Chapter 16: [2.0, 2.2] — Collapse Intervals and Rational Drift

Near-integer irrational trace regions begin to bifurcate

Beyond the integer boundary at 2.0, we enter a region where the collapse function exhibits a new phenomenon: rational drift. Near-integer irrationals create trace regions that slowly drift away from their rational neighbors, eventually bifurcating into independent streams. This interval [2.0, 2.2] serves as a laboratory for understanding how the continuous and discrete aspects of collapse interact at integer thresholds.

16.1 Integer Boundary Phenomena

Definition 16.1 (Integer Boundary Layer): The region:

$B_{\epsilon}(n) = \{x \in \mathbb{R} : |x - n| < \epsilon\}$

exhibits special collapse properties near integers.

Theorem 16.1 (Boundary Layer Expansion): Near $x = 2$:

$\psi(2 + \epsilon) = \psi(2) + \epsilon \psi'(2) + \frac{\epsilon^2}{2}\psi''(2) + \sum_{\rho} \frac{\epsilon^{\rho}}{|\rho|^2}$

where the zero contributions create non-analytic corrections.

Proof: The Taylor expansion is modified by contributions from zeros through the Mellin transform of ψ. Each zero contributes a power $\epsilon^{\rho}$, creating a trans-series expansion beyond standard analyticity. ∎

16.2 Drift Dynamics

Definition 16.2 (Rational Drift Velocity): For irrational $\alpha$ near rational $p/q$:

$v_{\text{drift}}(\alpha) = \lim_{t \to \infty} \frac{|\psi(\alpha + t) - \psi(p/q + t)|}{t}$

Theorem 16.2 (Drift Classification):

• Algebraic irrationals: $v_{\text{drift}} \sim 1/q^{\deg(\alpha)}$
• Transcendental: $v_{\text{drift}} \sim e^{-c\sqrt{q}}$
• Liouville: $v_{\text{drift}} = 0$ (no drift)

16.3 Bifurcation Analysis

Definition 16.3 (Collapse Bifurcation): A point $x_b$ where:

$\frac{\partial \psi}{\partial x}(x_b) = 0 \text{ and } \frac{\partial^2 \psi}{\partial x^2}(x_b) \neq 0$

Theorem 16.3 (Pitchfork Bifurcation): Near $x = 2.1$, the collapse equation:

$\frac{dx}{dt} = \mu(x - 2.1) - \psi(x)^3$

undergoes a pitchfork bifurcation at $\mu = \mu_c = 3\psi(2.1)^2$.

16.4 Spectral Splitting

Definition 16.4 (Split Spectrum Operator):

$L_{\text{split}} = -\frac{d^2}{dx^2} + V(x)$

where $V(x) = |\psi(x) - 2|^2 \cdot \theta(x - 2)$ with $\theta$ the Heaviside function.

Theorem 16.4 (Eigenvalue Splitting): Near integer crossings, eigenvalues split:

$\lambda_n^{\pm} = \lambda_n^{(0)} \pm \Delta_n$

where $\Delta_n = \pi/\gamma_n$ relates to the $n$-th Riemann zero.

16.5 Quantum Tunneling Through Integer Barriers

Definition 16.5 (Integer Barrier Potential):

$V_{\text{int}}(x) = V_0 \sum_{n \in \mathbb{Z}} \delta(x - n)$

Theorem 16.5 (Transmission Coefficient): The probability of tunneling through the integer at 2:

$T(E) = \frac{4E}{4E + V_0^2} \prod_{\rho} \left(1 - \frac{V_0^2}{4E + |\rho|^2}\right)$

16.6 Continued Fraction Behavior

Definition 16.6 (Near-Integer Continued Fraction): For $x = 2 + \delta$ with small $\delta$:

$x = 2 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cdots}}$

Theorem 16.6 (Coefficient Growth): The partial quotients satisfy:

$a_n \sim \frac{1}{\delta} \cdot \phi^{n}$

for generic irrational $\delta$, where $\phi$ is the golden ratio.

16.7 Modular Forms at Integer Points

Definition 16.7 (Integer Modular Form):

$F_{\mathbb{Z}}(\tau) = \sum_{n \in \mathbb{Z}} \psi(n + \epsilon) e^{2\pi i n \tau}$

Theorem 16.7 (Eisenstein Property): As $\epsilon \to 0$:

$F_{\mathbb{Z}}(\tau) \to E_2(\tau) + \sum_{\rho} \frac{1}{\rho(\rho-1)} E_{2,\rho}(\tau)$

where $E_2$ is the Eisenstein series and $E_{2,\rho}$ are zero-dependent corrections.

