# Question 33721: One Fixed Affine Transformation Preserves the Same-Filtration Martingale [Return to catalogue page 169](https://ocme.evemisslab.com/en/questions/page/169/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261031-r07-martingale-foundations/#q-20261014-advance-20261031-r07-martingale-0071) 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. R0066-33721 · q-20261014-advance-20261031-r07-martingale-0071 ## Original question and givens Use the declared equiprobable one-step martingale with constant initial=8 and complete terminal values=[7, 9, 8, 8]. The initial information is trivial and the terminal information reveals these outcomes. Fixed scale=-2, shift=23. Give (initial,terminal,scale,shift,transformed_initial,transformed_terminal,martingale?,affine_transform_preserves_same_filtration_martingale). - G001 — Use the declared equiprobable one-step martingale with constant initial=8 and complete terminal values=[7, 9, 8, 8]. The initial information is trivial and the terminal information reveals these outcomes. Fixed scale=-2, shift=23. Give (initial,terminal,scale,shift,transformed_initial,transformed_terminal,martingale?,affine_transform_preserves_same_filtration_martingale). ## 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-affine-stability, [domain, initial, operation, scale, shift, terminal]] After derived = [martdeep-affine-stability, 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 Affine ORIGINAL terminal sum := sum(7, 9, 8, 8) After Affine ORIGINAL terminal sum = 32 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 Affine ORIGINAL initial times outcome count := mul(8, 4) After Affine ORIGINAL initial times outcome count = 32 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 = [8, [7, 9, 8, 8], 32, 32] After derived = [Yes] Conditions: none FIRST qualify the COMPLETE nonempty equiprobable terminal mean=initial,with trivial F0 and terminal-revealing F1.A transformed mean alone cannot repair a failed original process. Step identifiers and dependenciesRule: Martingale_AFFINE_original_mean_premise_before_transform · Uses: G001, S003 S005 · calculationBefore Affine ORIGINAL scale times initial := mul(-2, 8) After Affine ORIGINAL scale times initial = -16 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 Affine ORIGINAL scaled initial plus fixed shift := sum(-16, 23) After Affine ORIGINAL scaled initial plus fixed shift = 7 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 · calculationBefore Affine ORIGINAL terminal product0 := mul(-2, 7) After Affine ORIGINAL terminal product0 = -14 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, S006 S008 · calculationBefore Affine ORIGINAL transformed terminal0 := sum(-14, 23) After Affine ORIGINAL transformed terminal0 = 9 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, S007 S009 · calculationBefore Affine ORIGINAL terminal product1 := mul(-2, 9) After Affine ORIGINAL terminal product1 = -18 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, S008 S010 · calculationBefore Affine ORIGINAL transformed terminal1 := sum(-18, 23) After Affine ORIGINAL transformed terminal1 = 5 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, S009 S011 · calculationBefore Affine ORIGINAL terminal product2 := mul(-2, 8) After Affine ORIGINAL terminal product2 = -16 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, S010 S012 · calculationBefore Affine ORIGINAL transformed terminal2 := sum(-16, 23) After Affine ORIGINAL transformed terminal2 = 7 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, S011 S013 · calculationBefore Affine ORIGINAL terminal product3 := mul(-2, 8) After Affine ORIGINAL terminal product3 = -16 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, S012 S014 · calculationBefore Affine ORIGINAL transformed terminal3 := sum(-16, 23) After Affine ORIGINAL transformed terminal3 = 7 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, S013 S015 · calculationBefore Affine ORIGINAL transformed sum := sum(9, 5, 7, 7) After Affine ORIGINAL transformed sum = 28 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, S014 S016 · calculationBefore Affine ORIGINAL transformed initial times count := mul(7, 4) After Affine ORIGINAL transformed initial times count = 28 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, S015 S017 · impliesBefore source roles = [-2, 23, 8, [7, 9, 8, 8]] After derived = [7, [9, 5, 7, 7], Yes] Conditions: none Apply the SAME fixed scale/shift at both times.Under the SAME filtration,conditional-expectation linearity preserves the martingale.Zero or negative scale is allowed;time-dependent/random coefficients are not supplied.Derived integers are not recapped at the RAW million bound. Step identifiers and dependenciesRule: Martingale_FIXED_same_filtration_affine_linearity · Uses: G001, S016 ## Reverse checks - S017 · 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. - S016 · premise_and_result_audit — Recompute Affine ORIGINAL transformed initial times count with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Affine ORIGINAL transformed sum with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Affine ORIGINAL transformed terminal3 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Affine ORIGINAL terminal product3 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Affine ORIGINAL transformed terminal2 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Affine ORIGINAL terminal product2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Affine ORIGINAL transformed terminal1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Affine ORIGINAL terminal product1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Affine ORIGINAL transformed terminal0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Affine ORIGINAL terminal product0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Affine ORIGINAL scaled initial plus fixed shift with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Affine ORIGINAL scale times initial 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 Affine ORIGINAL initial times outcome count with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Affine ORIGINAL terminal sum 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 (8, [7, 9, 8, 8], -2, 23, 7, [9, 5, 7, 7], Yes, affine_transform_preserves_same_filtration_martingale) 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: 900cee38f93b6ae01f776a5826605081bc5890933c23e3b316b58ffd6a66e94b - candidate record SHA-256: 5778e2f9cc7c554b00fa330fc269219dda223d130ca0337d9b693614e9043f2b - semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267 - question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261031-r07-martingale-0071.json)