Autaxys Ontological Closure: When Is a Physical Quantity Computationally Real?

In the paper's words

We propose an operational criterion for physical reality: a quantity is physically real only if there exists a finite Turing-machine protocol that approximates it with a computable modulus of convergence, yielding measurement-distinguishable predictions. This criterion is not a metaphysical claim about what "exists" — it is a boundary condition on measurability: if no finite computation can converge to a quantity's value, then no finite measurement protocol can distinguish that quantity from a nearby alternative, and the quantity is operationally indistinguishable from nonexistent. We show that standard physical quantities (mass, charge, spin, energy levels) satisfy the criterion, while continuum-valued fields at arbitrarily small scales, uncomputable real numbers as physical constants, and non-measurable sets fail it — and argue that this is a feature, not a bug.

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

Numbers it reports

1. A measurement protocol that might take 1 second or $10^{100}$ years to reach the desired precision, with no way to know which, is not a measurement protocol — it is a gamble.

   from 2.4 The Modulus of Convergence Requirement

2. The Bekenstein bound [12] states that the information content $\mathcal{I}$ of any bounded physical system, expressed in dimensionless Planck units ($\hbar = c = G = k_B = 1$), satisfies $\mathcal{I} \leq 2\pi R E / \ln 2$, where $R \equiv R_{\text{phys}}/\ell_P$ (spatial extent in Planck lengths $\ell_P \equiv \sqrt{\hbar G / c^3}$) and $E \equiv E_{\text{phys}}/E_P$ (total energy in Planck energies $E_P \equiv \sqrt{\hbar c^5 / G}$).

   from 4.2 Continuum Fields at Arbitrarily Small Scales

3. The Bekenstein bound forces the apparatus's energy density to diverge in response, and at sufficiently high density the apparatus collapses into a black hole before the measurement completes [13].

   from 4.2 Continuum Fields at Arbitrarily Small Scales

4. Risk: "This was a null prediction — the absence of hypercomputation is the default, not a discovery." [CHECK: 2028-06] OC-9: Landauer bound holds.

   from 6.2 Calibration Register

How it is done

1. We propose an operational criterion for physical reality: a quantity is physically real only if there exists a finite Turing-machine protocol that approximates it with a computable modulus of convergence, yielding measurement-distinguishable predictions.

   from Abstract

2. The central insight is that measurement is a form of computation: a finite physical protocol that takes a system as input and produces a numerical output.

   from 1 Introduction

3. Landauer's principle [2] established that information is physical — erasing a bit costs $k_B T \ln 2$ in thermodynamic entropy; in dimensionless Planck units ($\hbar = c = G = k_B = 1$): $T \ln 2$, where both the energy and temperature are pure numbers — Archimedean norms of quantities whose completions exist at every place per Ostrowski's theorem.

   from 1.1 Historical Context

4. A quantity $Q$ associated with a physical system $S$ is physically real if and only if there exists: A finite Turing-machine protocol $\mathcal{P}$ that, given a specification of the measurement apparatus and a desired precision $\varepsilon \gt 0$, halts after a finite number of steps and outputs a rational approximation $\tilde{Q}$ such that $|\tilde{Q} - Q| \lt \varepsilon$.

   from 2.1 Formal Statement

What would falsify it

1. If a theory predicted that $c_\Omega$ had a specific value, no finite measurement protocol could verify this prediction beyond a finite number of bits.

   from 4.1 Uncomputable Real Numbers as Physical Constants

2. For sufficiently high precision, the prediction would be indistinguishable from any other uncomputable constant within the same precision bounds.

   from 4.1 Uncomputable Real Numbers as Physical Constants

3. No physical theory currently predicts an uncomputable constant, and if one did, the OC criterion would correctly identify it as physically vacuous — it makes predictions that cannot be verified by any finite protocol.

   from 4.1 Uncomputable Real Numbers as Physical Constants

Where the claim stops

1. The Autaxys Ontological Closure criterion proposes a precise boundary for physical reality: a quantity is physically real if and only if it is computably approximable with a computable modulus of convergence, yielding measurement-distinguishable predictions.

   from 9 Conclusion

2. The interesting outcome would be a counterexample: a physically measured quantity that cannot be computably approximated.

   from 9 Conclusion

3. We propose an operational criterion for physical reality: a quantity is physically real only if there exists a finite Turing-machine protocol that approximates it with a computable modulus of convergence, yielding measurement-distinguishable predictions.

   from Abstract

Where to read

1. Abstract150 words · 1 min
2. 1 Introduction369 words · 2 min
3. 2 The Operational Definition484 words · 2 min
4. 3 What Survives: Quantities That Pass the Criterion372 words · 2 min
5. 4 What Fails: Quantities That Do Not Pass675 words · 3 min
6. 5 The Eleven Sub-Papers665 words · 3 min
7. 6 Judgment Sensitivity and Calibration591 words · 3 min
8. 7 Practical Applications214 words · 1 min
9. 8 Counterfactual Backcasting274 words · 1 min
10. 9 Conclusion245 words · 1 min
11. Declarations155 words · 1 min