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Question 18218: Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number

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

R0035-18218 · q-20261014-advance-20261104-r07-lp-convergence-0043

Original question and givens

Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[18, -19], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/27 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).

  • G001 — Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[18, -19], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/27 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, 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=lp-epsilon-witness;original snapshot and mathematics retained

After

explicit ordered source output fields=[p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate]

Conditions: none

Keep the historical interface marker in the original snapshot while giving ordered fields here,without JavaScript or an external table.Labels are not mathematical evidence,and each No marker retains the operation's claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · calculation

Before

Lp complete atomic probability mass := sum(1/2, 1/2)

After

Lp complete atomic probability mass = 1

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Lp coefficient atom0 absolute power product stage1 := mul(1, 18)

After

Lp coefficient atom0 absolute power product stage1 = 18

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Lp coefficient atom0 absolute power product stage2 := mul(18, 18)

After

Lp coefficient atom0 absolute power product stage2 = 324

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Lp coefficient atom0 absolute power product stage3 := mul(324, 18)

After

Lp coefficient atom0 absolute power product stage3 = 5832

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Lp constant weighted coefficient atom0 := mul(1/2, 5832)

After

Lp constant weighted coefficient atom0 = 2916

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

original atom=0;mass=1/2;SIGNED coefficient=18;fixed integer p=3

After

absolute coefficient power=5832;weighted coefficient power=2916

Conditions: none

Take each signed coefficient's absolute value before its p-th power,then multiply by the original atom mass.Do not use signed odd powers or an unweighted average.Keep null-atom data visible.

Step identifiers and dependencies

Rule: lp_atomic_absolute_power_weight_not_signed_or_unweighted · Uses: G001, S006

S008 · calculation

Before

Lp coefficient atom1 absolute power product stage1 := mul(1, 19)

After

Lp coefficient atom1 absolute power product stage1 = 19

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Lp coefficient atom1 absolute power product stage2 := mul(19, 19)

After

Lp coefficient atom1 absolute power product stage2 = 361

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Lp coefficient atom1 absolute power product stage3 := mul(361, 19)

After

Lp coefficient atom1 absolute power product stage3 = 6859

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Lp constant weighted coefficient atom1 := mul(1/2, 6859)

After

Lp constant weighted coefficient atom1 = 6859/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

original atom=1;mass=1/2;SIGNED coefficient=-19;fixed integer p=3

After

absolute coefficient power=6859;weighted coefficient power=6859/2

Conditions: none

Take each signed coefficient's absolute value before its p-th power,then multiply by the original atom mass.Do not use signed odd powers or an unweighted average.Keep null-atom data visible.

Step identifiers and dependencies

Rule: lp_atomic_absolute_power_weight_not_signed_or_unweighted · Uses: G001, S011

S013 · calculation

Before

Lp full weighted absolute coefficient power sum := sum(2916, 6859/2)

After

Lp full weighted absolute coefficient power sum = 12691/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

SAME complete probability space;masses=[1/2, 1/2];signed coefficients=[18, -19];shift=2;fixed p=3;weighted constant=12691/2

After

for EVERY n>=0,p-moment E|error_n|^p=(12691/2)/(n+2)^3;all moments finite and tend0;this is NOT the p-th-root norm value

Conditions: none

The common space,fixed p and whole-tail amplitudes are declared.Finitely many finite coefficients give finite moments;the positive denominator grows with n and p≥1,so moments tend0.The field is the moment,not its p-th-root norm,and is not inferred from finite observations.

Step identifiers and dependencies

Rule: lp_common_space_full_moment_formula_not_norm_or_samples · Uses: G001, S013

S015 · calculation

Before

Lp strict moment comparison numerator := mul(12691, 27)

After

Lp strict moment comparison numerator = 342657

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Lp strict moment comparison coefficient := mul(2, 1)

After

Lp strict moment comparison coefficient = 2

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Lp least cutoff positive denominator := sum(54, 2)

After

Lp least cutoff positive denominator = 56

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Lp cutoff denominator power product stage1 := mul(1, 56)

After

Lp cutoff denominator power product stage1 = 56

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Lp cutoff denominator power product stage2 := mul(56, 56)

After

Lp cutoff denominator power product stage2 = 3136

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

Lp cutoff denominator power product stage3 := mul(3136, 56)

After

Lp cutoff denominator power product stage3 = 175616

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

Lp strict cutoff integer right side := mul(2, 175616)

After

Lp strict cutoff integer right side = 351232

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Lp previous denominator := sub(56, 1)

After

Lp previous denominator = 55

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Lp previous denominator power product stage1 := mul(1, 55)

After

Lp previous denominator power product stage1 = 55

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

Lp previous denominator power product stage2 := mul(55, 55)

After

Lp previous denominator power product stage2 = 3025

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

Lp previous denominator power product stage3 := mul(3025, 55)

After

Lp previous denominator power product stage3 = 166375

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Lp previous integer right side := mul(2, 166375)

After

Lp previous integer right side = 332750

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

cutoff N=54;denominator=56;integer left=342657;current right=351232;previous right=332750

After

strict comparison holds at N and fails at N-1;all smaller allowed indices fail

Conditions: none

Cross-multiply positive factors and compare exact integer p-th powers.Failure at the previous index and monotonicity prove the least N,without rounding a floating-point root.

Step identifiers and dependencies

Rule: lp_least_moment_cutoff_integer_power_not_float_root · Uses: G001, S026

S028 · calculation

Before

Lp cutoff evaluation common error denominator := sum(54, 2)

After

Lp cutoff evaluation common error denominator = 56

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

Lp cutoff evaluation atom0 absolute error amplitude := div(18, 56)

After

Lp cutoff evaluation atom0 absolute error amplitude = 9/28

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

Lp cutoff evaluation atom0 error power product stage1 := mul(1, 9/28)

After

Lp cutoff evaluation atom0 error power product stage1 = 9/28

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

Lp cutoff evaluation atom0 error power product stage2 := mul(9/28, 9/28)

After

Lp cutoff evaluation atom0 error power product stage2 = 81/784

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

Lp cutoff evaluation atom0 error power product stage3 := mul(81/784, 9/28)

After

Lp cutoff evaluation atom0 error power product stage3 = 729/21952

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Lp cutoff evaluation atom0 weighted moment term := mul(1/2, 729/21952)

After

Lp cutoff evaluation atom0 weighted moment term = 729/43904

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Lp cutoff evaluation atom1 absolute error amplitude := div(19, 56)

After

Lp cutoff evaluation atom1 absolute error amplitude = 19/56

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

Lp cutoff evaluation atom1 error power product stage1 := mul(1, 19/56)

After

Lp cutoff evaluation atom1 error power product stage1 = 19/56

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

Lp cutoff evaluation atom1 error power product stage2 := mul(19/56, 19/56)

After

Lp cutoff evaluation atom1 error power product stage2 = 361/3136

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

Lp cutoff evaluation atom1 error power product stage3 := mul(361/3136, 19/56)

After

Lp cutoff evaluation atom1 error power product stage3 = 6859/175616

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

Lp cutoff evaluation atom1 weighted moment term := mul(1/2, 6859/175616)

After

Lp cutoff evaluation atom1 weighted moment term = 6859/351232

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

Lp cutoff evaluation full p-moment := sum(729/43904, 6859/351232)

After

Lp cutoff evaluation full p-moment = 37/1024

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · implies

Before

Lp cutoff evaluation:original n=54;fixed p=3;all weighted absolute-power terms=[729/43904, 6859/351232]

After

exact p-moment=37/1024;neither its p-th root nor unweighted atom mean

Conditions: none

Sum all mass-weighted absolute p-th-power amplitudes,without taking a p-th root or dividing by the atom count.

Step identifiers and dependencies

Rule: lp_selected_exact_moment_not_root_norm · Uses: G001, S039

S041 · implies

Before

moment_tolerance epsilon=1/27;fixed p=3;least N=54;EXACT moment at N=37/1024

After

for EVERY later n,moment=K/(n+shift)^p<=moment_at_N<epsilon;norm bound with SAME epsilon would instead require moment<epsilon^3

Conditions: none

The source epsilon directly bounds E|error|^p,not the norm by the same number.For fixed p both tend-to-zero notions agree,but their numerical thresholds differ.Keep the source moment criterion and its all-later guarantee.

Step identifiers and dependencies

Rule: lp_moment_tolerance_not_same_numeric_norm_tolerance · Uses: G001, S040

Reverse checks

  • S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S039 · premise_and_result_audit — Recompute Lp cutoff evaluation full p-moment with exact rational arithmetic, not rounded decimals.
  • S038 · premise_and_result_audit — Recompute Lp cutoff evaluation atom1 weighted moment term with exact rational arithmetic, not rounded decimals.
  • S037 · premise_and_result_audit — Recompute Lp cutoff evaluation atom1 error power product stage3 with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Recompute Lp cutoff evaluation atom1 error power product stage2 with exact rational arithmetic, not rounded decimals.
  • S035 · premise_and_result_audit — Recompute Lp cutoff evaluation atom1 error power product stage1 with exact rational arithmetic, not rounded decimals.
  • S034 · premise_and_result_audit — Recompute Lp cutoff evaluation atom1 absolute error amplitude with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Recompute Lp cutoff evaluation atom0 weighted moment term with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute Lp cutoff evaluation atom0 error power product stage3 with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute Lp cutoff evaluation atom0 error power product stage2 with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute Lp cutoff evaluation atom0 error power product stage1 with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute Lp cutoff evaluation atom0 absolute error amplitude with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute Lp cutoff evaluation common error denominator with exact rational arithmetic, not rounded decimals.
  • 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 — Recompute Lp previous integer right side with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute Lp previous denominator power product stage3 with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute Lp previous denominator power product stage2 with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute Lp previous denominator power product stage1 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute Lp previous denominator with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute Lp strict cutoff integer right side with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute Lp cutoff denominator power product stage3 with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute Lp cutoff denominator power product stage2 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Lp cutoff denominator power product stage1 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Lp least cutoff positive denominator with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Lp strict moment comparison coefficient with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Lp strict moment comparison numerator with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Recompute Lp full weighted absolute coefficient power sum with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Recompute Lp constant weighted coefficient atom1 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Lp coefficient atom1 absolute power product stage3 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Lp coefficient atom1 absolute power product stage2 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Lp coefficient atom1 absolute power product stage1 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Recompute Lp constant weighted coefficient atom0 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Lp coefficient atom0 absolute power product stage3 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Lp coefficient atom0 absolute power product stage2 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Lp coefficient atom0 absolute power product stage1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Lp complete atomic probability 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

(3, 1/27, 54, 37/1024, Yes, forall_n_ge_N_symbolic_monotone_tail)

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: 8845e7b202d062031720422c83927bb386a894d855129103c054e53254a3babf
  • candidate record SHA-256: 8f1925b4981b8d2992266d0001655a934e64db5d468bdd50a6da8c808b9f8700
  • semantic review receipt SHA-256: 713db6309b22bf8b31f0253d4aaa7aca9011ab908587ae0e9d4a365edce1f9bc
  • question-first witness SHA-256: 0338a3db344ab237113fc63be29153c3684da96e388109e9dcd913a0a693aed2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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