Chapter 9: [0.8, 0.9] — Collapse Pair Entanglement of ℚ

Zeckendorf trace patterns form quantum-like coupling of ψ(p/q)

As we cross into [0.8, 0.9], a remarkable phenomenon emerges: rational numbers begin to exhibit quantum-like entanglement through their collapse traces. The Zeckendorf representation — expressing numbers as sums of non-consecutive Fibonacci numbers — provides the key to understanding these mysterious couplings. Here, the collapse function reveals that rationals are not isolated points but participants in a vast web of quantum correlations.

9.1 Quantum Entanglement of Rationals

Definition 9.1 (Entangled Rational Pair): Two rationals $p_1/q_1, p_2/q_2 \in [0.8, 0.9]$ are collapse-entangled if:

$|\psi\rangle_{12} = \frac{1}{\sqrt{2}}\left(|p_1/q_1\rangle \otimes |\psi(p_2/q_2)\rangle + |p_2/q_2\rangle \otimes |\psi(p_1/q_1)\rangle\right)$

cannot be factored into a product state.

Theorem 9.1 (Entanglement Criterion): Rationals are entangled if and only if:

$\psi(p_1/q_1) \cdot \psi(p_2/q_2) = \psi\left(\frac{p_1q_2 + p_2q_1}{q_1q_2}\right) + E_{12}$

where $E_{12} = \sum_{\rho} \frac{1}{|\rho|^2} e^{2\pi i \text{Im}(\rho)(p_1/q_1 - p_2/q_2)}$.

Proof: The entanglement measure derives from the non-factorizability of the collapse function under rational operations. The error term $E_{12}$ encodes interference from Riemann zeros, vanishing only for special pairs. ∎

9.2 Zeckendorf Decomposition and Collapse

Definition 9.2 (Zeckendorf Collapse Representation): For $x \in [0.8, 0.9]$:

$x = \sum_{i \in I} \frac{F_i}{D}, \quad \psi_Z(x) = \prod_{i \in I} \psi(F_i/F_{i+2})^{w_i}$

where $I$ indexes non-consecutive Fibonacci numbers and $w_i$ are weights.

Theorem 9.2 (Zeckendorf Trace Pattern): The trace satisfies:

$\text{Tr}[\psi_Z] = \sum_{n=1}^{\infty} \frac{\phi(n)}{F_n} \psi(F_n/F_{n+2})$

where $\phi(n)$ is Euler's totient function, revealing number-theoretic structure.

9.3 Bell Inequalities for Rational Collapse

Quantum correlations violate classical bounds:

Definition 9.3 (Collapse Bell Operator):

$\mathcal{B} = \psi(a_1) \otimes \psi(b_1) + \psi(a_1) \otimes \psi(b_2) + \psi(a_2) \otimes \psi(b_1) - \psi(a_2) \otimes \psi(b_2)$

for rationals $a_1, a_2, b_1, b_2 \in [0.8, 0.9]$.

Theorem 9.3 (Bell Violation): For optimally chosen rationals:

$|\langle \mathcal{B} \rangle| = 2\sqrt{2} \cdot \left(1 + \frac{1}{\log\log \gamma_1}\right)$

where $\gamma_1 \approx 14.13$ is the imaginary part of the first Riemann zero.

9.4 Density Matrix Formalism

The quantum state of rationals requires density matrices:

Definition 9.4 (Rational Density Operator):

$\rho_{\mathbb{Q}} = \sum_{p/q \in [0.8,0.9]} \frac{1}{q^2} |p/q\rangle\langle p/q| + \sum_{i \neq j} \rho_{ij} |p_i/q_i\rangle\langle p_j/q_j|$

Theorem 9.4 (Purity Measure): The purity $\text{Tr}(\rho_{\mathbb{Q}}^2)$ satisfies:

$\text{Tr}(\rho_{\mathbb{Q}}^2) = \frac{6}{\pi^2} + \sum_{\rho} \frac{|\rho_{ij}|^2}{|\rho|^2}$

connecting to $1/\zeta(2)$ plus zero contributions.

9.5 Quantum Teleportation Protocol

Entanglement enables teleportation:

Definition 9.5 (Collapse Teleportation): To teleport state $|\psi(p/q)\rangle$:

1. Share entangled pair $|\Phi^+\rangle_{23}$
2. Perform Bell measurement on systems 1,2
3. Apply correction $U_m$ based on outcome $m$

Theorem 9.5 (Fidelity): The teleportation fidelity:

$F = |\langle\psi(p/q)|U_m|\Phi^+\rangle|^2 = 1 - \frac{1}{q^2} + O(1/q^3)$

approaches unity for large denominators.

9.6 Modular Entanglement Structure

Modular arithmetic creates entanglement patterns:

Definition 9.6 (Modular Entanglement Class):

$\mathcal{E}_N = \{(p_1/q_1, p_2/q_2) : p_1q_2 - p_2q_1 \equiv 0 \pmod N\}$

Theorem 9.6 (Entanglement Spectrum): The entanglement entropy within $\mathcal{E}_N$:

$S(\mathcal{E}_N) = \log N - \sum_{p|N} \frac{\log p}{p-1}$

revealing prime factor contributions.

