Question 29182: Merge All Masses with a Common Image

On this page

Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

← Question derivation index · Original question batch

R0057-29182 · q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0082

Original question and givens

Use the explicit finite discrete outcome list=[6, 7, 9] 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=[0, 1, 0] 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=[6, 7, 9] 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=[0, 1, 0] 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

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001

S002 · implies

Before

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 dependencies

Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S001

S003 · implies

Before

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 dependencies

Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S003

S005 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

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 dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [PMF ORIGINAL normalization running sum1, sum, [1/4, 1/4]]

After

derived = [[4], [2, 4], 1/2]

Conditions: none

Step identifiers and dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S007

S009 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [PMF ORIGINAL normalization running sum2, sum, [1/2, 1/2]]

After

derived = [[2], [2, 2], 1]

Conditions: none

Step identifiers and dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S009

S011 · implies

Before

source roles = [[6, 7, 9], [1/4, 1/4, 1/2]]

After

derived = [[[6, 1/4], [7, 1/4], [9, 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 dependencies

Rule: PMF_ALL_original_outcome_mass_alignment_NONNEG_AND_EXACTsum1 · Uses: G001, S010

S012 · calculation

Before

PMF ORIGINAL pushforward slot0 image0 := sum(0, 1/4)

After

PMF ORIGINAL pushforward slot0 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

source roles = [PMF ORIGINAL pushforward slot0 image0, sum, [0, 1/4]]

After

derived = [[1], [1, 4], 1/4]

Conditions: none

Step identifiers and dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S012

S014 · calculation

Before

PMF ORIGINAL pushforward slot1 image1 := sum(0, 1/4)

After

PMF ORIGINAL pushforward slot1 image1 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

source roles = [PMF ORIGINAL pushforward slot1 image1, sum, [0, 1/4]]

After

derived = [[1], [1, 4], 1/4]

Conditions: none

Step identifiers and dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S014

S016 · calculation

Before

PMF ORIGINAL pushforward slot2 image0 := sum(1/4, 1/2)

After

PMF ORIGINAL pushforward slot2 image0 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [PMF ORIGINAL pushforward slot2 image0, sum, [1/4, 1/2]]

After

derived = [[2], [3, 4], 3/4]

Conditions: none

Step identifiers and dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S016

S018 · implies

Before

source roles = [[6, 7, 9], [1/4, 1/4, 1/2], [0, 1, 0]]

After

derived = [[[0, 3/4], [1, 1/4]]]

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 dependencies

Rule: 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 image0 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 image1 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 image0 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

([6, 7, 9], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])

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: b6d77a93f942dd0a9414a74b5c68452a1bde23368c5f52d2e77aa8d415039d00
candidate record SHA-256: 37d91bdb7d664131870d53ff92abfa3041055e321a4fe140be571097b9386009
semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6
question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON