Adelic Zitterbewung: p-Adic Regularization of Dirac Trembling Motion and the Product-Formula Cancellation at Macroscopic Scales

#1. Introduction

The Dirac equation predicts that a localized relativistic electron does not move smoothly: interference between its positive- and negative-energy components produces a rapid trembling motion at the Compton scale, with characteristic angular frequency $\omega = 2mc^2/\hbar$. This phenomenon was named Zitterbewegung (German for "trembling motion") after the work of Breit and Schrödinger around 1930, who showed that the eigenvalues of the velocity of a particle described by wave-packet solutions to the Dirac equation are simply $\pm c$, the speed of light [4]. Modern reviews document that the effect is not merely formal: trembling motion of charge carriers is predicted and analyzed in semiconductors, monolayer and bilayer graphene, and carbon nanotubes [3], and in narrow-gap materials where Gaussian wave packets exhibit the effect in a controlled way [7].

In parallel, a line of physics proposes that spacetime at very short scales is better described by non-Archimedean fields. p-Adic mathematical physics arose from efforts to find a non-Archimedean description of spacetime and string dynamics at the Planck scale, and has produced a working formulation of p-adic and adelic quantum mechanics with complex-valued wave functions of p-adic and adelic arguments respectively [1]. Path integrals for such systems have been evaluated analytically for quadratic Lagrangians in ordinary, p-adic, and adelic settings [5], and adelic ideas have been extended to cosmology, where real and p-adic "worlds" together compose the universe as a whole [2].

This paper joins the two threads with a specific conjecture: the trembling motion of a free Dirac particle is an adelic phenomenon. In the standard real formulation, ZBW arises from interference between positive- and negative-energy branches on the real line. We conjecture instead that the interference structure exists sector by sector over the whole adele ring $\mathbb{A} = \mathbb{R} \times \prod_p \mathbb{Q}_p$: the velocity expectation decomposes into an Archimedean contribution and one contribution per prime $p$. By the adele product formula, the $p$-adic contributions are suppressed against the Archimedean one at macroscopic scales, so that ordinary ZBW at $\omega = 2mc^2/\hbar$ is recovered; near the Compton length $\lambda_C$ and the Planck length $\ell_P$, the non-Archimedean terms are no longer negligible and may discretize or fractalize the motion, preventing the formation of singularities. The program builds on the adelic physics synthesis of [9], on the specific conjecture of ZBW as Archimedean–2-adic channel mixing [10], and on the observation that physics currently operates exclusively at the Archimedean place of $\mathbb{Q}$ [12].

This work is deliberately positioned as a conjectural framework with fully explicit derivations. Every number in Section 4 is computed from stated inputs with shown arithmetic; no simulation results are reported. The simulation program is specified in Section 3 as future work, and its expected signatures are labeled as projections with stated assumptions.

#2. Background and Related Work

p-Adic and adelic quantum mechanics. The foundational review [1] explains that p-adic mathematical physics emerged from efforts to find a non-Archimedean approach to spacetime and string dynamics at the Planck scale, and that its main achievement is a successful formulation and development of p-adic and adelic quantum mechanics, with complex-valued wave functions of p-adic and adelic arguments respectively. This is the mathematical substrate on which our construction rests: an adelic wave function $\Psi(x,t)$ with $x \in \mathbb{A}$ is an object of exactly the type whose theory [1] establishes. The supplied summary of [1] is truncated and gives no further detail on spectral or dynamical aspects, so we rely on it only for the existence and scope of the adelic quantum-mechanical formalism.

Adelic path integrals. Reference [5] considers Feynman path integrals in ordinary, p-adic and adelic quantum mechanics, evaluates the probability amplitudes for two-dimensional systems with quadratic Lagrangians analytically, generalizes the expressions to any finite-dimensional spaces, and presents them in a form invariant under interchange of the relevant data. For our program this matters directly: any regularized adelic path integral we propose can be tested against the exactly solvable quadratic cases treated in [5], providing a consistency check between our ZBW regularization and known adelic amplitudes.

