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R0020-10729 · q-20261013-advance-20261023-r06-discrete-stochastic-process-0054
Use one common finite probability space specified by the complete distinct equal-horizon path list=[[3, 3, 4], [3, 4, 4], [4, 4, 3]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0, event input=[4]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.
G001 — Use one common finite probability space specified by the complete distinct equal-horizon path list=[[3, 3, 4], [3, 4, 4], [4, 4, 3]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0, event input=[4]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.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
original support index=0;path=[3, 3, 4];raw mass=[1, 4]
After
aligned reduced mass=1/4
Conditions: none
Complete paths are distinct and share one positive horizon.Masses remain paired to original support indices.Reduce fractions only:do not sort paths or divide by a total to repair an unqualified candidate.
Rule: path_mass_original_support_pairing · Uses: G001
Before
original support index=1;path=[3, 4, 4];raw mass=[1, 4]
After
aligned reduced mass=1/4
Conditions: none
Complete paths are distinct and share one positive horizon.Masses remain paired to original support indices.Reduce fractions only:do not sort paths or divide by a total to repair an unqualified candidate.
Rule: path_mass_original_support_pairing · Uses: G001, S001
Before
original support index=2;path=[4, 4, 3];raw mass=[1, 2]
After
aligned reduced mass=1/2
Conditions: none
Complete paths are distinct and share one positive horizon.Masses remain paired to original support indices.Reduce fractions only:do not sort paths or divide by a total to repair an unqualified candidate.
Rule: path_mass_original_support_pairing · Uses: G001, S002
Before
whole path-law mass sum := sum(1/4, 1/4, 1/2)
After
whole path-law mass sum = 1
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
horizon=3;normalized aligned masses=[1/4, 1/4, 1/2]
After
all nonnegative=Yes;total=1;qualified joint law=Yes
Conditions: none
The qualification task may retain negative or unnormalized masses to report failure.Marginal/event/joint queries require the whole law nonnegative with total exactly1.A finite horizon establishes neither Markov behavior nor stationarity.
Rule: whole_finite_joint_law_qualification · Uses: G001, S004
Before
support index=0;time=0;state=3;aligned mass=1/4;event=[4]
After
event contains state=No;contribution=0
Conditions: none
Normalize the event as a set while retaining labels outside support.Each path contributes its original mass once;outside-support event labels add no invented mass.
Rule: time_event_each_original_path_once · Uses: G001, S005
Before
support index=1;time=0;state=3;aligned mass=1/4;event=[4]
After
event contains state=No;contribution=0
Conditions: none
Normalize the event as a set while retaining labels outside support.Each path contributes its original mass once;outside-support event labels add no invented mass.
Rule: time_event_each_original_path_once · Uses: G001, S006
Before
support index=2;time=0;state=4;aligned mass=1/2;event=[4]
After
event contains state=Yes;contribution=1/2
Conditions: none
Normalize the event as a set while retaining labels outside support.Each path contributes its original mass once;outside-support event labels add no invented mass.
Rule: time_event_each_original_path_once · Uses: G001, S007
Before
complete time-event probability := sum(0, 0, 1/2)
After
complete time-event probability = 1/2
Conditions: none
Compute exactly with this step's original roles and operands,without floating approximation or reordering source data.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
requested time=0;raw event=[4];normalized event=[4]
After
event probability=1/2
Conditions: none
This is a time-event under the same complete path law,not a joint law reconstructed by assuming independence from marginals.
Rule: time_event_not_joint_inference · Uses: G001, S009
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Recompute complete time-event probability with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Recompute whole path-law mass sum with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(0, [4], 1/2)
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: 64948723de474cec64b97a848fbd2eeb2b7d0332336c685b3cab6eee1f8e3e55candidate record SHA-256: ed7affe6fe6e195034179941daa78d42effb5e5e7827048d7732a092a95eb5ccsemantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598fquestion-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4Lean: NOT_RUN · human review: NOT_PERFORMED