The Computable Real Boundary: Where Physics Ends and Cognitive Fiction Begins

In the paper's words

Every experiment reduces to a finite act of distinction. This paper argues that the boundary between physics and what lies beyond it coincides exactly with the computable reals. Drawing on Spencer-Brown's Laws of Form, its QNFO extension, and independent work on the physical Church-Turing thesis (Leshem 2019, Bolotin 2015), we define Cognitive Fiction as any formal system that is internally consistent but cannot be reduced by any finite sequence of the two primitive operations that generate all physically accessible distinctions.

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

Numbers it reports

1. Spencer-Brown's Laws of Form (1969) provides a calculus of indications built from two primitive gestures: Primitive Symbol Meaning Mark $\texttt{\#}$ The act of indicating: "I distinguish this from what it is not" Enclosure $[\;]$ The boundary created by a distinction: what is inside vs. what is outside Two reduction rules govern how forms composed of these primitives simplify: Calling: $\texttt{\#}\texttt{\#} = \texttt{\#}$.

   from 2. The Laws of Form Calculus

How it is done

1. Every experiment reduces to a finite act of distinction: this reading, not that; this frequency range, not that; this bit string, not its negation.

   from Abstract

2. But physical experience is pre-filtered by the mathematics we inherit: we count "seven sheep" rather than "a bunch," we measure distance in $\mathbb{R}$ rather than $\mathbb{Q}_p$, we model dynamics with Archimedean differential equations rather than ultrametric jump processes.

   from 1. Introduction — The Strange Loop

3. The QNFO extension maps these primitives onto two physical interpretations: Primitive Autaxys Vocabulary Physical Interpretation $\texttt{\#}$ D (Distinction) A measurement outcome: "this, not that" $[\;]$ R (Relation) The physical procedure connecting states Calling $(\texttt{\#}\texttt{\#} = \texttt{\#})$ Resonance/Coherence Two identical measurements produce one distinction — not two Crossing $([\texttt{\#}] = \text{void})$ OC (Ontological Closure) A distinction that can be made and unmade by finite procedures is not a new physical entity

   from 2. The Laws of Form Calculus

4. What cannot be distinguished by any finite physical procedure cannot be physically real.

   from 4. The Boundary at the Computable Reals

What would falsify it

1. Keywords: computable reals, Laws of Form, Church-Turing thesis, falsifiability, cognitive fiction, Monna-map, ultrametric physics, ontological closure

   from Abstract

2. No physical theory has ever required a non-computable real number to produce a measurement-distinguishable prediction.

   from 4. The Boundary at the Computable Reals

3. This is stronger than C1: it asserts that non-computable reals cannot enter physics without breaking falsifiability, not only that they haven't .

   from 4. The Boundary at the Computable Reals

Where the claim stops

1. They are forms that cannot be reduced to void by any finite sequence of the two primitive laws.

   from 8. Conclusion — The Fixed Point

2. Yet the mathematical architecture physics inherited — the real number field $\mathbb{R}$ — is uncountable, containing elements that no finite procedure can approximate or discriminate.

   from Abstract

3. Drawing on Spencer-Brown's Laws of Form , its QNFO extension, and independent work on the physical Church-Turing thesis, we define Cognitive Fiction as any formal system that is internally consistent but cannot be reduced to void — cannot be distinguished by any finite measurement procedure — under the two primitive operations that generate all physically accessible distinctions: the mark and the enclosure.

   from Abstract

Where to read

1. Abstract216 words · 1 min
2. 1. Introduction — The Strange Loop236 words · 1 min
3. 2. The Laws of Form Calculus161 words · 1 min
4. 3. Number Systems from Distinctions164 words · 1 min
5. 4. The Boundary at the Computable Reals355 words · 2 min
6. 5. Cognitive Fiction: A Formal Definition184 words · 1 min
7. 6. The Monna-Map and the Archimedean as Anthropic Projection207 words · 1 min
8. 7. The Stratigraphy of Measurement154 words · 1 min
9. 8. Conclusion — The Fixed Point261 words · 1 min
10. Declarations133 words · 1 min