Question 33782: A Declared Fair Independent-Increment Model Supplies a Specific Martingale

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0066-33782 · q-20261014-advance-20261031-r07-martingale-0082

Original question and givens

Use steps=7 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=7 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

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, 7)

After

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

S004 · calculation

Before

Fair ORIGINAL scale times signed count k=0 := mul(1, -7)

After

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

S005 · implies

Before

source roles = [7, 1, 0]

After

derived = [-7, 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, 7)

After

Fair ORIGINAL binomial previous times remaining 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, S005

S007 · calculation

Before

Fair ORIGINAL binomial recurrence division k=1 := div(7, 1)

After

Fair ORIGINAL binomial recurrence division 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, 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, 7)

After

Fair ORIGINAL positive minus negative count 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, S008

S010 · calculation

Before

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

After

Fair ORIGINAL scale times signed count 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, S009

S011 · implies

Before

source roles = [7, 1, 1]

After

derived = [-5, 7]

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(7, 6)

After

Fair ORIGINAL binomial previous times remaining k=2 = 42

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(42, 2)

After

Fair ORIGINAL binomial recurrence division k=2 = 21

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, 7)

After

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

S016 · calculation

Before

Fair ORIGINAL scale times signed count k=2 := mul(1, -3)

After

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

S017 · implies

Before

source roles = [7, 1, 2]

After

derived = [-3, 21]

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(21, 5)

After

Fair ORIGINAL binomial previous times remaining k=3 = 105

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(105, 3)

After

Fair ORIGINAL binomial recurrence division k=3 = 35

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, 7)

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 = [7, 1, 3]

After

derived = [-1, 35]

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(35, 4)

After

Fair ORIGINAL binomial previous times remaining k=4 = 140

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(140, 4)

After

Fair ORIGINAL binomial recurrence division k=4 = 35

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, 7)

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(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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

source roles = [7, 1, 4]

After

derived = [1, 35]

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(35, 3)

After

Fair ORIGINAL binomial previous times remaining k=5 = 105

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(105, 5)

After

Fair ORIGINAL binomial recurrence division k=5 = 21

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, 7)

After

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

S034 · calculation

Before

Fair ORIGINAL scale times signed count k=5 := mul(1, 3)

After

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

S035 · implies

Before

source roles = [7, 1, 5]

After

derived = [3, 21]

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(21, 2)

After

Fair ORIGINAL binomial previous times remaining k=6 = 42

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(42, 6)

After

Fair ORIGINAL binomial recurrence division k=6 = 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, 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, 7)

After

Fair ORIGINAL positive minus negative count k=6 = 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, S038

S040 · calculation

Before

Fair ORIGINAL scale times signed count k=6 := mul(1, 5)

After

Fair ORIGINAL scale times signed count k=6 = 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, S039

S041 · implies

Before

source roles = [7, 1, 6]

After

derived = [5, 7]

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(7, 1)

After

Fair ORIGINAL binomial previous times remaining k=7 = 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, S041

S043 · calculation

Before

Fair ORIGINAL binomial recurrence division k=7 := div(7, 7)

After

Fair ORIGINAL binomial recurrence division k=7 = 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, 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, 7)

After

Fair ORIGINAL positive minus negative count k=7 = 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, S044

S046 · calculation

Before

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

After

Fair ORIGINAL scale times signed count k=7 = 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, S045

S047 · implies

Before

source roles = [7, 1, 7]

After

derived = [7, 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, S046

S048 · calculation

Before

Fair ORIGINAL all binomial multiplicities := sum(1, 7, 21, 35, 35, 21, 7, 1)

After

Fair ORIGINAL all binomial multiplicities = 128

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 · implies

Before

source roles = [7, 1, 128]

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]]

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, S048

Reverse checks

S049 · 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.
S048 · premise_and_result_audit — Recompute Fair ORIGINAL all binomial multiplicities 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

(7, 1, [-7, -5, -3, -1, 1, 3, 5, 7], [1, 7, 21, 35, 35, 21, 7, 1], 2^7, 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: d735a732ccc651c35bc9b6ddf11a86fd28949c151f6637a246572b2f84241fe1
candidate record SHA-256: a7c304ecf56c67c0abd3055648d37fecd028044309a68d8750e98c2fecdada80
semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON