# Question 33784: A Declared Fair Independent-Increment Model Supplies a Specific 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-0084) 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. R0066-33784 · q-20261014-advance-20261031-r07-martingale-0084 ## Original question and givens Use steps=9 independent equiprobable signs, each increment ±scale with scale=1>0, on the full finite sign-product space with its natural revealed-sign filtration. Give (steps,scale,support,multiplicities,total_paths,martingale?,independent_fair_signed_increments_have_conditional_mean_zero). List support by increasing number of positive signs; retain total_paths as exact 2^steps text. - G001 — Use steps=9 independent equiprobable signs, each increment ±scale with scale=1>0, on the full finite sign-product space with its natural revealed-sign filtration. Give (steps,scale,support,multiplicities,total_paths,martingale?,independent_fair_signed_increments_have_conditional_mean_zero). List support by increasing number of positive signs; retain total_paths as exact 2^steps text. ## 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-fair-partial-sum, [domain, operation, scale, steps]] After derived = [martdeep-fair-partial-sum, 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 Fair ORIGINAL two positive signs k=0 := mul(2, 0) After Fair ORIGINAL two positive signs k=0 = 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 Fair ORIGINAL positive minus negative count k=0 := sub(0, 9) After Fair ORIGINAL positive minus negative count k=0 = -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, S002 S004 · calculationBefore Fair ORIGINAL scale times signed count k=0 := mul(1, -9) After Fair ORIGINAL scale times signed count k=0 = -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, S003 S005 · impliesBefore source roles = [9, 1, 0] After derived = [-9, 1] Conditions: none k positive and n−k negative signs give scale*(2k−n),with choose(n,k)choices of positive positions.Retain EVERY k=0..n support value and multiplicity. Step identifiers and dependenciesRule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S004 S006 · calculationBefore Fair ORIGINAL binomial previous times remaining k=1 := mul(1, 9) After Fair ORIGINAL binomial previous times remaining k=1 = 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, S005 S007 · calculationBefore Fair ORIGINAL binomial recurrence division k=1 := div(9, 1) After Fair ORIGINAL binomial recurrence division k=1 = 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, S006 S008 · calculationBefore Fair ORIGINAL two positive signs k=1 := mul(2, 1) After Fair ORIGINAL two positive signs k=1 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Fair ORIGINAL positive minus negative count k=1 := sub(2, 9) After Fair ORIGINAL positive minus negative count k=1 = -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, S008 S010 · calculationBefore Fair ORIGINAL scale times signed count k=1 := mul(1, -7) After Fair ORIGINAL scale times signed count k=1 = -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, S009 S011 · impliesBefore source roles = [9, 1, 1] After derived = [-7, 9] Conditions: none k positive and n−k negative signs give scale*(2k−n),with choose(n,k)choices of positive positions.Retain EVERY k=0..n support value and multiplicity. Step identifiers and dependenciesRule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S010 S012 · calculationBefore Fair ORIGINAL binomial previous times remaining k=2 := mul(9, 8) After Fair ORIGINAL binomial previous times remaining k=2 = 72 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 Fair ORIGINAL binomial recurrence division k=2 := div(72, 2) After Fair ORIGINAL binomial recurrence division k=2 = 36 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 Fair ORIGINAL two positive signs k=2 := mul(2, 2) After Fair ORIGINAL two positive signs k=2 = 4 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 Fair ORIGINAL positive minus negative count k=2 := sub(4, 9) After Fair ORIGINAL positive minus negative count k=2 = -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, S014 S016 · calculationBefore Fair ORIGINAL scale times signed count k=2 := mul(1, -5) After Fair ORIGINAL scale times signed count k=2 = -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, S015 S017 · impliesBefore source roles = [9, 1, 2] After derived = [-5, 36] Conditions: none k positive and n−k negative signs give scale*(2k−n),with choose(n,k)choices of positive positions.Retain EVERY k=0..n support value and multiplicity. Step identifiers and dependenciesRule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S016 S018 · calculationBefore Fair ORIGINAL binomial previous times remaining k=3 := mul(36, 7) After Fair ORIGINAL binomial previous times remaining k=3 = 252 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, S017 S019 · calculationBefore Fair ORIGINAL binomial recurrence division k=3 := div(252, 3) After Fair ORIGINAL binomial recurrence division k=3 = 84 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, S018 S020 · calculationBefore Fair ORIGINAL two positive signs k=3 := mul(2, 3) After Fair ORIGINAL two positive signs k=3 = 6 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, S019 S021 · calculationBefore Fair ORIGINAL positive minus negative count k=3 := sub(6, 9) After Fair ORIGINAL positive minus negative count k=3 = -3 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, S020 S022 · calculationBefore Fair ORIGINAL scale times signed count k=3 := mul(1, -3) After Fair ORIGINAL scale times signed count k=3 = -3 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, S021 S023 · impliesBefore source roles = [9, 1, 3] After derived = [-3, 84] Conditions: none k positive and n−k negative signs give scale*(2k−n),with choose(n,k)choices of positive positions.Retain EVERY k=0..n support value and multiplicity. Step identifiers and dependenciesRule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S022 S024 · calculationBefore Fair ORIGINAL binomial previous times remaining k=4 := mul(84, 6) After Fair ORIGINAL binomial previous times remaining k=4 = 504 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, S023 S025 · calculationBefore Fair ORIGINAL binomial recurrence division k=4 := div(504, 4) After Fair ORIGINAL binomial recurrence division k=4 = 126 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, S024 S026 · calculationBefore Fair ORIGINAL two positive signs k=4 := mul(2, 4) After Fair ORIGINAL two positive signs k=4 = 8 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, S025 S027 · calculationBefore Fair ORIGINAL positive minus negative count k=4 := sub(8, 9) After Fair ORIGINAL positive minus negative count k=4 = -1 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, S026 S028 · calculationBefore Fair ORIGINAL scale times signed count k=4 := mul(1, -1) After Fair ORIGINAL scale times signed count k=4 = -1 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, S027 S029 · impliesBefore source roles = [9, 1, 4] After derived = [-1, 126] Conditions: none k positive and n−k negative signs give scale*(2k−n),with choose(n,k)choices of positive positions.Retain EVERY k=0..n support value and multiplicity. Step identifiers and dependenciesRule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S028 S030 · calculationBefore Fair ORIGINAL binomial previous times remaining k=5 := mul(126, 5) After Fair ORIGINAL binomial previous times remaining k=5 = 630 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, S029 S031 · calculationBefore Fair ORIGINAL binomial recurrence division k=5 := div(630, 5) After Fair ORIGINAL binomial recurrence division k=5 = 126 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, S030 S032 · calculationBefore Fair ORIGINAL two positive signs k=5 := mul(2, 5) After Fair ORIGINAL two positive signs k=5 = 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, S031 S033 · calculationBefore Fair ORIGINAL positive minus negative count k=5 := sub(10, 9) After Fair ORIGINAL positive minus negative count k=5 = 1 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, S032 S034 · calculationBefore Fair ORIGINAL scale times signed count k=5 := mul(1, 1) After Fair ORIGINAL scale times signed count k=5 = 1 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, S033 S035 · impliesBefore source roles = [9, 1, 5] After derived = [1, 126] Conditions: none k positive and n−k negative signs give scale*(2k−n),with choose(n,k)choices of positive positions.Retain EVERY k=0..n support value and multiplicity. Step identifiers and dependenciesRule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S034 S036 · calculationBefore Fair ORIGINAL binomial previous times remaining k=6 := mul(126, 4) After Fair ORIGINAL binomial previous times remaining k=6 = 504 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, S035 S037 · calculationBefore Fair ORIGINAL binomial recurrence division k=6 := div(504, 6) After Fair ORIGINAL binomial recurrence division k=6 = 84 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, S036 S038 · calculationBefore Fair ORIGINAL two positive signs k=6 := mul(2, 6) After Fair ORIGINAL two positive signs k=6 = 12 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, S037 S039 · calculationBefore Fair ORIGINAL positive minus negative count k=6 := sub(12, 9) After Fair ORIGINAL positive minus negative count k=6 = 3 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, S038 S040 · calculationBefore Fair ORIGINAL scale times signed count k=6 := mul(1, 3) After Fair ORIGINAL scale times signed count k=6 = 3 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, S039 S041 · impliesBefore source roles = [9, 1, 6] After derived = [3, 84] Conditions: none k positive and n−k negative signs give scale*(2k−n),with choose(n,k)choices of positive positions.Retain EVERY k=0..n support value and multiplicity. Step identifiers and dependenciesRule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S040 S042 · calculationBefore Fair ORIGINAL binomial previous times remaining k=7 := mul(84, 3) After Fair ORIGINAL binomial previous times remaining k=7 = 252 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, S041 S043 · calculationBefore Fair ORIGINAL binomial recurrence division k=7 := div(252, 7) After Fair ORIGINAL binomial recurrence division k=7 = 36 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, S042 S044 · calculationBefore Fair ORIGINAL two positive signs k=7 := mul(2, 7) After Fair ORIGINAL two positive signs k=7 = 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, S043 S045 · calculationBefore Fair ORIGINAL positive minus negative count k=7 := sub(14, 9) After Fair ORIGINAL positive minus negative count k=7 = 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, S044 S046 · calculationBefore Fair ORIGINAL scale times signed count k=7 := mul(1, 5) After Fair ORIGINAL scale times signed count k=7 = 