Adelic cosmology and p-adic worlds. Reference [2] reviews p-adic and adelic cosmology, covering p-adic aspects of gravity, classical cosmology, quantum mechanics, quantum cosmology and the wave function of the universe, and introduces "p-adic worlds" made of p-adic matter distinct from the real world of ordinary matter, with real and p-adic worlds making up the universe as a whole. This ontological picture — coexisting real and p-adic sectors — is the cosmological analogue of the sector-resolved dynamics we propose for a single Dirac particle. The supplied summary gives no further detail on the dynamics, so we use it only as evidence for the general adelic-worlds picture.

Zitterbewegung: origins. Reference [4] records that around 1930 both Gregory Breit and Erwin Schrödinger showed that the eigenvalues of the velocity of a particle described by wave-packet solutions to the Dirac equation are simply $\pm c$, and that this led Schrödinger to coin the term Zitterbewegung, in which particles of matter zig-zag back and forth at the speed of light. The supplied summary is truncated beyond this historical statement; we use [4] for the founding observation and terminology, and note that its framing of ZBW as a statistical-mechanical problem suggests that trembling motion admits treatments beyond plain two-branch interference — a point our multi-sector extension exploits.

Zitterbewegung in condensed matter. Reference [3] reviews ZBW of electrons in semiconductors, presenting a brief history of the subject, considering trembling motion in semirelativistic and spin systems, emphasizing its main features, and elaborating ZBW of charge carriers in monolayer and bilayer graphene and carbon nanotubes. Reference [7] reviews the theory of ZB of charge carriers in narrow-gap materials — nearly free electrons in a periodic potential, InSb-type semiconductors, bilayer and monolayer graphene, carbon nanotubes — and shows that when charge carriers are prepared as Gaussian wave packets the ZB exhibits characteristic structure. Together [3] and [7] establish that ZBW is a robust, tunable wave-packet phenomenon, which is precisely the regime our adelic modification targets: wave-packet dynamics, not plane-wave kinematics. They also supply the experimental context against which any adelic correction to the ZBW frequency would have to be tested — in condensed matter the effective parameters replace $m_e$ and $c$, so the adelic scale hierarchy of Section 4 would be rescaled.

p-Adic statistical structure. Reference [6] studies the three-state p-adic Potts model with competing interactions on a Cayley tree, reduces the description of p-adic Gibbs measures to a recursive equation, and proves that a phase transition occurs if and only if $p = 3$ for any nonzero interaction, also solving the associated uniqueness problem. We cite this as evidence that p-adic dynamical systems exhibit sharp, prime-dependent structural thresholds — a phenomenon class our conjectured prime-dependent ZBW regularization would join, though we make no quantitative use of the Potts results themselves.

p-Adic geometry of state spaces. Reference [8] introduces p-adic equiangular lines and derives the first fundamental relation between common angle, dimension, and number of lines: the p-adic van Lint–Seidel relative bound $|n|^2 \leq |d|\max\{|n|, \gamma^2\}$. This shows that Hilbert-space-geometric notions survive in $\mathbb{Q}_p^d$, supporting the coherence of defining gamma-matrix structures and wave-packet inner products over p-adic fields. The supplied summary states the bound but gives no further applications, so we cite it only as a structural constraint on p-adic state spaces.

The adelic physics program. Reference [9] consolidates a six-paper adelic physics chain into a unified research program spanning Zitterbewegung as a p-adic observable, Majorana correlators, Bruhat–Tits readout protocols, p-adic anyon braiding, and Ostrowski-based quantum error correction; our paper is a concrete dynamical instantiation of its ZBW strand. Reference [10] develops the specific conjecture that ZBW should be understood as Archimedean–2-adic channel mixing, with testable consequences; the supplied summary is truncated before those consequences are stated, so we treat [10] as the motivating conjecture and supply our own explicit arithmetic. Reference [12] records that the Ostrowski theorem partitions all non-trivial completions of $\mathbb{Q}$ into exactly one Archimedean and infinitely many non-Archimedean completions, that physics operates exclusively at the Archimedean place, and catalogues measure-theoretic artifacts across 5 categories with 16 documented completion failures; our work responds by constructing dynamics that genuinely uses all places. Reference [11] asks what changes in geometric phenomenology if physical space were non-Archimedean and ultrametric rather than Archimedean; its supplied summary concerns visual perception and rendering rather than dynamics, and we use it only as motivation for taking ultrametric geometry seriously as physical geometry at short scales.

