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Question 20703: Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence

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

R0040-20703 · q-20261014-advance-20261108-r04-chebyshev-inequality-0053

Original question and givens

Use complete normalized finite law masses=[1/8, 2/8, 5/8] and aligned values X=[0, 3, 6]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).

  • G001 — Use complete normalized finite law masses=[1/8, 2/8, 5/8] and aligned values X=[0, 3, 6]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).

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 operation=cheb-bound-comparison;original snapshot and mathematics retained

After

explicit ordered source output fields=[tail_probability, variance, raw_upper_bound, bound_holds, certificate]

Conditions: none

Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · calculation

Before

Chebyshev complete probability law mass := sum(1/8, 1/4, 5/8)

After

Chebyshev complete probability law mass = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Chebyshev original atom0 weighted SIGNED X := mul(1/8, 0)

After

Chebyshev original atom0 weighted SIGNED X = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Chebyshev original atom1 weighted SIGNED X := mul(1/4, 3)

After

Chebyshev original atom1 weighted SIGNED X = 3/4

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Chebyshev original atom2 weighted SIGNED X := mul(5/8, 6)

After

Chebyshev original atom2 weighted SIGNED X = 15/4

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Chebyshev complete SAME-law mean := sum(0, 3/4, 15/4)

After

Chebyshev complete SAME-law mean = 9/2

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Chebyshev original atom0 X minus SAME mean := sub(0, 9/2)

After

Chebyshev original atom0 X minus SAME mean = -9/2

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Chebyshev original atom0 centered SQUARE := mul(-9/2, -9/2)

After

Chebyshev original atom0 centered SQUARE = 81/4

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Chebyshev original atom0 weighted centered square := mul(1/8, 81/4)

After

Chebyshev original atom0 weighted centered square = 81/32

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Chebyshev original atom1 X minus SAME mean := sub(3, 9/2)

After

Chebyshev original atom1 X minus SAME mean = -3/2

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Chebyshev original atom1 centered SQUARE := mul(-3/2, -3/2)

After

Chebyshev original atom1 centered SQUARE = 9/4

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Chebyshev original atom1 weighted centered square := mul(1/4, 9/4)

After

Chebyshev original atom1 weighted centered square = 9/16

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Chebyshev original atom2 X minus SAME mean := sub(6, 9/2)

After

Chebyshev original atom2 X minus SAME mean = 3/2

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Chebyshev original atom2 centered SQUARE := mul(3/2, 3/2)

After

Chebyshev original atom2 centered SQUARE = 9/4

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Chebyshev original atom2 weighted centered square := mul(5/8, 9/4)

After

Chebyshev original atom2 weighted centered square = 45/32

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Chebyshev full variance E centered square := sum(81/32, 9/16, 45/32)

After

Chebyshev full variance E centered square = 9/2

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

ORIGINAL complete normalized masses=[1/8, 1/4, 5/8];ALL aligned SIGNED values X=[0, 3, 6];SAME-law mean=9/2;all deviations=[-9/2, -3/2, 3/2]

After

ALL nonnegative squared deviations Y=[81/4, 9/4, 9/4];variance=E[Y]=9/2;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms

Conditions: none

This is one random variable under a complete law.X may be negative at positive mass;do not add X≥0 or independence.Each squared Y is nonnegative,and zero variance forces zero deviations only at positive-mass atoms;retain null outliers.

Step identifiers and dependencies

Rule: cheb_same_law_finite_signed_variable_no_independence_premise · Uses: G001, S016

S018 · calculation

Before

Chebyshev original atom0 absolute-event probability mass := mul(1/8, 1)

After

Chebyshev original atom0 absolute-event probability mass = 1/8

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

ORIGINAL atom=0;SIGNED centered deviation=-9/2;absolute deviation=9/2;positive epsilon=3

After

event abs(deviation)>=epsilon=Yes;EQUALITY INCLUDED;probability contribution=1/8

Conditions: none

Center at this same law's mean before taking the absolute value.Use ≥ for the event,not >;strict positivity of epsilon does not make the event inequality strict.

Step identifiers and dependencies

Rule: cheb_every_absolute_event_atom_including_equality · Uses: G001, S018

S020 · calculation

Before

Chebyshev original atom1 absolute-event probability mass := mul(1/4, 0)

After

Chebyshev original atom1 absolute-event probability mass = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

ORIGINAL atom=1;SIGNED centered deviation=-3/2;absolute deviation=3/2;positive epsilon=3

After

event abs(deviation)>=epsilon=No;EQUALITY INCLUDED;probability contribution=0

Conditions: none

Center at this same law's mean before taking the absolute value.Use ≥ for the event,not >;strict positivity of epsilon does not make the event inequality strict.

