Step-by-step derivation
S001 · implies
Before
source roles = [qf-surjective-first-preimage, [A, B, operation, pairs, t]]
After
derived = [first_preimage, 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, [-7, -6, -5, -4, -3, -2, -1]]
After
derived = [[-7, -6, -5, -4, -3, -2, -1]]
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, -1]]
After
derived = [[-4, -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, S004
S006 · implies
Before
source roles = [function ORIGINAL table, 2, [-3, -2]]
After
derived = [[-3, -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, S005
S007 · implies
Before
source roles = [function ORIGINAL table, 3, [-2, -2]]
After
derived = [[-2, -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, S006
S008 · implies
Before
source roles = [function ORIGINAL table, 4, [-1, -3]]
After
derived = [[-1, -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, S007
S009 · implies
Before
source roles = [function ORIGINAL table, 5, [0, -3]]
After
derived = [[0, -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, S008
S010 · implies
Before
source roles = [[-5, -4, -3, -2, -1, 0], [-7, -6, -5, -4, -3, -2, -1], [[-5, -1], [-4, -1], [-3, -2], [-2, -2], [-1, -3], [0, -3]]]
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 = [ORIGINAL B, -5, -1, -7]
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, S010
S012 · implies
Before
source roles = [ORIGINAL B, -4, -1, -7]
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, S011
S013 · implies
Before
source roles = [ORIGINAL B, -3, -2, -7]
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 = [ORIGINAL B, -2, -2, -7]
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 = [ORIGINAL B, -1, -3, -7]
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 = [ORIGINAL B, 0, -3, -7]
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 = [ORIGINAL B, -5, -1, -6]
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 = [ORIGINAL B, -4, -1, -6]
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, S017
S019 · implies
Before
source roles = [ORIGINAL B, -3, -2, -6]
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, S018
S020 · implies
Before
source roles = [ORIGINAL B, -2, -2, -6]
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 = [ORIGINAL B, -1, -3, -6]
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 = [ORIGINAL B, 0, -3, -6]
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 = [ORIGINAL B, -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, S022
S024 · implies
Before
source roles = [ORIGINAL B, -4, -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, S023
S025 · implies
Before
source roles = [ORIGINAL B, -3, -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, S024
S026 · implies
Before
source roles = [ORIGINAL B, -2, -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, S025
S027 · implies
Before
source roles = [ORIGINAL B, -1, -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, S026
S028 · implies
Before
source roles = [ORIGINAL B, 0, -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, S027
S029 · implies
Before
source roles = [ORIGINAL B, -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, S028
S030 · implies
Before
source roles = [ORIGINAL B, -4, -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, S029
S031 · implies
Before
source roles = [ORIGINAL B, -3, -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, S030
S032 · implies
Before
source roles = [ORIGINAL B, -2, -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, S031
S033 · implies
Before
source roles = [ORIGINAL B, -1, -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, S032
S034 · implies
Before
source roles = [ORIGINAL B, 0, -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, S033
S035 · implies
Before
source roles = [ORIGINAL B, -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, S034
S036 · implies
Before
source roles = [ORIGINAL B, -4, -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, S035
S037 · implies
Before
source roles = [ORIGINAL B, -3, -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, S036
S038 · implies
Before
source roles = [ORIGINAL B, -2, -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, S037
S039 · implies
Before
source roles = [ORIGINAL B, -1, -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, S038
S040 · implies
Before
source roles = [ORIGINAL B, 0, -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, S039
S041 · implies
Before
source roles = [ORIGINAL B, -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, S040
S042 · implies
Before
source roles = [ORIGINAL B, -4, -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, S041
S043 · implies
Before
source roles = [ORIGINAL B, -3, -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, S042
S044 · implies
Before
source roles = [ORIGINAL B, -2, -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, S043
S045 · implies
Before
source roles = [ORIGINAL B, -1, -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, S044
S046 · implies
Before
source roles = [ORIGINAL B, 0, -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, S045
S047 · implies
Before
source roles = [ORIGINAL B, -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, S046
S048 · implies
Before
source roles = [ORIGINAL B, -4, -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, S047
S049 · implies
Before
source roles = [ORIGINAL B, -3, -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, S048
S050 · implies
Before
source roles = [ORIGINAL B, -2, -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, S049
S051 · implies
Before
source roles = [ORIGINAL B, -1, -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, S050
S052 · implies
Before
source roles = [ORIGINAL B, 0, -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, S051
S053 · implies
Before
source roles = [-1, [[-7, []], [-6, []], [-5, []], [-4, []], [-3, [-1, 0]], [-2, [-3, -2]], [-1, [-5, -4]]]]
After
derived = [[-5, -4], -5]
Conditions: none
FIRST require t∈declared B.Then choose the NUMERICALLY LEAST input in its COMPLETE fiber,not the first raw-list position.Empty fiber gives none;existence does not imply uniqueness.
Step identifiers and dependencies
Rule: Surjective_FIRST_preimage_NUMERIC_least_or_none_NOT_unique · Uses: G001, S052