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Question 5327: Disjoint Tagged Choices and Prefix Counts

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0009-05327 · q-20261008-advance-20261012-r02-counting-principle-0002

Original question and givens

Mutually exclusive tagged categories have ordered sizes=[0]. Give (total choices, each prefix’s total), retaining an empty list.

  • G001 — Mutually exclusive tagged categories have ordered sizes=[0]. Give (total choices, each prefix’s total), retaining an empty list.

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.

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: finite_counting_protocol_scope · Uses: G001

S002 · implies

Before

ordered sizes=[0]

After

initial accumulator=0

Conditions: none

The empty disjoint sum is0;the empty full product is1,representing one empty choice sequence.

Step identifiers and dependencies

Rule: counting_prefix_initial · Uses: G001, S001

S003 · calculation

Before

prefix count1 := sum(0, 0)

After

prefix count1 = 0

Conditions: none

Multiply in a stage of the declared full product,or add a disjoint tagged class,retaining every prefix.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

prefix through position0;new size=0

After

prefix total=0

Conditions: none

A full product stays0 after a zero factor;the disjoint sum continues adding in order.Do not interchange these protocols.

Step identifiers and dependencies

Rule: counting_prefix_update · Uses: G001, S003

Reverse checks

  • 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.

Result

(0, [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: 519dbd6b92defc1446fe79ce1b70fb863b8d171d80693373099bf9251f30ee43
  • candidate record SHA-256: 12798eb60e76dfae8b357dcda169028f665a82e48701f3bf4475428fbeb1b5ec
  • semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
  • question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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