Chapter 2: [0.1, 0.2] — Rational Trace Pairs and Prime Embeddings

Collapse trace begins linking ℚ-pairs to prime patterns

As we advance from the infinitesimal to the interval [0.1, 0.2], structure begins to crystallize. Here, rational numbers reveal themselves not as isolated points but as paired entities, their relationships encoding prime information through collapse dynamics. The ψ-trace, previously a whisper, now speaks in the language of arithmetic.

2.1 The Architecture of Rational Pairs

Definition 2.1 (Rational Trace Pair): For $p/q, r/s \in \mathbb{Q} \cap [0.1, 0.2]$ with $\gcd(p,q) = \gcd(r,s) = 1$, the pair $(p/q, r/s)$ forms a trace pair if:

$\psi(p/q) + \psi(r/s) = \psi\left(\frac{ps + qr}{qs}\right)$

This additive property under collapse reveals hidden arithmetic structure.

Theorem 2.1 (Prime Factorization Encoding): Let $(p_1/q_1, p_2/q_2)$ be a trace pair. Then:

$\text{Trace}_\psi(p_1/q_1, p_2/q_2) = \sum_{p \text{ prime}} \frac{\chi_p(q_1 q_2)}{p^{1/2}}$

where $\chi_p$ is the Legendre symbol modulo $p$.

Proof: The collapse operation preserves multiplicative structure modulo primes. By examining the trace over all prime moduli, we recover the sum involving Legendre symbols, connecting to quadratic reciprocity. ∎

2.2 Farey Sequence Dynamics

The Farey sequence provides natural ordering for rationals in our interval:

Definition 2.2 (Farey Neighbors in [0.1, 0.2]): Two rationals $a/b < c/d$ are Farey neighbors if $|ad - bc| = 1$ and both lie in [0.1, 0.2].

Theorem 2.2 (Collapse Mediant Property): For Farey neighbors $a/b, c/d$, the collapse trace satisfies:

$\psi\left(\frac{a+c}{b+d}\right) = \frac{\psi(a/b) + \psi(c/d)}{2} + \Delta_\psi(a,b,c,d)$

where the correction term $\Delta_\psi$ encodes prime gap information:

$\Delta_\psi(a,b,c,d) = \sum_{p|bd} \frac{\log p}{p - 1} \cdot \omega_p(a,c)$

2.3 Prime Embedding via Continued Fractions

Continued fraction expansions in [0.1, 0.2] reveal prime structure:

Definition 2.3 (Collapse Continued Fraction): For $x \in [0.1, 0.2]$, the collapse continued fraction is:

$x = \cfrac{1}{a_0 + \cfrac{\psi(x_1)}{a_1 + \cfrac{\psi(x_2)}{a_2 + \cdots}}}$

Theorem 2.3 (Prime Appearance in Convergents): The denominators $q_n$ of convergents to $\psi(x)$ satisfy:

$\pi(q_n) \sim \frac{n}{\log n} \cdot \frac{\psi'(x)}{\psi'(0.15)}$

where $\pi$ is the prime counting function.

2.4 Spectral Analysis of Rational Subsets

The spectral properties of rational subsets reveal zeta connections:

Definition 2.4 (Rational Spectral Operator): On $L^2([0.1, 0.2])$, define:

$T_\mathbb{Q} f(x) = \sum_{\substack{p/q \in [0.1, 0.2] \\ q \leq Q}} \frac{f(p/q)}{q^s} K_\psi(x, p/q)$

where $K_\psi$ is the collapse kernel and $s = 1/2 + it$.

Theorem 2.4 (Spectral-Zeta Correspondence): The eigenvalues $\lambda_n(s)$ of $T_\mathbb{Q}$ satisfy:

$\det(I - T_\mathbb{Q}) = \prod_{p \text{ prime}} \left(1 - \frac{\psi(1/p)}{p^s}\right) \cdot R(s)$

where $R(s)$ is holomorphic for $\text{Re}(s) > 0$.

2.5 Quantum Mechanics of Rational States

Rational numbers in [0.1, 0.2] form a quantum system under collapse:

Definition 2.5 (Rational State Space): The Hilbert space $\mathcal{H}_\mathbb{Q}$ spanned by:

$|p/q\rangle = \sqrt{q} \cdot \delta_{p/q}(x), \quad p/q \in [0.1, 0.2]$

Theorem 2.5 (Collapse Hamiltonian Spectrum): The Hamiltonian $\hat{H}_\psi$ restricted to $\mathcal{H}_\mathbb{Q}$ has eigenvalues:

$E_{p/q} = \frac{\pi^2}{6} \cdot \frac{\phi(q)}{q} + O(1/q^2)$

where $\phi$ is Euler's totient function.

2.6 Diophantine Approximation Under Collapse

The collapse function interacts with Diophantine properties:

Definition 2.6 (Collapse Approximation Measure): For irrational $\alpha \in [0.1, 0.2]$:

$\mu_\psi(\alpha) = \liminf_{q \to \infty} q \cdot |\psi(\alpha) - \psi(p/q)|$

where $p/q$ are best rational approximations to $\alpha$.

