORIGINAL f, -9, [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S013

S015 · implies

Before

source roles = [ORIGINAL f, -8, [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S014

S016 · implies

Before

source roles = [ORIGINAL f, -7, [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S015

S017 · implies

Before

source roles = [ORIGINAL f, -6, [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S016

S018 · implies

Before

source roles = [ORIGINAL f, -5, [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S017

S019 · implies

Before

source roles = [ORIGINAL f, -4, [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S018

S020 · implies

Before

source roles = [ORIGINAL f, -3, [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S019

S021 · implies

Before

source roles = [ORIGINAL f, -2, [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S020

S022 · implies

Before

source roles = [ORIGINAL f, -1, [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S021

S023 · implies

Before

source roles = [ORIGINAL f, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17], [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [1, 1, 1, 1, 1, 1, 1, 1, 1], Yes]

Conditions: none

Permit lookups ONLY after BOTH all-row typing and all-input totality.Empty-set vacuity cannot conceal an invalid row.

Step identifiers and dependencies

Rule: FiniteMap_TOTAL_qualification_BEFORE_lookups_even_empty_domain · Uses: G001, S022

S024 · implies

Before

source roles = [ORIGINAL f, 1, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [[-8], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S023

S025 · implies

Before

source roles = [ORIGINAL f, 3, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [[-7], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S024

S026 · implies

Before

source roles = [ORIGINAL f, 5, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [[-6], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S025

S027 · implies

Before

source roles = [ORIGINAL f, 7, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [[-5], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S026

S028 · implies

Before

source roles = [ORIGINAL f, 9, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [[-4], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S027

S029 · implies

Before

source roles = [ORIGINAL f, 11, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [[-3], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S028

S030 · implies

Before

source roles = [ORIGINAL f, 13, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [[-2], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S029

S031 · implies

Before

source roles = [ORIGINAL f, 15, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [[-1], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S030

S032 · implies

Before

source roles = [ORIGINAL f, 17, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [[-9], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S031

S033 · implies

Before

source roles = [ORIGINAL f, [-9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17], [[-8], [-7], [-6], [-5], [-4], [-3], [-2], [-1], [-9]]]

After

derived = [Yes, Yes, Yes]

Conditions: none

Injective means every fiber≤1,surjective every fiber≥1,and BOTH mean bijective.Unattained B does not defeat injectivity.Empty→empty is bijective;empty→nonempty is only injective.

Step identifiers and dependencies

Rule: FiniteMap_injective_surjective_BOTH_needed_for_inverse · Uses: G001, S032

S034 · implies

Before

source roles = [[-9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17], [[-9, 17], [-8, 1], [-7, 3], [-6, 5], [-5, 7], [-4, 9], [-3, 11], [-2, 13], [-1, 15]]]

After

derived = [[[1, -8], [3, -7], [5, -6], [7, -5], [9, -4], [11, -3], [13, -2], [15, -1], [17, -9]], {(1, -8), (3, -7), (5, -6), (7, -5), (9, -4), (11, -3), (13, -2), (15, -1), (17, -9)}]

Conditions: none

AFTER actual bijectivity,swap EVERY(x,y)to(y,x)and sort numericLEX for the COMPLETE B→A inverse table.The empty bijection has inverse∅;an arbitrary converse is not automatically an inverse function.

Step identifiers and dependencies

Rule: Inverse_TABLE_actual_bijection_complete_swapped_numeric_LEX_pairs · Uses: G001, S033

Reverse checks

S034 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S033 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S032 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S031 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S030 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S029 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S028 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S027 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S026 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S025 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S024 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S023 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S022 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S021 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S020 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S019 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S018 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S017 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S016 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S015 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S014 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S013 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S012 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S011 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S010 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S009 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S008 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S007 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S006 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S005 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S004 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S003 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S002 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

{(1, -8), (3, -7), (5, -6), (7, -5), (9, -4), (11, -3), (13, -2), (15, -1), (17, -9)}

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: f3ec7bf346bacb0a475c1116410c44e5e848f8befd365bb9c7d70659985c5098
candidate record SHA-256: 4f244ed25d743c657d9e4462d2c88668bbe8ef9608e54834683ac7105aae4908
semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON