Scale-Invariant Reality

In the paper's words

The foundational principles of a scale-free, epistemically-grounded reality establish the philosophical and mathematical bedrock upon which the entire Scale-Invariant Epistemic Framework is constructed. This framework posits that physical reality is fundamentally scale-invariant, with no intrinsic minimum or maximum physical scales, and that our knowledge of this reality is inherently limited by epistemic constraints. The principle of universal scale invariance asserts that the laws of physics maintain consistent mathematical form across all observational scales, from the quantum to the cosmological, with dimensional quantities transforming predictably under scale transformations rather than possessing absolute values.

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

Numbers it reports

1. For example, a fair coin flip has entropy 1 bit per flip, meaning it cannot be compressed below 1 bit per flip on average, while a biased coin with probability p of heads has entropy H(p) = -p $\log_{2} p - (1-p) \log_{2}(1-p) \lt 1$, allowing for compression.

   from 1.2. The Principle of

2. Current LHC measurements of Higgs couplings to vector bosons and fermions constrain the Higgs-dilaton mixing angle to |sin $\theta | \lt 0.3$ at 95% confidence level, which translates to constraints on the ratio of vacuum expectation values $\xi _{\sigma }\langle \sigma \rangle ^{2}/\xi _{H}\langle H\rangle ^{2}$.

   from 4.2. Scale-Invariant

3. Current upper limits on the tensor-to-scalar ratio from Planck and BICEP/Keck Array constrain $r \lt 0.032$ (95% CL), providing important constraints on inflationary models and the energy scale of inflation.

   from 5.1.

4. This systematic approach ensures that when a manufacturer states that a component measures 10.000 ± 0.002 mm, the uncertainty value represents not merely instrumental precision but a rigorously evaluated assessment of all known error sources throughout the measurement process, providing a realistic assessment of what can and cannot be known about the component’s true dimension.

   from 1.2. The Principle of

5. The GUM framework requires that measurement uncertainty be expressed as a standard uncertainty (the standard deviation of the measurement result) or as an expanded uncertainty (a coverage interval with a specified level of confidence, typically 95%).

   from 1.2. The Principle of

6. For scale-free networks with $\gamma \approx 2.5$, numerical simulations show that removing just 5-10% of the highest-degree nodes can fragment the network, compared to the near-100% removal required for random failures.

   from 3.1.

How it is done

1. This approach eliminates the hierarchy problem in particle physics by recognizing that the apparent vast disparity between the electroweak scale and the Planck scale arises from logarithmic running of coupling constants rather than from fundamental differences in scale.

   from 1.1. The Principle

2. Physical interpretation: The Weyl tensor represents the “free gravitational field” that can exist in vacuum regions, while the Ricci tensor represents the “bound gravitational field” directly coupled to matter through the Einstein equations.

   from 1.1.2.

3. Mathematically, these constraints are encoded in the structure of quantum mechanics and general relativity, with the uncertainty principle arising from the non-commutativity of quantum operators, the horizon entropy from the area law of black hole thermodynamics, and the measurement limits from the quantum Cramér-Rao bound.

   from 1.2. The Principle of

4. For a continuous random variable X with probability density function p(x) defined over a domain $\Omega \subseteq \mathbb{R}^{n}$, the differential entropy h(X) is formally defined as h(X) = -∫_Ω p(x) log p(x) dx, where the logarithm is typically taken to base 2 (yielding units of bits) or the natural logarithm (yielding units of nats).

   from 2.1.

What would falsify it

1. This principle asserts that there exists no intrinsic minimum or maximum physical scale in nature, challenging conventional approaches that introduce fundamental scales such as the Planck length or the size of the observable universe as absolute boundaries.

   from 1.1. The Principle

2. The Postulate of No Intrinsic Minimum or Maximum Physical Scale The postulate of no intrinsic minimum or maximum physical scale represents a radical departure from conventional approaches to physics that posit fundamental boundaries such as the Planck scale or the size of the observable universe.

   from 1.1. The Principle

3. The Invalidation of an Absolute Maximum Scale in Cosmology The invalidation of an absolute maximum scale in cosmology represents a critical challenge to the conventional wisdom that the observable universe defines an absolute boundary beyond which physical reality cannot be meaningfully described.

   from 1.1. The Principle

Where the claim stops

1. Similarly, in cosmology, the postulate rejects the idea of an absolute maximum scale by recognizing that cosmological horizons are observer-dependent and that the global structure of the universe cannot be determined from within any single causal patch.

   from 1.1. The Principle

2. This postulate does not deny that physical systems may exhibit characteristic scales under specific conditions, but rather asserts that these scales are emergent properties of particular configurations rather than absolute boundaries of physical reality.

   from 1.1. The Principle

3. The process begins with a quantum field theory defined with a momentum cutoff $\Lambda$, where the path integral is restricted to field configurations with momenta $k \lt \Lambda$.

   from 1.1. The Principle

Where to read

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