Question 33790: A Declared Fair Independent-Increment Model Supplies a Specific Martingale

Original question and givens

Use steps=3 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=3 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.

• mko-martingale

Step-by-step derivation

source roles = [qf-mod31-martdeep-fair-partial-sum, [domain, operation, scale, steps]]

derived = [martdeep-fair-partial-sum, Exact_finite_martingale_construction]

Fair ORIGINAL two positive signs k=0 := mul(2, 0)

Fair ORIGINAL two positive signs k=0 = 0

Fair ORIGINAL positive minus negative count k=0 := sub(0, 3)

Fair ORIGINAL positive minus negative count k=0 = -3

Fair ORIGINAL scale times signed count k=0 := mul(2, -3)

Fair ORIGINAL scale times signed count k=0 = -6

source roles = [3, 2, 0]

derived = [-6, 1]

Fair ORIGINAL binomial previous times remaining k=1 := mul(1, 3)

Fair ORIGINAL binomial previous times remaining k=1 = 3

Fair ORIGINAL binomial recurrence division k=1 := div(3, 1)

Fair ORIGINAL binomial recurrence division k=1 = 3

Fair ORIGINAL two positive signs k=1 := mul(2, 1)

Fair ORIGINAL two positive signs k=1 = 2

Fair ORIGINAL positive minus negative count k=1 := sub(2, 3)

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

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

Fair ORIGINAL scale times signed count k=1 = -2

source roles = [3, 2, 1]

derived = [-2, 3]

Fair ORIGINAL binomial previous times remaining k=2 := mul(3, 2)

Fair ORIGINAL binomial previous times remaining k=2 = 6

Fair ORIGINAL binomial recurrence division k=2 := div(6, 2)

Fair ORIGINAL binomial recurrence division k=2 = 3

Fair ORIGINAL two positive signs k=2 := mul(2, 2)

Fair ORIGINAL two positive signs k=2 = 4

Fair ORIGINAL positive minus negative count k=2 := sub(4, 3)

Fair ORIGINAL positive minus negative count k=2 = 1

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

Fair ORIGINAL scale times signed count k=2 = 2

source roles = [3, 2, 2]

derived = [2, 3]

Fair ORIGINAL binomial previous times remaining k=3 := mul(3, 1)

Fair ORIGINAL binomial previous times remaining k=3 = 3

Fair ORIGINAL binomial recurrence division k=3 := div(3, 3)

Fair ORIGINAL binomial recurrence division k=3 = 1

Fair ORIGINAL two positive signs k=3 := mul(2, 3)

Fair ORIGINAL two positive signs k=3 = 6

Fair ORIGINAL positive minus negative count k=3 := sub(6, 3)

Fair ORIGINAL positive minus negative count k=3 = 3

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

Fair ORIGINAL scale times signed count k=3 = 6

source roles = [3, 2, 3]

derived = [6, 1]

Fair ORIGINAL all binomial multiplicities := sum(1, 3, 3, 1)

Fair ORIGINAL all binomial multiplicities = 8

source roles = [3, 2, 8]

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]]

Reverse checks

• S025 · 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.
• S024 · premise_and_result_audit — Recompute Fair ORIGINAL all binomial multiplicities 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

(3, 2, [-6, -2, 2, 6], [1, 3, 3, 1], 2^3, 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.