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R0009-05349 · q-20261008-advance-20261012-r02-counting-principle-0024
Mutually exclusive tagged categories have ordered sizes=[3, 4]. Give (total choices, each prefix’s total), retaining an empty list.
G001 — Mutually exclusive tagged categories have ordered sizes=[3, 4]. Give (total choices, each prefix’s total), retaining an empty list.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.
Before
declared finite tagged choices or full-product protocol
After
combinatorial counts,not probabilities or independence
Conditions: none
Addition uses disjoint tags and multiplication uses the declared full cross-stage protocol;neither is a probability-independence assumption.
Rule: finite_counting_protocol_scope · Uses: G001
Before
ordered sizes=[3, 4]
After
initial accumulator=0
Conditions: none
The empty disjoint sum is0;the empty full product is1,representing one empty choice sequence.
Rule: counting_prefix_initial · Uses: G001, S001
Before
prefix count1 := sum(0, 3)
After
prefix count1 = 3
Conditions: none
Multiply in a stage of the declared full product,or add a disjoint tagged class,retaining every prefix.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
prefix through position0;new size=3
After
prefix total=3
Conditions: none
A full product stays0 after a zero factor;the disjoint sum continues adding in order.Do not interchange these protocols.
Rule: counting_prefix_update · Uses: G001, S003
Before
prefix count2 := sum(3, 4)
After
prefix count2 = 7
Conditions: none
Multiply in a stage of the declared full product,or add a disjoint tagged class,retaining every prefix.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
prefix through position1;new size=4
After
prefix total=7
Conditions: none
A full product stays0 after a zero factor;the disjoint sum continues adding in order.Do not interchange these protocols.
Rule: counting_prefix_update · Uses: G001, S005
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 — Recompute prefix count2 with exact rational arithmetic, not rounded decimals.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 — Recompute prefix count1 with exact rational arithmetic, not rounded decimals.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.(7, [3, 7])
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: d08e4dfd9558e76ae89af4c2a6b00ee85f86bae931492463717892fbebd2e655candidate record SHA-256: d21f8928845a81b0a400c5afe1531bc09e9991a4a8830eb34050871b0e52171esemantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462Lean: NOT_RUN · human review: NOT_PERFORMED