Arbitrary comparison tournaments with Gaussian moments

Research note prepared on 14 September 2026, in response to Tito's request for a novel discovery.

Status: A mathematical result derived in this session, with a self-contained proof and an exactly checked example. Its historical novelty is a candidate claim. A targeted search did not identify the combined statement below, but priority and independent expert verification have not been established.

The result concerns independent numerical outcomes. Write “A beats B” when a fresh draw from A exceeds a fresh draw from B with probability greater than one half. The familiar nontransitive-dice phenomenon allows A to beat B, B to beat C, and C to beat A. Here the proposed extension imposes much stronger distributional regularity and statistical agreement.

Main result. Any finite pattern of pairwise wins can occur among real analytic, strongly log-concave densities that are arbitrarily close to the standard normal density, have the same median and mode, and match any prescribed finite number of its moments. Any specified finite list of its quantiles can also be matched exactly.

“Any finite number” means that a new construction can be made for each chosen number of moments. It does not mean that one collection matches all infinitely many Gaussian moments.

Precise theorem. Let

\phi(x)=\frac{e^{-x^2/2}}{\sqrt{2\pi}},\qquad \Phi(x)=\int_{-\infty}^{x}\phi(y)\,dy.

Fix integers $n\geq2$ , $m\geq2$ , a real skew-symmetric $n\times n$ matrix $A=(a_{ij})$ , and a relative tolerance $\eta>0$ . There are independent random variables $X_1,\ldots,X_n$ , strictly positive real analytic densities $f_i$ , and a number $t>0$ such that:

$\mathbb E[X_i^r]=\mathbb E[Z^r]$ for every $0\leq r\leq m$ , where $Z\sim N(0,1)$ .
Each $X_i$ has median zero and unique mode zero.
$(\log f_i)''(x)\leq-1/2$ for every real $x$ .
$\sup_x|f_i(x)/\phi(x)-1|\leq\eta$ .
For every $i\ne j$ ,

\boxed{\mathbb P(X_i>X_j)=\frac12+t^2a_{ij}.}

If finitely many probability levels $\alpha_1,\ldots,\alpha_s\in(0,1)$ are specified in advance, the construction can additionally satisfy

F_i\bigl(\Phi^{-1}(\alpha_\ell)\bigr)=\alpha_\ell \quad\text{for all }i,\ell.

The distributions and $t$ may depend on every specified input. In particular, the theorem makes no lower bound on the achievable winning margin that is uniform in the number of moments, competitors, or quantiles.

To realize a tournament, choose $a_{ij}=1$ for each desired win and $a_{ji}=-1$ . All its winning probabilities then equal $1/2+t^2$ . Each constructed distribution also ties against an independent standard normal draw.

Proof. All integrals without limits in the following argument are over $\mathbb R$ .

Step 1: Choose perturbations satisfying the constraints.

Consider the real vector space of functions

g(x)=e^{-x^2}p(x),\qquad p\text{ an even polynomial},

subject to the homogeneous linear constraints

\int x^rg(x)\,dx=0\quad(0\leq r\leq m),\qquad \int\phi(x)g(x)\,dx=0,\qquad g(0)=g''(0)=0. \tag{1}

This space is infinite-dimensional: the even polynomials are infinite-dimensional, multiplication by $e^{-x^2}$ is injective, and only finitely many linear constraints have been imposed. Choose $n$ linearly independent members and apply Gram–Schmidt in $L^2(\mathbb R)$ . Their resulting functions $g_1,\ldots,g_n$ remain in the space and satisfy

\int g_i(x)g_j(x)\,dx=\delta_{ij}. \tag{2}

All these functions and their derivatives decay faster than any reciprocal polynomial.

Each derivative $g_i'$ has zero moments through order $m$ , since integration by parts gives, for $r\geq1$ ,

\int x^r g_i'(x)\,dx=-r\int x^{r-1}g_i(x)\,dx=0. \tag{3}

The zeroth derivative moment is also zero by its vanishing boundary values. Evenness, $\Phi(x)+\Phi(-x)=1$ , and (1) give

\int\Phi g_i=\frac12\int g_i=0, \qquad \int\Phi g_i'=-\int\phi g_i=0. \tag{4}

Step 2: Identify the comparison bilinear form.

For zero-integral functions $u,v$ of the above type, define

B(u,v)=\int u(x)\left(\int_{-\infty}^{x}v(y)\,dy\right)dx. \tag{5}

Integration by parts shows $B(u,v)=-B(v,u)$ . The antiderivative of an even zero-integral function is odd. Therefore parity and (2) imply

B(g_i,g_j)=0,\qquad B(g_i',g_j')=0, \qquad B(g_i,g_j')=\delta_{ij}, \qquad B(g_i',g_j)=-\delta_{ij}. \tag{6}

For example, $B(g_i,g_j')=\int g_i g_j$ , because $\int_{-\infty}^{x}g_j'(y)\,dy=g_j(x)$ .

Define

q_i=g_i-\frac12\sum_{k=1}^{n}a_{ik}g_k'. \tag{7}

Equations (6) and skew-symmetry give

B(q_i,q_j)=-\frac12a_{ji}+\frac12a_{ij}=a_{ij}. \tag{8}

Step 3: Obtain exact pairwise probabilities.

Set $f_i=\phi+tq_i$ and $Q_i(x)=\int_{-\infty}^{x}q_i(y)\,dy$ . Positivity for small $t$ is verified below. For independent draws,

\begin{aligned} \mathbb P(X_i>X_j) &=\int(\phi+tq_i)(\Phi+tQ_j)\,dx\\ &=\frac12+t\left(\int\Phi q_i+\int\phi Q_j\right)+t^2B(q_i,q_j)\\ &=\frac12+t^2a_{ij}. \end{aligned} \tag{9}

Here (4) kills $\int\Phi q_i$ , and integration by parts gives $\int\phi Q_j=-\int\Phi q_j=0$ . Equation (9) is exact; no asymptotic remainder is omitted. Equation (4) also gives $\mathbb P(X_i>Z)=1/2$ for an independent $Z\sim N(0,1)$ .

Step 4: Verify normalization, moments, median, and shape.

Equations (1) and (3) show that each $q_i$ has zero moments through order $m$ , establishing normalization and the moment claims.

Evenness gives $\int_{-\infty}^{0}g_i=0$ . Also $\int_{-\infty}^{0}g_i'=g_i(0)=0$ . Thus $F_i(0)=1/2$ . Because $g_i'(0)=0$ by evenness and $g_i''(0)=0$ by construction, $q_i'(0)=0$ , and hence $f_i'(0)=0$ .

Let $v_i=q_i/\phi$ . Each $v_i$ is a polynomial times $e^{-x^2/2}$ , up to a constant, so $v_i$ , $v_i'$ , and $v_i''$ are bounded. Put

M_0=\max_i\|v_i\|_\infty,\qquad M_2=\max_i\|v_i''\|_\infty.

Choose $t>0$ sufficiently small that

tM_0\leq\min\{\eta,1/2\},\qquad tM_2\leq1/4. \tag{10}

Then $f_i=\phi(1+tv_i)>0$ , and the relative-closeness claim follows. Moreover,

\begin{aligned} (\log f_i)'' &=-1+\frac{tv_i''}{1+tv_i} -\frac{t^2(v_i')^2}{(1+tv_i)^2}\\ &\leq-1+2tM_2\leq-\frac12. \end{aligned} \tag{11}

This proves strong log-concavity. Together with $f_i'(0)=0$ , it makes zero the unique mode. Strict positivity makes the median unique as well. Real analyticity follows directly from the construction.

Step 5: Optional prescribed quantiles.

For each $z_\ell=\Phi^{-1}(\alpha_\ell)$ , add to (1) the two linear constraints

g(z_\ell)=0,\qquad \int_{-\infty}^{z_\ell}g(x)\,dx=0. \tag{12}

Only finitely many constraints have been added, so the space remains infinite-dimensional. They survive Gram–Schmidt and imply $Q_i(z_\ell)=0$ . Thus $F_i(z_\ell)=\Phi(z_\ell)$ , as required. This completes the proof.

An explicit example. Define

R(x)=x^4\bigl(3465-6237x^2+2970x^4-484x^6+24x^8\bigr), \qquad g(x)=e^{-x^2}R(x),\qquad \varepsilon=10^{-7}.

Take three independent variables with densities

f_A=\phi+\varepsilon g,\qquad f_B=\phi+\varepsilon g',\qquad f_C=\phi-\varepsilon(g+g'). \tag{13}

Every density is positive and real analytic on the whole real line. Each has the following exact statistics:

Statistic	A	B	C
Mean	0	0	0
Variance	1	1	1
Median	0	0	0
Unique mode	0	0	0
Skewness	0	0	0
Kurtosis	3	3	3
Fifth raw moment	0	0	0

Despite this agreement,

\begin{aligned} \mathbb P(A>B)=\mathbb P(B>C)=\mathbb P(C>A) &=\frac12+\varepsilon^2\int g(x)^2\,dx\\ &=\frac12+\frac{9439930269}{5242880000000000000}\sqrt{\frac\pi2}\\ &\approx0.5000000022566219. \end{aligned} \tag{14}

The probability advantage is deliberately small: about $2.2566\times10^{-9}$ . This is an exact existence example, not evidence of a practically large advantage. No optimality or minimal-degree claim is made for this polynomial.

How the explicit example was verified. The accompanying verify_gaussian_tournaments.py uses only the Python standard library. Run:

python verify_gaussian_tournaments.py

It derives $R$ from four physicists' Hermite polynomials and linear constraints; verifies normalization and all moments through order five; checks the median and stationary point at zero; and evaluates every pairwise comparison coefficient exactly. The moment and comparison integrals reduce to rational Gaussian moments. In particular,

\int g(x)^2\,dx=\frac{47199651345}{262144}\sqrt{\frac\pi2}.

The shape check is global, rather than a grid-based numerical inference. For a polynomial $p$ , its Bernstein coefficients on a rational interval bound its absolute value there. The verifier partitions each of $[-10,0]$ and $[0,10]$ into 160 intervals and uses the exact inequality

e^{-y}\leq\left(\sum_{k=0}^{80}\frac{y^k}{k!}\right)^{-1} \qquad(y\geq0).

For the tails, each $x^ke^{-x^2/2}$ decreases for $x\geq10$ when $k\leq100$ ; the actual polynomials have smaller degrees. Using $\sqrt{2\pi}<3$ , every bound is obtained with rational arithmetic.

For $q_A=g$ , $q_B=g'$ , $q_C=-g-g'$ , the resulting certified integer upper bounds are:

Bound	A	B	C
$\\|q_i/\phi\\|_\infty$	87,124	411,322	491,753
$\\|(q_i/\phi)''\\|_\infty$	357,750	1,922,276	2,172,225

Consequently, for every real $x$ and every density in (13),

\frac{f_i(x)}{\phi(x)}\geq\frac{9508247}{10000000}>0, \qquad (\log f_i)''(x)\leq-\frac{7336022}{9508247}<-\frac12. \tag{15}

These bounds certify the shape and the uniqueness of the mode. The verifier was executed successfully in this session. Floating-point arithmetic is used only for the displayed decimal probability; every verification assertion and enclosure uses exact rational arithmetic. This is an executable mathematical certificate for the explicit example, not a formal proof-assistant verification of the full theorem.

Relationship to existing work and novelty limits. Arbitrary tournaments represented by independent dice are established mathematics. Akin proves such a realization, including a continuous construction with a common mean. That general phenomenon is not claimed as new here. Ethan Akin, Generalized intransitive dice: Mimicking an arbitrary tournament, Journal of Dynamics and Games 8 (2021), 1–20; author's longer preprint.

Equal means and variances in nontransitive discrete dice are also established. Yakusheva studies finite sets under those constraints. Alexandra N. Yakusheva, Nontransitive dice with equal means and variances, Matematicheskaya Teoriya Igr i Ee Prilozheniya 14:3 (2022), 101–120.

The use of a skew comparison kernel also has clear precedent. Sah and Sawhney study the kernel governing random intransitive dice and obtain asymptotic results. The basic bilinear viewpoint is therefore not claimed as new. Ashwin Sah and Mehtaab Sawhney, The intransitive dice kernel, Probability Theory and Related Fields 189 (2024), 1073–1128.

The candidate contribution is the simultaneous realization under all of the theorem's constraints: any finite comparison pattern, any prescribed finite number of Gaussian moments, real analytic and uniformly strongly log-concave densities, arbitrarily small relative deviation from a Gaussian, common median and mode, and optional finitely many specified Gaussian quantiles. The derivative identity $B(g,h')=\int gh$ provides a short construction.

The search on 14 September 2026 included combinations of “nontransitive/intransitive dice,” “same/equal/arbitrary moments,” “moment matching,” “log-concave,” “strongly log-concave,” “smooth densities,” “Gaussian perturbation,” “median,” “mode,” and “tournament.” The relevant theorem statements and the accessible Akin preprint were inspected. No matching combined statement was identified in this search. It was not an exhaustive review of all papers, books, languages, unpublished results, or implications of existing general theorems. Nothing in this note establishes that no human has previously known the result.

Interpretation. Agreement of finitely many moments and quantiles, even among very regular distributions close to a Gaussian, cannot force pairwise winning probabilities into a transitive ranking. The result concerns exact ranking signs at sufficiently small margins. It does not contradict approximate prediction from many moments, or the fact that all Gaussian moments together determine the Gaussian distribution.