# Question 27308: Every Legal Split of a^p b^p Is Defeated by Pumping Down

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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.

R0053-27308 · q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0058



## Original question and givens

Use positive pumping length p=8 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).


- G001 — Use positive pumping length p=8 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).





## 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-pumping-lemma-regular-languages](https://ocme.evemisslab.com/en/mko/mko-pumping-lemma-regular-languages/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [pump-anbn-all-splits]

After

derived = [[p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope]]

Conditions: none

List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings.

Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [8]

After

derived = [a^8b^8, 16, ALL x>=0,y>=1,x+y<=p]

Conditions: none

Under assumed regularity take its pumping length p and choose a^p b^p of length2p>=p.Any legal xy lies in the initial p a-symbols,so y consists of a symbols and is nonempty.Do not preselect a convenient split.

Step identifiers and dependenciesRule: Pumping_adversary_word_and_ALL_legal_split_domain · Uses: G001, S001





S003 · calculationBefore

Pumping ORIGINAL remaining a after i0 split0,1 := sub(8, 1)

After

Pumping ORIGINAL remaining a after i0 split0,1 = 7

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

source roles = [8, 0, 1]

After

derived = [7, 15, [0, 1, 7, a^7b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S003





S005 · calculationBefore

Pumping ORIGINAL remaining a after i0 split0,2 := sub(8, 2)

After

Pumping ORIGINAL remaining a after i0 split0,2 = 6

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

source roles = [8, 0, 2]

After

derived = [6, 14, [0, 2, 6, a^6b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S005





S007 · calculationBefore

Pumping ORIGINAL remaining a after i0 split0,3 := sub(8, 3)

After

Pumping ORIGINAL remaining a after i0 split0,3 = 5

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

source roles = [8, 0, 3]

After

derived = [5, 13, [0, 3, 5, a^5b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S007





S009 · calculationBefore

Pumping ORIGINAL remaining a after i0 split0,4 := sub(8, 4)

After

Pumping ORIGINAL remaining a after i0 split0,4 = 4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

source roles = [8, 0, 4]

After

derived = [4, 12, [0, 4, 4, a^4b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S009





S011 · calculationBefore

Pumping ORIGINAL remaining a after i0 split0,5 := sub(8, 5)

After

Pumping ORIGINAL remaining a after i0 split0,5 = 3

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · impliesBefore

source roles = [8, 0, 5]

After

derived = [3, 11, [0, 5, 3, a^3b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S011





S013 · calculationBefore

Pumping ORIGINAL remaining a after i0 split0,6 := sub(8, 6)

After

Pumping ORIGINAL remaining a after i0 split0,6 = 2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · impliesBefore

source roles = [8, 0, 6]

After

derived = [2, 10, [0, 6, 2, a^2b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S013





S015 · calculationBefore

Pumping ORIGINAL remaining a after i0 split0,7 := sub(8, 7)

After

Pumping ORIGINAL remaining a after i0 split0,7 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · impliesBefore

source roles = [8, 0, 7]

After

derived = [1, 9, [0, 7, 1, a^1b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S015





S017 · calculationBefore

Pumping ORIGINAL remaining a after i0 split0,8 := sub(8, 8)

After

Pumping ORIGINAL remaining a after i0 split0,8 = 0

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · impliesBefore

source roles = [8, 0, 8]

After

derived = [0, 8, [0, 8, 0, a^0b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S017





S019 · calculationBefore

Pumping ORIGINAL remaining a after i0 split1,1 := sub(8, 1)

After

Pumping ORIGINAL remaining a after i0 split1,1 = 7

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · impliesBefore

source roles = [8, 1, 1]

After

derived = [6, 14, [1, 1, 6, a^7b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S019





S021 · calculationBefore

Pumping ORIGINAL remaining a after i0 split1,2 := sub(8, 2)

After

Pumping ORIGINAL remaining a after i0 split1,2 = 6

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · impliesBefore

source roles = [8, 1, 2]

