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Question 33780: 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

R0066-33780 · q-20261014-advance-20261031-r07-martingale-0080

Original question and givens

Use steps=5 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=5 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.

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, 5)

After

Fair ORIGINAL positive minus negative count k=0 = -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, S002

S004 · calculation

Before

Fair ORIGINAL scale times signed count k=0 := mul(1, -5)

After

Fair ORIGINAL scale times signed count k=0 = -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, S003

S005 · implies

Before

source roles = [5, 1, 0]

After

derived = [-5, 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, 5)

After

Fair ORIGINAL binomial previous times remaining k=1 = 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, S005

S007 · calculation

Before

Fair ORIGINAL binomial recurrence division k=1 := div(5, 1)

After

Fair ORIGINAL binomial recurrence division k=1 = 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, 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, 5)

After

Fair ORIGINAL positive minus negative count k=1 = -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, S008

S010 · calculation

Before

Fair ORIGINAL scale times signed count k=1 := mul(1, -3)

After

Fair ORIGINAL scale times signed count k=1 = -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, S009

S011 · implies

Before

source roles = [5, 1, 1]

After

derived = [-3, 5]

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(5, 4)

After

Fair ORIGINAL binomial previous times remaining k=2 = 20

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(20, 2)

After

Fair ORIGINAL binomial recurrence division 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, 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, 5)

After

Fair ORIGINAL positive minus negative count k=2 = -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, S014

S016 · calculation

Before

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

After

Fair ORIGINAL scale times signed count k=2 = -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, S015

S017 · implies

Before

source roles = [5, 1, 2]

After

derived = [-1, 10]

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(10, 3)

After

Fair ORIGINAL binomial previous times remaining k=3 = 30

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(30, 3)

After

Fair ORIGINAL binomial recurrence division k=3 = 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, 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, 5)

After

Fair ORIGINAL positive minus negative count k=3 = 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, S020

S022 · calculation

Before

Fair ORIGINAL scale times signed count k=3 := mul(1, 1)

After

Fair ORIGINAL scale times signed count k=3 = 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, S021

S023 · implies

Before

source roles = [5, 1, 3]

After

derived = [1, 10]

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(10, 2)

After

Fair ORIGINAL binomial previous times remaining k=4 = 20

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(20, 4)

After

Fair ORIGINAL binomial recurrence division k=4 = 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, 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, 5)

After

Fair ORIGINAL positive minus negative count k=4 = 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, S026

S028 · calculation

Before

Fair ORIGINAL scale times signed count k=4 := mul(1, 3)

After

Fair ORIGINAL scale times signed count k=4 = 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, S027

S029 · implies

Before

source roles = [5, 1, 4]

After

derived = [3, 5]

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(5, 1)

After

Fair ORIGINAL binomial previous times remaining k=5 = 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, S029

S031 · calculation

Before

Fair ORIGINAL binomial recurrence division k=5 := div(5, 5)

After

Fair ORIGINAL binomial recurrence division 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, 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, 5)

After

Fair ORIGINAL positive minus negative count k=5 = 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, S032

S034 · calculation

Before

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

After

Fair ORIGINAL scale times signed count k=5 = 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, S033

S035 · implies

Before

source roles = [5, 1, 5]

After

derived = [5, 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, S034

S036 · calculation

Before

Fair ORIGINAL all binomial multiplicities := sum(1, 5, 10, 10, 5, 1)

After

Fair ORIGINAL all binomial multiplicities = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · implies

Before

source roles = [5, 1, 32]

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]]

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, S036

Reverse checks

  • S037 · 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.
  • S036 · premise_and_result_audit — Recompute Fair ORIGINAL all binomial multiplicities 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

(5, 1, [-5, -3, -1, 1, 3, 5], [1, 5, 10, 10, 5, 1], 2^5, 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
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  • candidate record SHA-256: 64c77d001e3b52badc97c8556f38e65d0b659d0ca34eb1b907f09660482ecdb0
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