Category-Theoretic Unification of Marginalised Formalisms: Translating Invariants across Six Systems

In the paper's words

The marginalisation of six historically rich diagrammatic and logical formalisms—Laws of Form, Existential Graphs, Viable System Model, Catastrophe Theory, Pattern Language, and Polycontextural Logic—prevents researchers in mathematics, computer science, and systems engineering from exploiting their invariant structures. The specific gap is the absence of a rigorous, category‑theoretic translation protocol that can certify whether marginalisation is merely notational. We construct canonical categorical representations for each formalism, prove faithful translation theorems for five of them, and identify a single obstruction in the sheaf‑theoretic approach to Catastrophe Theory.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Solving this recurrence yields the odd‑index Fibonacci numbers: $$ E_{n}=F_{2n+1}. $$ Evaluating: $$ \begin{array}{c|c} n & E_{n}=F_{2n+1} \\ \hline 1 & F_{3}=2 \\ 2 & F_{5}=5 \\ 3 & F_{7}=13 \\ 4 & F_{9}=34 \\ 5 & F_{11}=89 \end{array} $$ Let $N_n$ denote the number of normal forms under the three published reduction rules.

   from 4.2 Fibonacci Enumeration of Mark‑Calculus Expressions

2. Hence $$ N_{n}=F_{n+2}. $$ Evaluating: $$ \begin{array}{c|c} n & N_{n}=F_{n+2} \\ \hline 1 & F_{3}=2 \\ 2 & F_{4}=3 \\ 3 & F_{5}=5 \\ 4 & F_{6}=8 \\ 5 & F_{7}=13 \end{array} $$ The compression factor between raw expressions and normal forms for a given $n$ is $$ \frac{E_{n}}{N_{n}}=\frac{F_{2n+1}}{F_{n+2}}\approx\varphi^{\,n-1}, $$ where $\varphi\approx1.618$ is the golden ratio.

   from 4.2 Fibonacci Enumeration of Mark‑Calculus Expressions

How it is done

1. The specific gap is the absence of a rigorous, category‑theoretic translation protocol that can certify whether marginalisation is merely notational.

   from Abstract

2. Practitioners lacking access to these invariants must reinvent analogous constructions, incurring additional development time, duplicated proof effort, and missed cross‑disciplinary insights.

   from 1. Introduction

3. Equivalence invariance of formal category theory discusses invariance properties of formal category theory, which we use as motivation for the PC translation (see Section 4.8).

   from 2. Background and Related Work

4. Our methodological pipeline consists of three stages for each formalism $F\in\{\text{LoF},\text{EG},\text{VSM},\text{CT},\text{PL},\text{PC}\}$: Identify the invariant core of $F$ as described in the original literature (e.g., the distinction operator in LoF, the negation enclosure in EG, the viability recursion in VSM, the unfolding stratification in CT, the pattern composition algebra in PL, the contexture Heyting lattice in PC).

   from 3. Methods

What would falsify it

1. If a concrete CT model with a non‑orientable cusp singularity can be shown to possess an invariant (e.g., a Thom polynomial) that is not recoverable from any sheaf $\mathcal{S}$ satisfying the oriented enrichment of [5], then the universal marginalisation‑as‑notation hypothesis is falsified.

   from 6.3 Falsification Criterion

2. These results reclassify marginalisation as a notational barrier and provide a reproducible, falsifiable criterion for future investigations.

   from Abstract

3. Where a translation fails, the hypothesis is falsified and the marginalisation gains substantive justification.

   from 1. Introduction

Where the claim stops

1. The ambient categories (augmented virtual double categories, $\mathcal{V}$‑normed categories, etc.) are assumed to be complete and cocomplete; without these properties the universal constructions may fail.

   from 6.1 Limitations

2. The CT translation presumes global orientability of stratified spaces; the identified obstruction shows sensitivity to this assumption.

   from 6.1 Limitations

3. Assumption A4 (Finite Pattern Set).

   from 6.1 Limitations

Where to read

1. Abstract150 words · 1 min
2. 1. Introduction271 words · 1 min
3. 2. Background and Related Work381 words · 2 min
4. 3. Methods265 words · 1 min
5. 4. Analysis654 words · 3 min
6. 5. Results145 words · 1 min
7. 6. Discussion273 words · 1 min
8. 7. Practical Implications and Engineering Pathways307 words · 1 min
9. 8. Conclusion117 words · 1 min
10. Correction Notice302 words · 1 min
11. Provenance and AI Involvement234 words · 1 min
12. Appendix A. Divergence report48 words · 1 min