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Question 32473: Lexicographically First Collision

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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R0063-32473 · q-20261004-advance-20261005-r01-injective-0048

Original question and givens

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

  • G001 — The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

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 roles = [qf-injective-first-collision, first_collision, no-required-domain, []]

After

derived = [[numeric_LEX_first_pair_STRING_OR_source_none]]

Conditions: none

BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds.

Step identifiers and dependencies

Rule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001

S002 · implies

Before

source roles = [A declared domain, legacy-injective, [-6, -5, -4, -3, -2, -1, 0]]

After

derived = [[-6, -5, -4, -3, -2, -1, 0]]

Conditions: none

RAW A/B/C are distinct SAFE integers.Legacy injective limits100,but PULSE has NO such cap.Sort ONLY a derived work table,retaining original lists.

Step identifiers and dependencies

Rule: Map_RAW_distinct_SAFE_set_BANK_specific_cardinality_bound · Uses: G001, S001

S003 · implies

Before

source roles = [B declared codomain, legacy-injective, [1, 3, 5, 7, 9, 11, 13, 15]]

After

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

Conditions: none

RAW A/B/C are distinct SAFE integers.Legacy injective limits100,but PULSE has NO such cap.Sort ONLY a derived work table,retaining original lists.

Step identifiers and dependencies

Rule: Map_RAW_distinct_SAFE_set_BANK_specific_cardinality_bound · Uses: G001, S002

S004 · implies

Before

source roles = [legacy-injective, [[-6, 1], [-5, 3], [-4, 5], [-3, 7], [-2, 9], [-1, 11], [0, 13]]]

After

derived = [[[-6, 1], [-5, 3], [-4, 5], [-3, 7], [-2, 9], [-1, 11], [0, 13]]]

Conditions: none

PULSE rejects EXACT duplicate RAW rows BEFORE totality,not as a legal No.Legacy injective FIRST checks raw length≤200,then normalizes identical rows as one pair.Do not exchange these policies;type-check EVERY raw cell first.

Step identifiers and dependencies

Rule: Map_RAW_pairs_arity_SAFE_DUPLICATE_policy_BEFORE_totality · Uses: G001, S003

S005 · implies

Before

source roles = [legacy-injective, 0, -6, 1, [-6, -5, -4, -3, -2, -1, 0], [1, 3, 5, 7, 9, 11, 13, 15]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S004

S006 · implies

Before

source roles = [legacy-injective, 1, -5, 3, [-6, -5, -4, -3, -2, -1, 0], [1, 3, 5, 7, 9, 11, 13, 15]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S005

S007 · implies

Before

source roles = [legacy-injective, 2, -4, 5, [-6, -5, -4, -3, -2, -1, 0], [1, 3, 5, 7, 9, 11, 13, 15]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S006

S008 · implies

Before

source roles = [legacy-injective, 3, -3, 7, [-6, -5, -4, -3, -2, -1, 0], [1, 3, 5, 7, 9, 11, 13, 15]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S007

S009 · implies

Before

source roles = [legacy-injective, 4, -2, 9, [-6, -5, -4, -3, -2, -1, 0], [1, 3, 5, 7, 9, 11, 13, 15]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S008

S010 · implies

Before

source roles = [legacy-injective, 5, -1, 11, [-6, -5, -4, -3, -2, -1, 0], [1, 3, 5, 7, 9, 11, 13, 15]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S009

S011 · implies

Before

source roles = [legacy-injective, 6, 0, 13, [-6, -5, -4, -3, -2, -1, 0], [1, 3, 5, 7, 9, 11, 13, 15]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input must belong to A and output to B,including unused/orphan rows.Even an empty A does not waive row typing.

Step identifiers and dependencies

Rule: Map_EACH_row_SOURCE_and_TARGET_membership_in_declared_sets · Uses: G001, S010

S012 · implies

Before

source roles = [legacy-injective, -6, [[-6, 1], [-5, 3], [-4, 5], [-3, 7], [-2, 9], [-1, 11], [0, 13]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S011

S013 · implies

Before

source roles = [legacy-injective, -5, [[-6, 1], [-5, 3], [-4, 5], [-3, 7], [-2, 9], [-1, 11], [0, 13]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S012

S014 · implies

Before

source roles = [legacy-injective, -4, [[-6, 1], [-5, 3], [-4, 5], [-3, 7], [-2, 9], [-1, 11], [0, 13]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S013

S015 · implies

Before

source roles = [legacy-injective, -3, [[-6, 1], [-5, 3], [-4, 5], [-3, 7], [-2, 9], [-1, 11], [0, 13]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S014

S016 · implies

Before

source roles = [legacy-injective, -2, [[-6, 1], [-5, 3], [-4, 5], [-3, 7], [-2, 9], [-1, 11], [0, 13]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S015

S017 · implies

Before

source roles = [legacy-injective, -1, [[-6, 1], [-5, 3], [-4, 5], [-3, 7], [-2, 9], [-1, 11], [0, 13]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S016

S018 · implies

Before

source roles = [legacy-injective, 0, [[-6, 1], [-5, 3], [-4, 5], [-3, 7], [-2, 9], [-1, 11], [0, 13]]]

After

derived = [1, Yes]

Conditions: none

Count rows for EACH DECLARED A input;EXACTLY one is required for a total function.Missing rows/multiple distinct outputs fail;do not conceal them with a first-match lookup.

