Adelic Quantum Arithmetic: Particles as Prime Factors

In the paper's words

Standard quantum mechanics is written in the Archimedean completion of the rational numbers: Hilbert space over the complex numbers uses only the real place, one among all completions classified by Ostrowski's theorem. The primes are the other places, and their structure is multiplicative — unique factorization — never additive. Two published results show that the statistics binary is the multiplicative readout of a single integer lattice: the unrestricted exponent rule gives the Riemann zeta function and Bose–Einstein occupation; the squarefree rule, each prime dividing at most once, gives a ratio of two zeta values and Fermi–Dirac occupation.

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

Numbers it reports

1. Findings on record that constrain the program rather than support it: the Fenna–Matthews–Olson coupling data are anti-ultrametric (cophenetic correlation $0.426, p = 0.984$); a strict-ultrametricity spectrum hypothesis for the Wheeler–DeWitt setting was falsified — no finite-dimensional energy spectrum produces strict ultrametricity; the cosmic microwave background shows no log-periodic oscillations above 0.3% amplitude; exact FMO clustering is null ($p = 0.598$); and a diagonal refinement in four dimensions is insufficient without a tree-structured clock spectrum.

   from 6. Proved, conjectured, disconfirmed

How it is done

1. Ostrowski's theorem says that $\mathbb{Q}$ has other completions — one for every prime $p$, the $p$-adic fields $\mathbb{Q}_p$, where a number is measured by how divisible it is by $p$ rather than by its size.

   from 1. The Archimedean shadow

2. A theorem about zeta ratios is a statement about occupation distributions; a measurement of a specific-heat deviation is a statement about prime gaps.

   from 5. What a practitioner can do

3. The Adelic Representation Theorem — that a system realizing a sufficiently rich hierarchy of distinctions is computationally equivalent to an adelic quantum computer — remains a conjecture with no formal proof, and it is labeled as such wherever it appears.

   from 6. Proved, conjectured, disconfirmed

4. The machinery: maximum-entropy reasoning on a lattice of modes, the standard method of statistical mechanics.

   from 7. Where the premises end

What would falsify it

1. The family carries no exchange phase for any cap: permuting occupation labels yields only the signs $+1$ and $-1$, every observable is invariant under any inserted phase, and the canonical symmetric reading assigns the phase $+1$ for every cap — including the cap that reproduces Fermi counting, where fermions carry $-1$.

   from 3. The anyonic continuation, adjudicated

2. The same record supplies the observable that separates the arithmetic structure from a smooth continuum.

   from 3. The anyonic continuation, adjudicated

3. The prime-gap structure then supplies a computable distinguishing observable: the specific heat deviates from the smooth-density-of-states ideal gas at every sampled temperature, by up to roughly three quarters at low temperature, in both statistics.

   from 5. What a practitioner can do

Where the claim stops

1. Species labels multiply — a state of two distinguishable subsystems carries a product of labels — and they do not add, exactly as primes multiply into integers and never add.

   from 4. The crosswalk: particles as prime factors

2. The algebra is exact; the dictionary is proposed; and the falsification conditions are written: the dictionary fails if it requires auxiliary assumptions beyond the published isomorphisms to reach any practitioner-measurable quantity, and the specific-heat signature of §5 is a concrete place to test it.

   from 7. Where the premises end

3. This line of work differs from those in emphasis: it does not study a $p$-adic or adelic theory as an alternative dynamics, but reads the standard statistics themselves as the multiplicative structure of the integers, with the product formula as the invariant linking the places.

   from 8. Related work

Where to read

1. 1. The Archimedean shadow251 words · 1 min
2. 2. Statistics from the product formula317 words · 1 min
3. 3. The anyonic continuation, adjudicated237 words · 1 min
4. 4. The crosswalk: particles as prime factors239 words · 1 min
5. 5. What a practitioner can do396 words · 2 min
6. 6. Proved, conjectured, disconfirmed303 words · 1 min
7. 7. Where the premises end168 words · 1 min
8. 8. Related work229 words · 1 min
9. 9. Limitations101 words · 1 min
10. 10. Reproducibility210 words · 1 min