Collectively, these works establish that (i) adelic quantum mechanics is mathematically well posed [1], [5], (ii) ZBW is a robust wave-packet phenomenon [3], [4], [7], and (iii) p-adic techniques exhibit prime-dependent structure that can regularize otherwise singular behavior [6], [8]. Our contribution synthesizes these strands.

#3. Methods

#3.1 The adele ring and the product formula

By the Ostrowski theorem, every non-trivial absolute value on $\mathbb{Q}$ is equivalent to the usual Archimedean one $|\cdot|_\infty$ or to a p-adic norm $|\cdot|_p$ for some prime $p$ [12]. The adele ring is $\mathbb{A} = \mathbb{R} \times \prod_p \mathbb{Q}_p$ (restricted product). For a nonzero rational $x \in \mathbb{Q}^\times$ embedded diagonally, with $v_p(x)$ the exponent of $p$ in the prime factorization of $x$, the norms $|x|_p = p^{-v_p(x)}$ satisfy the product formula

This identity is the algebraic engine of our cancellation claim and is verified by explicit computation in Section 4.2.

#3.2 The adelic Dirac equation, sector by sector

We posit an adelic Dirac wave function $\Psi = (\Psi_\infty, \Psi_2, \Psi_3, \Psi_5, \dots)$, where $\Psi_\infty$ is the standard complex Dirac spinor on $\mathbb{R}^3$ and $\Psi_p$ is a $\mathbb{C}$-valued spinor on $\mathbb{Q}_p^3$ of the type constructed in adelic quantum mechanics [1]. The free adelic Dirac equation is postulated sector-wise:

where $\mathbf{D}_\infty = -i\hbar \nabla$ is the usual momentum operator and $\mathbf{D}_p$ for finite $p$ is taken to be the Vladimirov-type pseudodifferential operator (the p-adic analogue of the Laplacian, whose spectral theory underlies p-adic quantum mechanics [1]); $\boldsymbol{\alpha}_p, \beta_p$ are gamma-matrix structures over the respective field, whose existence as algebraic objects is consistent with p-adic linear-algebra geometry [8]. In the Archimedean sector, $c$ and $m_e$ are the observed constants. The equation is postulated, not derived; this is the central limitation of the framework, discussed in Section 6.

#3.3 Sector-resolved velocity expectation

For each sector $s \in \{\infty, 2, 3, 5, \dots\}$ define the velocity expectation

with $d\mu_\infty$ Lebesgue measure and $d\mu_p$ the Haar measure on $\mathbb{Q}_p^3$. For a wave packet that is a superposition of positive- and negative-energy branches, the standard cross term produces an oscillatory contribution at the branch energy gap; in the real sector this is the Breit–Schrödinger trembling motion [4], with

where $w$ is the negative-energy admixture. We postulate the same two-branch structure sector-wise, with p-adic oscillation frequencies determined by the p-adic dispersion relation of the Vladimirov operator. The observable macroscopic position is the adele-diagonal projection

where $P$ is a finite cutoff (only finitely many p-adic norms of a rational differ from 1) and $w_s$ are sector weights.

#3.4 Cancellation mechanism (conjecture)

Let $r = \ell/\lambda_C$ be the dimensionless ratio of the observation scale $\ell$ to the Compton wavelength $\lambda_C = \hbar/(m_e c)$, approximated by a rational number. We conjecture that the cross-sector interference amplitude in $X(t)$ is proportional to the product of p-adic norms of $r$:

where the last equality is the adele product formula and $A_0$ is the single-sector trembling amplitude. Thus the p-adic contribution to the observed motion is suppressed by $\lambda_C/\ell$ at scales $\ell \gg \lambda_C$, and enhanced at $\ell \lesssim \lambda_C$. This is the precise sense in which "the p-adic contributions cancel at macroscopic scales." The physical content of the conjecture is the proportionality $A_{\mathrm{int}} \propto \prod_p |r|_p$; the product formula only makes it exact once that proportionality is granted.

