Ratio-Based Adelic Physics

In the paper's words

The tension between continuous Archimedean spacetime paradigms and discrete, non-Archimedean topological realities constitutes the central mathematical bottleneck in modern complexity science, hindering the unification of quantum mechanics, spin glass phase dynamics, and cognitive psychophysics. This manuscript resolves these disparities by introducing a ratio-based adelic framework, utilizing generalized scaling ratios (q) and Bruhat-Tits tree topologies to replace traditional integer-prime models. By formulating the Vladimirov fractional derivative operator natively for arbitrary transcendental values and deriving the corresponding q-adic product formulas, this methodology actively maps discrete states onto continuous metrics using the Monna projection.

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

Numbers it reports

1. Continuous Topology and Ultrametric Complexity across Spin Glasses, Linguistics, and Cosmology Author: Rowan Brad Quni-Gudzinas Contact: rowan.quni@outlook.com ORCID: 0009-0002-4317-5604 ISNI: 0000000526456062 DOI: https://papers.qnfo.org/papers/ratio-based-adelic-physics/ Date: 2026-04-06 Version: 1.0.2 Abstract : The tension between continuous Archimedean spacetime paradigms and discrete, non-Archimedean topological realities constitutes the central mathematical bottleneck in modern complexity science, hindering the unification of quantum mechanics, spin glass phase dynamics, and cognitive psychophysics.

   from Reconciling

2. Section 3.0 provides the newly generalized ratio-based mathematical methods, including transcendental adelic products and the continuous Monna projection mechanism.

   from 1.7 Structural Preview

3. Its eigenvalue spectrum fundamentally consists of discrete powers of the scaling ratio $q^{n\alpha}$, dictating that energy states are intrinsically quantized by the structural scale geometry itself (Dragovich et al., 2017).

   from 3.3 The Vladimirov

4. While the resulting equations lack the simple arithmetic elegance of base-10 classical physics, their operational fidelity across multiple distinct scale regimes compensates entirely for the abstraction.

   from 3.7 Methodological Validation

5. Empirical 3D short-range Ising models fail to show this symmetry breaking simply because they are operating mathematically below the critical scaling threshold required to stabilize the hierarchy against thermal decay (Hed et al., 2004).

   from 6.2 Resolving the

How it is done

1. By resolving the dimensionality dispute within 3D Ising models and formally deducing an emergent Lorentz symmetry directly from tree automorphisms, this generalized adelic ontology successfully projects discrete computational substrates onto continuous observables.

   from Reconciling

2. The study formally addresses how ultrametric topologies affect phase dynamics in disordered states, how scale ratios govern QEC thermodynamic limits, and how temporal dimensions emerge from static tree navigations.

   from 1.6 Research Questions and

3. Valuations can be formally constructed using arbitrary scaling ratios $q \gt 1$, establishing absolute values derived purely from fundamental constants rather than base-10 numerical representations.

   from 3.1 Ratio-Based Valuation

4. Consequently, the Vladimirov operator acts as the master kinetic algorithm for ratio-based physics.

   from 3.3 The Vladimirov

What would falsify it

1. Tension The fundamental architecture of physical reality exhibits a pervasive tension between the assumption of continuous Archimedean spacetime and the observable discrete, hierarchical nature of complex systems.

   from 1.1 Context and Foundational

2. This rigid constraint prevents the organic integration of $p$-adic mathematics into applied biophysics and observable cosmology.

   from 2.4 P-Adic Mathematics in

3. Product Formulas for Transcendental Ratios Ensuring that these discrete $q$-adic calculations map safely to the observable macro-universe requires a unification equation bridging all mathematical completions.

   from 3.4 Adelic

Where the claim stops

1. The framework natively struggles to model pure continuous dynamics without artificially executing infinite tree depths, inducing immense computational drag when approximating standard Newtonian physics.

   from 6.7 Limitations of the

2. Ending the discussion with responsible scientific skepticism does not weaken the core theorem, but rather maps the immediate territory for upcoming algorithmic refinement.

   from 6.7 Limitations of the

3. Tension The fundamental architecture of physical reality exhibits a pervasive tension between the assumption of continuous Archimedean spacetime and the observable discrete, hierarchical nature of complex systems.

   from 1.1 Context and Foundational

Where to read

1. Reconciling244 words · 1 min
2. 1.0 Introduction997 words · 4 min
3. 2.0 Literature1,029 words · 4 min
4. 3.0 Methodological Framework1,006 words · 4 min
5. 4.0 Results I: Physical1,105 words · 5 min
6. 5.0 Results1,128 words · 5 min
7. 6.0 Discussion1,078 words · 5 min
8. 7.0 Conclusion and Future709 words · 3 min
9. Appendices730 words · 3 min