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import torch
import torch.nn as nn
import torch.nn.functional as F
class CayleyDicksonEmbedding(nn.Module):
def __init__(self, num_embeddings: int, base_dim: int = 1, lifts: int = 3):
"""
num_embeddings : number of unique indices
base_dim : dimension of the seed embedding (usually 1)

Proposal: PROJECT GEMSTONE

joshuah.rainstar@gmail.com

Overview

This proposal outlines a method to augment an autoregressive Transformer (e.g., GPT) with multi-horizon probabilistic priors derived from external Markov models or a similar statistical basis system. Instead of modifying the architecture, the method uses auxiliary layer-wise losses to align each layer’s internal representation with a synthetic embedding derived from the Markov transition probabilities.

The idea is to teach the model how to utilize prior knowledge to arrive at the most likely futures at multiple temporal horizons and therefore to localize discovery to relevant layers while maintaining compatibility with standard next-token training.

The Fast Fourier Transform cannot be (presently) Learned

A forensic and mathematical analysis of why backpropagation fails to discover the radix‑2 RFFT factorization from data is provided showing a useful problem for the advancement of current optimizer and backpropagation algorithmic designs, aided by the target factorization being known in closed form.


0) Setup and exact object of study

We consider the 512‑point real FFT (RFFT), producing 257 complex outputs (DC through Nyquist). The butterfly network depth is:

Keybase proof

I hereby claim:

  • I am falseywinchnet on github.
  • I am rainstared (https://keybase.io/rainstared) on keybase.
  • I have a public key ASCcN4IBipZoaGV2nrfUuZnCeG-XqFcKyE2hOvZZRg_cRwo

To claim this, I am signing this object: