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Question 33796: A Declared Fair Independent-Increment Model Supplies a Specific Martingale

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0066-33796 · q-20261014-advance-20261031-r07-martingale-0096

Original question and givens

Use steps=9 independent equiprobable signs, each increment ±scale with scale=2>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=2>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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001

S002 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Fair ORIGINAL scale times signed count k=0 := mul(2, -9)

After

Fair ORIGINAL scale times signed count k=0 = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [9, 2, 0]

After

derived = [-18, 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 dependencies

Rule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S004

S006 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Fair ORIGINAL scale times signed count k=1 := mul(2, -7)

After

Fair ORIGINAL scale times signed count k=1 = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

source roles = [9, 2, 1]

After

derived = [-14, 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 dependencies

Rule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S010

S012 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Fair ORIGINAL scale times signed count k=2 := mul(2, -5)

After

Fair ORIGINAL scale times signed count k=2 = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [9, 2, 2]

After

derived = [-10, 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 dependencies

Rule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S016

S018 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Fair ORIGINAL scale times signed count k=3 := mul(2, -3)

After

Fair ORIGINAL scale times signed count 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · implies

Before

source roles = [9, 2, 3]

After

derived = [-6, 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 dependencies

Rule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S022

S024 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

Fair ORIGINAL scale times signed count k=4 := mul(2, -1)

After

Fair ORIGINAL scale times signed count k=4 = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

source roles = [9, 2, 4]

After

derived = [-2, 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 dependencies

Rule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S028

S030 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Fair ORIGINAL scale times signed count k=5 := mul(2, 1)

After

Fair ORIGINAL scale times signed count k=5 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · implies

Before

source roles = [9, 2, 5]

After

derived = [2, 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 dependencies

Rule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S034

S036 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

Fair ORIGINAL scale times signed count k=6 := mul(2, 3)

After

Fair ORIGINAL scale times signed count k=6 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · implies

Before

source roles = [9, 2, 6]

After

derived = [6, 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 dependencies

Rule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S040

S042 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

Fair ORIGINAL scale times signed count k=7 := mul(2, 5)

After

Fair ORIGINAL scale times signed count k=7 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · implies

Before

source roles = [9, 2, 7]

After

derived = [10, 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 dependencies

Rule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S046

S048 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

Fair ORIGINAL scale times signed count k=8 := mul(2, 7)

After

Fair ORIGINAL scale times signed count k=8 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · implies

Before

source roles = [9, 2, 8]

After

derived = [14, 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 dependencies

Rule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S052

S054 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

Fair ORIGINAL scale times signed count k=9 := mul(2, 9)

After

Fair ORIGINAL scale times signed count 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · implies

Before

source roles = [9, 2, 9]

After

derived = [18, 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 dependencies

Rule: Martingale_FAIR_EACH_complete_terminal_support_binomial_count · Uses: G001, S058

S060 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · implies

Before

source roles = [9, 2, 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 dependencies

Rule: 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, 2, [-18, -14, -10, -6, -2, 2, 6, 10, 14, 18], [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: a8e4870f69fa920a981e1376c0ac73490d06a39fb61bbf8e351687101c83865a
  • candidate record SHA-256: 7f5a57372a115685c05728553c56edd04027d1b21d6d3138a029d787c0754440
  • semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
  • question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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