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On the Existence of Anomalies, The Reals Case

Samuel Epstein

TL;DR

This work investigates anomalies in observations under the Independence Postulate by extending prior discrete-sequence analyses to observations modeled as infinite sequences of real numbers. It builds a framework with $K$, $d$, $I$, and the halting sequence $\mathcal{H}$, deriving bounds that relate randomness deficiencies and anomaly measures to mutual information with $\mathcal{H}$, including the real-valued extension of the main anomaly bound via $t_{\gamma,P}$ and $I(\langle\gamma\rangle;\mathcal{H})$. The key contributions are a central lemma bounding the maximal randomness deficiency within finite observation sets and a theorem bounding anomaly extent in real-valued observation streams, along with corollaries tightening these relationships. The results imply that observations with no anomalies are not realizable in nature and provide a bridge to broader physical and information-theoretic contexts, with potential extensions to quantum information and thermodynamics.

Abstract

The Independence Postulate (IP) is a finitary Church-Turing Thesis, saying mathematical sequences are independent from physical ones. Modelling observations as infinite sequences of real numbers, IP implies the existence of anomalies.

On the Existence of Anomalies, The Reals Case

TL;DR

This work investigates anomalies in observations under the Independence Postulate by extending prior discrete-sequence analyses to observations modeled as infinite sequences of real numbers. It builds a framework with , , , and the halting sequence , deriving bounds that relate randomness deficiencies and anomaly measures to mutual information with , including the real-valued extension of the main anomaly bound via and . The key contributions are a central lemma bounding the maximal randomness deficiency within finite observation sets and a theorem bounding anomaly extent in real-valued observation streams, along with corollaries tightening these relationships. The results imply that observations with no anomalies are not realizable in nature and provide a bridge to broader physical and information-theoretic contexts, with potential extensions to quantum information and thermodynamics.

Abstract

The Independence Postulate (IP) is a finitary Church-Turing Thesis, saying mathematical sequences are independent from physical ones. Modelling observations as infinite sequences of real numbers, IP implies the existence of anomalies.
Paper Structure (5 sections, 7 theorems, 10 equations)

This paper contains 5 sections, 7 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Sets with Low Randomness Deficiencies
  4. Observations as Reals
  5. Discussion

Key Result

Lemma 1

For probability $p$ over $\mathbb{N}$, $D{\subset}\mathbb{N}$, $|D|=2^s$, $s < \max_{a\in D}{\mathbf d}(a|p)+{\mathbf I}(D;{\mathcal{H}})+O(\log{\mathbf I}(D;{\mathcal{H}})+\log {\mathbf K}(p))$.

Theorems & Definitions (13)

  • Definition 1: Information
  • Lemma 1: EpsteinShort23
  • Lemma 2: EpsteinDerandom22
  • Definition 2
  • Remark 1
  • Lemma 3
  • proof
  • Proposition 1
  • Lemma 4
  • proof
  • ...and 3 more