# Question 18204: Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number [Return to catalogue page 92](https://ocme.evemisslab.com/en/questions/page/92/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261104-r07-lp-convergence-foundations/#q-20261014-advance-20261104-r07-lp-convergence-0029) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0035-18204 · q-20261014-advance-20261104-r07-lp-convergence-0029 ## 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=[4, -5], positive shift=2, and fixed integer p=1 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=1/13 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=[4, -5], positive shift=2, and fixed integer p=1 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=1/13 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. - [mko-lp-convergence](https://ocme.evemisslab.com/en/mko/mko-lp-convergence/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Lp coefficient atom0 absolute power product stage1 := mul(1, 4) After Lp coefficient atom0 absolute power product stage1 = 4 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Lp constant weighted coefficient atom0 := mul(1/2, 4) After Lp constant weighted coefficient atom0 = 2 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore original atom=0;mass=1/2;SIGNED coefficient=4;fixed integer p=1 After absolute coefficient power=4;weighted coefficient power=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 dependenciesRule: lp_atomic_absolute_power_weight_not_signed_or_unweighted · Uses: G001, S004 S006 · calculationBefore Lp coefficient atom1 absolute power product stage1 := mul(1, 5) After Lp coefficient atom1 absolute power product stage1 = 5 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Lp constant weighted coefficient atom1 := mul(1/2, 5) After Lp constant weighted coefficient atom1 = 5/2 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore original atom=1;mass=1/2;SIGNED coefficient=-5;fixed integer p=1 After absolute coefficient power=5;weighted coefficient power=5/2 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 dependenciesRule: lp_atomic_absolute_power_weight_not_signed_or_unweighted · Uses: G001, S007 S009 · calculationBefore Lp full weighted absolute coefficient power sum := sum(2, 5/2) After Lp full weighted absolute coefficient power sum = 9/2 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore SAME complete probability space;masses=[1/2, 1/2];signed coefficients=[4, -5];shift=2;fixed p=1;weighted constant=9/2 After for EVERY n>=0,p-moment E|error_n|^p=(9/2)/(n+2)^1;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 dependenciesRule: lp_common_space_full_moment_formula_not_norm_or_samples · Uses: G001, S009 S011 · calculationBefore Lp strict moment comparison numerator := mul(9, 13) After Lp strict moment comparison numerator = 117 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Lp least cutoff positive denominator := sum(57, 2) After Lp least cutoff positive denominator = 59 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Lp cutoff denominator power product stage1 := mul(1, 59) After Lp cutoff denominator power product stage1 = 59 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Lp strict cutoff integer right side := mul(2, 59) After Lp strict cutoff integer right side = 118 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Lp previous denominator := sub(59, 1) After Lp previous denominator = 58 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Lp previous denominator power product stage1 := mul(1, 58) After Lp previous denominator power product stage1 = 58 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Lp previous integer right side := mul(2, 58) After Lp previous integer right side = 116 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore cutoff N=57;denominator=59;integer left=117;current right=118;previous right=116 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 dependenciesRule: lp_least_moment_cutoff_integer_power_not_float_root · Uses: G001, S018 S020 · calculationBefore Lp cutoff evaluation common error denominator := sum(57, 2) After Lp cutoff evaluation common error denominator = 59 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Lp cutoff evaluation atom0 absolute error amplitude := div(4, 59) After Lp cutoff evaluation atom0 absolute error amplitude = 4/59 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Lp cutoff evaluation atom0 error power product stage1 := mul(1, 4/59) After Lp cutoff evaluation atom0 error power product stage1 = 4/59 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Lp cutoff evaluation atom0 weighted moment term := mul(1/2, 4/59) After Lp cutoff evaluation atom0 weighted moment term = 2/59 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Lp cutoff evaluation atom1 absolute error amplitude := div(5, 59) After Lp cutoff evaluation atom1 absolute error amplitude = 5/59 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Lp cutoff evaluation atom1 error power product stage1 := mul(1, 5/59) After Lp cutoff evaluation atom1 error power product stage1 = 5/59 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Lp cutoff evaluation atom1 weighted moment term := mul(1/2, 5/59) After Lp cutoff evaluation atom1 weighted moment term = 5/118 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Lp cutoff evaluation full p-moment := sum(2/59, 5/118) After Lp cutoff evaluation full p-moment = 9/118 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · impliesBefore Lp cutoff evaluation:original n=57;fixed p=1;all weighted absolute-power terms=[2/59, 5/118] After exact p-moment=9/118;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 dependenciesRule: lp_selected_exact_moment_not_root_norm · Uses: G001, S027 S029 · impliesBefore moment_tolerance epsilon=1/13;fixed p=1;least N=57;EXACT moment at N=9/118 After for EVERY later n,moment=K/(n+shift)^p<=moment_at_N