# Question 33730: Constant Total Expectations Can Fail Conditional Martingale Steps

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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.

R0066-33730 · q-20261014-advance-20261031-r07-martingale-0005



## Original question and givens

Use two equiprobable outcomes. X0=0, X1=[a,−a], X2=[−a,a] with amplitude a=5>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=5>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.


- [mko-martingale](https://ocme.evemisslab.com/en/mko/mko-martingale/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-mod31-martdeep-constant-expectation-counterexample, [amplitude, domain, operation]]

After

derived = [martdeep-constant-expectation-counterexample, Exact_finite_martingale_construction]

Conditions: none

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.

Step identifiers and dependenciesRule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001





S002 · calculationBefore

Martingale ORIGINAL X1 total := sum(5, -5)

After

Martingale ORIGINAL X1 total = 0

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

Martingale ORIGINAL X2 total := sum(-5, 5)

After

Martingale ORIGINAL X2 total = 0

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

source roles = [5]

After

derived = [[omega0, omega1], [F0=trivial, F1=F2=full], 0, [5, -5], [-5, 5], [0, 0, 0]]

Conditions: none

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.

Step identifiers and dependenciesRule: Martingale_EXPLICIT_equally_likely_space_filtration_adapted · Uses: G001, S003





S005 · calculationBefore

Martingale ORIGINAL E X2 given full information minus X1 outcome0 := sub(-5, 5)

After

Martingale ORIGINAL E X2 given full information minus X1 outcome0 = -10

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

Martingale ORIGINAL E X2 given full information minus X1 outcome1 := sub(5, -5)

After

Martingale ORIGINAL E X2 given full information minus X1 outcome1 = 10

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

source roles = [[5, -5], [-5, 5]]

After

derived = [E[X2|F1]=X2, [-10, 10], No]

Conditions: none

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.

Step identifiers and dependenciesRule: Martingale_COUNTEREXAMPLE_conditional_failure_despite_constant_means · Uses: G001, S006









## Reverse checks


- 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, [5, -5], [-5, 5], [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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 724e49dfa2d6432b7537eb6d2578bb7a81eb6c33e25a7e57bd041c6c2af772e4
- candidate record SHA-256: f9a0165ca313b00b30c1a98d5b473f2c2eef0c867c1b7344f24ba471b7300eb9
- semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
- question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
- Lean: NOT_RUN · human review: NOT_PERFORMED

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