Adelic Consistency Conditions Across Primes as a Candidate Selection Principle for Physical Constants: Formulation, Negative Control, and Toy Dynamics

In the paper's words

The three constants problem—why the fine-structure constant takes its observed value, why charged-lepton masses are hierarchically spaced, and what fixes the absolute mass scale—has no accepted resolution within the Standard Model. We develop a research program, stated as a testable conjecture rather than a result, in which the missing information is encoded not in the real continuum but in the non-archimedean (p-adic) places of an adelic structure over the rationals. The real line is treated as one leaf of an adelic foliation whose other leaves are ultrametric Bruhat–Tits trees; particle stability is conjectured to require consistency conditions satisfied at every prime simultaneously.

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

Numbers it reports

1. Finally, the Adelic Cross-Domain Program report [12] documents, in its v3.2 correction record dated 2026-07-25, that arithmetic errors in mass ratio triplets were corrected with 5 of 9 triplets verified independently, that an Efimov lambda derivation was replaced with an honest structural correspondence, that a constraining-literature section was added, and that a semigroup density epistemological risk was acknowledged.

   from 2. Background and Related Work

2. Following the correction history of [12], in which arithmetic errors in mass-ratio triplets were found and only partially independently verified, we adopt: every numerical claim is computed in Section 4 with all steps shown; no empirical physics data are used except where explicitly labeled; structural analogies are labeled as such.

   from 3.4 The three checks and the honesty protocol

How it is done

1. We construct a toy cross-prime closure functional whose stationary point can be expressed analytically, and we note that a direct quantitative comparison with the observed fine-structure constant is deferred to future work, and exhibit a Newton iteration for $\sqrt{137}$ as a template exposing how naive dynamics leave the $p$-adic places passive.

   from Abstract

2. The entry for quasi-adelic measures and equidistribution on $\mathbb{P}^1$ [1] records that Baker-Rumely and Favre-Rivera-Letelier independently proved an arithmetic equidistribution theorem for points of small height on the Berkovich compactification of the projective line with respect to an adelic measure, and that Chambert-Loir proved a more general version for curves.

   from 2. Background and Related Work

3. We posit a foliation $\mathcal{F}$ of the adelic configuration space whose leaves $\mathcal{L}_\infty, \mathcal{L}_2, \mathcal{L}_3, \ldots$ correspond to the places; $\mathcal{L}_\infty$ is the real line and $\mathcal{L}_p$ is (a subtree of) the Bruhat–Tits tree $\mathcal{T}_p$ of $\mathbb{Q}_p$, the contractible simplicial complex whose boundary is $\mathbb{P}^1(\mathbb{Q}_p)$, the space on which the equidistribution theorem of [1] is formulated.

   from 3.2 Foliation and stability conjecture

4. Any implementation must pass three checks: (1) a product-formula audit determining what constraint the identity actually imposes (Section 4.1–4.2); (2) a cross-prime closure check of whether the product formula, supplemented by quantization conditions, yields discrete admissible sets matching observed spectra (Section 4.3 gives a toy version; the full check is open); (3) an ultrametric simulation on the trees $\mathcal{T}_p$ (specified in Section 3.5 but not executed in this paper).

   from 3.4 The three checks and the honesty protocol

What would falsify it

1. It claims (a) a precise mathematical skeleton for the conjecture, (b) explicit verifications of the ingredients that can be verified by hand, (c) a negative control showing the product formula is constraint-free, (d) a toy closure model whose unique stationary point is computable and which is empirically falsified in its simplest form—an instructive negative result—and (e) a map of the literature and prior project reports against which progress can be measured.

   from 1. Introduction

2. Phase 3 Synthesis [11] investigated the central open question left by Phase 2—whether the adelic framework makes a falsifiable numerical prediction differing from the Standard Model—targeting a "suggestive coincidence" discovered in Phase 2; the summary truncates before naming the coincidence, so we treat the existence of such a target as the only established fact.

   from 2. Background and Related Work

Where the claim stops

1. We construct a toy cross-prime closure functional whose stationary point can be expressed analytically, and we note that a direct quantitative comparison with the observed fine-structure constant is deferred to future work, and exhibit a Newton iteration for $\sqrt{137}$ as a template exposing how naive dynamics leave the $p$-adic places passive.

   from Abstract

2. A functional on $\mathbb{A}_{\mathbb{Q}}$ whose stationary points reproduce $\alpha$ and the lepton mass ratios should exist, or the program fails.

   from 1. Introduction

3. We emphasize what this paper does and does not claim.

   from 1. Introduction

Where to read

1. Abstract229 words · 1 min
2. 1. Introduction418 words · 2 min
3. 2. Background and Related Work944 words · 4 min
4. 3. Methods504 words · 2 min
5. 4. Analysis627 words · 3 min