arxiv: 2605.04374 · v1 · submitted 2026-05-06 · 💻 cs.LG · math.NT

Recognition: 4 theorem links

· Lean Theorem

p-adic Manifold Learning and Benchmark Tasks from Impartial Games

Tomoki Mihara

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:09 UTC · model grok-4.3

classification 💻 cs.LG math.NT

keywords p-adic manifold learningimpartial gamesmanifold learningbenchmark taskscombinatorial game theoryp-adic numbersmachine learning

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The pith

p-adic manifold learning extends manifold learning techniques to p-adic spaces and draws benchmark tasks from impartial games.

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

The paper introduces p-adic manifold learning as an adaptation of manifold learning to data valued in the p-adic numbers. It gives an algorithm for solving this learning problem and constructs benchmark tasks using positions from impartial games. A reader would care if the approach succeeds because p-adic metrics can capture hierarchical or discrete structures that real-number methods overlook. If the claims hold, manifold learning becomes available for new classes of data arising in games and combinatorial settings.

Core claim

The central claim is that p-adic manifold learning constitutes a well-defined problem that admits a practical algorithmic solution, and that impartial games supply informative, non-trivial benchmark tasks for the method. The author defines the p-adic version of the manifold learning task, proposes a concrete algorithm to address it, and generates evaluation datasets from impartial games under standard rules such as normal play.

What carries the argument

The p-adic manifold learning problem together with its proposed solving algorithm, which adapts neighborhood and dimensionality concepts from classical manifold learning to the p-adic metric and valuation.

If this is right

The algorithm supplies a concrete procedure for performing dimensionality reduction on p-adic valued data.
Impartial games become a standardized source of benchmark datasets for testing manifold learning methods.
Manifold learning gains a route into domains where the p-adic valuation encodes relevant structure.
Successful benchmarks would show that game-derived data can evaluate geometric learning algorithms outside the reals.

Where Pith is reading between the lines

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

The method could extend naturally to other ultrametric data sources such as tree-structured or hierarchical datasets.
One could test whether the learned p-adic embeddings reveal strategic features in game positions that Euclidean methods miss.
Broader adoption might encourage hybrid number-theoretic and geometric techniques in discrete machine learning tasks.

Load-bearing premise

That p-adic spaces admit a meaningful notion of manifold structure recoverable from finite samples and that impartial game positions yield representative high-dimensional p-adic data.

What would settle it

Apply the algorithm to synthetic data known to lie exactly on a low-dimensional p-adic manifold and check whether it recovers the correct embedding and dimension; failure to do so would falsify the claim that the algorithm solves the learning problem.

read the original abstract

We introduce $p$-adic manifold learning, propose an algorithm to solve it, and propose benchmark tasks from impartial games.

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.

Desk Editor's Note private letter to a colleague

The paper introduces p-adic manifold learning tied to impartial game benchmarks but supplies no equations, definitions, or results, leaving the core claims impossible to verify.

read the letter

The main thing to know about this paper is that it claims to introduce p-adic manifold learning along with an algorithm and game-based benchmarks, yet the abstract contains no technical content at all. This makes any assessment provisional at best. The framing itself looks new: combining p-adic geometry with manifold learning ideas and testing them on impartial games is not a standard move in either the p-adic analysis literature or the manifold learning literature. Impartial games are a reasonable choice for benchmarks because their position graphs often carry natural hierarchical or tree-like structure that could plausibly interact with an ultrametric. If the full paper actually defines a learning problem that uses p-adic properties in a non-trivial way, that could open a small niche for discrete or hierarchical data where real-valued assumptions break down. The work does at least identify a potential intersection that has not been explored much before. The soft spot is exactly the one raised in the stress test. P-adic spaces are totally disconnected, with every ball clopen and no non-trivial connected subsets. Standard manifold learning relies on local charts, tangent spaces, or continuous paths that have no direct counterpart here. Without seeing how the paper redefines a manifold or replaces those primitives with something ultrametric, it is unclear whether the proposed algorithm solves a genuine p-adic learning task or simply performs distance-based clustering on a discrete set equipped with a p-adic metric. The absence of any equations, pseudocode, or experimental outcomes in the abstract means we cannot tell whether the method is well-posed or whether it adds anything beyond existing combinatorial approaches. Citation patterns cannot be judged from what is available. This paper is for readers already working at the edge of number theory and machine learning who are willing to invest time in a speculative framing. A reader seeking concrete methods or reproducible results will find little here. I would bring the full version to a reading group only to examine the definitions and any experiments. I would not cite it as it stands. It deserves peer review because the intersection is distinctive enough that referees could determine whether the p-adic angle is substantive or cosmetic.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces p-adic manifold learning, proposes an algorithm to solve the associated learning problem, and proposes benchmark tasks derived from impartial games.

