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Solving biquadratic equations determines local normal forms for every p-adic analytic integrable system in dimension 4.

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2026-06-25 21:49 UTC pith:TS67I7HI

load-bearing objection The paper gives concrete methods to compute 4D p-adic normal forms by reducing to biquadratic equations plus two new auxiliary notions, but the coverage claim rests on the authors' prior classification without re-checking it here. the 2 major comments →

arxiv 2606.24363 v1 pith:TS67I7HI submitted 2026-06-23 math.SG math-phmath.MP

p-adic integrable systems: from biquadratic equations to local models

Luis Crespo , \'Alvaro Pelayo This is my paper
classification math.SG math-phmath.MP
keywords p-adic integrable systemsnormal formsbiquadratic equationssymplectic geometryp-coupled angular momentumalmost eigenvectorsaligned symplectic coordinates
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper develops explicit techniques to compute local normal forms of p-adic analytic integrable systems, which are maps from a symplectic manifold to p-adic space. These techniques apply to every case in four dimensions by reducing the problem to solving biquadratic equations, and they rely on analytic estimates together with Galois theory over p-adic fields. The authors introduce almost eigenvectors and aligned symplectic coordinates as tools for the proofs, and they illustrate the approach on the p-coupled angular momentum system. A reader would care because the normal forms encode the geometric and dynamical structure of the systems, and the methods make this structure accessible without requiring specialized background in p-adic symplectic geometry.

Core claim

The central claim is that the local normal forms of p-adic analytic integrable systems in dimension 4 can be determined in all cases by solving biquadratic equations, using the prior classification of those forms together with analytic estimates and Galois theory of p-adic extensions; the new notions of almost eigenvectors and aligned symplectic coordinates support the computations and are of independent interest.

What carries the argument

Reduction of the normal-form problem to the solution of biquadratic equations, supported by the auxiliary notions of almost eigenvectors and aligned symplectic coordinates.

Load-bearing premise

The existing classification of normal forms in dimension 4 is correct and applies without exception to the integrable systems considered here.

What would settle it

An explicit integrable system in dimension 4 whose local normal form, computed via the biquadratic-equation procedure, fails to match the actual local behavior obtained from the p-adic classification would falsify the claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Normal forms of the p-coupled angular momentum system become computable by direct solution of biquadratic equations.
  • All cases in dimension 4 are covered by the same reduction procedure.
  • Geometrical and dynamical properties encoded in the normal forms can be read off once the biquadratic equations are solved.
  • The auxiliary notions of almost eigenvectors and aligned symplectic coordinates can be applied to other questions in p-adic symplectic geometry.

Where Pith is reading between the lines

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

  • The same biquadratic reduction might extend to higher dimensions once a classification of normal forms becomes available there.
  • The method could be tested numerically on concrete p-adic examples to produce explicit coordinate changes that align the system with the classified models.
  • If the biquadratic equations admit closed-form solutions in many cases, the approach would yield fully explicit local models rather than merely existence statements.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The paper introduces techniques based on almost eigenvectors, aligned symplectic coordinates, and solving biquadratic equations to compute explicit information about local normal forms of p-adic analytic integrable systems F:(M,ω)→(Q_p)^n. It claims these methods cover all cases in dimension 4, building on the authors' prior classification of normal forms, and illustrates their use for the p-coupled angular momentum system. Proofs combine analytic estimates with Galois theory over p-adic extensions, while main statements are presented as self-contained.

Significance. If the coverage claim holds, the work would make determination of normal forms in dimension 4 more algorithmic and practical, aiding geometric and dynamical analysis of p-adic integrable systems. The new notions of almost eigenvectors and aligned symplectic coordinates could have independent value in p-adic symplectic geometry.

major comments (2)
  1. [Introduction / abstract] Introduction and abstract: The central claim that the new techniques 'cover all cases in dimension 4' is stated to rely explicitly on the authors' previous classification of local normal forms. No independent verification, exhaustive check, or summary of that classification's completeness for p-adic analytic systems appears in the manuscript; if the prior classification misses cases or does not apply to examples such as p-coupled angular momentum, the coverage assertion fails.
  2. [Proofs] Proofs section (on analytic estimates and Galois theory steps): The dependence on the prior classification is load-bearing for the 'all cases' result, yet the manuscript presents main results as self-contained without re-deriving or citing specific theorems from the previous work that establish the normal-form list used here.
minor comments (1)
  1. [Abstract] Abstract: The phrasing 'the statements of the main results are essentially self-contained' could be clarified to distinguish between the statements themselves and the proofs, which rely on prior work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and the opportunity to clarify the manuscript's scope and dependencies. We address the two major comments point by point. The central claim of coverage in dimension 4 is explicitly conditional on our prior classification, as already stated in the abstract; we will strengthen the presentation by adding a concise summary of that classification and explicit theorem citations.

