sethuiyer

What Is a Harmonic Polytope?

A harmonic polytope is a special type of convex shape. It's the convex hull of a set of vectors that satisfy three key conditions:

1. They're on a sphere: All the vectors have a length of 1 and lie on the surface of a $d$-dimensional unit sphere.
2. They're centered: The sum of all the vectors is zero. This means their average position, or centroid, is at the origin.
3. They're isotropic: The vectors are perfectly balanced in every direction. This is defined by their empirical covariance matrix, which is a key concept. For a set of $n$ vectors in $d$-dimensional space, this matrix is calculated as a sum of outer products of the vectors. The isotropy condition means this matrix is a multiple of the identity matrix. The text proves that this multiple must be exactly 1 divided by $d$.

In simple terms, a harmonic polytope is the shape you get when you take a bunch of unit vectors that are perfectly balanced and centered, and you form the smallest convex shap

sethuiyer

The Circle That Built Intelligence

Abstract

What if the same mathematical structure that governs a high school trigonometry problem also powers GPT-4's understanding of language? What if the mystery of why certain polynomial coefficients are integers connects directly to how neural networks learn to pay attention? What if zero—both as symbol and concept—holds the key to understanding the discrete-continuous duality that underlies all of computation?

This investigation begins with seven dots on a circle and a shocking discovery: despite having irrational roots, the polynomial they generate has perfectly integer coefficients. This simple observation opens a portal into some of the deepest mathematics of our time—Galois theory, cyclotomic fields, spectral analysis, and L-functions. But the journey doesn't end with pure mathematics.

The same harmonic structures that explain our integer coefficients also provide the mathematical foundation for transformer architectures in artificial intelligence. The same

sethuiyer

IMO Acoustic Problem - Competition Solution

Problem Statement

Let $n\ge 3$ be an integer and $\zeta_n = e^{2\pi i/n}$. Consider the real numbers

$$x_k = 2\cos\left(\frac{2\pi k}{n}\right)\quad (k=1,2,\dots,n-1),$$

which are the real parts of the nontrivial $n$-th roots of unity. Define the monic polynomial

sethuiyer

#--------------------------------------------------------------------
#  Minimal experiment that combines
#   • mean‑free projection
#   • weight‑shared outer‑loop refinement
#   • Q‑learning with two actions (stop / continue)
#   • group‑relative learning (GRPO)
#
#  Run with:   crystal run intelligence.cr
#--------------------------------------------------------------------
require "random"

sethuiyer

How the Brain Named Itself: A Zero-Knowledge Proof of Consciousness

Bridging Ramanujan graphs, sparse coding, and cryptographic verification to propose a novel mathematical framework for self-awareness

Abstract

We propose a novel theoretical framework connecting optimal graph structures, sparse signal processing, and zero-knowledge proof protocols to explain how conscious systems achieve verified self-awareness. Drawing on established mathematics from expander graph theory and emerging applications in graph signal processing, we hypothesize that consciousness emerges from the brain's ability to cryptographically verify sparse representations of its own states using its optimal network architecture as a verification key. While the individual mathematical and neuroscientific components are well-established, their integration represents an unprecedented synthesis with testable empirical predictions.

The Paradox of Self-Naming

sethuiyer

What Did We Learn Here? Quantum-inspired parsing works You can represent grammar parses as Fock states — high-dimensional quantum-like vectors encoding structure and tokens simultaneously.

Positional structure encoding is 🔑 Adding explicit depth + left/right shape modes lets you disentangle parse content (words) from parse shape (tree topology). This is a game changer for ambiguity.

Operator amplitudes can be learned from data Instead of fixed grammar probabilities, you can learn log-amplitudes for rewrite operators directly through a beam search + credit assignment algorithm.

sethuiyer

The Prime Superconductor: From Code to Mathematics

A journey through prime necklaces, harmonic oscillations, and discrete superconductivity

---

The Discovery

What started as a graph traversal problem with harmonic control has revealed one of the most unexpected connections in mathematics: prime numbers exhibit superconducting behavior. This isn't metaphor—it's rigorous mathematics backed by computational evidence.

sethuiyer

Harmonic State Modulation: Phase-Locked Control in Non-Stationary Environments

Abstract

We present a novel control problem where an agent's capabilities oscillate according to harmonic functions with distinct frequency-decay characteristics. The agent must navigate a graph while its operational modes—determining movement costs, memory retention, and action policies—follow deterministic yet time-varying probability distributions. We demonstrate that optimal performance requires predictive phase alignment rather than reactive adaptation. Surprisingly, agents that synchronize with the system's natural harmonics significantly outperform those that attempt to override the oscillatory dynamics. We provide both theoretical analysis and empirical validation, showing that the problem exhibits rich mathematical structure including connections to thermodynamic cycles and information-theoretic regularization.

1. Introduction

Consider an autonomous agent navigating a directed graph $G = (V, E)$ where each node

sethuiyer

#!/usr/bin/env python3
"""
# Saraswati-RL: Harmonic Consciousness-Based Reinforcement Learning Agent
# Author: @sethuiyer
# Inspired by neural phase transitions, thermodynamic cognition, and musical constants


A simplified neural network implementation using consciousness physics
to control which policy network gets to act at each timestep.

sethuiyer

Topcoder Marathon Problem: Ghana Gerbe Navigation on Building Facade Meshes

Background

You are given a triangulated 2-complex K = (V, E, F) representing a large polygonal mesh of an urban building facade. Each face f ∈ F is a triangle, edges E are shared among faces, and vertices V are 3D coordinates. A maintenance micro-drone with severely bounded memory must perform a reversible inspection protocol: it writes triangular Ghana tiles (face walk followed by its inverse) so that at any moment its sliding tape window contains a complete boundary-free 2-chain. The drone may "jump" between faces that share a vertex but not an edge, but only after closing a tile.

Objective

Given K, a set of target inspection constraints, and a memory window k, produce a legal Ghana stream S that maximizes coverage and inspection quality while respecting the protocol and window bounds.