# Question 19406: Exact Chromatic Number Needs Both Lower and Upper Evidence [Return to catalogue page 98](https://ocme.evemisslab.com/en/questions/page/98/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261106-r01-chromatic-number-foundations/#q-20261014-advance-20261106-r01-chromatic-number-0056) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0037-19406 · q-20261014-advance-20261106-r01-chromatic-number-0056 ## Original question and givens Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate). - G001 — Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-chromatic-number](https://ocme.evemisslab.com/en/mko/mko-chromatic-number/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ACTUAL complete finite carrier Fin(6);ALL original canonical simple undirected edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [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 dependenciesRule: chromatic_complete_graph_not_semantic_color_relabeling · Uses: G001, S001 S003 · impliesBefore 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 dependenciesRule: chromatic_complete_lex_search_not_greedy_optimality · Uses: G001, S002 S004 · impliesBefore 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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S003 S005 · impliesBefore 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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S004 S006 · impliesBefore 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 dependenciesRule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S005 S007 · impliesBefore 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 dependenciesRule: chromatic_exhausted_branch_backtracking_not_one_bad_sample · Uses: G001, S006 S008 · impliesBefore 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 dependenciesRule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S007 S009 · impliesBefore 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 dependenciesRule: chromatic_actual_lower_palette_exhaustion · Uses: G001, S008 S010 · impliesBefore 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 dependenciesRule: chromatic_complete_lex_search_not_greedy_optimality · Uses: G001, S009 S011 · impliesBefore 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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S010 S012 · impliesBefore 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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S011 S013 · impliesBefore 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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S012 S014 · impliesBefore palette=2;original vertex=2;assigned ORIGINAL prefix=[0, 1];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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S013 S015 · impliesBefore palette=2;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=1 After ALL earlier adjacent vertices already using this label=[1];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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S014 S016 · impliesBefore palette=2;original vertex=2;original fixed prefix=[0, 1];ALL colors0..1 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 dependenciesRule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S015 S017 · impliesBefore palette=2;original prefix=[0];vertex=1;chosen color=1;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 dependenciesRule: chromatic_exhausted_branch_backtracking_not_one_bad_sample · Uses: G001, S016 S018 · impliesBefore palette=2;original vertex=1;original fixed prefix=[0];ALL colors0..1 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 dependenciesRule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S017 S019 · impliesBefore palette=2;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 dependenciesRule: chromatic_exhausted_branch_backtracking_not_one_bad_sample · Uses: G001, S018 S020 · impliesBefore palette=2;original vertex=0;assigned ORIGINAL prefix=[];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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S019 S021 · impliesBefore palette=2;original vertex=1;assigned ORIGINAL prefix=[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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S020 S022 · impliesBefore palette=2;original vertex=2;assigned ORIGINAL prefix=[1, 0];try next ascending color=0 After ALL earlier adjacent vertices already using this label=[1];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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S021 S023 · impliesBefore palette=2;original vertex=2;assigned ORIGINAL prefix=[1, 0];try next ascending color=1 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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S022 S024 · impliesBefore palette=2;original vertex=2;original fixed prefix=[1, 0];ALL colors0..1 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 dependenciesRule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S023 S025 · impliesBefore palette=2;original prefix=[1];vertex=1;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 dependenciesRule: chromatic_exhausted_branch_backtracking_not_one_bad_sample · Uses: G001, S024 S026 · impliesBefore palette=2;original vertex=1;assigned ORIGINAL prefix=[1];try next ascending color=1 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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S025 S027 · impliesBefore palette=2;original vertex=1;original fixed prefix=[1];ALL colors0..1 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 dependenciesRule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S026 S028 · impliesBefore palette=2;original prefix=[];vertex=0;chosen color=1;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 dependenciesRule: chromatic_exhausted_branch_backtracking_not_one_bad_sample · Uses: G001, S027 S029 · impliesBefore palette=2;original vertex=0;original fixed prefix=[];ALL colors0..1 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 dependenciesRule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S028 S030 · impliesBefore palette[0,2) 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 dependenciesRule: chromatic_actual_lower_palette_exhaustion · Uses: G001, S029 S031 · impliesBefore ACTUAL graph Fin(6);try palette[0,3);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 dependenciesRule: chromatic_complete_lex_search_not_greedy_optimality · Uses: G001, S030 S032 · impliesBefore palette=3;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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S031 S033 · impliesBefore palette=3;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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S032 S034 · impliesBefore palette=3;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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S033 S035 · impliesBefore palette=3;original vertex=2;assigned ORIGINAL prefix=[0, 1];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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S034 S036 · impliesBefore palette=3;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=1 After ALL earlier adjacent vertices already using this label=[1];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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S035 S037 · impliesBefore palette=3;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=2 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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S036 S038 · impliesBefore palette=3;original vertex=3;assigned ORIGINAL prefix=[0, 1, 2];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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S037 S039 · impliesBefore palette=3;original vertex=4;assigned ORIGINAL prefix=[0, 1, 2, 0];try next ascending color=0 After ALL earlier adjacent vertices already using this label=[3];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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S038 S040 · impliesBefore palette=3;original vertex=4;assigned ORIGINAL prefix=[0, 1, 2, 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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S039 S041 · impliesBefore palette=3;original vertex=5;assigned ORIGINAL prefix=[0, 1, 2, 0, 1];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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S040 S042 · impliesBefore palette=3;original vertex=5;assigned ORIGINAL prefix=[0, 1, 2, 0, 1];try next ascending color=1 After ALL earlier adjacent vertices already using this label=[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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S041 S043 · impliesBefore palette=3;original vertex=5;assigned ORIGINAL prefix=[0, 1, 2, 0, 1];try next ascending color=2 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 dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S042 S044 · impliesBefore FIRST completed vertex-order increasing-color search witness=[0, 1, 2, 0, 1, 2];palette size=3;ALL smaller palettes exhausted After every ORIGINAL edge endpoints different;constructive chi<= 3 and exhaustive chi>= 3;chi=3;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 dependenciesRule: chromatic_constructive_upper_and_exhausted_lower_tie_order · Uses: G001, S043 ## Reverse checks - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (3, [0, 1, 2, 0, 1, 2], Yes, exhaustive_lower_and_constructive_upper_certificates) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: c1a639716df76b38fd711f2885bd8dae0eb1ee49a830587f203c7f6fe99bd505 - candidate record SHA-256: 6f8a542914bf2e35f6736624ffadc52fead8638e55641a3c37c5e3189c4fbc82 - semantic review receipt SHA-256: 065507b602ceee60cf47a700d387a51940e0c6724c35a3336a7a7948a199ed80 - question-first witness SHA-256: 7203f33b030792559673b0fee520a8c0e70903c4dd970b35293aa40852f333d6 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261106-r01-chromatic-number-0056.json)