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R0009-05372 · q-20261008-advance-20261012-r02-counting-principle-0047
Ordered stage sizes=[3, 1]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.
G001 — Ordered stage sizes=[3, 1]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.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, 1]
After
initial accumulator=1
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 := mul(1, 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 := mul(3, 1)
After
prefix count2 = 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, S004
Before
prefix through position1;new size=1
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, 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.(3, [3, 3])
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: 3bfc5e8d907a2b56a5c257fe666cf1675e3a7ba7799e05d15a091a1e980c97bccandidate record SHA-256: a83b44262f61b1513a9dadc0352f49279ea80307608a63346393e1bd01208e3bsemantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462Lean: NOT_RUN · human review: NOT_PERFORMED