Step identifiers and dependencies

Rule: cheb_every_absolute_event_atom_including_equality · Uses: G001, S020

S022 · calculation

Before

Chebyshev original atom2 absolute-event probability mass := mul(5/8, 0)

After

Chebyshev original atom2 absolute-event probability mass = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · implies

Before

ORIGINAL atom=2;SIGNED centered deviation=3/2;absolute deviation=3/2;positive epsilon=3

After

event abs(deviation)>=epsilon=No;EQUALITY INCLUDED;probability contribution=0

Conditions: none

Center at this same law's mean before taking the absolute value.Use ≥ for the event,not >;strict positivity of epsilon does not make the event inequality strict.

Step identifiers and dependencies

Rule: cheb_every_absolute_event_atom_including_equality · Uses: G001, S022

S024 · calculation

Before

Chebyshev COMPLETE absolute-deviation tail probability := sum(1/8, 0, 0)

After

Chebyshev COMPLETE absolute-deviation tail probability = 1/8

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

Chebyshev POSITIVE epsilon SQUARED threshold := mul(3, 3)

After

Chebyshev POSITIVE epsilon SQUARED threshold = 9

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

positive original epsilon=3;consider ANY real centered deviation d;let r=abs(d)>=0

After

r+epsilon>0,so r>=epsilon iff(r-epsilon)*(r+epsilon)>=0 iff d^2>=epsilon^2;POINTWISE event identity,not just accidental probability equality

Conditions: none

Factoring the difference of squares with r+epsilon positive preserves the inequality,proving the events equivalent for every real d.This is the bridge from signed X to nonnegative Y,not an inference of event identity from equal numerical probabilities.

Step identifiers and dependencies

Rule: cheb_square_absolute_event_iff_positive_epsilon · Uses: G001, S025

S027 · implies

Before

ORIGINAL atom=0;deviation=-9/2;its SQUARE=81/4;EPSILON SQUARED=9

After

absolute event=Yes;squared event=Yes;same pointwise membership=Yes

Conditions: none

Check membership at every original atom,not just equality of weighted totals.Keep the mathematical event correspondence correct even on null atoms.

Step identifiers and dependencies

Rule: cheb_each_original_squared_event_same_membership · Uses: G001, S026

S028 · implies

Before

ORIGINAL atom=1;deviation=-3/2;its SQUARE=9/4;EPSILON SQUARED=9

After

absolute event=No;squared event=No;same pointwise membership=Yes

Conditions: none

Check membership at every original atom,not just equality of weighted totals.Keep the mathematical event correspondence correct even on null atoms.

Step identifiers and dependencies

Rule: cheb_each_original_squared_event_same_membership · Uses: G001, S027

S029 · implies

Before

ORIGINAL atom=2;deviation=3/2;its SQUARE=9/4;EPSILON SQUARED=9

After

absolute event=No;squared event=No;same pointwise membership=Yes

Conditions: none

Check membership at every original atom,not just equality of weighted totals.Keep the mathematical event correspondence correct even on null atoms.

Step identifiers and dependencies

Rule: cheb_each_original_squared_event_same_membership · Uses: G001, S028

S030 · calculation

Before

Chebyshev RAW variance over epsilon SQUARED := div(9/2, 9)

After

Chebyshev RAW variance over epsilon SQUARED = 1/2

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

Chebyshev Markov atom0 squared-threshold indicator := mul(9, 1)

After

Chebyshev Markov atom0 squared-threshold indicator = 9

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

Chebyshev Markov atom0 Y minus threshold indicator := sub(81/4, 9)

After

Chebyshev Markov atom0 Y minus threshold indicator = 45/4

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Chebyshev Markov atom0 weighted nonnegative slack := mul(1/8, 45/4)

After

Chebyshev Markov atom0 weighted nonnegative slack = 45/32

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Chebyshev Markov atom1 squared-threshold indicator := mul(9, 0)

After

Chebyshev Markov atom1 squared-threshold indicator = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

Chebyshev Markov atom1 Y minus threshold indicator := sub(9/4, 0)

After

Chebyshev Markov atom1 Y minus threshold indicator = 9/4

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

Chebyshev Markov atom1 weighted nonnegative slack := mul(1/4, 9/4)

After

Chebyshev Markov atom1 weighted nonnegative slack = 9/16

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

Chebyshev Markov atom2 squared-threshold indicator := mul(9, 0)

After

Chebyshev Markov atom2 squared-threshold indicator = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

Chebyshev Markov atom2 Y minus threshold indicator := sub(9/4, 0)