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, S045 S047 · impliesBefore source roles = [9, 1, 7] After derived = [5, 36] Conditions: none k positive and n−k negative signs give scale*(2k−n),with choose(n,k)choices of positive positions.Retain EVERY k=0..n support value and multiplicity. Step identifiers and dependenciesRule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S046 S048 · calculationBefore Fair ORIGINAL binomial previous times remaining k=8 := mul(36, 2) After Fair ORIGINAL binomial previous times remaining k=8 = 72 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, S047 S049 · calculationBefore Fair ORIGINAL binomial recurrence division k=8 := div(72, 8) After Fair ORIGINAL binomial recurrence division k=8 = 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, S048 S050 · calculationBefore Fair ORIGINAL two positive signs k=8 := mul(2, 8) After Fair ORIGINAL two positive signs k=8 = 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, S049 S051 · calculationBefore Fair ORIGINAL positive minus negative count k=8 := sub(16, 9) After Fair ORIGINAL positive minus negative count k=8 = 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, S050 S052 · calculationBefore Fair ORIGINAL scale times signed count k=8 := mul(1, 7) After Fair ORIGINAL scale times signed count k=8 = 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, S051 S053 · impliesBefore source roles = [9, 1, 8] After derived = [7, 9] Conditions: none k positive and n−k negative signs give scale*(2k−n),with choose(n,k)choices of positive positions.Retain EVERY k=0..n support value and multiplicity. Step identifiers and dependenciesRule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S052 S054 · calculationBefore Fair ORIGINAL binomial previous times remaining k=9 := mul(9, 1) After Fair ORIGINAL binomial previous times remaining k=9 = 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, S053 S055 · calculationBefore Fair ORIGINAL binomial recurrence division k=9 := div(9, 9) After Fair ORIGINAL binomial recurrence division k=9 = 1 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, S054 S056 · calculationBefore Fair ORIGINAL two positive signs k=9 := mul(2, 9) After Fair ORIGINAL two positive signs k=9 = 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, S055 S057 · calculationBefore Fair ORIGINAL positive minus negative count k=9 := sub(18, 9) After Fair ORIGINAL positive minus negative count k=9 = 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, S056 S058 · calculationBefore Fair ORIGINAL scale times signed count k=9 := mul(1, 9) After Fair ORIGINAL scale times signed count k=9 = 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, S057 S059 · impliesBefore source roles = [9, 1, 9] After derived = [9, 1] Conditions: none k positive and n−k negative signs give scale*(2k−n),with choose(n,k)choices of positive positions.Retain EVERY k=0..n support value and multiplicity. Step identifiers and dependenciesRule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S058 S060 · calculationBefore Fair ORIGINAL all binomial multiplicities := sum(1, 9, 36, 84, 126, 126, 84, 36, 9, 1) After Fair ORIGINAL all binomial multiplicities = 512 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, S059 S061 · impliesBefore source roles = [9, 1, 512] After derived = [[time=0, prefixes=1, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=1, prefixes=2, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=2, prefixes=4, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=3, prefixes=8, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=4, prefixes=16, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=5, prefixes=32, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=6, prefixes=64, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=7, prefixes=128, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=8, prefixes=256, next_signs=+s,-s equiprobable, conditional_increment_mean=0]] Conditions: none The PREMISE is the FULL equiprobable independent sign-product space and its revealed-sign filtration.At EVERY prefix,the next±sign extensions have equal count,so conditional increment mean=(s−s)/2=0.The adapted finite integrable partial sums form THIS martingale.This does not make independent increments necessary for arbitrary martingales or supply stopping/convergence theorems. Step identifiers and dependenciesRule: Martingale_FAIR_PRODUCT_space_natural_filtration_EVERY_prefix · Uses: G001, S060 ## Reverse checks - S061 · 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. - S060 · premise_and_result_audit — Recompute Fair ORIGINAL all binomial multiplicities with exact rational arithmetic, not rounded decimals. - S059 · 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. - S058 · premise_and_result_audit — Recompute Fair ORIGINAL scale times signed count k=9 with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute Fair ORIGINAL positive minus negative count k=9 with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute Fair ORIGINAL two positive signs k=9 with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute Fair ORIGINAL binomial recurrence division k=9 with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute Fair ORIGINAL binomial previous times remaining k=9 with exact rational arithmetic, not rounded decimals. - S053 · 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. - S052 · premise_and_result_audit — Recompute Fair ORIGINAL scale times