Use two equiprobable outcomes. X0=0, X1=[a,−a], X2=[−a,a] with amplitude a=1>0. F0 is trivial and F1=F2 is the full information partition revealed by X1. Give (X0,X1,X2,expectations,martingale?,constant_expectations_do_not_imply_conditional_martingale_steps).
G001 — Use two equiprobable outcomes. X0=0, X1=[a,−a], X2=[−a,a] with amplitude a=1>0. F0 is trivial and F1=F2 is the full information partition revealed by X1. Give (X0,X1,X2,expectations,martingale?,constant_expectations_do_not_imply_conditional_martingale_steps).
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.
THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged.
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Use TWO equiprobable outcomes,F0 trivial and F1=F2 full.The three-time process is adapted and integrable;all total expectations are0.That is not yet the conditional martingale equation.
Martingale ORIGINAL E X2 given full information minus X1 outcome0 := sub(-1, 1)
After
Martingale ORIGINAL E X2 given full information minus X1 outcome0 = -2
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
Martingale ORIGINAL E X2 given full information minus X1 outcome1 := sub(1, -1)
After
Martingale ORIGINAL E X2 given full information minus X1 outcome1 = 2
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
F1 reveals the outcome,so E[X2|F1]=X2.The difference from X1 is nonzero on EACH outcome,so the second step fails.This refutes sufficiency of constant expectations,not every constant-expectation process.
S007 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S006 · premise_and_result_audit — Recompute Martingale ORIGINAL E X2 given full information minus X1 outcome1 with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute Martingale ORIGINAL E X2 given full information minus X1 outcome0 with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S003 · premise_and_result_audit — Recompute Martingale ORIGINAL X2 total with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute Martingale ORIGINAL X1 total with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
Result
(0, [1, -1], [-1, 1], [0, 0, 0], No, constant_expectations_do_not_imply_conditional_martingale_steps)
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.