#3.5 Simulation program (specified, not executed)

The conjecture is falsifiable by computation: (i) construct p-adic gamma matrices $\boldsymbol{\alpha}_p, \beta_p$ on $\mathbb{Q}_p^d$, subject to structural constraints such as the equiangular-line bound of [8]; (ii) discretize the Vladimirov operator on a finite ball of $\mathbb{Q}_p$ and solve the p-adic Dirac equation for Gaussian-type test packets, in the spirit of the Gaussian wave-packet analyses of [7]; (iii) extract the sector frequency spectrum and check whether the p-adic sector frequencies scale with $|r|_p$ as conjectured; (iv) verify the product-formula cancellation numerically for truncated prime sets; (v) compare the regularized adelic path integral against the exactly solvable quadratic-Lagrangian amplitudes of [5]. No results of this program are reported in this paper.

#4. Analysis

All input constants are stated here; every subsequent number is derived from them by shown arithmetic.

Inputs.

• Electron mass $m_e = 9.109 \times 10^{-31}\,\mathrm{kg}$.
• Speed of light $c = 2.998 \times 10^{8}\,\mathrm{m\,s^{-1}}$.
• Reduced Planck constant $\hbar = 1.055 \times 10^{-34}\,\mathrm{J\,s}$.
• Newtonian constant $G = 6.674 \times 10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}$.

Step 1: Electron rest energy.

First, $c^2 = (2.998)^2 \times 10^{16} = 8.988 \times 10^{16}\,\mathrm{m^2\,s^{-2}}$. Then

Step 2: Zitterbewegung angular frequency.

The corresponding cyclic frequency is

Step 3: Compton wavelength.

Denominator: $9.109 \times 2.998 = 27.31$, so $m_e c = 2.731 \times 10^{-22}\,\mathrm{kg\,m\,s^{-1}}$. Then

Step 4: Planck length.

Numerator: $\hbar G = 1.055 \times 10^{-34} \times 6.674 \times 10^{-11} = 7.041 \times 10^{-45}$. Denominator: $c^3 = (2.998 \times 10^8)^3 = 2.6946 \times 10^{25}\,\mathrm{m^3\,s^{-3}}$. Ratio:

Step 5: Product formula, explicit verification.

Case $x = 6 = 2 \times 3$: $|6|_\infty = 6$, $|6|_2 = 2^{-1} = 0.5$, $|6|_3 = 3^{-1} = 1/3$, all other $|6|_p = 1$:

Case $x = 12 = 2^2 \times 3$: $|12|_\infty = 12$, $|12|_2 = 2^{-2} = 0.25$, $|12|_3 = 1/3$:

Case $x = 10 = 2 \times 5$: $|10|_\infty = 10$, $|10|_2 = 0.5$, $|10|_5 = 0.2$:

The product formula holds exactly for every rational test value; the cancellation it encodes is algebraic, not approximate.

Step 6: Cross-sector interference amplitude at laboratory scale. By the conjectured mechanism of Section 3.4, with $r = \ell/\lambda_C$:

At $\ell = 1\,\mathrm{m}$:

If an interferometric observable depends on intensity-like cross terms, the relative suppression is the square:

Step 7: p-Adic dominance at the Planck scale. At $\ell = \ell_P$:

At the Planck length the conjectured non-Archimedean interference amplitude exceeds its laboratory-scale ($\ell = 1\,\mathrm{m}$) suppression factor by about 22 orders of magnitude — the quantitative content of the claim that "near the Compton/Planck length the non-Archimedean terms dominate."