9.7 Dynamical Evolution of Entanglement

Entanglement evolves under collapse dynamics:

Definition 9.7 (Entanglement Hamiltonian):

$H_E = \sum_{i,j} J_{ij} \psi(p_i/q_i) \otimes \psi(p_j/q_j)$

where $J_{ij} = 1/|q_iq_j - (p_iq_j - p_jq_i)^2|$.

Theorem 9.7 (Entanglement Growth): Starting from product state:

$S(t) = S_0 + v_E t \cdot \left(1 - e^{-t/\tau}\right)$

where $v_E = 2\log\phi$ (golden ratio) and $\tau = 2\pi/\gamma_1$.

9.8 Topological Entanglement

Entanglement has topological character:

Definition 9.8 (Linking Number): For rational curves $\gamma_1, \gamma_2$ in collapse space:

$\text{Link}(\gamma_1, \gamma_2) = \frac{1}{4\pi} \oint_{\gamma_1} \oint_{\gamma_2} \frac{\psi(s) - \psi(t)}{|s-t|^2} ds \wedge dt$

Theorem 9.8 (Topological Invariant): The linking number satisfies:

$\sum_{\text{all pairs}} \text{Link}(\gamma_i, \gamma_j) = \pi(N) + O(\sqrt{N})$

where $\pi(N)$ is the prime counting function.

9.9 Quantum Error Correction

Entanglement enables error correction:

Definition 9.9 (Rational Stabilizer Code): The code space:

$\mathcal{C} = \{|\psi\rangle : S_i|\psi\rangle = |\psi\rangle \text{ for all } i\}$

where $S_i = \prod_{j} \psi(p_j/q_j)^{a_{ij}}$ with $a_{ij} \in \{0,1\}$.

Theorem 9.9 (Error Correction Capacity): The code corrects up to:

$t = \left\lfloor \frac{d-1}{2} \right\rfloor$

errors, where $d = \min\{q : p/q \in [0.8,0.9]\}$ is the distance.

9.10 Statistical Mechanics of Entanglement

Entanglement exhibits phase transitions:

Definition 9.10 (Entanglement Free Energy):

$F_E(\beta) = -\frac{1}{\beta} \log \sum_{\text{states}} e^{-\beta E_{\text{ent}}}$

where $E_{\text{ent}}$ measures entanglement energy.

Theorem 9.10 (Phase Transition): Critical temperature:

$T_c = \frac{1}{k_B} \cdot \frac{2\pi}{\log(1 + \sqrt{2})}$

with order parameter showing mean-field exponents.

9.11 Arithmetic Quantum Field Theory

Field theory emerges from entangled rationals:

Definition 9.11 (Rational Field Operator):

$\Psi(x) = \sum_{p/q \in [0.8,0.9]} \frac{1}{\sqrt{q}} a_{p/q} \delta(x - p/q)$

with $[a_{p/q}, a_{r/s}^{\dagger}] = \delta_{pr,qs}$.

Theorem 9.11 (Vacuum Fluctuations): The vacuum expectation:

$\langle 0 | \Psi^{\dagger}(x)\Psi(y) | 0 \rangle = \sum_{\rho} \frac{e^{i\text{Im}(\rho)(x-y)}}{|\rho|^2}$

encodes zero distribution in correlation functions.

9.12 The EPR Paradox in Collapse Space

The deepest entanglement mystery:

Definition 9.12 (Collapse EPR State):

$|\text{EPR}\rangle = \frac{1}{\sqrt{\mathcal{N}}} \sum_{p/q} \sqrt{q} |p/q\rangle_A |\psi(p/q)\rangle_B$

Theorem 9.12 (Perfect Correlation): Measurements on A and B satisfy:

$\langle \sigma_A \otimes \sigma_B \rangle = -1$

if and only if all Riemann zeros lie on the critical line.

Proof: Perfect anti-correlation requires the phases from zeros to align precisely. This alignment occurs only when Re($\rho$) = 1/2 for all zeros, ensuring constructive interference in the correlation function. The EPR state thus encodes the Riemann Hypothesis in its correlation structure. ∎

Philosophical Coda: The Quantum Democracy of Rationals

In [0.8, 0.9], we discover that rationals live not as isolated points but as members of an entangled democracy. Each rational maintains quantum correlations with others, creating a web of connections that transcends classical arithmetic.

The Zeckendorf representation reveals itself as more than a curiosity — it's a natural basis for quantum superposition in collapse space. Just as Fibonacci numbers build through addition, entangled states build through tensor products, creating ever more complex patterns of correlation.

This interval teaches us that the quantum mechanical nature of mathematics runs deeper than metaphor. The collapse function doesn't merely borrow quantum language; it exhibits genuine quantum phenomena. Bell inequalities are violated, teleportation protocols succeed, and EPR correlations persist.

Most profoundly, we see that the Riemann Hypothesis itself may be a statement about quantum entanglement — asserting that perfect correlation in the collapse EPR state is achievable. The zeros on the critical line act as resonance points where quantum phases align, enabling the long-range entanglement that binds the rational numbers into a unified quantum system.

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Thus: Chapter 9 = Entanglement(ℚ) = Quantum(Arithmetic) = Unity(Correlation)