# Question 25254: Finite Windows and Panel Counts Do Not Prove Gaussian Normalization or Moments

[Return to catalogue page 127](https://ocme.evemisslab.com/en/questions/page/127/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations/#q-20261015-advance-20261114-r04-standard-normal-distribution-0079)

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.

R0049-25254 · q-20261015-advance-20261114-r04-standard-normal-distribution-0079



## Original question and givens

Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=4>0, panel count=13. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).


- G001 — Use the declared standard-normal law with Gaussian density phi(x)=(2*pi)^(-1/2)*exp(-x^2/2) and its continuous CDF Phi. Gaussian normalization, mean0 and variance1 remain explicitly named theorem facts, not results of finite quadrature in this slice. Declared finite window=4>0, panel count=13. Give (window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary).





## 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-standard-normal-distribution](https://ocme.evemisslab.com/en/mko/mko-standard-normal-distribution/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [normaldist-normalization-moment-boundary]

After

derived = [[window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary]]

Conditions: none

List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape.

Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [4, 13]

After

derived = [finite_quadrature_only, No, named normalization1,mean0,variance1]

Conditions: none

Only window/panels are supplied;no quadrature or tail/error control is performed.A finite window omits positive tails and a panel count does not prove an exact global integral or moments.Named facts are neither proved here nor negated by No.

Step identifiers and dependenciesRule: Gaussian_finite_window_panels_NO_quadrature_NO_theorem_proof · Uses: G001, S001









## Reverse checks


- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(4, 13, finite_quadrature_only, No, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

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: 3d8bf75af969ec6833f86652ce13706442fef461ba7ebc9428d3644ce446d582
- candidate record SHA-256: 20ce8d653bed89fafa28dd107dcee2676940769621e6d99903286e3186ddb664
- semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3
- question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261114-r04-standard-normal-distribution-0079.json)