Original question and givens
Use four equiprobable outcomes with aligned increments=[11, -11, 12, -12] and current information block labels=[0, 0, 1, 1]. Give (conditional_rows,all_zero?,classification), with rows [block_label,increments_in_outcome_order,exact_block_mean] sorted by block label. Classification is conditional_mean_zero_difference or difference_fails_martingale_condition.
G001 — Use four equiprobable outcomes with aligned increments=[11, -11, 12, -12] and current information block labels=[0, 0, 1, 1]. Give (conditional_rows,all_zero?,classification), with rows [block_label,increments_in_outcome_order,exact_block_mean] sorted by block label. Classification is conditional_mean_zero_difference or difference_fails_martingale_condition.
Step-by-step derivation
S001 · implies
Before
source roles = [qf-mod31-martdeep-difference-mean, [blocks, domain, increments, operation]]
After
derived = [martdeep-difference-mean, 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
Difference ORIGINAL block0 sum := sum(11, -11)
After
Difference ORIGINAL block0 sum = 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 · implies
Before
source roles = [Difference block0 mean, 1000000, 0, 2]
After
derived = [[0, 1], 0]
Conditions: none
Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.
Step identifiers and dependencies
Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S002
S004 · calculation
Before
Difference block0 mean ORIGINAL numerator over denominator := div(0, 2)
After
Difference block0 mean ORIGINAL numerator over denominator = 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, S003
S005 · implies
Before
source roles = [0, [0, 1], [11, -11]]
After
derived = [0]
Conditions: none
Conditioning four equiprobable outcomes on THIS information block leaves equal weights,so its conditional mean is the ORIGINAL increment sum divided by this block’s size.Check EVERY nonempty block,not global cancellation.
Step identifiers and dependencies
Rule: Martingale_EVERY_information_block_conditional_mean · Uses: G001, S004
S006 · calculation
Before
Difference ORIGINAL block1 sum := sum(12, -12)
After
Difference ORIGINAL block1 sum = 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, S005
S007 · implies
Before
source roles = [Difference block1 mean, 1000000, 0, 2]
After
derived = [[0, 1], 0]
Conditions: none
Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.
Step identifiers and dependencies
Rule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S006
S008 · calculation
Before
Difference block1 mean ORIGINAL numerator over denominator := div(0, 2)
After
Difference block1 mean ORIGINAL numerator over denominator = 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, S007
S009 · implies
Before
source roles = [1, [2, 3], [12, -12]]
After
derived = [0]
Conditions: none
Conditioning four equiprobable outcomes on THIS information block leaves equal weights,so its conditional mean is the ORIGINAL increment sum divided by this block’s size.Check EVERY nonempty block,not global cancellation.
Step identifiers and dependencies
Rule: Martingale_EVERY_information_block_conditional_mean · Uses: G001, S008
S010 · implies
Before
source roles = [[11, -11, 12, -12], [0, 0, 1, 1], [[0, [11, -11], 0], [1, [12, -12], 0]]]
After
derived = [Yes]
Conditions: none
Pass the conditional-difference obligation ONLY if EVERY information-block mean is0.A globally zero mean with a nonzero block still fails.
Step identifiers and dependencies
Rule: Martingale_DIFFERENCE_all_block_means_zero_required · Uses: G001, S009
Result
([[0, [11, -11], 0], [1, [12, -12], 0]], Yes, conditional_mean_zero_difference)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.