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Primes Between Squares -- Commentary on Appendix 8 of Laws Of Form

J. M. Flagg, Louis H. Kauffman, Divyamaan Sahoo

TL;DR

This paper offers a structured, number-theoretic reading of Appendix 8 of Laws Of Form, reframing Spencer-Brown's primes-between-squares framework through precise notions like square segment seg$(n)$, $^ extcircled{\pi}\text{seg}(n)$, and the sects (R-sect, Q-sect). It formalizes the central constructions—the sect pyramid, Lemmas 3–4, and the short proof of Theorem 1—while extending the theory via ratchet points (Theorem 1A/1B), haploid/diploid prime-neighborhood corollaries, and Theorem 2 about primes in half segments. The exposition then connects modulators, especially the E4 reductor, to a binary sieving process and the Spencer-Brown sieve (GSB sieve), offering a constructive, circuit-based view of prime generation embedded in the Laws Of Form framework. Collectively, these ideas illuminate deep regularities in prime gaps and provide a bridge between classical conjectures (Legendre, Andrica, Oppermann, etc.) and a novel, algebraic-sieve perspective that blends rigorous argument with structural observations about numbers.

Abstract

This paper provides a commentary and guide to Appendix 8 of Laws Of Form, which is a chapter (appendix) on number theory in the book Laws of Form by Spencer-Brown. (Spencer-Brown,Laws Of Form,Revised Seventh English edition. Bohmeier Verlag. 2020) This chapter in the book provides Spencer-Brown's proofs of the conjecture that there are at least two prime numbers between any consecutive squared numbers. That there are primes between squares has been a conjecture in number theory since Legendre. In Spencer-Brown's appendix he gives his proofs of the conjecture. Those proofs are a highly original mixture of standard rigorous arguments and also some stated facts about the way numbers behave that would be considered conjectures by most number theorists. These phenomena are very interesting and constitute a deep observation about the nature of number itself. We intend that our guide will enable the reader to gain insight into Spencer-Brown's point of view and that our discussions will be of interest to anyone with curiosity about the theory of numbers.

Primes Between Squares -- Commentary on Appendix 8 of Laws Of Form

TL;DR

This paper offers a structured, number-theoretic reading of Appendix 8 of Laws Of Form, reframing Spencer-Brown's primes-between-squares framework through precise notions like square segment seg, , and the sects (R-sect, Q-sect). It formalizes the central constructions—the sect pyramid, Lemmas 3–4, and the short proof of Theorem 1—while extending the theory via ratchet points (Theorem 1A/1B), haploid/diploid prime-neighborhood corollaries, and Theorem 2 about primes in half segments. The exposition then connects modulators, especially the E4 reductor, to a binary sieving process and the Spencer-Brown sieve (GSB sieve), offering a constructive, circuit-based view of prime generation embedded in the Laws Of Form framework. Collectively, these ideas illuminate deep regularities in prime gaps and provide a bridge between classical conjectures (Legendre, Andrica, Oppermann, etc.) and a novel, algebraic-sieve perspective that blends rigorous argument with structural observations about numbers.

Abstract

This paper provides a commentary and guide to Appendix 8 of Laws Of Form, which is a chapter (appendix) on number theory in the book Laws of Form by Spencer-Brown. (Spencer-Brown,Laws Of Form,Revised Seventh English edition. Bohmeier Verlag. 2020) This chapter in the book provides Spencer-Brown's proofs of the conjecture that there are at least two prime numbers between any consecutive squared numbers. That there are primes between squares has been a conjecture in number theory since Legendre. In Spencer-Brown's appendix he gives his proofs of the conjecture. Those proofs are a highly original mixture of standard rigorous arguments and also some stated facts about the way numbers behave that would be considered conjectures by most number theorists. These phenomena are very interesting and constitute a deep observation about the nature of number itself. We intend that our guide will enable the reader to gain insight into Spencer-Brown's point of view and that our discussions will be of interest to anyone with curiosity about the theory of numbers.

Paper Structure

This paper contains 19 sections, 37 equations, 35 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Foundation
  4. Sects
  5. Odd sects
  6. What is the maximum sect that can appear in a square segment?
  7. Quadrantic and Reflexive sects
  8. A short proof of Theorem 1
  9. Generalizing Theorem 1
  10. The first ratchet
  11. Ratchet values and ratchet primes
  12. Haploid and Diploid numbers
  13. Half segments
  14. Elementals
  15. Spencer-Brown's proof of Theorem 1
  16. ...and 4 more sections

Figures (35)

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  • ...and 30 more figures