read point-by-point responses

  1. Referee: [Introduction / abstract] The central claim that the new techniques 'cover all cases in dimension 4' is stated to rely explicitly on the authors' previous classification of local normal forms. No independent verification, exhaustive check, or summary of that classification's completeness for p-adic analytic systems appears in the manuscript; if the prior classification misses cases or does not apply to examples such as p-coupled angular momentum, the coverage assertion fails.

    Authors: We agree that the coverage assertion is conditional on the completeness of the prior classification (arXiv reference to be added). The manuscript already notes this dependence in the abstract and introduction. To make the dependence transparent, we will insert a short paragraph summarizing the cases from the previous classification (the four normal-form types in 4D) together with explicit citations to the theorems establishing that list. This will also confirm applicability to the p-coupled angular momentum example treated in Section 5. No independent re-derivation of the classification is feasible within the present paper, as that would duplicate the earlier work. revision: yes

  2. Referee: [Proofs] Proofs section (on analytic estimates and Galois theory steps): The dependence on the prior classification is load-bearing for the 'all cases' result, yet the manuscript presents main results as self-contained without re-deriving or citing specific theorems from the previous work that establish the normal-form list used here.

    Authors: The phrase 'essentially self-contained' in the abstract refers to the statements of the main theorems, which are formulated without assuming prior knowledge of p-adic symplectic geometry. The proofs, however, do invoke the classification. We will revise the proofs section to include precise citations (e.g., Theorem X and Corollary Y of the prior paper) at each step where a normal-form case is invoked. This will clarify the logical structure without altering the analytic or Galois-theoretic arguments. revision: yes

Circularity Check

1 steps flagged

Reliance on authors' prior classification of dim-4 normal forms for coverage claim

specific steps
  1. self citation load bearing [Abstract]
    "The proofs use our previous classification of normal forms and rely on a combination of analytic estimates and Galois theory of p-adic extension fields. However, the statements of the main results are essentially self-contained and do not require prior knowledge of p-adic integrable systems or p-adic symplectic geometry."

    The assertion that the new techniques cover all cases in dimension 4 is justified by invoking the authors' prior classification of local normal forms; without independent re-derivation or external verification of that classification within this paper, the coverage claim rests on self-cited prior work by the same authors.

full rationale

The paper's main results introduce original techniques (almost eigenvectors, aligned symplectic coordinates, biquadratic solving) that are presented as self-contained. However, the proofs of coverage for all dim-4 cases explicitly depend on the authors' own previous classification of normal forms, which is not re-established here. This matches self-citation load-bearing but does not reduce the central claims to a definition or fit by construction. No other circular patterns are present.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no identifiable free parameters, axioms, or invented entities; the work references a prior classification but details are unavailable.

pith-pipeline@v0.9.1-grok · 5755 in / 1289 out tokens · 33917 ms · 2026-06-25T21:49:23.968600+00:00 · methodology

0 comments
read the original abstract

Let $p$ be a prime number and $n$ a positive integer. The study of normal forms of $p$-adic analytic integrable systems $F=(f_1,\ldots,f_n):(M,\omega)\to(\mathbb{Q}_p)^n$ is essential to understand their geometrical and dynamical properties. Even though in some cases, such as dimension $4$, there is a classification of the local normal forms, it can be a challenge to determine them explicitly. Our goal in this paper is to introduce techniques to compute information about these local normal forms. We then explain how this is useful for instance to study the $p$-coupled angular momentum. The techniques we introduce cover all cases in dimension $4$ and require solving biquadratic equations. Along the way we define two new notions: almost eigenvectors and aligned symplectic coordinates. They are useful to prove our results but also of independent interest. The proofs use our previous classification of normal forms and rely on a combination of analytic estimates and Galois theory of $p$-adic extension fields. However, the statements of the main results are essentially self-contained and do not require prior knowledge of $p$-adic integrable systems or $p$-adic symplectic geometry.