After

derived = [5, 13, [1, 2, 5, a^6b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S021





S023 · calculationBefore

Pumping ORIGINAL remaining a after i0 split1,3 := sub(8, 3)

After

Pumping ORIGINAL remaining a after i0 split1,3 = 5

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · impliesBefore

source roles = [8, 1, 3]

After

derived = [4, 12, [1, 3, 4, a^5b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S023





S025 · calculationBefore

Pumping ORIGINAL remaining a after i0 split1,4 := sub(8, 4)

After

Pumping ORIGINAL remaining a after i0 split1,4 = 4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · impliesBefore

source roles = [8, 1, 4]

After

derived = [3, 11, [1, 4, 3, a^4b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S025





S027 · calculationBefore

Pumping ORIGINAL remaining a after i0 split1,5 := sub(8, 5)

After

Pumping ORIGINAL remaining a after i0 split1,5 = 3

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · impliesBefore

source roles = [8, 1, 5]

After

derived = [2, 10, [1, 5, 2, a^3b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S027





S029 · calculationBefore

Pumping ORIGINAL remaining a after i0 split1,6 := sub(8, 6)

After

Pumping ORIGINAL remaining a after i0 split1,6 = 2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · impliesBefore

source roles = [8, 1, 6]

After

derived = [1, 9, [1, 6, 1, a^2b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S029





S031 · calculationBefore

Pumping ORIGINAL remaining a after i0 split1,7 := sub(8, 7)

After

Pumping ORIGINAL remaining a after i0 split1,7 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · impliesBefore

source roles = [8, 1, 7]

After

derived = [0, 8, [1, 7, 0, a^1b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S031





S033 · calculationBefore

Pumping ORIGINAL remaining a after i0 split2,1 := sub(8, 1)

After

Pumping ORIGINAL remaining a after i0 split2,1 = 7

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032





S034 · impliesBefore

source roles = [8, 2, 1]

After

derived = [5, 13, [2, 1, 5, a^7b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S033





S035 · calculationBefore

Pumping ORIGINAL remaining a after i0 split2,2 := sub(8, 2)

After

Pumping ORIGINAL remaining a after i0 split2,2 = 6

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · impliesBefore

source roles = [8, 2, 2]

After

derived = [4, 12, [2, 2, 4, a^6b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S035





S037 · calculationBefore

Pumping ORIGINAL remaining a after i0 split2,3 := sub(8, 3)

After

Pumping ORIGINAL remaining a after i0 split2,3 = 5

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · impliesBefore

source roles = [8, 2, 3]

After

derived = [3, 11, [2, 3, 3, a^5b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S037





S039 · calculationBefore

Pumping ORIGINAL remaining a after i0 split2,4 := sub(8, 4)

After

Pumping ORIGINAL remaining a after i0 split2,4 = 4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038





S040 · impliesBefore

source roles = [8, 2, 4]

After

derived = [2, 10, [2, 4, 2, a^4b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S039





S041 · calculationBefore

Pumping ORIGINAL remaining a after i0 split2,5 := sub(8, 5)

After

Pumping ORIGINAL remaining a after i0 split2,5 = 3

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040





S042 · impliesBefore

source roles = [8, 2, 5]

After

derived = [1, 9, [2, 5, 1, a^3b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S041





S043 · calculationBefore

Pumping ORIGINAL remaining a after i0 split2,6 := sub(8, 6)

After

Pumping ORIGINAL remaining a after i0 split2,6 = 2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042





S044 · impliesBefore

source roles = [8, 2, 6]

After

derived = [0, 8, [2, 6, 0, a^2b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S043





S045 · calculationBefore

Pumping ORIGINAL remaining a after i0 split3,1 := sub(8, 1)

After

Pumping ORIGINAL remaining a after i0 split3,1 = 7

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044





S046 · impliesBefore

source roles = [8, 3, 1]

After

derived = [4, 12, [3, 1, 4, a^7b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S045





