Ontological Re-Entry: What Can We Say Exists?

In the paper's words

This paper addresses the "re-entry" problem in ontology: given a theory, formalism, or model built over some primitive vocabulary, what can we legitimately say exists when we project that structure back onto the world, and what does answering the question cost a physically bounded observer? We develop a finite-distinction framework in which existence claims are admissible only when expressible as distinctions at finite resolution. A system of N binary distinctions supports 2^N states and exactly 2²^N extensionally distinct existence predicates, of which only a vanishing fraction is reachable by short formulas; we compute this syntactic bottleneck explicitly.

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

Numbers it reports

1. Its core commitment, taken from the finite-distinction program [11], is that a finite-entropy world is a finite-distinction world: uncountable precision is unphysical, while computable depth and $p$-adic valuation remain physically real.

   from 1. Introduction

2. Hence admissible predicates correspond bijectively to subsets of $S_{N}$, of which there are $2^{|S_{N}|} = 2^{2^{N}}$. $\square$ The bound is tight but syntactically unreachable: the number of short formulas is far smaller, and Section 4.1 quantifies the gap for a concrete budget.

   from 3.2 Admissible existence predicates and the counting bound

3. The re-entry condition fails when $V_{\mathrm{high}}$ contains predicates whose extension depends on precision beyond the $N$-bit budget — the "uncountable precision is unphysical" clause of [11].

   from 3.4 Ontological re-entry

4. At $10^{9}$ distinctions cycled at $1\ \mathrm{s^{-1}}$, the minimal erasure power is $\approx 2.87 \times 10^{-12}\ \mathrm{W}$ — negligible for silicon-scale hardware but a hard floor scaling linearly in both $N_{\mathrm{dist}}$ and $f_{\mathrm{cycle}}$.

   from 4.2 Erasure cost of a distinction

5. Power $P_{\mathrm{obs}} = 10^{-12}\ \mathrm{W}$ (illustrative nanowatt-scale bound, modeling choice), coherence time $\tau = 1\ \mathrm{s}$, $T = 300\ \mathrm{K}$.

   from 4.3 Distinction budget from a power budget

6. From Step 2 of Section 4.2, $k_{B} T \ln 2 = 2.8709 \times 10^{-21}\ \mathrm{J/bit}$. $$ N_{\max} = \frac{P_{\mathrm{obs}} \tau}{k_{B} T \ln 2} = \frac{10^{-12}}{2.8709 \times 10^{-21}} = 3.4832 \times 10^{8}. $$ So a nanowatt observer with $1\ \mathrm{s}$ coherence can in principle sustain at most $N_{\max} \approx 3.48 \times 10^{8}$ bit-distinctions per second against the erasure floor of [7].

   from 4.3 Distinction budget from a power budget

How it is done

1. A system of $N$ binary distinctions supports $2^{N}$ states and exactly $2^{2^{N}}$ extensionally distinct existence predicates, of which only a vanishing fraction is reachable by short formulas; we compute this syntactic bottleneck explicitly.

   from Abstract

2. It is a request for a criterion : a procedure that, given a theory and an observer with finite resources, outputs the set of existence claims the observer is entitled to make, together with the cost of making them.

   from 1. Introduction

3. At the applied end, [8] presents an ontology-driven software engineering approach for industrial robotics control software, introducing the ReApp architecture that synthesizes model-driven engineering with semantic technologies for the development and reuse of ROS-based components and applications, showing how different ontological classification systems for hardware can be combined.

   from 2. Background and Related Work

4. Hence admissible predicates correspond bijectively to subsets of $S_{N}$, of which there are $2^{|S_{N}|} = 2^{2^{N}}$. $\square$ The bound is tight but syntactically unreachable: the number of short formulas is far smaller, and Section 4.1 quantifies the gap for a concrete budget.

   from 3.2 Admissible existence predicates and the counting bound

What would falsify it

1. We situate the framework against database query rewriting, primitive-ontology and relational readings of quantum mechanics, false-premise failures in multimodal models, and the definitional stagnation of decentralization, and we state falsification conditions.

   from Abstract

2. The tension the summary reports — segmentation ability bought at the price of presence-detection — is a resource trade-off of exactly the kind our budget formalism predicts.

   from 2. Background and Related Work

Where the claim stops

1. We develop a finite-distinction framework in which existence claims are admissible only when expressible as distinctions at finite resolution.

   from Abstract

2. We define re-entry as the rewriting of a theory's high-level existence claims into predicates over its own base distinctions, modeled on ontological query compilation, and derive the conditions under which re-entry fails: false-premise capture, resolution collapse, and unpayable erasure cost.

   from Abstract

3. The logical question: when can an ontology's high-level existence claims be rewritten into statements over its own base distinctions without loss — the re-entry condition — and when does rewriting fail?

   from 1. Introduction

Where to read

1. Abstract218 words · 1 min
2. 1. Introduction689 words · 3 min
3. 2. Background and Related Work1,035 words · 5 min
4. 3. Methods576 words · 3 min
5. 4. Analysis1,216 words · 5 min