# Question 5316: Actual Branch Widths versus a Candidate Width [Return to catalogue page 27](https://ocme.evemisslab.com/en/questions/page/27/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261012-r02-counting-principle-foundations/#q-20261008-advance-20261012-r02-counting-principle-0066) 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. R0009-05316 · q-20261008-advance-20261012-r02-counting-principle-0066 ## Original question and givens Ordered second-stage branch counts=[6, 7], 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=[6, 7], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total). ## 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-counting-principle](https://ocme.evemisslab.com/en/mko/mko-counting-principle/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: finite_counting_protocol_scope · Uses: G001 S002 · calculationBefore actual branch total := sum(6, 7) After actual branch total = 13 Conditions: none Add the actual widths of the conditional branches. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore branch0 width=6;candidate h=1 After matches=No Conditions: none Audit every branch width;matching aggregate totals can still hide unequal widths. Step identifiers and dependenciesRule: each_branch_common_width_audit · Uses: G001, S003 S005 · impliesBefore branch1 width=7;candidate h=1 After matches=No Conditions: none Audit every branch width;matching aggregate totals can still hide unequal widths. Step identifiers and dependenciesRule: each_branch_common_width_audit · Uses: G001, S004 S006 · calculationBefore assumed minus actual total := sub(2, 13) After assumed minus actual total = -11 Conditions: none The signed defect is assumed n·h minus the actual total. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore branch comparisons=[No, No] 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. Step identifiers and dependenciesRule: uniform_width_not_aggregate_equality · Uses: G001, S006 ## Reverse checks - 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. ## Result (13, 2, No, -11) 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: 764026a0e6fb03d855fef58a56bc1b98643563c1a1342e3e9ac70a2213dec5b7 - candidate record SHA-256: 70ba3112e2f98460605dcd1e545bf472b4a22006e7388de1288bf4952e557eab - semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56 - question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r02-counting-principle-0066.json)