S047 · calculationBefore

Pumping ORIGINAL remaining a after i0 split3,2 := sub(8, 2)

After

Pumping ORIGINAL remaining a after i0 split3,2 = 6

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046





S048 · impliesBefore

source roles = [8, 3, 2]

After

derived = [3, 11, [3, 2, 3, a^6b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S047





S049 · calculationBefore

Pumping ORIGINAL remaining a after i0 split3,3 := sub(8, 3)

After

Pumping ORIGINAL remaining a after i0 split3,3 = 5

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048





S050 · impliesBefore

source roles = [8, 3, 3]

After

derived = [2, 10, [3, 3, 2, a^5b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S049





S051 · calculationBefore

Pumping ORIGINAL remaining a after i0 split3,4 := sub(8, 4)

After

Pumping ORIGINAL remaining a after i0 split3,4 = 4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050





S052 · impliesBefore

source roles = [8, 3, 4]

After

derived = [1, 9, [3, 4, 1, a^4b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S051





S053 · calculationBefore

Pumping ORIGINAL remaining a after i0 split3,5 := sub(8, 5)

After

Pumping ORIGINAL remaining a after i0 split3,5 = 3

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052





S054 · impliesBefore

source roles = [8, 3, 5]

After

derived = [0, 8, [3, 5, 0, a^3b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S053





S055 · calculationBefore

Pumping ORIGINAL remaining a after i0 split4,1 := sub(8, 1)

After

Pumping ORIGINAL remaining a after i0 split4,1 = 7

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054





S056 · impliesBefore

source roles = [8, 4, 1]

After

derived = [3, 11, [4, 1, 3, a^7b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S055





S057 · calculationBefore

Pumping ORIGINAL remaining a after i0 split4,2 := sub(8, 2)

After

Pumping ORIGINAL remaining a after i0 split4,2 = 6

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056





S058 · impliesBefore

source roles = [8, 4, 2]

After

derived = [2, 10, [4, 2, 2, a^6b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S057





S059 · calculationBefore

Pumping ORIGINAL remaining a after i0 split4,3 := sub(8, 3)

After

Pumping ORIGINAL remaining a after i0 split4,3 = 5

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058





S060 · impliesBefore

source roles = [8, 4, 3]

After

derived = [1, 9, [4, 3, 1, a^5b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S059





S061 · calculationBefore

Pumping ORIGINAL remaining a after i0 split4,4 := sub(8, 4)

After

Pumping ORIGINAL remaining a after i0 split4,4 = 4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060





S062 · impliesBefore

source roles = [8, 4, 4]

After

derived = [0, 8, [4, 4, 0, a^4b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S061





S063 · calculationBefore

Pumping ORIGINAL remaining a after i0 split5,1 := sub(8, 1)

After

Pumping ORIGINAL remaining a after i0 split5,1 = 7

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062





S064 · impliesBefore

source roles = [8, 5, 1]

After

derived = [2, 10, [5, 1, 2, a^7b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S063





S065 · calculationBefore

Pumping ORIGINAL remaining a after i0 split5,2 := sub(8, 2)

After

Pumping ORIGINAL remaining a after i0 split5,2 = 6

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064





S066 · impliesBefore

source roles = [8, 5, 2]

After

derived = [1, 9, [5, 2, 1, a^6b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S065





S067 · calculationBefore

Pumping ORIGINAL remaining a after i0 split5,3 := sub(8, 3)

After

Pumping ORIGINAL remaining a after i0 split5,3 = 5

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066





S068 · impliesBefore

source roles = [8, 5, 3]

After

derived = [0, 8, [5, 3, 0, a^5b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S067





S069 · calculationBefore

Pumping ORIGINAL remaining a after i0 split6,1 := sub(8, 1)

After

Pumping ORIGINAL remaining a after i0 split6,1 = 7

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068





S070 · impliesBefore

source roles = [8, 6, 1]

After

derived = [1, 9, [6, 1, 1, a^7b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S069





S071 · calculationBefore

Pumping ORIGINAL remaining a after i0 split6,2 := sub(8, 2)

