The Ostrowski Dimensionless Reformulation: A Systematic Compilation of Fundamental Physics Equations in Planck Units

In the paper's words

Systematic compilation and dimensionless reformulation of 53 fundamental physics equations across ten disciplines, motivated by the Ostrowski Dimensionless Mandate. Dimensional formulations implicitly privilege the Archimedean completion of the rationals per Ostrowski theorem (1916). All quantities expressed as pure-number ratios to Planck-scale counterparts (ħ=c=G=k_B=1).

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

Numbers it reports

1. The critical distinction: the Ostrowski/Tate mandate requires dimensionless ratios, but a dimensionless ratio may still be a FUNCTION of scale. $\alpha (\tilde{Q}^{2})$ is dimensionless at every scale — but its numerical value changes from 1/137.036 at Q̃ → 0 (the IR limit) to 1/127.9 at Q̃ = $M_{Z}/E_{P}$ (the Z-boson scale), a 7.14% variation [established — PDG: α⁻¹(0) = 137.036, α⁻¹(M_Z) = 127.9] .

   from 2.5 Running Couplings and the Scale-Dependence of Dimensionless Quantities

2. Every formula in §3 that writes "$\alpha$" implicitly means "$\alpha (\tilde{Q}^{2})$ at the formula's characteristic scale": Atomic physics (Bohr radius, Rydberg, hydrogen levels, Thomson): Q̃ ~ $\alpha$ m̃_e → IR value 1/137.036 Quantum Hall / Josephson / conductance: Q̃ at the Landau-level scale → IR value (0.01% correction negligible) Coulomb law: Q̃ ~ 1/r̃ → running with distance Z-pole / electroweak: Q̃ = $M_{Z}/E_{P} \to 1/127.9$ Planck-scale: Q̃ ~ 1 → determined by the running equation The same applies to $\sin^{2}\theta _{W}(\tilde{Q}^{2})$ and G̃_F(Q̃²): the Weinberg angle runs from ~0.23 (low energy) to 0.2312 ($M_{Z}$, MS-bar); the Fermi constant is scale-dependent.

   from 2.5 Running Couplings and the Scale-Dependence of Dimensionless Quantities

3. A Planck-mass black hole would have ${\tilde{T}}_{H}$ ≈ $1/(8\pi ) \approx 0.04$, corresponding to approximately 4% of the Planck temperature.

   from 3.3 General Relativity

4. The ODR alternative: Represent every physical quantity as a Hensel code — a tuple of residues modulo prime powers: x∈Q↦(x mod p1k1, x mod p2k2, …, x mod pnkn)x∈Q↦(xmodp1k1​​,xmodp2k2​​,…,xmodpnkn​​) Choose primes pipi​ such that the product P=∏pikiP=∏piki​​ exceeds the maximum denominator and numerator you expect.

   from Program II: The Computational Revolution — Exact Rational Simulations

5. Concrete step: For the lepton sector, construct a 3×3 matrix MijMij​ whose entries are the cross-ratios of the Compton frequencies.

   from Program III: The Mechanism Problem — Fixed Points on the Bruhat-Tits Tree

6. The special case of \pi: The paradigmatic dimensionless number -- the circumference-to-diameter ratio -- illustrates the same principle. pi as the real number 3.14159... exists only at the Archimedean place; its decimal expansion has no p-adic meaning.

   from 2.1 Ostrowski's Theorem and Place-Democracy

How it is done

1. The central claim is refined from the original v2.0 compilation: dimensional formulations of physical laws implicitly privilege the Archimedean completion of the rational numbers, while dimensionless ratios preserve place-democracy under Ostrowski's theorem (1916).

   from Abstract

2. The mandate is grounded in Ostrowski's theorem (1916), which classifies every non-trivial absolute value on $\mathbb{Q}$ as equivalent either to the standard real absolute value |·|_$\infty$ or to a p-adic absolute value |·|_p for some prime p [established -- Ostrowski, 1916, Acta Mathematica 41:271-284] .

   from 1.1 The Ostrowski Dimensionless Mandate

3. Theorem (Ostrowski, 1916): Every non-trivial absolute value on $\mathbb{Q}$ is equivalent either to the standard real absolute value |·|_$\infty$ or to a p-adic absolute value |·|_p for some prime p. [established] Proof sketch: Let |·| be a non-trivial absolute value on $\mathbb{Q}$.

   from 2.1 Ostrowski's Theorem and Place-Democracy

4. Theorem: Let F(x₁,..., $x_{n}$; $\hbar , c, G$, $k_{B}$) = 0 be a dimensionally homogeneous physical law.

   from 2.2 The Dimensional-Dimensionless Correspondence Theorem

What would falsify it

1. Key $C_{2}$ cross-formula connections: Spin → B-T vertex: The Zitterbewegung as a p-Adic Observable paper (https://papers.qnfo.org/papers/zbw-majorana-tqc/) proposes that Majorana zero modes have ultrametric signatures readable at $C_{2}$ vertices.

   from 2.4 The Completion Lattice and Cross-Formula Bridges

2. Falsifiability condition: The adelic extension would be disconfirmed if any dimensionally homogeneous physics formula, when expressed in dimensionless Planck units, takes a functional form that cannot be expressed as a rational function of ratios — i.e., if F(x̃_1,..., x̃_n) = 0 is not a member of ℚ(x̃_1,..., x̃_n).

   from 2.4 The Completion Lattice and Cross-Formula Bridges

3. Such a formula would be Archimedean-privileging in a sense stronger than mere dimensional convention: its very functional form would require the analytic structure of $\mathbb{R}$. [not yet falsifiable — all known physics formulas reduce to rational relations of dimensionless ratios]

   from 2.4 The Completion Lattice and Cross-Formula Bridges

Where the claim stops

1. It covers 53 fundamental equations but does not include every formula in every subfield.

   from 4.3 Limitations

2. Future work could extend the inventory to these areas. [speculative] Pedagogical precedent: The external literature provides strong support for the pedagogical value of the dimensionless reformulation.

   from 4.3 Limitations

3. While Humpherys's approach is explicitly pedagogical (the "hidden role" is a teaching insight, not an ontological claim), ODR's Ostrowski rationale provides the mathematical justification for why the pedagogical insight has ontological force: the dimensional form literally cannot be evaluated at non-Archimedean places, making the dimensionless form not only clearer but uniquely well-defined across all completions of $\mathbb{Q}$. [established] Additionally, the dimensionless reformulation does not address the question of whether physical laws ARE place-democratic -- it only makes the formulas compatible with such an interpretation if one chooses to adopt it.

   from 4.3 Limitations

Where to read

1. The Research Roadmap: From Mathematical Hygiene to Physical Prediction1,468 words · 6 min
2. Appendix A: Detailed Formula Derivations1,971 words · 9 min
3. Appendix B: Calibration Register405 words · 2 min