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arxiv: 2604.13137 · v1 · submitted 2026-04-14 · 📊 stat.CO · math.NT· math.ST· stat.TH

Recognition: unknown

p-adic Linear Regression for Random Sampling with Digitwise Noise

Tomoki Mihara
Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:51 UTC · model grok-4.3

classification 📊 stat.CO math.NTmath.STstat.TH
keywords p-adic linear regressionprobabilistic algorithmdigitwise noiserandom samplingmodulo p regressioncomputational statistics
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The pith

A new probabilistic algorithm performs p-adic linear regression on samples corrupted by digitwise noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a probabilistic method for linear regression in the p-adic numbers when data arrives through random sampling subject to noise that alters individual digits. It extends the same approach to the simpler case of modulo p regression. A sympathetic reader would care because standard real-number regression tools break down or lose efficiency under this structured noise, while p-adic arithmetic appears in cryptography, coding theory, and certain computational number problems. The algorithm is offered as a direct way to recover linear relations without first converting the data to ordinary reals.

Core claim

The paper proposes a new probabilistic algorithm of p-adic linear regression for random sampling with digitwise noise. This includes a new probabilistic algorithm of modulo p linear regression.

What carries the argument

The proposed probabilistic algorithm that operates directly on p-adic expansions while treating digitwise noise as a random process.

If this is right

  • Linear models can be fitted directly in p-adic arithmetic rather than after conversion to reals.
  • Modulo p regression becomes available as a special case of the same procedure.
  • Sampling schemes that produce digit-corrupted observations gain a regression tool matched to their noise structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method might extend to regression problems in other non-Archimedean fields where digit-like expansions exist.
  • It could be combined with existing p-adic cryptographic primitives to analyze noisy leaked data.
  • Practical validation would require testing on finite-precision p-adic approximations rather than infinite expansions.

Load-bearing premise

The algorithm correctly recovers the underlying linear relation from noisy p-adic samples without extra unstated restrictions on the noise or the sampling process.

What would settle it

Simulate linear data in p-adic numbers, apply controlled digitwise noise, run the algorithm, and check whether the recovered coefficients match the true ones within the expected probabilistic error bounds.

read the original abstract

We propose a new probabilistic algorithm of $p$-adic linear regression for random sampling with digitwise noise. This includes a new probabilistic algorithm of modulo $p$ linear regression.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a new probabilistic algorithm for p-adic linear regression applicable to random sampling with digitwise noise, along with a special case consisting of a new probabilistic algorithm for modulo p linear regression.

Significance. If a correct and efficient algorithm were provided along with supporting analysis, the work could represent a novel contribution to computational statistics by extending linear regression to p-adic settings under a specific noise model, potentially relevant to modular or discrete data problems in cryptography or number-theoretic computations.

major comments (1)
  1. The manuscript contains no definition of the proposed algorithm, no mathematical model for digitwise noise, no estimator or update rule, no derivation establishing properties such as unbiasedness or consistency, and no complexity or convergence analysis. The central claim therefore consists solely of an unsupported assertion of existence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report on our manuscript. We acknowledge the validity of the concerns regarding the absence of detailed definitions, models, derivations, and analyses in the current version. We will revise the manuscript substantially to address these issues.

read point-by-point responses

  1. Referee: The manuscript contains no definition of the proposed algorithm, no mathematical model for digitwise noise, no estimator or update rule, no derivation establishing properties such as unbiasedness or consistency, and no complexity or convergence analysis. The central claim therefore consists solely of an unsupported assertion of existence.

    Authors: We agree that the submitted manuscript is limited to a concise proposal statement and does not include the requested mathematical details. In the revised version, we will add: (1) a precise probabilistic model for digitwise noise in the p-adic setting, (2) a complete definition of the algorithm including the estimator, update rules, and the special case for modulo p linear regression, (3) derivations establishing unbiasedness and consistency, and (4) analysis of computational complexity and convergence properties. These additions will provide the necessary support for the claims. revision: yes

Circularity Check

0 steps flagged

No derivation or equations present; proposal is an unelaborated assertion

full rationale

The manuscript consists solely of a one-sentence proposal to introduce a probabilistic p-adic linear regression algorithm (and its modulo-p case) for digitwise noise. No model, estimator, update rule, derivation, proof of correctness, or fitted quantities appear in the provided text. Because no derivation chain, self-citation, ansatz, or prediction step exists to inspect, none of the enumerated circularity patterns can be triggered. The absence of any technical content means the work is self-contained against external benchmarks only in the trivial sense that it makes no claims requiring verification.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is available from the abstract.

pith-pipeline@v0.9.0 · 5307 in / 1040 out tokens · 25371 ms · 2026-05-10T13:51:27.371089+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. $p$-adic Manifold Learning and Benchmark Tasks from Impartial Games

    cs.LG 2026-05 unverdicted novelty 6.0

    p-adic manifold learning is introduced with a solving algorithm and impartial-game benchmark tasks.

Reference graph

Works this paper leans on

22 extracted references · 3 canonical work pages · cited by 1 Pith paper

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