signed count k=8 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute Fair ORIGINAL positive minus negative count k=8 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Fair ORIGINAL two positive signs k=8 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Fair ORIGINAL binomial recurrence division k=8 with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Fair ORIGINAL binomial previous times remaining k=8 with exact rational arithmetic, not rounded decimals. - S047 · 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. - S046 · premise_and_result_audit — Recompute Fair ORIGINAL scale times signed count k=7 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Fair ORIGINAL positive minus negative count k=7 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Fair ORIGINAL two positive signs k=7 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Fair ORIGINAL binomial recurrence division k=7 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Fair ORIGINAL binomial previous times remaining k=7 with exact rational arithmetic, not rounded decimals. - S041 · 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. - S040 · premise_and_result_audit — Recompute Fair ORIGINAL scale times signed count k=6 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Fair ORIGINAL positive minus negative count k=6 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Fair ORIGINAL two positive signs k=6 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Fair ORIGINAL binomial recurrence division k=6 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Fair ORIGINAL binomial previous times remaining k=6 with exact rational arithmetic, not rounded decimals. - S035 · 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. - S034 · premise_and_result_audit — Recompute Fair ORIGINAL scale times signed count k=5 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Fair ORIGINAL positive minus negative count k=5 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Fair ORIGINAL two positive signs k=5 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Fair ORIGINAL binomial recurrence division k=5 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Fair ORIGINAL binomial previous times remaining k=5 with exact rational arithmetic, not rounded decimals. - S029 · 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. - S028 · premise_and_result_audit — Recompute Fair ORIGINAL scale times signed count k=4 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Fair ORIGINAL positive minus negative count k=4 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Fair ORIGINAL two positive signs k=4 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Fair ORIGINAL binomial recurrence division k=4 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Fair ORIGINAL binomial previous times remaining k=4 with exact rational arithmetic, not rounded decimals. - S023 · 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. - S022 · premise_and_result_audit — Recompute Fair ORIGINAL scale times signed count k=3 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Fair ORIGINAL positive minus negative count k=3 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Fair ORIGINAL two positive signs k=3 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Fair ORIGINAL binomial recurrence division k=3 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Fair ORIGINAL binomial previous times remaining k=3 with exact rational arithmetic, not rounded decimals. - 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 Fair ORIGINAL scale times signed count k=2 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Fair ORIGINAL positive minus negative count k=2 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Fair ORIGINAL two positive signs k=2 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Fair ORIGINAL binomial recurrence division k=2 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Fair ORIGINAL binomial previous times remaining k=2 with exact rational arithmetic, not rounded decimals. - S011 · 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. - S010 · premise_and_result_audit — Recompute Fair ORIGINAL scale times signed count k=1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Fair ORIGINAL positive minus negative count k=1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Fair ORIGINAL two positive signs k=1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Fair ORIGINAL binomial recurrence division k=1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Fair ORIGINAL binomial previous times remaining k=1 with exact rational arithmetic, not rounded decimals. - S005 · 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. - S004 · premise_and_result_audit — Recompute Fair ORIGINAL scale times signed count k=0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Fair ORIGINAL positive minus negative count k=0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Fair ORIGINAL two positive signs k=0 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 (9, 1, [-9, -7, -5, -3, -1, 1, 3, 5, 7, 9], [1, 9, 36, 84, 126, 126, 84, 36, 9, 1], 2^9, Yes, independent_fair_signed_increments_have_conditional_mean_zero) 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: f543e9b75ee806a029d9afb186d5761ca674f4933f7af005131c5c487457df08 - candidate record SHA-256: e31f4ada09aa133e6b6189bf41153201bbdbc2c962f88d55001aece329761ce4 - 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-0084.json)