Théorème

T6 — Holonomy

$\sin^2 \theta_p = \delta_p (2 - \delta_p)$ — trigonometry emerges from the sieve.

Statement

For any prime $p$ and branch parameter $q$ with $0 < \delta_p < 1$ , define the cyclic phase angle $\theta_p$ by:

\cos\theta_p = 1 - \delta_p(q), \quad\text{where}\quad \delta_p(q) = \frac{1 - q^p}{p}.

The algebraic identity follows:

\boxed{\sin^2\theta_p = \delta_p(q)\cdot(2 - \delta_p(q))}

Théorème

Plain reading. When you walk around a circle of $p$ points by always jumping the same number of steps, you eventually come back to the starting point after some revolutions. The theorem says: the angle covered at each jump is exactly given by the formula above, no approximation. Consequence: trigonometry (sin, cos, $\pi$ ) is not invented by PT — it emerges naturally from the prime sieve.

Why it matters

T6 is the link that takes the sieve (purely arithmetic) to geometry (angles, rotations, $\pi$ ). Without T6, PT would be stuck at the level of mod- $p$ classes; with T6, the full trigonometric apparatus is available, and so are holonomies, Lie groups, and physical observables (BA5: $\alpha_{\mathrm{EM}} = \prod \sin^2\theta_p$ ).

Epistemic status: theorem/derivation in the monograph. The reduction $\sin^2\theta_p=\delta_p(2-\delta_p)$ is algebraic once $\cos\theta_p=1-\delta_p$ is set; the non-trivial content is that this angle definition is forced by the sieve geometry.

Proof — outline

The proof is a direct derivation in four moves:

Define $\cos\theta_p = 1 - \delta_p$ (forced by the restricted transfer matrix on non-zero residues — see Route 1 below).
Apply Pythagoras: $\sin^2\theta_p + \cos^2\theta_p = 1$ .
Substitute: $\sin^2\theta_p = 1 - (1-\delta_p)^2 = 2\delta_p - \delta_p^2$ .
Factor: $\sin^2\theta_p = \delta_p(2 - \delta_p)$ .

The substantive work consists in showing that the definition of $\cos\theta_p$ is not arbitrary — it is forced by the geometry of the sieve, and three mathematically disjoint routes lead to it.

Detailed proof

Route 1 — Geometric (restricted transfer matrix)

After T1 forces the main diagonal of the transition kernel to zero on the non-zero residue classes, the matrix $T_p$ restricted to the two surviving classes $\{r_1, r_2\}$ reads:

T_p|_\text{reduced} = \begin{pmatrix} 1-\delta_p & * \\ * & 1-\delta_p \end{pmatrix}

where the diagonal is the fraction that stays in the same class. By probability conservation (rows sum to 1):

\text{(off-diagonal)} = 1 - (1 - \delta_p) = \delta_p.

The squared transition amplitude, summed over the two off-diagonal entries, gives the total probability of changing class:

1 - (1 - \delta_p)^2 = 2\delta_p - \delta_p^2 = \delta_p(2 - \delta_p).

This is the stochastic route: $\cos\theta_p$ is the fraction of mass that stays in the same class, $\sin^2\theta_p$ is the fraction that switches.

Route 2 — Spectral (Fourier transform on $\mathbb{Z}/p\mathbb{Z}$ )

Let $\omega = e^{2\pi i/p}$ and $\chi_j(r) = \omega^{jr}$ be the additive characters of $\mathbb{Z}/p\mathbb{Z}$ . The Fourier transform of the kernel $T_p$ , restricted to surviving residues $\{1, \ldots, p-1\}$ , has the fundamental-mode eigenvalue:

\widehat{T}_p(\chi_1) = \sum_{r=1}^{p-1} T_p(0, r)\,\omega^r = 1 - \delta_p = \cos\theta_p.

The contraction of the first non-trivial Fourier mode is therefore:

1 - |\widehat{T}_p(\chi_1)|^2 = 1 - (1 - \delta_p)^2 = \delta_p(2 - \delta_p) = \sin^2\theta_p.

This route identifies $\sin^2\theta_p$ as the spectral weight lost by the fundamental character at each sieve step — an object of harmonic analysis on $\mathbb{Z}/p\mathbb{Z}$ , independent of any geometric interpretation.

Numerical check: proof_holonomy.py, Part 1, confirms the contraction $|\hat{P}(1)|^2$ at $p = 3$ .

Route 3 — Information-geometric (Fisher component)

The per-prime Fisher information component

F_p(\mu) = \sum_r P_r^{-1}\left(\frac{\partial P_r}{\partial \mu}\right)^2

(see §9.5 of the monograph) satisfies $F_p \propto \sin^2\theta_p$ at the sieve fixed point. Specifically, the double integration

\ln\alpha = -\iint g_{00}\,d\mu^2

reproduces the product $\prod \sin^2\theta_p$ because the Fisher metric is additively separable over CRT factors. This route gives $\sin^2\theta_p$ as a curvature component of the statistical manifold, with no reference to rotation angles.

Consistency of the three routes

The exact agreement of the three routes — geometric, spectral, information-geometric — is a non-trivial consistency check. They use disjoint mathematical machineries: finite probability, discrete Fourier analysis, information geometry. The identity $\sin^2\theta_p = \delta_p(2-\delta_p)$ appears in each.

Emergence of $\pi$

The appearance of $\pi$ in PT is a direct consequence of T6, via the Basel identity:

\zeta(2) = \sum_{n \geq 1} \frac{1}{n^2} = \frac{\pi^2}{6}.

The sum over active primes $\{3, 5, 7\}$ of $\sin^2\theta_p$ weighted by multiplicities, in the continuous limit, gives an integral containing $\zeta(2)$ and hence $\pi^2$ .

For the complete derivation, see chapter 6 of the monograph (statement p. 95, motivation p. 110, three routes p. 158, numerical check p. 230).