Automorphisms and Scaling Transformations: Conjugation Action, Parameter Rescaling, and Growth of Associative Structure

In the paper's words

Automorphism groups encode the symmetries of algebraic varieties, non-associative algebras, parabolic geometries, group splittings, and holomorphic dynamical systems, but each setting is treated with its own tools, and practitioners lack a uniform test for when a rescaling of coordinates, dimensions, or metrics acts nontrivially on automorphism parameters. Normalization and reduction routines in computer algebra, and classification arguments that quotient by scaling symmetries, silently assume that dilations either preserve or trivially commute with the automorphisms at hand. We introduce a three-part conjugation test — normalization, centralization, and parameter rescaling — and carry out its quantitative branch with explicit arithmetic.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. This paper shows the existence of automorphisms $F$ of $\mathbb{C}^2$ with a non-recurrent Fatou component $\Omega$ biholomorphic to $\mathbb{C}\times\mathbb{C}^*$ that is the basin of attraction to an invariant entire curve on which $F$ acts as an irrational rotation, and that the biholomorphism can be chosen to conjugate $F$ to the translation $(z,w)\mapsto(z+1,w)$.

   from 2. Background and Related Work

2. In the Cayley–Dickson tower, the octonion-to-sedenion step multiplies the count of associative quaternion rings from $7$ to $35$, a factor of $5$, i.e., $\log_2 5\approx 2.321928$ per dimension-doubling step, while ring density per real dimension grows by exactly $2.5$ (from $0.875$ to $2.1875$).

   from Abstract

3. We show that conjugation by the anisotropic dilation $D=\operatorname{diag}(2,3)$ rescales the shear coefficient of the planar unipotent map $\phi(x,y)=(x+y,y)$ by exactly $2/3$, that scalar dilations act trivially on both shears and rotations, and that in the Cayley–Dickson tower the count of associative quaternion rings grows from $7$ (octonions) to $35$ (sedenions) — a factor of $5$, or $\log_2 5\approx 2.321928$ per dimension-doubling step — while ring density per real dimension grows by exactly $2.5$.

   from 1. Introduction

4. Our contributions are: A derivation of the conjugation test's central computation, showing that anisotropic dilations rescale shear parameters by the ratio $\lambda_1/\lambda_2$ of dilation factors while scalar dilations centralize shears and rotations exactly (Sections 3 and 4.2–4.3).

   from 1. Introduction

5. A growth analysis of associative structure in the Cayley–Dickson tower, with the factor $5$, the exponent $\log_2 5\approx 2.321928$, and the density ratio $2.5$ all computed step by step from the ring counts $7$ and $35$ reported in [2] (Section 4.4).

   from 1. Introduction

6. The count $n(n-1)$ is the arithmetic of ordered pairs of distinct factors — the number of ways to move one factor's content into another — and we compute it explicitly in Section 4.6 as the permutation-scope arithmetic of the symmetric condition. [6] Induced Automorphisms on Irreducible Symplectic Manifolds (arXiv:1405.5706v2).

   from 2. Background and Related Work

How it is done

1. Mathematicians and computational scientists who classify symmetries face a recurring operational problem: before quotienting a configuration space by scalings, or running a normalization algorithm that rescales coordinates, they must know whether the rescaling changes the automorphisms of the object.

   from 1. Introduction

2. We discuss the related works in their fixed numbering, citing each for a specific method, result, or number its full text states. [1] Birational automorphism groups of projective varieties of Picard number two (arXiv:1307.5490v2).

   from 2. Background and Related Work

3. We use two concrete families: Scalar dilations $S_\lambda(x)=\lambda x$ on a vector space or algebra, with $\lambda\in\mathbb{R}_{\gt 0}$ or $\mathbb{C}^*$.

   from 3.1 The conjugation test

4. Two planar linear maps serve as canonical test automorphisms of $\mathbb{R}^2$: $$ \phi_{\mathrm{uni}}(x,y)=(x+y,\;y), \qquad \phi_{\mathrm{rot}}(x,y)=(\cos\theta\, x-\sin\theta\, y,\;\sin\theta\, x+\cos\theta\, y), $$ a unipotent shear with coefficient $c=1$ and a rotation by angle $\theta$.

   from 3.2 Model automorphisms

What would falsify it

1. The results give a computable, falsifiable rule — ratio rescaling for anisotropic dilations, triviality for isotropic ones — that practitioners can apply before invoking scaling arguments in classification or normalization pipelines.

   from Abstract

2. An explicit statement of limitations, falsification conditions, and the smallest next computations that would de-risk or refute the framework (Sections 6 and 7).

   from 1. Introduction

3. Uncertainty is dominated entirely by the unverified linearity assumption; we assign no error bar and present the projection solely to make the framework's predictive content explicit and falsifiable.

   from 5. Results

Where the claim stops

1. Specifically: (i) the ratio rule $c'=\frac{\lambda_1}{\lambda_2}c$ is proved only for linear shears on $\mathbb{R}^2$; whether it has a counterpart for the nonlinear automorphisms of $\mathbb{C}^2$ in [8], whose Fatou component carries an irrational rotation and a conjugacy to a translation, is open, and [8] does not address dilations.

   from 6. Discussion

2. (iii) The binomial identities of R5 are unexplained: [2] does not address whether the associative quaternion rings are indexed by subsets of a $7$-element set, and a demonstration that the match $\binom{7}{3}=35$ is coincidental would falsify any structural reading of R5.

   from 6. Discussion

3. P1 rests on an unverified linearity assumption.

   from 6. Discussion

Where to read

1. Abstract242 words · 1 min
2. 1. Introduction628 words · 3 min
3. 2. Background and Related Work1,279 words · 6 min
4. 3. Methods507 words · 2 min
5. 4. Analysis1,453 words · 6 min
6. 5. Results417 words · 2 min
7. 6. Discussion519 words · 2 min
8. 7. Practical Implications and Engineering Pathways478 words · 2 min
9. 8. Conclusion382 words · 2 min
10. Provenance and AI Involvement225 words · 1 min