Non-Anthropocentric Natural Units: From the Bekenstein Bound to Ostrowski's Theorem

In the paper's words

We strip away all anthropocentric constructions from the Bekenstein-Hawking entropy bound. Area as dimensionless geometric invariant; exponential bound defined without logarithmic base; integer Hilbert space dimension as universal invariant. Ostrowski's theorem reveals the Archimedean completion is merely one choice — an adelic formulation is proposed.

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

Numbers it reports

1. The bound $N \leq \exp(A/4)$ appears to smuggle in a base: the constant $e \approx 2.71828\dots$.

   from 2.3 The Base-Independence of the Exponential

2. The dark-matter-to-baryon ratio $\Omega_c/\Omega_b \approx 5.36$ is close to $27/5 = 5.4$, and the muon-to-electron mass ratio is close to $3 \cdot 7 \cdot 11 / 2$ [my conjecture]. p-Adic structure in entanglement entropy: If holographic screens carry $p$-adic structure, the entanglement entropy of a region should exhibit deviations from the smooth area law at scales comparable to the Planck length, with a hierarchical pattern characteristic of ultrametric spaces.

   from 6.2 Falsifiable Consequences

3. The adelic proposal would be disconfirmed if no log-periodic signal is found in the CMB at any amplitude above $10^{-3}$ of the primary power spectrum; if all particle mass ratios are consistent with random real numbers (no rational approximation); or if entanglement entropy measurements at the Planck scale show no ultrametric clustering.

   from 6.2 Falsifiable Consequences

4. Following the 9-stage Bayesian cascade methodology, we register the following calibration entries: [CHECK: 2035] By 2035, log-periodic oscillations in the CMB power spectrum should be detected at the $3\sigma$ level if the adelic model with active primes $S = \{2,3,5,7,\dots\}$ is correct. [CHECK: 2035] By 2035, the dark-matter-to-baryon ratio should be measured with sufficient precision to distinguish $5.36 \pm 0.05$ from $5.400$ (the rational $27/5$) — requiring a $\sim 1\%$ improvement in precision.

   from 6.3 Calibration Register

How it is done

1. The Bekenstein-Hawking entropy bound $S \leq A/4\ell_P^2$ is a pillar of quantum gravity, typically expressed in Planck units with entropy measured in bits or nats.

   from Abstract

2. Every physical law is expressed in numbers.

   from 1. Introduction: What Is a Measurement?

3. Together, these results establish that a spacetime region bounded by a horizon of area $A$ contains at most $\exp(A/4)$ independent quantum states — the Bekenstein bound in its strongest form.

   from 2.2 The Bekenstein-Hawking Entropy in Natural Units

4. The bound $N \leq \exp(A/4)$ appears to smuggle in a base: the constant $e \approx 2.71828\dots$.

   from 2.3 The Base-Independence of the Exponential

What would falsify it

1. For the proposal to be scientific, it must make falsifiable predictions.

   from 6.2 Falsifiable Consequences

2. This prediction is testable with existing Planck data and is independent of the specific adelic model [speculative].

   from 6.2 Falsifiable Consequences

3. Each of these predictions carries a specific falsifiability condition.

   from 6.2 Falsifiable Consequences

Where the claim stops

1. The rational numbers are democratic: they do not distinguish between Archimedean and $p$-adic completions.

   from 7. Conclusion: The Open Question

2. What is needed now is a systematic exploration of the $p$-adic and adelic analogues of black hole thermodynamics — a project this paper initiates but does not complete.

   from 7. Conclusion: The Open Question

3. If nature does not choose, then our laws must be reformulated to reflect that democratic structure.

   from 7. Conclusion: The Open Question

Where to read

1. Abstract197 words · 1 min
2. 1. Introduction: What Is a Measurement?357 words · 2 min
3. 2. The Bekenstein Bound Without a Base692 words · 3 min
4. 3. Ostrowski's Theorem: Why the Real Numbers?485 words · 2 min
5. 4. p-Adic and Adelic Quantum Mechanics436 words · 2 min
6. 5. Connection to QNFO Ultrametric Physics384 words · 2 min
7. 6. Proposal: An Adelic Bekenstein Bound505 words · 2 min
8. 7. Conclusion: The Open Question360 words · 2 min