Standard-normal symmetry with explicit named facts

Goal and scope

Separate Gaussian prerequisites from derived CDF symmetry.

No finite-quadrature proof of the Gaussian integral or moments.

Givens and symbols

• work

• G1 — phi(x)=(2pi)^(-1/2)exp(-x²/2), Z has this density, and Phi(t)=P(Z≤t).
• G2 — Explicit named theorem facts: the Gaussian integral is sqrt(2pi), hence normalization; E[Z]=0 and E[Z²]=1. These are premises, not proved here.

• φ — standard-normal density
• Φ — standard-normal CDF
• Z — real random variable

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• mko_definition · cited — Gaussian-integral normalization, E[Z]=0 and Var(Z)=1 are explicitly named theorem facts, not derivations manufactured from finite quadrature. Exact families handle symbolic density, evenness and phi(x)/phi(0)=exp(−x^2/2), plus Phi(0)=1/2 and Phi(−x)=1−Phi(x) from declared normalization and symmetry. No general nonzero elementary CDF value is computed. Finite numerical integration proves neither normalization nor moments; a continuous CDF need not have a density or be normal. · mko-standard-normal-distribution
• work · prerequisite — The Gaussian integral and moments are explicit named theorem prerequisites; also assume integral linearity, substitution, atomlessness of density laws and the second-moment variance identity.

Main derivation

\varphi(x)=\frac{e^{-x^2/2}}{\sqrt{2\pi}}

\varphi(x)>0\quad(x\in\mathbb R)

\int_{\mathbb R}e^{-x^2/2}\,dx=\sqrt{2\pi}\quad\text{(assumed)}

\int_{\mathbb R}\varphi(x)\,dx=1

(-x)^2=x^2

\varphi(-x)=\varphi(x)

\varphi(0)=1/\sqrt{2\pi}

\varphi(x)/\varphi(0)=e^{-x^2/2}

\Phi(-t)=\int_{-\infty}^{-t}\varphi(u)\,du

\Phi(-t)=\int_t^\infty\varphi(v)\,dv=1-\Phi(t)

t=0,\quad\Phi(0)=1-\Phi(0)

\Phi(0)=1/2

E[Z]=0,\quad E[Z^2]=1\quad\text{(assumed)}

\operatorname{Var}(Z)=E[Z^2]-E[Z]^2=1

t=1

\Phi(-1)=1-\Phi(1),\quad\varphi(1)/\varphi(0)=e^{-1/2}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, G2, S07, work — Phi(1) has not been evaluated rationally, and finite integration proves none of the premises.
• S07 · premise_audit · G1, G2, S06, work — Do not relabel the moment premises as an integral proof completed here.
• S06 · premise_audit · G1, G2, S05, work — This exact value follows from symmetry and normalization, not approximation.
• S05 · premise_audit · G1, G2, S04, work — The endpoint discrepancy in the complement has zero mass.
• S04 · premise_audit · G1, G2, S03, work — This is a density ratio, not a CDF ratio.
• S03 · premise_audit · G1, G2, S02, work — The center is 0; no unknown mu has been introduced.
• S02 · premise_audit · G1, G2, S01, work — No finite quadrature points are being used to infer the integral.
• S01 · premise_audit · G1, G2, work — Nonnegativity alone does not prove normalization.

Error controls and negative controls

Premise labeling

Keep normalization and moments labeled as named facts, not newly proved results.

Event and threshold

Phi(t)=P({omega:Z(omega)≤t}); t is a scalar threshold.

• A few quadrature values prove normalization. → reject — Finite values do not prove an integral over the whole line.
• Phi(1)=phi(1). → reject — Positive phi gives Phi(1)>Phi(0)=1/2, whereas phi(1)<phi(0)=1/sqrt(2pi)<1/2 using pi>2, so they differ.
• Every continuous CDF is standard normal. → reject — A uniform distribution has a different continuous CDF.

Result and limitations

Under the explicit Gaussian prerequisites, Phi(-t)=1-Phi(t), Phi(0)=1/2 and Var(Z)=1.

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: ce6e79a0422b1524888abb53a308a69a70552c942a0768637ab742837af70ae2
• original source SHA-256: 9be2b5a4eeeb9649020eca8625ccd6a76eef2d3e32963e884d8eadc5a4490a58
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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