# Question 29184: Merge All Masses with a Common Image [Return to catalogue page 146](https://ocme.evemisslab.com/en/questions/page/146/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations/#q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0084) 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. R0057-29184 · q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0084 ## Original question and givens Use the explicit finite discrete outcome list=[8, 9, 11] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero. - G001 — Use the explicit finite discrete outcome list=[8, 9, 11] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero. ## 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-finite-discrete-probability-distribution](https://ocme.evemisslab.com/en/mko/mko-finite-discrete-probability-distribution/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-str22-distribution-pushforward, distribution-pushforward] After derived = [[outcomes, masses, images, pushforward_rows]] Conditions: none Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning. Step identifiers and dependenciesRule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001 S002 · impliesBefore source roles = [PMF raw mass slot0, [1, 4]] After derived = [1/4, 1, 4] Conditions: none A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds. Step identifiers and dependenciesRule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S001 S003 · impliesBefore source roles = [PMF raw mass slot1, [1, 4]] After derived = [1/4, 1, 4] Conditions: none A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds. Step identifiers and dependenciesRule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S002 S004 · impliesBefore source roles = [PMF raw mass slot2, [1, 2]] After derived = [1/2, 1, 2] Conditions: none A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds. Step identifiers and dependenciesRule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S003 S005 · calculationBefore PMF ORIGINAL normalization running sum0 := sum(0, 1/4) After PMF ORIGINAL normalization running sum0 = 1/4 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [PMF ORIGINAL normalization running sum0, sum, [0, 1/4]] After derived = [[1], [1, 4], 1/4] Conditions: none SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate. Step identifiers and dependenciesRule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S005 S007 · calculationBefore PMF ORIGINAL normalization running sum1 := sum(1/4, 1/4) After PMF ORIGINAL normalization running sum1 = 1/2 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [PMF ORIGINAL normalization running sum1, sum, [1/4, 1/4]] After derived = [[4], [2, 4], 1/2] Conditions: none SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate. Step identifiers and dependenciesRule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S007 S009 · calculationBefore PMF ORIGINAL normalization running sum2 := sum(1/2, 1/2) After PMF ORIGINAL normalization running sum2 = 1 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [PMF ORIGINAL normalization running sum2, sum, [1/2, 1/2]] After derived = [[2], [2, 2], 1] Conditions: none SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate. Step identifiers and dependenciesRule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S009 S011 · impliesBefore source roles = [[8, 9, 11], [1/4, 1/4, 1/2]] After derived = [[[8, 1/4], [9, 1/4], [11, 1/2]], Yes, 1, Yes, Yes] Conditions: none Preserve ORIGINAL outcome-mass slot alignment and sum every mass in original order.Nonnegativity and total1 are separate obligations.Qualification may legally be No;do not renormalize or clip negative masses. Step identifiers and dependenciesRule: PMF_ALL_original_outcome_mass_alignment_NONNEG_AND_EXACTsum1 · Uses: G001, S010 S012 · calculationBefore PMF ORIGINAL pushforward slot0 image2 := sum(0, 1/4) After PMF ORIGINAL pushforward slot0 image2 = 1/4 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore source roles = [PMF ORIGINAL pushforward slot0 image2, sum, [0, 1/4]] After derived = [[1], [1, 4], 1/4] Conditions: none SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate. Step identifiers and dependenciesRule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S012 S014 · calculationBefore PMF ORIGINAL pushforward slot1 image0 := sum(0, 1/4) After PMF ORIGINAL pushforward slot1 image0 = 1/4 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore source roles = [PMF ORIGINAL pushforward slot1 image0, sum, [0, 1/4]] After derived = [[1], [1, 4], 1/4] Conditions: none SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate. Step identifiers and dependenciesRule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S014 S016 · calculationBefore PMF ORIGINAL pushforward slot2 image2 := sum(1/4, 1/2) After PMF ORIGINAL pushforward slot2 image2 = 3/4 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [PMF ORIGINAL pushforward slot2 image2, sum, [1/4, 1/2]] After derived = [[2], [3, 4], 3/4] Conditions: none SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate. Step identifiers and dependenciesRule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S016 S018 · impliesBefore source roles = [[8, 9, 11], [1/4, 1/4, 1/2], [2, 0, 2]] After derived = [[[0, 1/4], [2, 3/4]]] Conditions: none P(f(Y)=z)=sum_{i:f(y_i)=z}p_i.Add each ORIGINAL slot mass exactly once into its COMPLETE fibre,then sort image rows numerically.Retain a row even if its combined mass is0;do not alter pairing or keep only positive support. Step identifiers and dependenciesRule: PMF_pushforward_ALL_fibres_ORIGINAL_alignment_ZERO_rows_retained · Uses: G001, S017 ## Reverse checks - S018 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S017 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S016 · premise_and_result_audit — Recompute PMF ORIGINAL pushforward slot2 image2 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S014 · premise_and_result_audit — Recompute PMF ORIGINAL pushforward slot1 image0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S012 · premise_and_result_audit — Recompute PMF ORIGINAL pushforward slot0 image2 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S010 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S009 · premise_and_result_audit — Recompute PMF ORIGINAL normalization running sum2 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S007 · premise_and_result_audit — Recompute PMF ORIGINAL normalization running sum1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S005 · premise_and_result_audit — Recompute PMF ORIGINAL normalization running sum0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S003 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S002 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S001 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. ## Result ([8, 9, 11], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]]) 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: 1a26215be7c5054ddade175f1cb09af9759047bc14d1a3bf4209141998db8564 - candidate record SHA-256: bab46a40bc68b477bdbfcbc03567961dce5613891fba3cc7fd1d621f01df155c - semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6 - question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0084.json)