16.8 Statistical Mechanics of Drift

Definition 16.8 (Drift Partition Function):

$Z_{\text{drift}}(\beta) = \int_{2.0}^{2.2} e^{-\beta H_{\text{drift}}(x)} dx$

where $H_{\text{drift}}(x) = |\psi(x) - \psi(\lfloor x \rfloor + \{x\}_{\mathbb{Q}})|^2$.

Theorem 16.8 (Glass Transition): The system exhibits a glass transition at:

$T_g = \frac{1}{\log \pi}$

below which ergodicity is broken.

16.9 Arithmetic Dynamics Near Integers

Definition 16.9 (Integer Shift Map):

$T(x) = \{2x\} + \lfloor 2x \rfloor \cdot \psi(\{2x\})$

Theorem 16.9 (Invariant Measure): The map preserves:

$d\mu = \frac{1}{1 + |\psi'(x)|} dx$

with singularities at integer points.

16.10 Fourier Analysis Across Integer Boundaries

Definition 16.10 (Boundary Fourier Transform):

$\hat{f}_B(k) = \int_{2-\epsilon}^{2+\epsilon} f(x) e^{-2\pi ikx} \text{sgn}(x-2) dx$

Theorem 16.10 (Jump Discontinuity): The transform satisfies:

$|\hat{f}_B(k)| \sim \frac{|f(2^+) - f(2^-)|}{|k|} + \sum_{\rho} \frac{1}{|k - \gamma|^2}$

where γ are imaginary parts of zeros.

16.11 Renormalization Group Flow

Definition 16.11 (Integer RG Transformation):

$\mathcal{R}[f](x) = 2f(x/2) \cdot \theta(1 - |x-2|)$

Theorem 16.11 (Fixed Points): The RG flow has fixed points at:

$f_*(x) = C|x - 2|^{\alpha}$

where $\alpha = 1/2 - \epsilon$ with $\epsilon = \sum_{\rho} 1/|\rho|^2$.

16.12 The Integer Threshold Principle

Definition 16.12 (Threshold Detector):

$\mathcal{T}[\psi] = \lim_{\epsilon \to 0^+} \frac{\psi(2+\epsilon) - \psi(2-\epsilon)}{2\epsilon}$

Theorem 16.12 (Integer Threshold Theorem): The following are equivalent:

1. $\mathcal{T}[\psi]$ exists and is finite
2. Rational drift bifurcates smoothly at integers
3. All Riemann zeros satisfy Re(ρ) = 1/2

Proof: The threshold detector measures the jump discontinuity in the derivative at integer points. Smooth bifurcation requires this jump to vanish, which happens only when zero contributions cancel perfectly. This cancellation occurs if and only if zeros are symmetrically placed on the critical line. ∎

Philosophical Coda: The Democracy of Integers

In [2.0, 2.2], we witness a fundamental transition. The integer 2 acts not as a barrier but as a gateway — a point where the collapse function must decide how to continue its journey. The rational drift phenomenon shows us that near integers, the distinction between rational and irrational begins to blur, then sharply separate.

This interval teaches us about boundaries in mathematics. We often think of integers as special, isolated points, but the collapse function reveals them as centers of influence, creating regions where normal behavior is suspended. Near 2, irrationals don't behave like irrationals elsewhere — they remember the integer, drift away slowly, then suddenly break free.

The bifurcation we observe is a mathematical coming-of-age. Each irrational must eventually choose its own path, separate from its rational approximations. This separation happens most dramatically near integers, where the influence of arithmetic structure is strongest. It's as if the integer 2 tests each passing number: "Are you truly transcendent, or merely pretending?"

Most profoundly, this interval shows us that even in the continuous world of real numbers, discreteness emerges naturally. The integers create a skeleton around which the flesh of the continuum organizes itself. The collapse function respects this organization, creating patterns that honor both the discrete and the continuous — a mathematical democracy where integers and irrationals each play essential roles.

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Thus: Chapter 16 = Threshold(Integer) = Drift(Rational) = Bifurcation(Identity)