Treehouse of Horror: The LaTeX Massacre

Segment 1: The Formatting

Homer works as a LaTeX typesetter at the nuclear plant. After Mr. Burns demands perfectly aligned equations, Homer goes insane trying to format complex mathematical expressions, eventually snapping when his equations run off the page. In a parody of "The Shinning," Homer chases his family around with a mechanical keyboard while screaming "All work and no proper alignment makes Homer go crazy!"

Segment 2: Time and Compilation

In a nod to "Time and Punishment", Homer accidentally breaks his LaTeX compiler and tries to fix it, but ends up creating a time paradox where every document compiles differently in parallel universes. He desperately tries to find his way back to a reality where his equations render properly.

Segment 3: The Cursed Code

Bart discovers an ancient LaTeX document that contains forbidden mathematics. When he compiles it, it summons an eldrich horror made entirely of misaligned integrals and malformed matrices. Lisa must save Springfield by finding the one perfect alignment that will banish the mathematical monster back to its dimension.

The episode ends with a meta-joke about how even the credits won't compile properly.

A Modest Proposal: Statistical Token Prediction Is No Replacement for Syntactic Construction

A Modest Proposal: Statistical Token Prediction Is No Replacement for Syntactic Construction by Stephen Crowley October 25, 2024

1Current Generative-Pretrained-Transformer Architecture

Given vocabulary $V$, $|V| = v$, current models map token sequences to vectors:

$\displaystyle (t_1, \ldots, t_n) \mapsto X \in \mathbb{R}^{n \times d}$

Through layers of transformations:

$\displaystyle \text{softmax} (QK^T / \sqrt{d}) V$

where $Q = XW_Q$, $K = XW_K$, $V = XW_V$

Optimizing:

$\displaystyle \max_{\theta} \sum \log P (t_{n + 1} |t_1, \ldots, t_n ; \theta)$

2Required Reformulation

Instead, construct Abstract Syntax Trees where each node $\eta$ must satisfy:

$\displaystyle \eta \in \{ \text{Noun}, \text{Verb}, \text{Adjective}, \text{Conjunction}, \ldots\}$

With composition rules $R$ such that for nodes $\eta_1, \eta_2$:

$\displaystyle R (\eta_1, \eta_2) = \left\{ \begin{array}{ll} \text{valid\_subtree} & \text{if grammatically valid}\\ \emptyset & \text{otherwise} \end{array} \right.$

And logical constraints $L$ such that for any subtree $T$:

$\displaystyle L (T) = \left\{ \begin{array}{ll} T & \text{if logically consistent}\\ \emptyset & \text{if contradictory} \end{array} \right.$

3Parsing and Generation

Input text $s$ maps to valid AST $T$ or error $E$:

$\displaystyle \text{parse} (s) = \left\{ \begin{array}{ll} T & \text{if } \exists \text{valid AST}\\ E (\text{closest\_valid}, \text{violation}) & \text{otherwise} \end{array} \right.$

Generation must traverse only valid AST constructions:

$\displaystyle \text{generate} (c) = \{T|R (T) \neq \emptyset \wedge L (T) \neq \emptyset\}$

where $c$ is the context/prompt.

4Why Current GPT Fails

The statistical model:

$\displaystyle \text{softmax} (QK^T / \sqrt{d}) V$

Has no inherent conception of:

• Syntactic validity

• Logical consistency

• Conceptual preservation

It merely maximizes:

$\displaystyle P (t_{n + 1} |t_1, \ldots, t_n)$

Based on training patterns, with no guaranteed constraints on:

$\displaystyle \prod_{i = 1}^n P (t_i |t_1, \ldots, t_{i - 1})$

This allows generation of:

• Grammatically invalid sequences

• Logically contradictory statements

• Conceptually inconsistent responses

5Conclusion

The fundamental flaw is attempting to learn syntax and logic from data rather than building them into the architecture. An AST-based approach with formal grammar rules and logical constraints must replace unconstrained statistical token prediction.

Uniformly Convergent Expansions of Positive Definite Functions

Uniformly Convergent Expansions of Positive Definite Functions by Stephen Crowley <stephencrowley214@gmail.com> October 22, 2024

Theorem 1. The covariance function $K (t)$ of a stationary Gaussian process has a uniformly convergent expansion in terms of functions from the orthogonal complement of the null space of the inner product defined by $K$. This uniform convergence holds initially on the real line and extends to the entire complex plane.

Proof. Let $\{P_n (\omega)\}_{n = 0}^{\infty}$ be the orthogonal polynomials with respect to the spectral density $S (\omega)$ of a stationary Gaussian process, and $\{f_n (t)\}_{n = 0}^{\infty}$ their Fourier transforms defined as:

$\displaystyle f_n (t) = \int P_n (\omega) e^{i \omega t} d \omega$

Let $K (t)$ be the covariance function of the Gaussian process.

1) First, the orthogonality of the polynomials $P_n (\omega)$ is established:

a) By definition of orthogonal polynomials, for $m \neq n$:

$\displaystyle \int P_m (\omega) P_n (\omega) S (\omega) d \omega = 0$

b) The spectral density and covariance function form a Fourier transform pair:

$\displaystyle K (t) = \int S (\omega) e^{i \omega t} d \omega$

2) The null space property of $\{f_n (t)\}_{n = 1}^{\infty}$ is proven:

a) Consider the inner product $\langle f_n, K \rangle$ for $n \geq 1$:

$\displaystyle \langle f_n, K \rangle = \int f_n (t) K (t) dt = \int f_n (t) \left( \int S (\omega) e^{i \omega t} d \omega \right) dt$

b) Applying Fubini's theorem:

$\displaystyle \langle f_n, K \rangle = \int S (\omega) \left( \int f_n (t) e^{i \omega t} dt \right) d \omega = \int S (\omega) P_n (\omega) d \omega = 0$

Thus, $\{f_n (t)\}_{n = 1}^{\infty}$ are in the null space of the inner product defined by $K$.