After

Pumping ORIGINAL remaining a after i0 split6,2 = 6

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070





S072 · impliesBefore

source roles = [8, 6, 2]

After

derived = [0, 8, [6, 2, 0, a^6b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S071





S073 · calculationBefore

Pumping ORIGINAL remaining a after i0 split7,1 := sub(8, 1)

After

Pumping ORIGINAL remaining a after i0 split7,1 = 7

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072





S074 · impliesBefore

source roles = [8, 7, 1]

After

derived = [0, 8, [7, 1, 0, a^7b^8, No], Yes]

Conditions: none

The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S073





S075 · calculationBefore

Pumping ORIGINAL triangular numerator := mul(8, 9)

After

Pumping ORIGINAL triangular numerator = 72

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S074





S076 · calculationBefore

Pumping ORIGINAL triangular all-splits count := div(72, 2)

After

Pumping ORIGINAL triangular all-splits count = 36

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075





S077 · impliesBefore

source roles = [8, [[0, 1, 7, a^7b^8, No], [0, 2, 6, a^6b^8, No], [0, 3, 5, a^5b^8, No], [0, 4, 4, a^4b^8, No], [0, 5, 3, a^3b^8, No], [0, 6, 2, a^2b^8, No], [0, 7, 1, a^1b^8, No], [0, 8, 0, a^0b^8, No], [1, 1, 6, a^7b^8, No], [1, 2, 5, a^6b^8, No], [1, 3, 4, a^5b^8, No], [1, 4, 3, a^4b^8, No], [1, 5, 2, a^3b^8, No], [1, 6, 1, a^2b^8, No], [1, 7, 0, a^1b^8, No], [2, 1, 5, a^7b^8, No], [2, 2, 4, a^6b^8, No], [2, 3, 3, a^5b^8, No], [2, 4, 2, a^4b^8, No], [2, 5, 1, a^3b^8, No], [2, 6, 0, a^2b^8, No], [3, 1, 4, a^7b^8, No], [3, 2, 3, a^6b^8, No], [3, 3, 2, a^5b^8, No], [3, 4, 1, a^4b^8, No], [3, 5, 0, a^3b^8, No], [4, 1, 3, a^7b^8, No], [4, 2, 2, a^6b^8, No], [4, 3, 1, a^5b^8, No], [4, 4, 0, a^4b^8, No], [5, 1, 2, a^7b^8, No], [5, 2, 1, a^6b^8, No], [5, 3, 0, a^5b^8, No], [6, 1, 1, a^7b^8, No], [6, 2, 0, a^6b^8, No], [7, 1, 0, a^7b^8, No]]]

After

derived = [36, 36, Yes]

Conditions: none

For each x there are p-x choices of y,total p(p+1)/2,so every legal split is included.The general argument repeats for ANY proposed p,with i=0 defeating every split.A table at one fixed p alone is not the general proof.

Step identifiers and dependenciesRule: Pumping_ALL_splits_and_GENERAL_every_proposed_p_contradiction · Uses: G001, S076









