arxiv: 2605.01385 · v1 · submitted 2026-05-02 · 🧮 math-ph · math.MP

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The Haar measure of the p-adic rotation group textrm{SO}(3)_p via nautical angles

Ilaria Svampa, Lorenzo Guglielmi, Stefano Mancini, Vincenzo Parisi

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Pith reviewed 2026-05-08 19:29 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords Haar measurep-adic rotation groupSO(3)_pnautical anglesquaternionschange of variablesJacobiannon-Archimedean

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The pith

The normalized Haar measure on the p-adic rotation group SO(3)_p is expressed explicitly as a factorized density in its three nautical angles.

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

The paper establishes an explicit formula for the normalized Haar measure on the compact p-adic rotation group SO(3)_p by using nautical angle parametrization. It leverages the topological group isomorphism to the multiplicative group of p-adic quaternions modulo scalars, derives the change of variables formulas along with the associated Jacobian in the p-adic numbers, and combines this with the known Haar measure on the quaternion group. The result is a density that factors into independent contributions from each of the three angles. A reader would care because this gives a practical way to perform invariant integration over the group in settings that use non-Archimedean numbers and benefit from angular coordinates.

Core claim

Exploiting its topological group isomorphism with the quotient of p-adic quaternions by scalars, the corresponding change of variables formulas and the associated Jacobian are derived in the p-adic setting. This is combined with the known Haar measure on the multiplicative group of p-adic quaternions to yield an explicit formula for the normalized Haar measure on SO(3)_p in nautical coordinates with a factorized density in the three angles.

What carries the argument

The topological group isomorphism SO(3)_p ≅ H_p^x / Q_p^x together with the p-adic change-of-variables Jacobian for the nautical parametrization, which transports the Haar measure from the quaternion group.

If this is right

The Haar integral over SO(3)_p reduces to an ordinary triple integral over the angle ranges with the explicit product density.
Any continuous function on the group can be integrated using this coordinate expression of the measure.
The normalization ensures the total measure is 1.
This holds for every prime p, giving a uniform expression across all p-adic cases.

Where Pith is reading between the lines

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

Analogous constructions might work for other p-adic compact groups admitting quaternion-like descriptions.
The factored form could simplify the computation of characters or representation theory integrals on this group.
Implementations in computer algebra systems could use this to sample from the invariant measure on p-adic rotations.
Connections to p-adic special functions or orthogonal polynomials might be explored using this measure.

Load-bearing premise

The isomorphism between SO(3)_p and the quotient of the multiplicative p-adic quaternions by the scalars correctly transports the Haar measure when combined with the computed Jacobian.

What would settle it

Directly integrating the constant function 1 over the proposed measure and checking whether the result equals 1, or computing the measure of a known subset in two different ways.

read the original abstract

We study the explicit construction of the Haar measure on the compact $p$-adic rotation group $\textrm{SO}(3)_p$ by nautical (Cardano) parametrization. Exploiting its topological group isomorphism with $\mathbb{H}_p^\times/\mathbb{Q}_p^\times$ of $p$-adic quaternions modulo scalars, we derive the corresponding change of variables formulas and compute the associated Jacobian in the $p$-adic setting, which we combine with the known Haar measure on the multiplicative group of $p$-adic quaternions $\mathbb{H}_p^\times$. This yields an explicit formula for the normalized Haar measure on $\textrm{SO}(3)_p$ in nautical coordinates, with a factorized density in the three angles. Our construction provides a concrete tool suited for applications of non-Archimedean models where an explicit angular description of invariant integration is required.

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 supplies an explicit factorized formula for the normalized Haar measure on p-adic SO(3) in nautical angles by transporting the standard measure from p-adic quaternions.

read the letter

The main takeaway is that this work produces a concrete, ready-to-use expression for the Haar measure on SO(3)_p in Cardano coordinates. It starts from the known topological isomorphism with H_p^x / Q_p^x, derives the p-adic change-of-variables map, computes the Jacobian using the p-adic absolute value, and multiplies by the known Haar measure on the quaternion group to obtain a density that factors across the three angles and integrates to one after normalization.

