Step-by-step derivation
S001 · implies
Before
source roles = [qf-density17-bij-output-transposition, bij-output-transposition, Fin_total_table]
After
derived = [[original_images, next_images, ALL_next_fibers, original_bij, next_bij, flags_agree]]
Conditions: none
density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status.
Step identifiers and dependencies
Rule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001
S002 · implies
Before
source roles = [Bijection ORIGINAL complete table, 5, 3, [2, 2, 0, 0, 0]]
After
derived = [[2, 2, 0, 0, 0]]
Conditions: none
Require the WHOLE declared table length to equal its source size and EVERY cell to lie in Fin(m),including unused cells.Even an empty outer source cannot waive this complete table typing obligation.
Step identifiers and dependencies
Rule: Composition_WHOLE_typed_table_ALL_cells_including_unused · Uses: G001, S001
S003 · calculation
Before
Bijection ORIGINAL reverse ALL fiber sizes through target0 := sum(0, 3)
After
Bijection ORIGINAL reverse ALL fiber sizes through target0 = 3
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
source roles = [Bijection ORIGINAL, [2, 2, 0, 0, 0], 3, 0]
After
derived = [[2, 3, 4], 3]
Conditions: none
For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order.
Step identifiers and dependencies
Rule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S003
S005 · calculation
Before
Bijection ORIGINAL reverse ALL fiber sizes through target1 := sum(3, 0)
After
Bijection ORIGINAL reverse ALL fiber sizes through target1 = 3
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
source roles = [Bijection ORIGINAL, [2, 2, 0, 0, 0], 3, 1]
After
derived = [[], 0]
Conditions: none
For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order.
Step identifiers and dependencies
Rule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S005
S007 · calculation
Before
Bijection ORIGINAL reverse ALL fiber sizes through target2 := sum(3, 2)
After
Bijection ORIGINAL reverse ALL fiber sizes through target2 = 5
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
source roles = [Bijection ORIGINAL, [2, 2, 0, 0, 0], 3, 2]
After
derived = [[0, 1], 2]
Conditions: none
For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order.
Step identifiers and dependencies
Rule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S007
S009 · implies
Before
source roles = [Bijection ORIGINAL, [2, 2, 0, 0, 0], 3]
After
derived = [[[2, 3, 4], [], [0, 1]], [[0, 1], [2, 3], [2, 4], [3, 4]], [1], No, No, No, 5]
Conditions: none
A collision has x<y and equal images.List ALL pairs in GLOBAL numeric LEX order,not codomain-block order or first repeated output.No collision gives injectivity;no missing target gives surjectivity;BOTH give bijectivity.Empty->empty has all three;empty->nonempty only injectivity.
Step identifiers and dependencies
Rule: Bijection_ALL_collisions_GLOBAL_LEX_missing_targets_THREE_distinct_properties · Uses: G001, S008
S010 · implies
Before
source roles = [5, 3, 0, 2, 0, 2]
After
derived = [0]
Conditions: none
Apply the SAME codomain-label transposition to EVERY original source output,without reordering sources or changing only the first occurrence.Equal a,b is a legal no-op.
Step identifiers and dependencies
Rule: Bijection_OUTPUT_label_transposition_EVERY_original_slot · Uses: G001, S009
S011 · implies
Before
source roles = [5, 3, 0, 2, 1, 2]
After
derived = [0]
Conditions: none
Apply the SAME codomain-label transposition to EVERY original source output,without reordering sources or changing only the first occurrence.Equal a,b is a legal no-op.
Step identifiers and dependencies
Rule: Bijection_OUTPUT_label_transposition_EVERY_original_slot · Uses: G001, S010
S012 · implies
Before
source roles = [5, 3, 0, 2, 2, 0]
After
derived = [2]
Conditions: none
Apply the SAME codomain-label transposition to EVERY original source output,without reordering sources or changing only the first occurrence.Equal a,b is a legal no-op.
Step identifiers and dependencies
Rule: Bijection_OUTPUT_label_transposition_EVERY_original_slot · Uses: G001, S011
S013 · implies
Before
source roles = [5, 3, 0, 2, 3, 0]
After
derived = [2]
Conditions: none
Apply the SAME codomain-label transposition to EVERY original source output,without reordering sources or changing only the first occurrence.Equal a,b is a legal no-op.
Step identifiers and dependencies
Rule: Bijection_OUTPUT_label_transposition_EVERY_original_slot · Uses: G001, S012
S014 · implies
Before
source roles = [5, 3, 0, 2, 4, 0]
After
derived = [2]
Conditions: none
Apply the SAME codomain-label transposition to EVERY original source output,without reordering sources or changing only the first occurrence.Equal a,b is a legal no-op.
Step identifiers and dependencies
Rule: Bijection_OUTPUT_label_transposition_EVERY_original_slot · Uses: G001, S013
S015 · calculation
Before
Bijection TRANSPOSED reverse ALL fiber sizes through target0 := sum(0, 2)
After
Bijection TRANSPOSED reverse ALL fiber sizes through target0 = 2
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · implies
Before
source roles = [Bijection TRANSPOSED, [0, 0, 2, 2, 2], 3, 0]
After
derived = [[0, 1], 2]
Conditions: none
For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order.
Step identifiers and dependencies
Rule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S015
S017 · calculation
Before
Bijection TRANSPOSED reverse ALL fiber sizes through target1 := sum(2, 0)
After
Bijection TRANSPOSED reverse ALL fiber sizes through target1 = 2
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · implies
Before
source roles = [Bijection TRANSPOSED, [0, 0, 2, 2, 2], 3, 1]
After
derived = [[], 0]
Conditions: none
For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order.
Step identifiers and dependencies
Rule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S017
S019 · calculation
Before
Bijection TRANSPOSED reverse ALL fiber sizes through target2 := sum(2, 3)
After
Bijection TRANSPOSED reverse ALL fiber sizes through target2 = 5
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · implies
Before
source roles = [Bijection TRANSPOSED, [0, 0, 2, 2, 2], 3, 2]
After
derived = [[2, 3, 4], 3]
Conditions: none
For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order.
Step identifiers and dependencies
Rule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S019
S021 · implies
Before
source roles = [Bijection TRANSPOSED, [0, 0, 2, 2, 2], 3]
After
derived = [[[0, 1], [], [2, 3, 4]], [[0, 1], [2, 3], [2, 4], [3, 4]], [1], No, No, No, 5]
Conditions: none
A collision has x<y and equal images.List ALL pairs in GLOBAL numeric LEX order,not codomain-block order or first repeated output.No collision gives injectivity;no missing target gives surjectivity;BOTH give bijectivity.Empty->empty has all three;empty->nonempty only injectivity.
Step identifiers and dependencies
Rule: Bijection_ALL_collisions_GLOBAL_LEX_missing_targets_THREE_distinct_properties · Uses: G001, S020
S022 · implies
Before
source roles = [5, 3, [2, 2, 0, 0, 0], 0, 2]
After
derived = [[0, 0, 2, 2, 2], [[0, 1], [], [2, 3, 4]], No, No, Yes, [0, 1, 2]]
Conditions: none
A TYPED codomain transposition restores EVERY label when applied twice,so it preserves bijectivity eligibility.The original map need not be bijective.False/False still has agreementTrue;do not confuse preservation with original bijectivity.At m0 no valid a,b exist for THIS family.
Step identifiers and dependencies
Rule: Bijection_transposition_is_codomain_PERMUTATION_flags_AGREEMENT · Uses: G001, S021