Original question and givens
Use the complete total mapping=[2, 2, 2, 2, 2, 2, 2, 2] from source indices Fin(8)={0,…,7} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).
G001 — Use the complete total mapping=[2, 2, 2, 2, 2, 2, 2, 2] from source indices Fin(8)={0,…,7} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).
Step-by-step derivation
S001 · implies
Before
source operation=pigeon-generalized-load;original snapshot and mathematics retained
After
explicit ordered source output fields=[loads, maximum_load, ceiling_lower_bound, bound_holds, 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=[2, 2, 2, 2, 2, 2, 2, 2];source Fin(8);ACTUAL target Fin(3)
After
every source has ONE in-range image;all target labels0..2 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 · calculation
Before
pigeon target0 complete fibre load := sum()
After
pigeon target0 complete fibre load = 0
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
ACTUAL original target=0;ALL source fibre indices=[]
After
complete load=0;unused targets remain with load0
Conditions: none
List every codomain fibre,including load0 for unused targets.Do not count only occupied bins and thereby change the average's denominator.
Step identifiers and dependencies
Rule: pigeon_every_target_load_null_bins_retained · Uses: G001, S003
S005 · calculation
Before
pigeon target1 complete fibre load := sum()
After
pigeon target1 complete fibre load = 0
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
ACTUAL original target=1;ALL source fibre indices=[]
After
complete load=0;unused targets remain with load0
Conditions: none
List every codomain fibre,including load0 for unused targets.Do not count only occupied bins and thereby change the average's denominator.
Step identifiers and dependencies
Rule: pigeon_every_target_load_null_bins_retained · Uses: G001, S005
S007 · calculation
Before
pigeon target2 complete fibre load := sum(1, 1, 1, 1, 1, 1, 1, 1)
After
pigeon target2 complete fibre load = 8
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
ACTUAL original target=2;ALL source fibre indices=[0, 1, 2, 3, 4, 5, 6, 7]
After
complete load=8;unused targets remain with load0
Conditions: none
List every codomain fibre,including load0 for unused targets.Do not count only occupied bins and thereby change the average's denominator.
Step identifiers and dependencies
Rule: pigeon_every_target_load_null_bins_retained · Uses: G001, S007
S009 · calculation
Before
pigeon complete sum of all target loads := sum(0, 0, 8)
After
pigeon complete sum of all target loads = 8
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
pigeon exact average load := div(8, 3)
After
pigeon exact average load = 8/3
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
pigeon exact ceiling lower bound := sum(2, 1)
After
pigeon exact ceiling lower bound = 3
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
ALL original-order loads=[0, 0, 8];sum=8;nonempty target count=3;exact average=8/3;division m=n*2+2
After
ceiling average=3;actual max load=8 attained at target2;maximum>=ceiling;not necessarily equality
Conditions: none
If every integer load were below ceil(m/n),all would be at most ceil−1 and their total would be less than m,a contradiction.For m=0 all loads are0 and max=ceil=0 directly.This lower bound need not equal the maximum load.
Step identifiers and dependencies
Rule: pigeon_generalized_ceiling_full_load_argument · Uses: G001, S011