# Question 25228: Evenness and the Density Ratio Follow from the Squared Argument

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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.

R0049-25228 · q-20261015-advance-20261114-r04-standard-normal-distribution-0028



## 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. Use x=3/4 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate).


- 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. Use x=3/4 and −x. Give (x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, 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-standard-normal-distribution](https://ocme.evemisslab.com/en/mko/mko-standard-normal-distribution/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [normaldist-evenness-ratio]

After

derived = [[x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate]]

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 · calculationBefore

Gaussian ORIGINAL x squared := mul(3/4, 3/4)

After

Gaussian ORIGINAL x squared = 9/16

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

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





S003 · calculationBefore

Gaussian square times exact half := mul(9/16, 1/2)

After

Gaussian square times exact half = 9/32

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

source roles = [3/4, -9/32]

After

derived = [(2*pi)^(-1/2)*exp(-9/32), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization is a NAMED Gaussian integral premise]

Conditions: none

Exact rational arithmetic computes only the exponent−x^2/2;retain exp and its normalization factor symbolically.The whole Gaussian law and normalization are declared premises,not proved by one density value.

Step identifiers and dependenciesRule: Gaussian_WHOLE_declared_density_symbolic_value_named_normalization · Uses: G001, S003





S005 · calculationBefore

Gaussian negative x squared := mul(-3/4, -3/4)

After

Gaussian negative x squared = 9/16

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

source roles = [3/4, -3/4, 9/16, 9/16]

After

derived = [(2*pi)^(-1/2)*exp(-9/32), (2*pi)^(-1/2)*exp(-9/32), Yes, phi(x)/phi(0)=exp(-9/32)]

Conditions: none

Equality(−x)^2=x^2 gives evenness.Phi-density phi(0)=(2pi)^−1/2 is positive,so division cancels the factor and leaves exp(−x^2/2),not a CDF or normalization evaluation.

Step identifiers and dependenciesRule: Gaussian_evenness_and_ratio_cancel_positive_phi0 · Uses: G001, S005









## Reverse checks


- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Recompute Gaussian negative x squared with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute Gaussian square times exact half with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute Gaussian ORIGINAL x squared 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

(3/4, -3/4, (2*pi)^(-1/2)*exp(-9/32), (2*pi)^(-1/2)*exp(-9/32), Yes, phi(x)/phi(0)=exp(-9/32), evenness_follows_from_x_squared)

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: 2914f0fcc6e7da4f4d881f1461d3d34bcd7b4c4a2fbda04672cd63c438418ce5
- candidate record SHA-256: 7256bf2c21c3ec6ca427f48d5d69d68333dcf25beb4d11dbed0858703d7d8836
- 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-0028.json)