Original question and givens
Use the complete total mapping=[3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0] from source indices Fin(12)={0,…,11} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).
G001 — Use the complete total mapping=[3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0] from source indices Fin(12)={0,…,11} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).
Step-by-step derivation
S001 · implies
Before
source operation=pigeon-collision-witness;original snapshot and mathematics retained
After
explicit ordered source output fields=[domain_count, codomain_count, collision, has_collision, certificate]
Conditions: none
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL complete source-indexed total map=[3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0];source Fin(12);ACTUAL target Fin(7)
After
every source has ONE in-range image;all target labels0..6 retained even if unused;empty source allowed
Conditions: none
Retain original source order and target labels rather than relabeling bins by first appearance.Complete total-map data support full fibres and collisions,including an empty source into a nonempty codomain.
Step identifiers and dependencies
Rule: pigeon_complete_total_map_not_sample_or_bin_relabeling · Uses: G001, S001
S003 · implies
Before
next LEX ordered original source pair=[0, 1] with left<right;original images=[3, 4]
After
same target image=No;every earlier LEFT-then-RIGHT pair already checked unequal
Conditions: none
Check left first and then right;the first equal pair is the source's first lexicographic pair,which may differ from the earliest repeat encountered by increasing right endpoint.
Step identifiers and dependencies
Rule: pigeon_first_collision_lex_pair_not_first_repeat_stream · Uses: G001, S002
S004 · implies
Before
next LEX ordered original source pair=[0, 2] with left<right;original images=[3, 5]
After
same target image=No;every earlier LEFT-then-RIGHT pair already checked unequal
Conditions: none
Check left first and then right;the first equal pair is the source's first lexicographic pair,which may differ from the earliest repeat encountered by increasing right endpoint.
Step identifiers and dependencies
Rule: pigeon_first_collision_lex_pair_not_first_repeat_stream · Uses: G001, S003
S005 · implies
Before
next LEX ordered original source pair=[0, 3] with left<right;original images=[3, 6]
After
same target image=No;every earlier LEFT-then-RIGHT pair already checked unequal
Conditions: none
Check left first and then right;the first equal pair is the source's first lexicographic pair,which may differ from the earliest repeat encountered by increasing right endpoint.
Step identifiers and dependencies
Rule: pigeon_first_collision_lex_pair_not_first_repeat_stream · Uses: G001, S004
S006 · implies
Before
next LEX ordered original source pair=[0, 4] with left<right;original images=[3, 0]
After
same target image=No;every earlier LEFT-then-RIGHT pair already checked unequal
Conditions: none
Check left first and then right;the first equal pair is the source's first lexicographic pair,which may differ from the earliest repeat encountered by increasing right endpoint.
Step identifiers and dependencies
Rule: pigeon_first_collision_lex_pair_not_first_repeat_stream · Uses: G001, S005
S007 · implies
Before
next LEX ordered original source pair=[0, 5] with left<right;original images=[3, 1]
After
same target image=No;every earlier LEFT-then-RIGHT pair already checked unequal
Conditions: none
Check left first and then right;the first equal pair is the source's first lexicographic pair,which may differ from the earliest repeat encountered by increasing right endpoint.
Step identifiers and dependencies
Rule: pigeon_first_collision_lex_pair_not_first_repeat_stream · Uses: G001, S006
S008 · implies
Before
next LEX ordered original source pair=[0, 6] with left<right;original images=[3, 2]
After
same target image=No;every earlier LEFT-then-RIGHT pair already checked unequal
Conditions: none
Check left first and then right;the first equal pair is the source's first lexicographic pair,which may differ from the earliest repeat encountered by increasing right endpoint.
Step identifiers and dependencies
Rule: pigeon_first_collision_lex_pair_not_first_repeat_stream · Uses: G001, S007
S009 · implies
Before
next LEX ordered original source pair=[0, 7] with left<right;original images=[3, 3]
After
same target image=Yes;every earlier LEFT-then-RIGHT pair already checked unequal
Conditions: none
Check left first and then right;the first equal pair is the source's first lexicographic pair,which may differ from the earliest repeat encountered by increasing right endpoint.
Step identifiers and dependencies
Rule: pigeon_first_collision_lex_pair_not_first_repeat_stream · Uses: G001, S008
S010 · implies
Before
ORIGINAL full domain size=12>target size=7;actual first collision=[0, 7, 3]
After
two DISTINCT original sources have the SAME original image;exact instance witness and finite size theorem agree
Conditions: none
The two source indices in the witness are distinct.This is an actual collision in a complete total map,consistent with the size argument for every map with m>n,not a global inference from partial samples.
Step identifiers and dependencies
Rule: pigeon_actual_distinct_source_collision_not_equal_indices · Uses: G001, S009