After

Chebyshev Markov atom2 Y minus threshold indicator = 9/4

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

Chebyshev Markov atom2 weighted nonnegative slack := mul(5/8, 9/4)

After

Chebyshev Markov atom2 weighted nonnegative slack = 45/32

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

Chebyshev Markov ALL weighted slacks sum := sum(45/32, 9/16, 45/32)

After

Chebyshev Markov ALL weighted slacks sum = 27/8

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

Chebyshev epsilon squared times SAME event probability := mul(9, 1/8)

After

Chebyshev epsilon squared times SAME event probability = 9/8

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

Chebyshev variance minus threshold-event mass := sub(9/2, 9/8)

After

Chebyshev variance minus threshold-event mass = 27/8

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · implies

Before

Y=ORIGINAL squared deviations=[81/4, 9/4, 9/4] nonnegative at EVERY atom;E[Y]=variance=9/2;positive threshold epsilon^2=9;complete weighted slacks=[45/32, 9/16, 45/32]

After

variance-epsilon^2*tail=27/8>=0;divide POSITIVE epsilon^2 to obtain actual tail=1/8<=RAW bound=1/2;no independence and no equality/clipping claim

Conditions: none

On the event Y≥epsilon squared;off it Y≥0.Weight by the complete original law,sum and divide by positive epsilon squared.Apply Markov to Y,not potentially signed X.No independence is needed.The raw bound may exceed1 and need not equal the tail;variance0 is included.

Step identifiers and dependencies

Rule: cheb_full_Markov_on_squared_variable_not_signed_X_or_IID · Uses: G001, S042

Reverse checks

  • S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S042 · premise_and_result_audit — Recompute Chebyshev variance minus threshold-event mass with exact rational arithmetic, not rounded decimals.
  • S041 · premise_and_result_audit — Recompute Chebyshev epsilon squared times SAME event probability with exact rational arithmetic, not rounded decimals.
  • S040 · premise_and_result_audit — Recompute Chebyshev Markov ALL weighted slacks sum with exact rational arithmetic, not rounded decimals.
  • S039 · premise_and_result_audit — Recompute Chebyshev Markov atom2 weighted nonnegative slack with exact rational arithmetic, not rounded decimals.
  • S038 · premise_and_result_audit — Recompute Chebyshev Markov atom2 Y minus threshold indicator with exact rational arithmetic, not rounded decimals.
  • S037 · premise_and_result_audit — Recompute Chebyshev Markov atom2 squared-threshold indicator with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Recompute Chebyshev Markov atom1 weighted nonnegative slack with exact rational arithmetic, not rounded decimals.
  • S035 · premise_and_result_audit — Recompute Chebyshev Markov atom1 Y minus threshold indicator with exact rational arithmetic, not rounded decimals.
  • S034 · premise_and_result_audit — Recompute Chebyshev Markov atom1 squared-threshold indicator with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Recompute Chebyshev Markov atom0 weighted nonnegative slack with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute Chebyshev Markov atom0 Y minus threshold indicator with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute Chebyshev Markov atom0 squared-threshold indicator with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute Chebyshev RAW variance over epsilon SQUARED with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Recompute Chebyshev POSITIVE epsilon SQUARED threshold with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute Chebyshev COMPLETE absolute-deviation tail probability with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Recompute Chebyshev original atom2 absolute-event probability mass with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Recompute Chebyshev original atom1 absolute-event probability mass with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Recompute Chebyshev original atom0 absolute-event probability mass with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Recompute Chebyshev full variance E centered square with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Chebyshev original atom2 weighted centered square with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Chebyshev original atom2 centered SQUARE with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Chebyshev original atom2 X minus SAME mean with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Chebyshev original atom1 weighted centered square with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Chebyshev original atom1 centered SQUARE with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Chebyshev original atom1 X minus SAME mean with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Chebyshev original atom0 weighted centered square with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Chebyshev original atom0 centered SQUARE with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Chebyshev original atom0 X minus SAME mean with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Chebyshev complete SAME-law mean with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Chebyshev original atom2 weighted SIGNED X with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Chebyshev original atom1 weighted SIGNED X with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Chebyshev original atom0 weighted SIGNED X with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Chebyshev complete probability law mass with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(1/8, 9/2, 1/2, Yes, upper_bound_not_tail_equality_and_no_independence_needed)

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: c40705e3bd2b2980f71d843c16312fb957e640f6ff5ec2604fa03339fe53689f
  • candidate record SHA-256: b128ab29185e78c3311c2db3e68e0294bed1fdb3c5de98556e842fca30cf5354
  • semantic review receipt SHA-256: c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d
  • question-first witness SHA-256: e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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