3) The Gram-Schmidt process is applied to the Fourier transforms $\{f_n (t)\}_{n = 0}^{\infty}$ to obtain an orthonormal basis $\{g_n (t)\}_{n = 0}^{\infty}$ for the orthogonal complement of the null space:

$\displaystyle \tilde{g}_0 (t) = f_0 (t)$

$\displaystyle g_0 (t) = \frac{\tilde{g}_0 (t)}{\| \tilde{g}_0 (t)\|}$

For $n \geq 1$:

$\displaystyle \tilde{g}_n (t) = f_n (t) - \sum_{k = 0}^{n - 1} \langle f_n, g_k \rangle g_k (t)$

$\displaystyle g_n (t) = \frac{\tilde{g}_n (t)}{\| \tilde{g}_n (t)\|}$

where $\| \cdot \|$ and $\langle \cdot, \cdot \rangle$ denote the norm and inner product induced by $K$, respectively.

4) $K (t)$ can be expressed in terms of this basis:

$\displaystyle K (t) = \sum_{n = 0}^{\infty} \alpha_n g_n (t)$

where $\alpha_n = \langle K, g_n \rangle$ are the projections of $K$ onto $g_n (t)$.

5) The partial sum is defined as:

$\displaystyle S_N (t) = \sum_{n = 0}^N \alpha_n g_n (t)$

6) The sequence of partial sums $S_N (t)$ converges uniformly to $K (t)$ in the canonical metric induced by the kernel as $N \to \infty$.

7) To realize this, recall that the canonical metric is defined as:

$\displaystyle d (f, g) = \sqrt{\int \int (f (t) - g (t)) (f (s) - g (s)) K (t - s) dtds}$

8) The error in this metric is considered:

$\displaystyle d (K, S_N)^2 = \int \int (K (t) - S_N (t)) (K (s) - S_N (s)) K (t - s) dtds$

9) As the kernel operator is compact in this metric:

For every positive epsilon, there exists an N (which depends on epsilon) less than n, such that the distance between K and S_n is less than epsilon.

$\displaystyle \exists N (\epsilon) < n : d (K, S_n) < \epsilon \quad \forall \epsilon > 0$

10) Extension to the Complex Plane:

a) The covariance function $K (t)$ of a stationary Gaussian process is positive definite and therefore analytic in the complex plane.

b) The partial sum $S_N (t)$ is a finite sum of analytic functions (as $g_n (t)$ are analytic), and is thus analytic in the complex plane.

c) The convergence of $S_N (t)$ to $K (t)$ on the real line is uniform, as shown in steps 1-9.

d) Consider any open disk D in the complex plane that intersects the real line. The intersection of D with the real line contains an accumulation point.

e) By the Identity Theorem for analytic functions, since $K (t)$ and $S_N (t)$ agree on a set with an accumulation point within D (namely, the intersection of D with the real line), they must agree on the entire disk D.

f) As this holds for any disk intersecting the real line, and such disks cover the entire complex plane, the uniform convergence of $S_N (t)$ to $K (t)$ extends to the entire complex plane.

Thus, it has been shown that the covariance function $K (t)$ has a uniformly convergent expansion in terms of functions from the orthogonal complement of the null space of the inner product defined by $K$. This uniform convergence holds initially on the real line and extends to the entire complex plane.$\Box$

Accomodation Ascension

In a convergence of accommodation and purpose, the journey began—a journey not unlike my own endeavor with the Riemann Hypothesis. With every insight, each approximation revealed a deeper understanding, like discovering the hidden higher-dimensional representations embedded in the seemingly one-dimensional solutions. What if this all ties back to the Hardy Z function and Bessel function J0, drawing a line between the elementary harmonic waves and, incredibly, the proof of the mass gap as described in Alexi Svcestikonov's 'Towards Nonperturbative Quantization of Yang-Mills Fields'? A coherence begins to emerge, a link between seemingly disparate domains—a bridge that feels almost inevitable now.

It's not just the universe's complex beauty that is at play here. It's the convergence of abstract mathematical landscapes into something tangible—a retrodiction, a rigorous Bayesian narrative that may very well give us the integer address of our universe itself. Every zero of the conformally transformed Hardy Z function, incorporating a timelike parameter in a transformation like tanh(log(1+alpha*x^2)), does describe the universe's expansion from zero volume to a maximum bound, as natural and bounded as the hyperbolic tangent's squash. The loci of zeros form intricate shapes like the lemniscate of Bernoulli, and the imaginary loci branch off into hyperbolas—the entire manifold reshapes into a compact origin, where geometry manifests its secrets.

And so, I found myself contemplating the origin, the very heart of coherence, where the phase lines diverge not into infinity but form elegant figure-eight lemniscates. Where asymmetry is born from the underlying warping of this mathematical space, the Z function's surface becomes a landscape of purpose. This is not merely science; it is a stunning composition of verses—a manifestation of something profound, where math becomes poetry and the universe itself becomes an anthem of ataraxia, waiting to be decoded. The synchronic and diachronic facets of the journey spoke in tandem, affirming the intermediate steps as intrinsic to the overarching resolution. In the pursuit of understanding, in the tenuous grasp of knowledge, the intrepid traveler found not only clarity but a resonance—an emblematic, unified ascension.

And so, the journey persisted, forever on the precipice of something profound, beckoning, both beguiling and benevolent—a true manifestation of the Pleroma—a profound, enigmatic totality, where all things become unified and whole.