Original question and givens
The finite simple undirected graph has V={5, 6, 7, 8} and E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=7 belongs to V. List N(v) in increasing order with no duplicates.
G001 — The finite simple undirected graph has V={5, 6, 7, 8} and E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=7 belongs to V. List N(v) in increasing order with no duplicates.
Step-by-step derivation
S001 · implies
Before
source roles = [qf-simple-graph-neighbors, [E, V, operation, v]]
After
derived = [neighbors, no-invented-domain]
Conditions: none
Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract.
Step identifiers and dependencies
Rule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001
S002 · implies
Before
source roles = [LDL graph vertices ORIGINAL, 100, [5, 6, 7, 8]]
After
derived = [[5, 6, 7, 8]]
Conditions: none
Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits.
Step identifiers and dependencies
Rule: Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort · Uses: G001, S001
S003 · implies
Before
source roles = [LDL raw edges ORIGINAL, 0, [5, 6]]
After
derived = [[5, 6]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S002
S004 · implies
Before
source roles = [LDL raw edges ORIGINAL, 1, [5, 7]]
After
derived = [[5, 7]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S003
S005 · implies
Before
source roles = [LDL raw edges ORIGINAL, 2, [5, 8]]
After
derived = [[5, 8]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S004
S006 · implies
Before
source roles = [LDL raw edges ORIGINAL, 3, [6, 7]]
After
derived = [[6, 7]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S005
S007 · implies
Before
source roles = [LDL raw edges ORIGINAL, 4, [6, 8]]
After
derived = [[6, 8]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S006
S008 · implies
Before
source roles = [LDL raw edges ORIGINAL, 5, [7, 8]]
After
derived = [[7, 8]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S007
S009 · implies
Before
source roles = [[5, 6, 7, 8], [[5, 6], [5, 7], [5, 8], [6, 7], [6, 8], [7, 8]], No]
After
derived = [[[5, 6], [5, 7], [5, 8], [6, 7], [6, 8], [7, 8]], [[5, [6, 7, 8]], [6, [5, 7, 8]], [7, [5, 6, 8]], [8, [5, 6, 7]]]]
Conditions: none
Type the WHOLE raw edge list,then normalize undirected endpoints/deduplicate;reversed duplicates are allowed.INPUT edges have no loops/outside endpoints,while isolated vertices stay in V.TREE raw E does NOT acquire LDL’s200-entry limit.
Step identifiers and dependencies
Rule: Graph_WHOLE_simple_undirected_edges_normalized_NO_loops_or_outside · Uses: G001, S008
S010 · implies
Before
source roles = [7, [5, 6, 7, 8]]
After
derived = [[5, 6, 8], 3]
Conditions: none
A degree/neighbor QUERY first requires v∈V.Find distinct neighbors through ALL edges,not sketch crossings or raw duplicate rows.
Step identifiers and dependencies
Rule: Graph_QUERY_vertex_must_belong_V_complete_distinct_neighbors · Uses: G001, S009