Step-by-step derivation
S001 · implies
Before
source roles = [qf-surjective-codomain-change, [A, B, B2, operation, pairs]]
After
derived = [changed_codomain, 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 = [function ORIGINAL A, 100, [-5, -4, -3, -2, -1, 0]]
After
derived = [[-5, -4, -3, -2, -1, 0]]
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 = [function ORIGINAL B, 100, [0, 1, 2, 3, 4, 5]]
After
derived = [[0, 1, 2, 3, 4, 5]]
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, S002
S004 · implies
Before
source roles = [function ORIGINAL table, 0, [-5, 1]]
After
derived = [[-5, 1]]
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 = [function ORIGINAL table, 1, [-4, 2]]
After
derived = [[-4, 2]]
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 = [function ORIGINAL table, 2, [-3, 3]]
After
derived = [[-3, 3]]
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 = [function ORIGINAL table, 3, [-2, 4]]
After
derived = [[-2, 4]]
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 = [function ORIGINAL table, 4, [-1, 5]]
After
derived = [[-1, 5]]
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 = [function ORIGINAL table, 5, [0, 0]]
After
derived = [[0, 0]]
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 = [[-5, -4, -3, -2, -1, 0], [0, 1, 2, 3, 4, 5], [[-5, 1], [-4, 2], [-3, 3], [-2, 4], [-1, 5], [0, 0]]]
After
derived = [Yes]
Conditions: none
FIRST qualify the WHOLE total function A→B:EXACTLY one output for EVERY A and all pairs in A×B.Identical repeated rows normalize in THIS legacy bank,unlike a different PULSE table rule.Qualify ORIGINAL B even for later codomain changes.
Step identifiers and dependencies
Rule: Function_WHOLE_qualified_total_AtoB_before_SURJECTIVE_claim · Uses: G001, S009
S011 · implies
Before
source roles = [function NEW codomain B2, 100, [1, 2, 3, 4, 5, 0, 117]]
After
derived = [[0, 1, 2, 3, 4, 5, 117]]
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, S010
S012 · implies
Before
source roles = [[0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5, 117], [[-5, 1], [-4, 2], [-3, 3], [-2, 4], [-1, 5], [0, 0]]]
After
derived = [Yes]
Conditions: none
FIRST require B2 to contain the ENTIRE attained range for the same table to remain a total function.It may add/remove unattained targets and need not contain ALL old B.Never discard an actual output to force surjectivity.
Step identifiers and dependencies
Rule: Surjective_NEW_codomain_contains_WHOLE_range_not_oldB_required · Uses: G001, S011
S013 · implies
Before
source roles = [NEW B2, -5, 1, 0]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S012
S014 · implies
Before
source roles = [NEW B2, -4, 2, 0]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S013
S015 · implies
Before
source roles = [NEW B2, -3, 3, 0]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S014
S016 · implies
Before
source roles = [NEW B2, -2, 4, 0]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S015
S017 · implies
Before
source roles = [NEW B2, -1, 5, 0]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S016
S018 · implies
Before
source roles = [NEW B2, 0, 0, 0]
After
derived = [Yes]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S017
S019 · implies
Before
source roles = [NEW B2, -5, 1, 1]
After
derived = [Yes]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S018
S020 · implies
Before
source roles = [NEW B2, -4, 2, 1]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S019
S021 · implies
Before
source roles = [NEW B2, -3, 3, 1]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S020
S022 · implies
Before
source roles = [NEW B2, -2, 4, 1]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S021
S023 · implies
Before
source roles = [NEW B2, -1, 5, 1]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S022
S024 · implies
Before
source roles = [NEW B2, 0, 0, 1]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S023
S025 · implies
Before
source roles = [NEW B2, -5, 1, 2]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S024
S026 · implies
Before
source roles = [NEW B2, -4, 2, 2]
After
derived = [Yes]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S025
S027 · implies
Before
source roles = [NEW B2, -3, 3, 2]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S026
S028 · implies
Before
source roles = [NEW B2, -2, 4, 2]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S027
S029 · implies
Before
source roles = [NEW B2, -1, 5, 2]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S028
S030 · implies
Before
source roles = [NEW B2, 0, 0, 2]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S029
S031 · implies
Before
source roles = [NEW B2, -5, 1, 3]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S030
S032 · implies
Before
source roles = [NEW B2, -4, 2, 3]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S031
S033 · implies
Before
source roles = [NEW B2, -3, 3, 3]
After
derived = [Yes]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S032
S034 · implies
Before
source roles = [NEW B2, -2, 4, 3]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S033
S035 · implies
Before
source roles = [NEW B2, -1, 5, 3]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S034
S036 · implies
Before
source roles = [NEW B2, 0, 0, 3]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S035
S037 · implies
Before
source roles = [NEW B2, -5, 1, 4]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S036
S038 · implies
Before
source roles = [NEW B2, -4, 2, 4]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S037
S039 · implies
Before
source roles = [NEW B2, -3, 3, 4]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S038
S040 · implies
Before
source roles = [NEW B2, -2, 4, 4]
After
derived = [Yes]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S039
S041 · implies
Before
source roles = [NEW B2, -1, 5, 4]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S040
S042 · implies
Before
source roles = [NEW B2, 0, 0, 4]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S041
S043 · implies
Before
source roles = [NEW B2, -5, 1, 5]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S042
S044 · implies
Before
source roles = [NEW B2, -4, 2, 5]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S043
S045 · implies
Before
source roles = [NEW B2, -3, 3, 5]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S044
S046 · implies
Before
source roles = [NEW B2, -2, 4, 5]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S045
S047 · implies
Before
source roles = [NEW B2, -1, 5, 5]
After
derived = [Yes]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S046
S048 · implies
Before
source roles = [NEW B2, 0, 0, 5]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S047
S049 · implies
Before
source roles = [NEW B2, -5, 1, 117]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S048
S050 · implies
Before
source roles = [NEW B2, -4, 2, 117]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S049
S051 · implies
Before
source roles = [NEW B2, -3, 3, 117]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S050
S052 · implies
Before
source roles = [NEW B2, -2, 4, 117]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S051
S053 · implies
Before
source roles = [NEW B2, -1, 5, 117]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S052
S054 · implies
Before
source roles = [NEW B2, 0, 0, 117]
After
derived = [No]
Conditions: none
Compare EVERY permitted input’s output with THIS target.List EACH fiber completely;do not confuse target/input or infer surjectivity from cardinalities alone.
Step identifiers and dependencies
Rule: Surjective_EVERY_source_target_fiber_comparison · Uses: G001, S053
S055 · implies
Before
source roles = [[-5, -4, -3, -2, -1, 0], [0, 1, 2, 3, 4, 5, 117], [[0, [0]], [1, [-5]], [2, [-4]], [3, [-3]], [4, [-2]], [5, [-1]], [117, []]]]
After
derived = [[117], No]
Conditions: none
Surjectivity requires a preimage for EVERY declared target;multiple preimages are allowed.For a QUALIFIED empty function,the empty-codomain condition is vacuous.The missing set is B(or B2)minus range,not a replacement of the ORIGINAL codomain.
Step identifiers and dependencies
Rule: Surjective_ALL_codomain_targets_covered_including_EMPTY_vacuity · Uses: G001, S054