## Reverse checks


- S077 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S076 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular all-splits count with exact rational arithmetic, not rounded decimals.
- S075 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular numerator with exact rational arithmetic, not rounded decimals.
- S074 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S073 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split7,1 with exact rational arithmetic, not rounded decimals.
- S072 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S071 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split6,2 with exact rational arithmetic, not rounded decimals.
- S070 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S069 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split6,1 with exact rational arithmetic, not rounded decimals.
- S068 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S067 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split5,3 with exact rational arithmetic, not rounded decimals.
- S066 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S065 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split5,2 with exact rational arithmetic, not rounded decimals.
- S064 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S063 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split5,1 with exact rational arithmetic, not rounded decimals.
- S062 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S061 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,4 with exact rational arithmetic, not rounded decimals.
- S060 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S059 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,3 with exact rational arithmetic, not rounded decimals.
- S058 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S057 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,2 with exact rational arithmetic, not rounded decimals.
- S056 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S055 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,1 with exact rational arithmetic, not rounded decimals.
- S054 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S053 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,5 with exact rational arithmetic, not rounded decimals.
- S052 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S051 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,4 with exact rational arithmetic, not rounded decimals.
- S050 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S049 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,3 with exact rational arithmetic, not rounded decimals.
- S048 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S047 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,2 with exact rational arithmetic, not rounded decimals.
- S046 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S045 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,1 with exact rational arithmetic, not rounded decimals.
- S044 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S043 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,6 with exact rational arithmetic, not rounded decimals.
- S042 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S041 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,5 with exact rational arithmetic, not rounded decimals.
- S040 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S039 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,4 with exact rational arithmetic, not rounded decimals.
- S038 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S037 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,3 with exact rational arithmetic, not rounded decimals.
- S036 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S035 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,2 with exact rational arithmetic, not rounded decimals.
- S034 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S033 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,1 with exact rational arithmetic, not rounded decimals.
- S032 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S031 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,7 with exact rational arithmetic, not rounded decimals.
- S030 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S029 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,6 with exact rational arithmetic, not rounded decimals.
- S028 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S027 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,5 with exact rational arithmetic, not rounded decimals.
- S026 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S025 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,4 with exact rational arithmetic, not rounded decimals.
- S024 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S023 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,3 with exact rational arithmetic, not rounded decimals.
- S022 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S021 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,2 with exact rational arithmetic, not rounded decimals.
- S020 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S019 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,1 with exact rational arithmetic, not rounded decimals.
- S018 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S017 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split0,8 with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S015 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split0,7 with exact rational arithmetic, not rounded decimals.
- S014 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S013 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split0,6 with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S011 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split0,5 with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S009 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split0,4 with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S007 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split0,3 with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S005 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split0,2 with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S003 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split0,1 with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
- S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.





## Result

(8, a^8b^8, [[0, 1, 7, a^7b^8, No], [0, 2, 6, a^6b^8, No], [0, 3, 5, a^5b^8, No], [0, 4, 4, a^4b^8, No], [0, 5, 3, a^3b^8, No], [0, 6, 2, a^2b^8, No], [0, 7, 1, a^1b^8, No], [0, 8, 0, a^0b^8, No], [1, 1, 6, a^7b^8, No], [1, 2, 5, a^6b^8, No], [1, 3, 4, a^5b^8, No], [1, 4, 3, a^4b^8, No], [1, 5, 2, a^3b^8, No], [1, 6, 1, a^2b^8, No], [1, 7, 0, a^1b^8, No], [2, 1, 5, a^7b^8, No], [2, 2, 4, a^6b^8, No], [2, 3, 3, a^5b^8, No], [2, 4, 2, a^4b^8, No], [2, 5, 1, a^3b^8, No], [2, 6, 0, a^2b^8, No], [3, 1, 4, a^7b^8, No], [3, 2, 3, a^6b^8, No], [3, 3, 2, a^5b^8, No], [3, 4, 1, a^4b^8, No], [3, 5, 0, a^3b^8, No], [4, 1, 3, a^7b^8, No], [4, 2, 2, a^6b^8, No], [4, 3, 1, a^5b^8, No], [4, 4, 0, a^4b^8, No], [5, 1, 2, a^7b^8, No], [5, 2, 1, a^6b^8, No], [5, 3, 0, a^5b^8, No], [6, 1, 1, a^7b^8, No], [6, 2, 0, a^6b^8, No], [7, 1, 0, a^7b^8, No]], 36, 36, Yes, every_legal_split_is_defeated_by_i=0)

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: 54ca70608caa6c6e184fe8010e6f96cc47ee7493284519871938e38730d7ed4d
- candidate record SHA-256: 0494a574f2daf3c6182989434cb1abcdffa804fd468b0b695496a4ef80175803
- semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
- question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
- Lean: NOT_RUN · human review: NOT_PERFORMED

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