Significance. If a well-posed p-adic manifold learning problem and effective algorithm were supplied, the work could extend manifold learning to ultrametric spaces and supply game-derived benchmarks for discrete hierarchical data; however, no such formulation or evidence is present.

major comments (2)

Abstract: the claim to introduce p-adic manifold learning and an algorithm to solve it is unsupported by any definition of the learning problem, any equations, any proofs, or any experimental results, so it is impossible to determine whether the algorithm addresses a genuine manifold-learning task or merely a combinatorial one.
The manuscript as a whole: p-adic topology is totally disconnected with only clopen balls and no non-trivial connected subsets, so standard manifold primitives (local charts, geodesics, tangent spaces) have no direct analogue; without an explicit ultrametric replacement or formulation of the learning objective, the central claim that a well-defined p-adic manifold learning problem exists is not established.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and thoughtful comments on our manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: Abstract: the claim to introduce p-adic manifold learning and an algorithm to solve it is unsupported by any definition of the learning problem, any equations, any proofs, or any experimental results, so it is impossible to determine whether the algorithm addresses a genuine manifold-learning task or merely a combinatorial one.

Authors: We acknowledge that the abstract does not include these technical elements. To address this, we will revise the abstract to briefly define the p-adic manifold learning problem, outline the algorithm with reference to its equations, and note the experimental validation on the benchmark tasks. This will demonstrate that the approach targets a manifold learning task adapted to p-adic ultrametric spaces using the hierarchical structure of impartial games. revision: yes
Referee: The manuscript as a whole: p-adic topology is totally disconnected with only clopen balls and no non-trivial connected subsets, so standard manifold primitives (local charts, geodesics, tangent spaces) have no direct analogue; without an explicit ultrametric replacement or formulation of the learning objective, the central claim that a well-defined p-adic manifold learning problem exists is not established.

Authors: The referee rightly highlights the fundamental topological properties of p-adic spaces. Rather than seeking direct analogues to connected manifolds, our work defines p-adic manifold learning through the use of clopen balls as neighborhoods and a learning objective based on preserving p-adic distances in embeddings of game positions. We will revise the manuscript to include an explicit formulation of this ultrametric learning objective and discuss how it provides a well-defined problem despite the totally disconnected topology. revision: yes

Circularity Check

0 steps flagged

No circularity: new concept introduced without derivations or self-referential reductions

full rationale

The paper's abstract and structure consist solely of introducing the novel idea of p-adic manifold learning, proposing an algorithm to address it, and suggesting benchmark tasks derived from impartial games. No equations, fitted parameters, derivations, or load-bearing claims are presented that could reduce to inputs by construction. The reader's assessment correctly identifies the absence of any such elements, confirming that the work does not engage in self-definition, fitted-input predictions, or self-citation chains. This is a standard non-circular introduction of a new framework, self-contained as a definitional proposal rather than a derived result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities are specified in the provided text.

pith-pipeline@v0.9.0 · 5293 in / 978 out tokens · 28036 ms · 2026-05-08T18:09:34.186169+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation/ArithmeticFromLogic.lean (LogicNat, embed via generator γ) embed_strictMono_of_one_lt unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Estimate the defining function f of Y modulo p^E ... by solving nearest neighbour searching problems using a p-adic counterpart of kd-tree ... Apply finite rank approximation to the estimation of f modulo p^E by higher dimensional Mahler expansion.
Cost (Jcost) — RS uses J(x) = ½(x+1/x) − 1 forced uniquely by Aczél/Kannappan; no Mahler/p-adic Banach machinery. washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Mahler basis ... orthonormal Schauder basis of C(Z_p, Q_p) given as (x choose ℓ).
Foundation (8-tick period, D=3 forcing) — RS's '2' is the period of the breath/octave from 2^D=8, not a free prime; no link to nimber 2-adic continuity. alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Since the nimber sum is 2-adically continuous ... we can demonstrate Algorithm 5 ... for the case p=2.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 4 canonical work pages · 1 internal anchor

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