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R0009-05310 · q-20261008-advance-20261012-r02-counting-principle-0060
Ordered second-stage branch counts=[0, 1], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).
G001 — Ordered second-stage branch counts=[0, 1], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).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
actual branch total := sum(0, 1)
After
actual branch total = 1
Conditions: none
Add the actual widths of the conditional branches.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
assumed common-width total := mul(2, 1)
After
assumed common-width total = 2
Conditions: none
Separately compute n times candidate width h without yet asserting equal branch widths.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
branch0 width=0;candidate h=1
After
matches=No
Conditions: none
Audit every branch width;matching aggregate totals can still hide unequal widths.
Rule: each_branch_common_width_audit · Uses: G001, S003
Before
branch1 width=1;candidate h=1
After
matches=Yes
Conditions: none
Audit every branch width;matching aggregate totals can still hide unequal widths.
Rule: each_branch_common_width_audit · Uses: G001, S004
Before
assumed minus actual total := sub(2, 1)
After
assumed minus actual total = 1
Conditions: none
The signed defect is assumed n·h minus the actual total.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
branch comparisons=[No, Yes]
After
all equal h=No
Conditions: none
With no first-stage branches,the width universal is vacuously true but the actual count is0;this differs from a zero-stage full product's single empty sequence.
Rule: uniform_width_not_aggregate_equality · Uses: G001, S006
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 — Recompute assumed minus actual total with exact rational arithmetic, not rounded decimals.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 — Recompute assumed common-width total with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute actual branch total with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(1, 2, No, 1)
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: 427dd8cfb809d915a892950982425e9d96883231d56c732f8f1bcd1f8089b444candidate record SHA-256: 4e16ef21dba4060a7c7b6258408d00f80db9c1c4feffef24f9e36460f38501b4semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462Lean: NOT_RUN · human review: NOT_PERFORMED