# Question 18254: The p-Moment Is a Weighted Absolute-Power Error

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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.

R0035-18254 · q-20261014-advance-20261104-r07-lp-convergence-0004



## 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. n=4. Give (p, n, p_moment, 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. n=4. Give (p, n, p_moment, 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-moment-at-n;original snapshot and mathematics retained

After

explicit ordered source output fields=[p, n, p_moment, 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

selected Lp index common error denominator := sum(4, 2)

After

selected Lp index common error denominator = 6

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

selected Lp index atom0 absolute error amplitude := div(4, 6)

After

selected Lp index atom0 absolute error amplitude = 2/3

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

selected Lp index atom0 error power product stage1 := mul(1, 2/3)

After

selected Lp index atom0 error power product stage1 = 2/3

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

selected Lp index atom0 weighted moment term := mul(1/2, 2/3)

After

selected Lp index atom0 weighted moment term = 1/3

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

selected Lp index atom1 absolute error amplitude := div(5, 6)

After

selected Lp index atom1 absolute error amplitude = 5/6

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

selected Lp index atom1 error power product stage1 := mul(1, 5/6)

After

selected Lp index atom1 error power product stage1 = 5/6

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

selected Lp index atom1 weighted moment term := mul(1/2, 5/6)

After

selected Lp index atom1 weighted moment term = 5/12

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

selected Lp index full p-moment := sum(1/3, 5/12)

After

selected Lp index full p-moment = 3/4

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, S017





S019 · impliesBefore

selected Lp index:original n=4;fixed p=1;all weighted absolute-power terms=[1/3, 5/12]

After

exact p-moment=3/4;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, S018









## Reverse checks


- 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 selected Lp index full p-moment with exact rational arithmetic, not rounded decimals.
- S017 · premise_and_result_audit — Recompute selected Lp index atom1 weighted moment term with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Recompute selected Lp index atom1 error power product stage1 with exact rational arithmetic, not rounded decimals.
- S015 · premise_and_result_audit — Recompute selected Lp index atom1 absolute error amplitude with exact rational arithmetic, not rounded decimals.
- S014 · premise_and_result_audit — Recompute selected Lp index atom0 weighted moment term with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Recompute selected Lp index atom0 error power product stage1 with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute selected Lp index atom0 absolute error amplitude with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Recompute selected Lp index common error denominator with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Recompute Lp full weighted absolute coefficient power sum with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — Recompute Lp constant weighted coefficient atom1 with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute Lp coefficient atom1 absolute power product stage1 with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute Lp constant weighted coefficient atom0 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

(1, 4, 3/4, exact_finite_atomic_moment)

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: 42909441adc3298240cc16c8875119d574e2e51a3101eb58a68a81862555a6f3
- candidate record SHA-256: f4b9168ea4054d427d29e70da06d5349b67261e2af25f302fe3c8d9c0e35e5d3
- semantic review receipt SHA-256: 713db6309b22bf8b31f0253d4aaa7aca9011ab908587ae0e9d4a365edce1f9bc
- question-first witness SHA-256: 0338a3db344ab237113fc63be29153c3684da96e388109e9dcd913a0a693aed2
- Lean: NOT_RUN · human review: NOT_PERFORMED

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