Mathematical Structures Underlying Physical Laws

In the paper's words

This work addresses the foundational schism in modern physics, which arises from the perceived incompatibility between classical continuous, deterministic reality and quantum discrete, probabilistic phenomena. The Post-Quantum Synthesis (PQS) framework resolves this by asserting that the universe is fundamentally continuous, local, and deterministic, while observation is intrinsically discrete. Quantum mechanics is presented as the unique calculus of inference bridging these domains.

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

Numbers it reports

1. According to Axiom III, it is an epistemic tool representing the observer’s 50% uncertainty in their knowledge of the cat’s physical state, given the causal disconnection from the box’s interior.

   from 4.2

2. 4.4.2.3 The Conclusion of Instrumental Dominance: 99.999816 Percent of Observed Width Originating from the Detector This high CEC value implies that the detector’s resolution overwhelmingly dominates the observed lineshape.

   from 4.4

3. 1.1.1.2.1 The Experimental Anomalies Defying Classical Wave Theory: The Existence of a Threshold Frequency and Instantaneous Emission Experiments showed that electron emission occurred only if the light’s frequency exceeded a specific threshold, regardless of intensity, and that emission was virtually instantaneous, even at very low light intensities.

   from 1.1

4. 2.2.3.2 The Wiener Measure for Diffusion as a Sum Over Paths Weighted by a Real Decaying Factor: $\int \mathcal{D}[x] e^{-S_E}$ Similarly, the propagator for a diffusion process (the heat kernel) is expressed as a sum over paths.

   from 2.2

5. 3.1.3 The Principle of Maximum Entropy: The Information-Theoretic Foundation for Statistical Neutrality From an information-theoretic perspective, the Gaussian represents the most “neutral” or “unbiased” probability distribution possible under certain constraints.

   from 3.1

6. 3.1.3.1 The Formulation of the Variational Problem for Shannon Entropy: Maximize $H(f) = -\int f(x) \log f(x) dx$ Subject to $\int F Dx = 1$ and $\int (x-\mu)^2 F Dx = \sigma^2$ (Shannon, 1948) Given a random variable for which only the mean ($\mu$) and variance ($\sigma^2$) are known, the principle of maximum entropy states that the most rational and least biased probability distribution to assume is the one that maximizes the Shannon information entropy, $H(f)$.

   from 3.1

How it is done

1. The spectral theorem for the self-adjoint operator $L = -\Delta$ allows all solutions to be expressed in a unified form.

   from 2.1

2. 2.2.2.1 The General Bandwidth Theorem for Fourier Pairs: $\Delta X \Delta K \ge \frac{1}{2}$ The bandwidth theorem is a rigorous mathematical result stating that for any function and its Fourier transform, the product of their standard deviations (a measure of their spread) cannot be less than a certain constant.

   from 2.2

3. 3.1.1.2 The Role as the Mathematical Basis for the Central Limit Theorem This convolutional stability is the mathematical engine that drives the Central Limit Theorem.

   from 3.1

4. 3.2.1 The Central Limit Theorem as the Law of Statistical Gravity for Aggregated Systems The Central Limit Theorem (CLT) is the primary mechanism by which the Gaussian manifests in the epistemic domain.

   from 3.2

What would falsify it

1. 1.1.1.1 The Misidentification of Statistical Binning in Planck’s Blackbody Solution At the close of the 19th century, classical physics, specifically the Rayleigh-Jeans law, failed to accurately predict the spectral radiance of blackbody radiation.

   from 1.1

2. 1.1.1.1.1 The Classical Rayleigh-Jeans Law and the Ultraviolet Catastrophe: The Divergence of the Energy Density Integral The classical prediction for spectral radiance $B_\nu(T)$ at frequency $\nu$ and temperature $T$ is: $$B_\nu(T) = \frac{2 \nu^2}{c^2} k_B T$$ where $k_B$ is the Boltzmann constant and $c$ is the speed of light (Jeans, 1905; Rayleigh, 1900).

   from 1.1

3. It meticulously encodes all information an observer possesses that can be used to predict future outcomes of measurements.

   from 1.2 The

Where the claim stops

1. 1.1.1.2.1 The Experimental Anomalies Defying Classical Wave Theory: The Existence of a Threshold Frequency and Instantaneous Emission Experiments showed that electron emission occurred only if the light’s frequency exceeded a specific threshold, regardless of intensity, and that emission was virtually instantaneous, even at very low light intensities.

   from 1.1

2. 2.2.2 The Inherent Trade-off in Localization: The Uncertainty Principle as a Universal Wave Property A fundamental property of the Fourier transform is that a function cannot be simultaneously localized in both its original and its conjugate domain.

   from 2.2

3. This mathematical trade-off is the origin of the uncertainty principle.

   from 2.2

Where to read

1. The231 words · 1 min
2. 1.01,532 words · 7 min
3. 2.01,278 words · 6 min
4. 3.01,753 words · 8 min
5. 4.02,316 words · 10 min
6. 5.01,190 words · 5 min
7. 6.01,858 words · 8 min
8. 7.05,160 words · 22 min