Figures

Figures reproduced from arXiv: 2606.24363 by \'Alvaro Pelayo, Luis Crespo.

Figure 1
Figure 1. Figure 1: Normal forms of regular and critical points of elliptic-elliptic, focus-focus and elliptic-regular type of an integrable system F : R 4 → R 2 . Some of these can be normal forms of Theorem B and Proposition 5.6. 1 arXiv:2606.24363v1 [math.SG] 23 Jun 2026 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The real Jaynes-Cummings model on S2 × R 2 . of F : (M, ω) → (Qp) n is a point at which the p-adic analytic 1-forms df1, . . . , dfn are linearly dependent. In this paper we are interested only in non-degenerate critical points in the Morse-Bott sense (see Definition 5.2 for the precise notion). The rank of a critical point m is the number of linearly independent 1-forms among df1, . . . , dfn at m, and th… view at source ↗
Figure 3
Figure 3. Figure 3: A representation of the polynomial p1,0,γ,c(r, s) for r, s ∈ Zp. Each figure corresponds to different values of p and γ, and the colors used in each figure correspond to the values of c in Xp. We see that, in each figure, only the points of one color reach z = 0; this color indicates the c which is needed in the normal form. In these particular cases, this c is precisely γ. The same figures work for Theore… view at source ↗
Figure 4
Figure 4. Figure 4: (A, λ, Ω)-almost eigenvectors for A = Ω =  0 1 −1 0 ∈ M2(Q3) and λ = i ∈ C3. The axes represent the “real” part of the first coordinate and the “imaginary” part of the second coordinate (so the point in the top right represents (8, 8i)). The vector (1, i) is the true eigenvector, and the blue points, such as (4, 7i), represent almost eigenvectors. Theorem B (Normal form computation for critical points ba… view at source ↗
Figure 5
Figure 5. Figure 5: The equation Av = 0 always has a solution in the ball B C p (v0, 1 c maxi∈I |Ai·v0|p ∥Ai·∥p ), according to Lemma 4.3. Lemma 4.3 (Approximation of solutions to p-adic equations). Let m, n, r be positive integers with 1 ⩽ r ⩽ m. Let p be a prime number. Let A ∈ Mm×n(Cp) with rank r. Let v0 ∈ (Cp) n . Let I ⊂ {1, . . . , m} be a set of size r. Suppose that Indp(AI·) ̸= 0. Then n x ∈ (Cp) n : Ax = 0o ∩ B C p … view at source ↗
Figure 6
Figure 6. Figure 6: Symbolic representation of 2-dimensional fiber of the case c = −1 of point (i) of Theorem B, which is symplectically equivalent to c = 1 if p ≡ 1 mod 4. The four “cones” are 2-dimensional planes in 4-dimensional space. See also Proposition 5.6. system; we just gave a list of normal forms and proved that every critical point was locally symplectomorphic to one and only one of these. The main results of the … view at source ↗
Figure 7
Figure 7. Figure 7: The intuition behind the idea of “independence number”. The points represent balls of radius 1. The two vectors v1, v2 are linearly in￾dependent, yet there are relatively long vectors, such as the ones marked in red, which are approximately orthogonal to both v1 and v2. This is re￾flected on v1 and v2 having low independence number (1/3 in this case). 6. Example and intuition behind almost eigenvectors: De… view at source ↗
Figure 8
Figure 8. Figure 8: The coupled angular momentum system is obtained by coupling two spin systems in a nontrivial way. 8. Application of Theorem B to study the coupled angular momentum We can also use Theorem B to classify the rank 0 critical points of the p-adic coupled angular momentum system. The system depends on three parameters t, R1, R2, with t ∈ Zp and R1, R2 ∈ Qp with |R1|p < |R2|p. It is defined in S2 p × S 2 p endow… view at source ↗
Figure 9
Figure 9. Figure 9: A real analog of [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: 3-adic balls in Qp in dimension 2. The purple point represents a ball of radius 1, the dark blue points form a ball of radius 3, the light blue points form a ball of radius 9 and the green points form a ball of radius 27. In this paper we are concerned with balls in Cp, which contain the corresponding balls in Qp, but no points in (Qp) 2 outside the ball. The field Cp is defined as the metric completion o… view at source ↗

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