Step-by-step derivation
S001 · implies
Before
source operation=chromatic-exact-number;original snapshot and mathematics retained
After
explicit ordered source output fields=[chromatic_number, canonical_coloring, lower_bound_exhausted, certificate]
Conditions: none
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ACTUAL complete finite carrier Fin(6);ALL original canonical simple undirected edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]
After
use full original vertex/edge data;exact search domain n<=9;empty carrier and isolated vertices RETAINED
Conditions: none
Retain every original vertex and edge.This family includes n=0 rather than importing a previous nonempty-carrier guard or deleting isolates.Actual palette labels must not be replaced by semantic relabeling.
Step identifiers and dependencies
Rule: chromatic_complete_graph_not_semantic_color_relabeling · Uses: G001, S001
S003 · implies
Before
ACTUAL graph Fin(6);try palette[0,1);original vertex order0..5
After
search ALL earlier lexicographic prefixes:vertex order fixed,each color tried increasingly;only an actual earlier-edge conflict may prune
Conditions: none
Search each k completely rather than treating a failed greedy run as non-colorability.Fixed original vertex order and increasing colors define the source's first lexicographic witness.Prune only an actual edge conflict;otherwise continue or exhaustively backtrack.
Step identifiers and dependencies
Rule: chromatic_complete_lex_search_not_greedy_optimality · Uses: G001, S002
S004 · implies
Before
palette=1;original vertex=0;assigned ORIGINAL prefix=[];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S003
S005 · implies
Before
palette=1;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[0];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S004
S006 · implies
Before
palette=1;original vertex=1;original fixed prefix=[0];ALL colors0..0 attempted
After
no valid completion of this prefix in this palette
Conditions: none
All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette.
Step identifiers and dependencies
Rule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S005
S007 · implies
Before
palette=1;original prefix=[];vertex=0;chosen color=0;ALL its later completions exhausted
After
restore this unassigned coordinate and try the next ascending color
Conditions: none
Backtrack only after the whole subtree fails.Every later completion of this branch was excluded by an actual conflict or a completely exhausted subtree,not one sampled attempt.
Step identifiers and dependencies
Rule: chromatic_exhausted_branch_backtracking_not_one_bad_sample · Uses: G001, S006
S008 · implies
Before
palette=1;original vertex=0;original fixed prefix=[];ALL colors0..0 attempted
After
no valid completion of this prefix in this palette
Conditions: none
All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette.
Step identifiers and dependencies
Rule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S007
S009 · implies
Before
palette[0,1) searched from EMPTY root prefix with full branch exclusions
After
NO proper coloring exists in this palette;any smaller palette is a subset of it
Conditions: none
Exhausting the complete root tree supplies the lower certificate.Any smaller palette embeds in this one and is excluded as well;no greedy path establishes this conclusion.
Step identifiers and dependencies
Rule: chromatic_actual_lower_palette_exhaustion · Uses: G001, S008
S010 · implies
Before
ACTUAL graph Fin(6);try palette[0,2);original vertex order0..5
After
search ALL earlier lexicographic prefixes:vertex order fixed,each color tried increasingly;only an actual earlier-edge conflict may prune
Conditions: none
Search each k completely rather than treating a failed greedy run as non-colorability.Fixed original vertex order and increasing colors define the source's first lexicographic witness.Prune only an actual edge conflict;otherwise continue or exhaustively backtrack.
Step identifiers and dependencies
Rule: chromatic_complete_lex_search_not_greedy_optimality · Uses: G001, S009
S011 · implies
Before
palette=2;original vertex=0;assigned ORIGINAL prefix=[];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S010
S012 · implies
Before
palette=2;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[0];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S011
S013 · implies
Before
palette=2;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=1
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S012
S014 · implies
Before
palette=2;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S013
S015 · implies
Before
palette=2;original vertex=3;assigned ORIGINAL prefix=[0, 1, 0];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[0, 2];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S014
S016 · implies
Before
palette=2;original vertex=3;assigned ORIGINAL prefix=[0, 1, 0];try next ascending color=1
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S015
S017 · implies
Before
palette=2;original vertex=4;assigned ORIGINAL prefix=[0, 1, 0, 1];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S016
S018 · implies
Before
palette=2;original vertex=5;assigned ORIGINAL prefix=[0, 1, 0, 1, 0];try next ascending color=0
After
ALL earlier adjacent vertices already using this label=[0, 4];admissible extension=No
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S017
S019 · implies
Before
palette=2;original vertex=5;assigned ORIGINAL prefix=[0, 1, 0, 1, 0];try next ascending color=1
After
ALL earlier adjacent vertices already using this label=[];admissible extension=Yes
Conditions: none
List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.
Step identifiers and dependencies
Rule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S018
S020 · implies
Before
FIRST completed vertex-order increasing-color search witness=[0, 1, 0, 1, 0, 1];palette size=2;ALL smaller palettes exhausted
After
every ORIGINAL edge endpoints different;constructive chi<= 2 and exhaustive chi>= 2;chi=2;first LEX witness retained
Conditions: none
The complete proper original-edge coloring gives the upper bound;the preceding exhaustive failures give the lower bound.Their equality establishes the exact minimum.Fixed DFS order preserves the source's first lexicographic witness,not an arbitrary optimum.A nonempty edgeless graph succeeds first at k=1 with all zeros;no fictitious0-color search is needed.
Step identifiers and dependencies
Rule: chromatic_constructive_upper_and_exhausted_lower_tie_order · Uses: G001, S019