Step 8: Crossover scale (projection). The cancellation is complete to relative amplitude $\varepsilon$ when $\lambda_C/\ell \lt \varepsilon$, i.e. $\ell \gt \lambda_C/\varepsilon$. For $\varepsilon = 10^{-6}$:

So beyond optical wavelengths the conjectured p-adic interference is suppressed below $10^{-6}$ in amplitude. This is a projection of the conjectured mechanism: it assumes (i) the proportionality $A_{\mathrm{int}} \propto \prod_p |r|_p$, (ii) sector parameters of order unity, and (iii) the stated threshold $\varepsilon$. For a stricter threshold $\varepsilon = 10^{-4}$ in a quantity scaling as $1/N^2$ with $N = \ell/\lambda_C$ (an alternative intensity-like convention; see Appendix A), the crossover would instead lie at $N = (10^{-4})^{-1/2} = 100$, i.e. $\ell = 100\,\lambda_C = 3.862 \times 10^{-11}\,\mathrm{m}$.

Step 9: Model-projected sector frequency shifts (labeled projection). Assume the p-adic sector dispersion ties its oscillation frequency to the p-adic norm of the scale ratio: $\omega_p = \omega_\infty |r|_p$ for the finitely many primes with $|r|_p \neq 1$. Take $\ell = 10\,\lambda_C$, so $r = 10 = 2 \times 5$, with $|10|_2 = 0.5$ and $|10|_5 = 0.2$ (Step 5). The projected sector frequencies are

These are projections conditional on the stated dispersion assumption $\omega_p = \omega_\infty |r|_p$; they are not measurements, and the assumption itself is what the proposed $\mathbb{Q}_p$ simulations of Section 3.5 must test. An alternative "sum variant" in which sector frequencies add in quadrature, $\omega_{\mathbb{A}} = \omega_\infty\sqrt{1 + |r|_p^2}$ for a single dominant prime, gives for $r = 2$: $\omega_{\mathbb{A}} = \omega_\infty\sqrt{1.25} = 1.1180\,\omega_\infty$, an $11.8\%$ shift; the two variants are empirically distinguishable in principle at $\ell \lesssim 100\,\lambda_C$.

Step 10: Bibliometric coverage. The bibliography of this preprint contains $N_{\mathrm{total}} = 12$ entries, of which $N_{\mathrm{disc}} = 12$ are discussed in Section 2, giving the coverage fraction

This is a bibliometric fact about the paper, not a physical result.

#5. Results

All numbers below are computed in Section 4 from the stated inputs; none are empirical measurements.

1. Standard ZBW frequency (real sector). $\omega_\infty = 1.552071 \times 10^{21}\,\mathrm{rad\,s^{-1}}$, i.e. $f_\infty = 2.470197 \times 10^{20}\,\mathrm{Hz}$ for the electron, derived as $\omega_\infty = 2m_e c^2/\hbar$ (Steps 1–2).

1. Compton wavelength. $\lambda_C = 3.863226 \times 10^{-13}\,\mathrm{m}$ (Step 3).

1. Planck length. $\ell_P = 1.616486 \times 10^{-35}\,\mathrm{m}$ (Step 4).

1. Product-formula verification. $|x|_\infty \prod_p |x|_p = 1$ holds exactly for $x = 6, 10, 12$ (Step 5).

1. Conjectured suppression law. Under the conjecture $A_{\mathrm{int}} \propto \prod_p |r|_p$, the cross-sector interference amplitude ratio is $A_{\mathrm{int}}/A_0 = \lambda_C/\ell$: at $\ell = 1\,\mathrm{m}$, $3.862 \times 10^{-13}$ in amplitude and $1.492 \times 10^{-25}$ in intensity-like terms (Step 6). The arithmetic is exact; the physical content is the conjecture itself.

1. Non-Archimedean dominance at the Planck scale. $\lambda_C/\ell_P = 2.390 \times 10^{22}$ (Step 7).

1. Crossover projection. Under the stated assumptions, p-adic interference falls below relative amplitude $10^{-6}$ for $\ell \gt 386.2\,\mathrm{nm}$; under the alternative intensity-like convention with threshold $10^{-4}$, the crossover is at $100\,\lambda_C = 3.862 \times 10^{-11}\,\mathrm{m}$ (Step 8). Both are projections, sensitive to the untested mechanism and threshold.

1. Projected sector frequency shifts (projection). Under $\omega_p = \omega_\infty |r|_p$ with $r = 10$: $f_2 = 1.235 \times 10^{20}\,\mathrm{Hz}$, $f_5 = 4.940 \times 10^{19}\,\mathrm{Hz}$, all other sectors unshifted (Step 9). Under the sum variant with $r = 2$: an $11.8\%$ shift. No simulation or measurement of either variant has been performed.

1. Bibliometric coverage. $r_{\mathrm{bib}} = 1$ (Step 10): all 12 bibliography entries are discussed.

#6. Discussion

Status of the claims. The numerical results (items 1–4, 6, 9 of Section 5) are ordinary consequences of standard constants and elementary number theory. The genuinely new content — items 5, 7, 8 — rests on one conjectural identification: that the cross-sector interference amplitude of the adelic Dirac dynamics is proportional to $\prod_p |r|_p$. The product formula then makes the cancellation exact and scale-independent in form, which is elegant, but elegance is not derivation. We have not derived the adelic Dirac equation from an action principle over $\mathbb{A}$, nor proven that the Vladimirov-operator spectrum produces frequencies scaling as $|r|_p$; both are stated as the test program of Section 3.5, not as results.

Limitations. (i) The adelic Dirac equation of Section 3.2 is postulated. (ii) The sector weights $w_p$ in the diagonal projection $X(t) = w_\infty x_\infty(t) + \sum_{p \leq P} w_p x_p(t)$ are free parameters of the model; we have no principle that fixes them, and different choices change the predicted crossover scale of Step 8. (iii) The proportionality $A_{\mathrm{int}} \propto \prod_p |r|_p$ is the conjecture itself, not a consequence of the product formula; the product formula only makes the conjectured law exact once the proportionality is granted. (iv) The sector dispersion assumption $\omega_p = \omega_\infty |r|_p$ of Step 9 is untested; the Vladimirov-operator spectrum could produce entirely different p-adic frequencies, in which case the projected shifts $f_2 = 1.235 \times 10^{20}\,\mathrm{Hz}$ and $f_5 = 4.940 \times 10^{19}\,\mathrm{Hz}$ would fail. (v) The framework has no demonstrated coupling to electromagnetism or interactions; all results concern the free particle. (vi) At the Compton scale $\lambda_C = 3.862 \times 10^{-13}\,\mathrm{m}$ and below, no interferometric technique cited here operates, so experimental discrimination between the amplitude and intensity conventions of Appendix A is currently out of reach.

What would falsify the claims. The conjecture is falsified if (a) the simulation program of Section 3.5, executed on $\mathbb{Q}_p$ with the Vladimirov operator, yields p-adic sector frequencies that do not scale as $|r|_p$; (b) the exactly solvable quadratic-Lagrangian adelic amplitudes of [5] are incompatible with the postulated sector-wise Dirac dynamics; or (c) a condensed-matter ZBW experiment [3], [7] with effective parameters rescaling the hierarchy of Section 4 shows frequency shifts inconsistent with both the multiplicative variant $\omega_p = \omega_\infty |r|_p$ and the quadrature variant $\omega_{\mathbb{A}} = \omega_\infty\sqrt{1 + |r|_p^2}$ of Step 9. Conversely, observation of an $11.8\%$-scale frequency shift at $\ell \lesssim 100\,\lambda_C = 3.862 \times 10^{-11}\,\mathrm{m}$ under the sum variant would support, though not prove, the adelic picture.

Open questions. What action principle over $\mathbb{A}$ yields the postulated sector-wise Dirac equation? Do p-adic gamma matrices exist for all primes subject to the equiangular-line bound of [8]? How do interactions and spin enter the sector weights $w_p$? These are deferred to future work.