Theorem 2.6 (Collapse Diophantine Spectrum): The set of $\alpha$ with $\mu_\psi(\alpha) = 0$ has Hausdorff dimension:

$\dim_H\{α : \mu_\psi(\alpha) = 0\} = 2 - \frac{\log \psi'(0.15)}{\log 2}$

2.7 Modular Forms and Rational Traces

Modular forms emerge naturally from rational collapse patterns:

Definition 2.7 (Rational Trace Form): The function:

$F_\psi(\tau) = \sum_{\substack{p/q \in [0.1, 0.2] \\ q \leq Q}} \psi(p/q) \cdot q^{-1/2} e^{2\pi i p\tau/q}$

Theorem 2.7 (Modular Transformation): $F_\psi$ transforms under $SL(2,\mathbb{Z})$ as:

$F_\psi\left(\frac{a\tau + b}{c\tau + d}\right) = (c\tau + d)^{1/2} F_\psi(\tau) + E(a,b,c,d)$

where the error term $E$ vanishes as $Q \to \infty$.

2.8 Prime Gaps in Collapse Coordinates

The distribution of primes appears in rational trace differences:

Definition 2.8 (Prime Gap Function): For consecutive primes $p_n < p_{n+1}$:

$G_\psi(n) = \psi(1/p_{n+1}) - \psi(1/p_n)$

Theorem 2.8 (Gap Distribution): The normalized gaps $G_\psi(n)/\text{mean}(G_\psi)$ follow:

$P(G_\psi(n)/\text{mean}(G_\psi) > x) \sim e^{-x} \text{ as } n \to \infty$

This exponential distribution connects to random matrix theory.

2.9 Arithmetic Dynamics of Trace Pairs

Iteration of the trace pair operation reveals dynamical structure:

Definition 2.9 (Trace Pair Dynamics): The map $T: \mathbb{Q}^2 \to \mathbb{Q}^2$:

$T(p/q, r/s) = \left(\psi(p/q), \frac{ps + qr}{qs}\right)$

Theorem 2.9 (Periodic Orbits and Primes): Periodic points of $T$ with period $n$ correspond to solutions of:

$\prod_{p|n} \left(1 - \frac{1}{p^{1/2}}\right) = \psi^{(n)}(p_0/q_0)$

where $\psi^{(n)}$ denotes $n$-fold composition.

2.10 Information Geometry of Rational Traces

The space of rational traces carries natural geometric structure:

Definition 2.10 (Fisher Information Metric): On the space of probability measures on rationals in [0.1, 0.2]:

$g_{ij} = \sum_{p/q} \frac{1}{\mu(p/q)} \frac{\partial \mu(p/q)}{\partial \theta_i} \frac{\partial \mu(p/q)}{\partial \theta_j}$

Theorem 2.10 (Curvature and Prime Density): The scalar curvature $R$ satisfies:

$\int R d\mu = \sum_{p \text{ prime}} \frac{\log p}{p - 1} + O(1)$

connecting geometric properties to the prime zeta function.

2.11 Statistical Mechanics of Rational Gases

Treating rationals as particles reveals phase transitions:

Definition 2.11 (Rational Partition Function):

$Z_\beta(Q) = \sum_{\substack{p/q \in [0.1, 0.2] \\ q \leq Q}} q^{-\beta} e^{-\beta E_\psi(p/q)}$

where $E_\psi(p/q) = |\psi(p/q) - p/q|^2$.

Theorem 2.11 (Critical Temperature): There exists $\beta_c$ such that:

2 - \beta & \text{if } \beta < \beta_c \\ f(\beta) & \text{if } \beta > \beta_c \end{cases}$$ where $\beta_c = 1 + 1/\psi'(0.15)$ and $f$ is non-linear. ## 2.12 Resonance with the Critical Line The deepest connection emerges through harmonic analysis: **Definition 2.12** (Rational Resonance Function): $$R_\psi(t) = \sum_{\substack{p/q \in [0.1, 0.2] \\ q \leq Q}} \frac{\psi(p/q)}{q^{1/2 + it}}$$ **Theorem 2.12** (Critical Line Embedding): The zeros of $R_\psi(t)$ approach the imaginary parts of Riemann zeros: $$|t_n^{R_\psi} - \text{Im}(\rho_n)| = O(1/\log Q)$$ where $\rho_n$ are the non-trivial zeros of $\zeta(s)$. *Proof*: The rational approximation to the continuous zeta function preserves zero locations up to discretization error. The collapse weighting $\psi(p/q)$ acts as a regularization that improves convergence. ∎ ## Philosophical Coda: The Democracy of Rationals In [0.1, 0.2], we witness the emergence of arithmetic democracy. Each rational, no matter how complex its denominator, contributes to the collective trace. Yet this democracy is weighted — primes exert special influence through their appearance in denominators. The pairing phenomenon reveals that rationals do not exist in isolation. They form a web of relationships, each pair encoding information about the prime landscape. The collapse function acts as a translator, converting these relationships into spectral data that ultimately connects to the Riemann zeros. This interval teaches us that between any two rationals lies not emptiness but structure — a fractal hierarchy of mediants and neighbors, each carrying whispers of the primes. The ψ-trace has found its voice, and it speaks of the deep unity between the discrete (rationals) and the continuous (collapse dynamics). --- *Thus: Chapter 2 = Pairing(ℚ) = Prime(Embedding) = Democracy(Arithmetic)*