Referee Report

1 major / 3 minor

Summary. The manuscript derives an explicit formula for the normalized Haar measure on the compact p-adic rotation group SO(3)_p in nautical (Cardano) angles. It exploits the topological group isomorphism SO(3)_p ≅ H_p^x / Q_p^x, obtains the corresponding p-adic change-of-variables formulas and Jacobian, and transports the known Haar measure on the multiplicative group H_p^x to produce a factorized density in the three angles.

Significance. If the explicit Jacobian and normalization are correctly computed, the result supplies a concrete angular parametrization of invariant integration on SO(3)_p. This is a useful tool for non-Archimedean models that require explicit angular descriptions, extending classical rotation-group techniques to the p-adic setting.

major comments (1)

The central claim requires that the derived density integrate to one with respect to the p-adic measure on the parameter domain. The manuscript should supply this explicit normalization integral (or a reference to its evaluation) to confirm the measure is properly normalized rather than merely proportional.

minor comments (3)

The abstract states that the Jacobian is computed but does not display the final expression; including the explicit p-adic Jacobian determinant in the abstract or a dedicated equation would improve immediate readability.
Notation for the p-adic absolute value |·|_p and the domain of the nautical angles should be introduced with a brief reminder of their ranges in the p-adic topology.
A short reference to the standard proof of the topological isomorphism SO(3)_p ≅ H_p^x / Q_p^x would help readers who are not already familiar with the quaternion construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. The manuscript constructs the normalized Haar measure on SO(3)_p by transporting the known normalized Haar measure on H_p^x via the group isomorphism and the p-adic change-of-variables formula. We address the single major comment below.

read point-by-point responses

Referee: The central claim requires that the derived density integrate to one with respect to the p-adic measure on the parameter domain. The manuscript should supply this explicit normalization integral (or a reference to its evaluation) to confirm the measure is properly normalized rather than merely proportional.

Authors: We agree that an explicit verification strengthens the presentation. The normalization is inherited from the normalized Haar measure on H_p^x together with the Jacobian factor arising from the change of variables, but the manuscript does not display the resulting integral over the nautical-angle domain. In the revised version we will add a short subsection that evaluates this integral directly, using the factorized form of the density and the product structure of the p-adic parameter domain, thereby confirming that the total mass equals one. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from known Haar measure and isomorphism

full rationale

The paper begins with the established topological group isomorphism SO(3)_p ≅ H_p^x / Q_p^x and the known Haar measure on the multiplicative group H_p^x. It then derives the p-adic change-of-variables formulas and Jacobian to transport the measure to nautical coordinates, yielding an explicit factorized density. No load-bearing step reduces the final formula to a fitted input, self-definition, or self-citation chain; the central result follows from standard p-adic group theory and analysis without internal circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on two background facts: the topological isomorphism between SO(3)_p and the projective p-adic quaternion group, and the existence of a standard (bi-invariant) Haar measure on the multiplicative group of p-adic quaternions. No free parameters or new entities are introduced.

axioms (2)

domain assumption SO(3)_p is topologically isomorphic to H_p^x / Q_p^x as compact groups
Invoked in the abstract to reduce the problem to quaternion coordinates
standard math A normalized Haar measure on H_p^x is known and bi-invariant
Used as the starting point for the change-of-variables computation

pith-pipeline@v0.9.0 · 5465 in / 1401 out tokens · 16925 ms · 2026-05-08T19:29:21.300263+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.

IndisputableMonolith.Foundation.AlexanderDuality (D=3 forcing in the Archimedean spacetime chain) alexander_duality_circle_linking unclear
We study the explicit construction of the Haar measure on the compact p-adic rotation group SO(3)_p by nautical (Cardano) parametrization. Exploiting its topological group isomorphism with H_p^× / Q_p^× ...
IndisputableMonolith.Cost (J-cost ½(x+x⁻¹)−1) Jcost / washburn_uniqueness_aczel unclear
dμ_SO(3)_p(R(α,β,γ)) = |dα dβ dγ / ((1−vα²)(1+pβ²)(1−(p/v)γ²))|_p

Reference graph

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