Step-by-step derivation
S001 · implies
Before
source roles = [qf-simple-graph-handshaking, [E, V, operation]]
After
derived = [handshaking, 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, [11, 12, 13, 14, 15, 16]]
After
derived = [[11, 12, 13, 14, 15, 16]]
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, [11, 12]]
After
derived = [[11, 12]]
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, [11, 13]]
After
derived = [[11, 13]]
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, [11, 14]]
After
derived = [[11, 14]]
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, [11, 15]]
After
derived = [[11, 15]]
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, [11, 16]]
After
derived = [[11, 16]]
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, [12, 13]]
After
derived = [[12, 13]]
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 = [LDL raw edges ORIGINAL, 6, [12, 14]]
After
derived = [[12, 14]]
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, S008
S010 · implies
Before
source roles = [LDL raw edges ORIGINAL, 7, [12, 15]]
After
derived = [[12, 15]]
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, S009
S011 · implies
Before
source roles = [LDL raw edges ORIGINAL, 8, [12, 16]]
After
derived = [[12, 16]]
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, S010
S012 · implies
Before
source roles = [LDL raw edges ORIGINAL, 9, [13, 14]]
After
derived = [[13, 14]]
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, S011
S013 · implies
Before
source roles = [LDL raw edges ORIGINAL, 10, [13, 15]]
After
derived = [[13, 15]]
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, S012
S014 · implies
Before
source roles = [LDL raw edges ORIGINAL, 11, [13, 16]]
After
derived = [[13, 16]]
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, S013
S015 · implies
Before
source roles = [LDL raw edges ORIGINAL, 12, [14, 15]]
After
derived = [[14, 15]]
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, S014
S016 · implies
Before
source roles = [LDL raw edges ORIGINAL, 13, [14, 16]]
After
derived = [[14, 16]]
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, S015
S017 · implies
Before
source roles = [LDL raw edges ORIGINAL, 14, [15, 16]]
After
derived = [[15, 16]]
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, S016
S018 · implies
Before
source roles = [[11, 12, 13, 14, 15, 16], [[11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [12, 13], [12, 14], [12, 15], [12, 16], [13, 14], [13, 15], [13, 16], [14, 15], [14, 16], [15, 16]], No]
After
derived = [[[11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [12, 13], [12, 14], [12, 15], [12, 16], [13, 14], [13, 15], [13, 16], [14, 15], [14, 16], [15, 16]], [[11, [12, 13, 14, 15, 16]], [12, [11, 13, 14, 15, 16]], [13, [11, 12, 14, 15, 16]], [14, [11, 12, 13, 15, 16]], [15, [11, 12, 13, 14, 16]], [16, [11, 12, 13, 14, 15]]]]
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, S017
S019 · implies
Before
source roles = [11, [12, 13, 14, 15, 16]]
After
derived = [5]
Conditions: none
For EVERY V vertex,count distinct neighbors;isolated degree is0.Include vertices absent from the edge list.
Step identifiers and dependencies
Rule: Graph_DEGREE_EACH_vertex_including_isolated · Uses: G001, S018
S020 · implies
Before
source roles = [12, [11, 13, 14, 15, 16]]
After
derived = [5]
Conditions: none
For EVERY V vertex,count distinct neighbors;isolated degree is0.Include vertices absent from the edge list.
Step identifiers and dependencies
Rule: Graph_DEGREE_EACH_vertex_including_isolated · Uses: G001, S019
S021 · implies
Before
source roles = [13, [11, 12, 14, 15, 16]]
After
derived = [5]
Conditions: none
For EVERY V vertex,count distinct neighbors;isolated degree is0.Include vertices absent from the edge list.
Step identifiers and dependencies
Rule: Graph_DEGREE_EACH_vertex_including_isolated · Uses: G001, S020
S022 · implies
Before
source roles = [14, [11, 12, 13, 15, 16]]
After
derived = [5]
Conditions: none
For EVERY V vertex,count distinct neighbors;isolated degree is0.Include vertices absent from the edge list.
Step identifiers and dependencies
Rule: Graph_DEGREE_EACH_vertex_including_isolated · Uses: G001, S021
S023 · implies
Before
source roles = [15, [11, 12, 13, 14, 16]]
After
derived = [5]
Conditions: none
For EVERY V vertex,count distinct neighbors;isolated degree is0.Include vertices absent from the edge list.
Step identifiers and dependencies
Rule: Graph_DEGREE_EACH_vertex_including_isolated · Uses: G001, S022
S024 · implies
Before
source roles = [16, [11, 12, 13, 14, 15]]
After
derived = [5]
Conditions: none
For EVERY V vertex,count distinct neighbors;isolated degree is0.Include vertices absent from the edge list.
Step identifiers and dependencies
Rule: Graph_DEGREE_EACH_vertex_including_isolated · Uses: G001, S023
S025 · calculation
Before
Graph ORIGINAL all vertex degrees sum := sum(5, 5, 5, 5, 5, 5)
After
Graph ORIGINAL all vertex degrees sum = 30
Conditions: none
Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S024
S026 · calculation
Before
Graph ORIGINAL two endpoints per normalized edge := mul(2, 15)
After
Graph ORIGINAL two endpoints per normalized edge = 30
Conditions: none
Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S025
S027 · implies
Before
source roles = [[11, 12, 13, 14, 15, 16], [[11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [12, 13], [12, 14], [12, 15], [12, 16], [13, 14], [13, 15], [13, 16], [14, 15], [14, 16], [15, 16]], [5, 5, 5, 5, 5, 5]]
After
derived = [30, 30]
Conditions: none
Every undirected edge contributes TWO endpoint incidences,so the COMPLETE degree sum is2|E|.For the empty graph both sides are0.This is finite full recomputation/double counting,not a sketch inference.
Step identifiers and dependencies
Rule: Graph_HANDSHAKE_complete_incidence_double_count · Uses: G001, S026