# Question 10760: Aggregate a Marginal from the Complete Joint Path Law [Return to catalogue page 54](https://ocme.evemisslab.com/en/questions/page/54/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations/#q-20261013-advance-20261023-r06-discrete-stochastic-process-0035) 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. R0020-10760 · q-20261013-advance-20261023-r06-discrete-stochastic-process-0035 ## Original question and givens Use one common finite probability space specified by the complete distinct equal-horizon path list=[[9, 9, 10], [9, 10, 10], [10, 10, 9]] 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. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero. - G001 — Use one common finite probability space specified by the complete distinct equal-horizon path list=[[9, 9, 10], [9, 10, 10], [10, 10, 9]] 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. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated 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-discrete-stochastic-process](https://ocme.evemisslab.com/en/mko/mko-discrete-stochastic-process/) ## Step-by-step derivation S001 · impliesBefore original support index=0;path=[9, 9, 10];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. Step identifiers and dependenciesRule: path_mass_original_support_pairing · Uses: G001 S002 · impliesBefore original support index=1;path=[9, 10, 10];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. Step identifiers and dependenciesRule: path_mass_original_support_pairing · Uses: G001, S001 S003 · impliesBefore original support index=2;path=[10, 10, 9];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. Step identifiers and dependenciesRule: path_mass_original_support_pairing · Uses: G001, S002 S004 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore 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. Step identifiers and dependenciesRule: whole_finite_joint_law_qualification · Uses: G001, S004 S006 · calculationBefore marginal path0 bucket update := sum(0, 1/4) After marginal path0 bucket update = 1/4 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore support index=0;time=0;state=9;original mass=1/4;previous bucket=0 After updated state mass=1/4 Conditions: none Aggregate paths with the same state at this time without reassigning masses.Retain a listed state's bucket even when its total is0. Step identifiers and dependenciesRule: marginal_all_paths_including_zero_mass · Uses: G001, S006 S008 · calculationBefore marginal path1 bucket update := sum(1/4, 1/4) After marginal path1 bucket update = 1/2 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore support index=1;time=0;state=9;original mass=1/4;previous bucket=1/4 After updated state mass=1/2 Conditions: none Aggregate paths with the same state at this time without reassigning masses.Retain a listed state's bucket even when its total is0. Step identifiers and dependenciesRule: marginal_all_paths_including_zero_mass · Uses: G001, S008 S010 · calculationBefore marginal path2 bucket update := sum(0, 1/2) After marginal path2 bucket update = 1/2 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore support index=2;time=0;state=10;original mass=1/2;previous bucket=0 After updated state mass=1/2 Conditions: none Aggregate paths with the same state at this time without reassigning masses.Retain a listed state's bucket even when its total is0. Step identifiers and dependenciesRule: marginal_all_paths_including_zero_mass · Uses: G001, S010 S012 · impliesBefore requested time=0;all state buckets=[[9, 1/2], [10, 1/2]] After marginal only;no joint or Markov conclusion Conditions: none Order marginal states numerically.A marginal table determines neither two-time dependence,the joint law nor Markov behavior. Step identifiers and dependenciesRule: marginal_not_joint_or_Markov_certificate · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute marginal path2 bucket update with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute marginal path1 bucket update with exact rational arithmetic, not rounded decimals. - 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 marginal path0 bucket update 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 — 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. ## Result (0, [[9, 1/2], [10, 1/2]]) 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: 7986c577fc82015c0f913b4b9768b44cbac6041a36113640566b7e17485df1a7 - candidate record SHA-256: c810d0b0c595b319db9bc5d5ffc8c6d57618bad08a11827b5b9f20275b17a8c9 - semantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f - question-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261023-r06-discrete-stochastic-process-0035.json)