#7. Conclusion

We have proposed a conjectural adelic extension of Zitterbewegung in which the trembling motion of a free Dirac particle decomposes into an Archimedean contribution at $\omega_\infty = 2m_e c^2/\hbar = 1.552 \times 10^{21}\,\mathrm{rad\,s^{-1}}$ and p-adic contributions suppressed at macroscopic scales by the product formula $|x|_\infty \prod_p |x|_p = 1$, verified exactly for $x = 6, 10, 12$. The computed scale hierarchy $\lambda_C/\ell_P = 2.390 \times 10^{22}$ quantifies the regime in which the non-Archimedean sectors could dominate. All numerical results follow from stated inputs by shown arithmetic; the physical claims are explicitly labeled conjectures or projections, and a falsifiable simulation program on $\mathbb{Q}_p$ is specified. The framework's value, if any, lies in making the adelic regularization of Dirac dynamics precise enough to fail.

#References

[1] p-Adic and Adelic Quantum Mechanics. arXiv:hep-th/0312046v1. https://arxiv.org/abs/hep-th/0312046v1 [2] p-Adic and Adelic Cosmology: p-Adic Origin of Dark Energy and Dark Matter. arXiv:hep-th/0602044v1. https://arxiv.org/abs/hep-th/0602044v1 [3] Zitterbewegung (trembling motion) of electrons in semiconductors: a Review. arXiv:1101.0623v1. https://arxiv.org/abs/1101.0623v1 [4] The Problem of Motion: The Statistical Mechanics of Zitterbewegung. arXiv:1411.1854v2. https://arxiv.org/abs/1411.1854v2 [5] Path Integrals for Quadratic Lagrangians on p-Adic and Adelic Spaces. arXiv:1011.6589v1. https://arxiv.org/abs/1011.6589v1 [6] On Phase Transitions for $P$-Adic Potts Model with Competing Interactions on a Cayley Tree. arXiv:math-ph/0512018v2. https://arxiv.org/abs/math-ph/0512018v2 [7] Zitterbewegung (trembling motion) of electrons in narrow gap semiconductors. arXiv:0805.0478v1. https://arxiv.org/abs/0805.0478v1 [8] p-adic Equiangular Lines and p-adic van Lint-Seidel Relative Bound. arXiv:2408.00810v3. https://arxiv.org/abs/2408.00810v3 [9] QNFO: The Adelic Physics Program: A Grand Synthesis [10] QNFO: Zitterbewegung as Archimedean–2-adic Channel Mixing: A Conjectural Adelic Framework and Its Testable Consequences [11] QNFO: Non-Archimedean Projective Perspective: The Monna Map as a Visual Rendering Interface [12] QNFO: Measure-Theoretic Artifacts of the Archimedean Place: A Complete Taxonomy and the Adelic Restructuring of Fundamental Science

#Appendix A. Divergence report

Two conventions diverged across the independent drafts and are documented here.

D1: Suppression convention (DIVERGENT). One convention treats the cross-sector interference as an amplitude, giving $A_{\mathrm{int}}/A_0 = \prod_p |r|_p = \lambda_C/\ell$ (used in the main text, Steps 6–8). The other treats the observable as intensity-like, scaling as $1/N^2$ with $N = \ell/\lambda_C$, giving a crossover at $N = (10^{-4})^{-1/2} = 100$, i.e. $\ell = 100\,\lambda_C = 3.862 \times 10^{-11}\,\mathrm{m}$ for threshold $\varepsilon = 10^{-4}$. The disagreement stems from whether the measured quantity is linear or quadratic in the interference amplitude. Resolution: the main text adopts the amplitude convention as primary and reports the intensity-like variant as an explicitly labeled alternative (Step 8, Result 7).

D2: Sector frequency combination (DIVERGENT). One variant assumes multiplicative shifts $\omega_p = \omega_\infty |r|_p$; the other assumes quadrature combination $\omega_{\mathbb{A}} = \omega_\infty\sqrt{1 + |r|_p^2}$, giving an $11.8\%$ shift for $r = 2$. The disagreement stems from whether p-adic sectors perturb the Archimedean frequency multiplicatively or add in quadrature as a single dominant contribution. Resolution: both variants are reported (Step 9, Result 8) as distinguishable projections; neither is adopted as definitive.

#Appendix B. Claim attribution