A Number That Changes When the Ruler Changes

In the paper's words

In the QNFO ultrametric physics program, the fine-structure constant α has been reframed as a cross-ratio of two measurable electron length scales, and adelic extensions on Bruhat–Tits buildings have been proposed. A natural but unresolved question is how the characteristic "depth" associated with the α-gap behaves when the underlying ultrametric hierarchy is re-parameterized to an arbitrary scaling ratio q rather than the privileged ratio q = α⁻¹. This paper adopts the convention D_q^α = ln(1/α)/ln q: the number of hierarchy levels, measured in units of the ratio q, needed to span the interval between the classical electron radius and the reduced Compton wavelength.

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

Numbers it reports

1. For $q = 2$ and a 1% uncertainty in $q (\delta q = 0.02)$: $\delta D = 4.9202436 \times 0.02/(2 \times 0.480453) = 0.0984048/0.960906 = 0.10241$, i.e., $D_{2}^\alpha = 7.098 \pm 0.102$ under that assumption.

   from 5. Results

2. Within that program, the fine-structure constant $\alpha \approx 1/137.036$ has been reframed as a cross-ratio of two measurable electron length scales, the classical electron radius and the reduced Compton wavelength [11].

   from 1. Introduction

3. The ultrametric physics research corpus [9] states the general program; the research plan [10] organizes its open problems, of which arbitrary-q re-parameterization of characteristic depths is one; the cross-ratio reframing of the fine-structure constant [11] demonstrates the methodological stance — reframing a physical constant as a geometric invariant with explicit falsifiability conditions — that we adopt here; and the adelic core synthesis [12] assembles p-adic analysis, Bruhat–Tits geometry, Ostrowski completions, and information-theoretic foundations into a single structure against which any relaxation of the base q must be measured.

   from 2. Background and Related Work

4. (a) Valuation depth on buildings: on a Bruhat–Tits building for a group over a discretely valued field, horoball layers are indexed by valuation; $D_{q}^\alpha$ counts layers between the two electron scales when the local parameter has ratio q [12].

   from 3. Methods

5. All numerical work uses: $\alpha = 1/137.035999$ (CODATA-style value as adopted in the QNFO cross-ratio reframing [11]); reduced Compton wavelength $\lambda _{C} = 3.861593 \times 10^{-13} m$; classical electron radius $r_{e} = 2.817940 \times 10^{-15} m$; Planck length $\ell _{P} = 1.616255 \times 10^{-35} m$.

   from 3. Methods

6. The $\alpha$-gap ratio is $R_{\alpha } = 1/\alpha = 137.035999$, with $\ln R_{\alpha } = 4.9202436$.

   from 5. Results

How it is done

1. (Two alternative conventions for the symbol $D_{q}^\alpha$ — a fractional q-derivative and a Rényi multifractal exponent — were proposed by independent drafts; per the reconciliation policy, they are documented in Appendix A rather than silently merged.) We derive $D_{q}^\alpha$ in closed form, prove its monotonicity, base covariance, and multiplicativity, and evaluate it numerically for $q \in {2, 3, \varphi , 10, \alpha ^{-1}}$: $D_{2}^\alpha = 7.09841, D_{3}^\alpha = 4.47860, D_{\varphi }^\alpha = 10.22469, D_{10}^\alpha = 2.13683$, and $D_{\alpha ^{-1}}^\alpha = 1$ exactly.

   from Abstract

2. We define $D_{q}^\alpha$, prove its elementary but load-bearing properties, compute it exactly for a family of bases, and show how the same machinery handles the vastly larger Planck-to-electron gap.

   from 1. Introduction

3. The view of ultrametrics as a zero-dimensional analogue of ordinary metrics, with the expectation that metric-space theorems admit ultrametric parallels, is systematized in the embedding–extension–interpolation program [3], which provides ultrametric versions of the Arens–Eells isometric embedding theorem, the Hausdorff extension theorem, and the Niemytzki–Tychonoff compactness characterization.

   from 2. Background and Related Work

4. 3.1 Setup and definitions.

   from 3. Methods

What would falsify it

1. Falsifiability criteria, failure modes, and convention-dependences are stated explicitly.

   from Abstract

2. The ultrametric physics research corpus [9] states the general program; the research plan [10] organizes its open problems, of which arbitrary-q re-parameterization of characteristic depths is one; the cross-ratio reframing of the fine-structure constant [11] demonstrates the methodological stance — reframing a physical constant as a geometric invariant with explicit falsifiability conditions — that we adopt here; and the adelic core synthesis [12] assembles p-adic analysis, Bruhat–Tits geometry, Ostrowski completions, and information-theoretic foundations into a single structure against which any relaxation of the base q must be measured.

   from 2. Background and Related Work

3. That invariance is the deepest precedent for our base-covariance property ($D_{q}^\alpha \cdot \ln q$ = const): if the valuation of [6] survives reparametrization, then invariance under change of ruler is a natural demand on ultrametric observables, and our finite, computable cousin of that invariance inherits its legitimacy.

   from 2. Background and Related Work

Where the claim stops

1. The central limitation is definitional: $D_{q}^\alpha$ is a bookkeeping quantity, not a new physical prediction.

   from 6. Discussion

2. Second, the assumption of a single uniform ratio q per level is idealized: real dendrograms [5], [7] and real contact hierarchies [8] have non-uniform branching, and a level-counting functional presupposes the homogeneous-lattice idealization underlying Bruhat–Tits buildings [12].

   from 6. Discussion

3. If the two electron scales were not taken in the exact ratio $\alpha$ — e.g., if one used the unreduced Compton wavelength $2\pi \lambda _{C} = 2.426310 \times 10^{-12} m$ — the identity $r_{e}/\lambda _{C} = \alpha$ would fail: $r_{e}/\lambda _{C}^{unreduced} = 2.817940/2426.310 \times 10^{-3} = 0.00116141 = \alpha /2\pi$, and the "$\alpha$-gap" would instead be a $(2\pi /\alpha )$-gap with $\ln = 4.9202436 + 1.8378771 = 6.7581207$, giving $D_{2} = 9.74997$.

   from 6. Discussion

Where to read

1. Abstract278 words · 1 min
2. 1. Introduction395 words · 2 min
3. 2. Background and Related Work672 words · 3 min
4. 3. Methods425 words · 2 min
5. 4. Analysis450 words · 2 min
6. 5. Results236 words · 1 min
7. 6. Discussion575 words · 3 min
8. 7. Conclusion141 words · 1 min
9. Appendix A. Divergence report421 words · 2 min
10. Appendix B. Claim attribution199 words · 1 min