Step identifiers and dependencies

Rule: Map_EACH_declared_input_EXACTLY_one_normalized_row · Uses: G001, S017

S019 · implies

Before

source roles = [legacy-injective, [-6, -5, -4, -3, -2, -1, 0], [1, 3, 5, 7, 9, 11, 13, 15], [[-6, 1], [-5, 3], [-4, 5], [-3, 7], [-2, 9], [-1, 11], [0, 13]], Yes]

After

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

Conditions: none

A total function needs BOTH every declared input AND every row qualified.PULSE validity may legally return No;other queries and legacy injective REQUIRE actual totality BEFORE lookup.Empty A/table can be valid;unattained B labels remain.

Step identifiers and dependencies

Rule: Map_TOTAL_all_declared_inputs_AND_ALL_rows_before_any_lookup · Uses: G001, S018

S020 · implies

Before

source roles = [ORIGINAL A, -6, -5, 1, 3]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S019

S021 · implies

Before

source roles = [ORIGINAL A, -6, -4, 1, 5]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S020

S022 · implies

Before

source roles = [ORIGINAL A, -6, -3, 1, 7]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S021

S023 · implies

Before

source roles = [ORIGINAL A, -6, -2, 1, 9]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S022

S024 · implies

Before

source roles = [ORIGINAL A, -6, -1, 1, 11]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S023

S025 · implies

Before

source roles = [ORIGINAL A, -6, 0, 1, 13]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S024

S026 · implies

Before

source roles = [ORIGINAL A, -5, -4, 3, 5]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S025

S027 · implies

Before

source roles = [ORIGINAL A, -5, -3, 3, 7]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S026

S028 · implies

Before

source roles = [ORIGINAL A, -5, -2, 3, 9]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S027

S029 · implies

Before

source roles = [ORIGINAL A, -5, -1, 3, 11]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S028

S030 · implies

Before

source roles = [ORIGINAL A, -5, 0, 3, 13]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S029

S031 · implies

Before

source roles = [ORIGINAL A, -4, -3, 5, 7]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S030

S032 · implies

Before

source roles = [ORIGINAL A, -4, -2, 5, 9]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S031

S033 · implies

Before

source roles = [ORIGINAL A, -4, -1, 5, 11]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S032

S034 · implies

Before

source roles = [ORIGINAL A, -4, 0, 5, 13]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S033

S035 · implies

Before

source roles = [ORIGINAL A, -3, -2, 7, 9]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S034

S036 · implies

Before

source roles = [ORIGINAL A, -3, -1, 7, 11]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S035

S037 · implies

Before

source roles = [ORIGINAL A, -3, 0, 7, 13]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S036

S038 · implies

Before

source roles = [ORIGINAL A, -2, -1, 9, 11]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S037

S039 · implies

Before

source roles = [ORIGINAL A, -2, 0, 9, 13]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S038

S040 · implies

Before

source roles = [ORIGINAL A, -1, 0, 11, 13]

After

derived = [No]

Conditions: none

Take EVERY distinct permitted pair x<y in numeric order,then compare its full outputs.This is not a stream-first repeated value or duplicate copies of one row.

Step identifiers and dependencies

Rule: Injective_EVERY_distinct_permitted_pair_numeric_LEX_test · Uses: G001, S039

S041 · implies

Before

source roles = [[-6, -5, -4, -3, -2, -1, 0], [[-6, 1], [-5, 3], [-4, 5], [-3, 7], [-2, 9], [-1, 11], [0, 13]]]

After

derived = [[], does not exist]

Conditions: none

After the COMPLETE collision inventory,select the GLOBAL numeric-LEX first pair.With no collision retain the source absence tag,displayed as none in THIS English context.Keep original data.

Step identifiers and dependencies

Rule: Injective_FIRST_collision_GLOBAL_numeric_LEX_complete_inventory · Uses: G001, S040

Reverse checks

  • S041 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S040 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S039 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S038 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S037 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S036 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S035 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S034 · premise_and_result_audit — Reverse-check ORIGINAL inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,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 inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

none

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: 3dc76cf33849ddd6801879785724f92863efa2f4085699d05230195130f83be2
  • candidate record SHA-256: 299addc48defa3c1141f39568839d9fd05f0e32f8788efb0264a949f00c543a6
  • semantic review receipt SHA-256: acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0
  • question-first witness SHA-256: eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON