Théorème

L0 — Unique geometric distribution

The unique memoryless maximum-entropy distribution on even gaps.

Statement

Let $X$ be a random variable on the even gaps $\{2, 4, 6, \ldots\}$ satisfying:

(a) memorylessness: $P(X > a + b \mid X > a) = P(X > b)$ ,
(b) fixed mean: $\mathbb{E}[X] = \mu$ ,
(c) maximum (Shannon) entropy among distributions satisfying (a) and (b).

Then $X$ necessarily follows a geometric law:

\boxed{P(X = 2k) = (1 - q)\, q^{k-1}, \qquad q = 1 - 2/\mu, \qquad k \geq 1.}

Théorème

Plain reading. If we look for the “simplest” probability law to describe gaps between primes — one that invents no hidden structure — there’s no choice: it’s the geometric law. Like flipping a coin to decide when the next prime arrives, except the bias is fixed by the mean $\mu$ .

Why it matters

L0 is the second link of the chain (after T0). Once T0 has identified $\{g_n\}$ as the dynamical field, L0 fixes its statistical law without any arbitrary choice. This is what makes PT parameter-free: no alternative distribution can be inserted in place of the geometric law without violating one of the three conditions (a)(b)(c).

Direct consequence: everything that follows (T1, T2, T6, etc.) inherits the parametrisation $q = 1 - 2/\mu$ , and $q$ becomes the unique free variable of the system — fixed itself at the fixed point by T5.

Proof — outline

Reformulate: “memorylessness” on $\mathbb{N}$ implies $\log P(X = 2k)$ is affine in $k$ .
Lagrange multipliers: max entropy under $\mathbb{E}[X] = \mu$ gives an exponential family.
Identify the law: geometric with parameter $q$ .
Compute $q$ from $\mu$ : the relation $\mu = 2/(1-q)$ inverts to $q = 1 - 2/\mu$ .

Detailed proof

Step 1 — Consequence of memorylessness

Memorylessness $P(X > a + b \mid X > a) = P(X > b)$ on the discrete support $\{2, 4, 6, \ldots\}$ implies:

P(X > 2(k+1)) = P(X > 2) \cdot P(X > 2k).

By induction: $P(X > 2k) = q^k$ with $q = P(X > 2) \in (0, 1)$ . And so:

P(X = 2k) = P(X > 2(k-1)) - P(X > 2k) = q^{k-1}(1 - q).

This is already the geometric form. Uniqueness among memoryless laws is immediate.

Step 2 — Entropy maximisation

Without condition (a), the class of distributions on $\{2, 4, 6, \ldots\}$ with $\sum_k 2k \cdot p_k = \mu$ and $\sum_k p_k = 1$ is broader. By Lagrange multipliers:

\mathcal{L} = -\sum_k p_k \log p_k - \alpha\!\left(\sum_k 2k\, p_k - \mu\right) - \beta\!\left(\sum_k p_k - 1\right).

The condition $\partial \mathcal{L} / \partial p_k = 0$ gives $\log p_k = -1 - 2 \alpha k - \beta$ , so $p_k \propto e^{-2 \alpha k}$ . We recover a geometric family with $q = e^{-2\alpha}$ .

Argmax uniqueness (entropy is strictly concave) ensures only one solution in this family at fixed $\mu$ .

Step 3 — Conjunction (a) + (c)

Step 1 shows: (a) alone ⇒ geometric. Step 2 shows: (b) + (c) alone ⇒ geometric.

Theorem L0 asserts: (a) ∧ (b) ∧ (c) determine the same geometric distribution with forced consistency: the value of $q$ is fixed both by (a) (free parameter of the exponential model) and by (b) (relation to $\mu$ ). Compatibility between the two is guaranteed by the property $\mathbb{E}[X] = \frac{2}{1 - q}$ for $X$ geometric on the evens.

Step 4 — Computing $q$

For $X$ geometric on $\{2, 4, 6, \ldots\}$ with parameter $q$ :

\mathbb{E}[X] = \sum_{k \geq 1} 2k \cdot (1 - q) q^{k-1} = 2(1 - q) \cdot \frac{1}{(1 - q)^2} = \frac{2}{1 - q}.

Imposing $\mathbb{E}[X] = \mu$ inverts to:

q = 1 - \frac{2}{\mu}.

This is the $q_+$ branch of PT. The branch $q_- = e^{-1/\mu}$ appears in the continuous version (high-energy limit) and gives geometry / quarks rather than couplings / leptons.

Why no other law?

The only other distributions on $\mathbb{N}$ with maximum entropy at fixed mean are:

the exponential law (on $\mathbb{R}_+$ ) — excluded because $\{g_n\}$ is discrete,
the Poisson law — excluded because it does not satisfy memorylessness (a) on the evens.

Geometric is the unique candidate on $\{2, 4, 6, \ldots\}$ satisfying all three conditions simultaneously.

For the complete derivation, see chapter 3 of the monograph (L0 proved in section “Maximum entropy under mean constraint”).