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arxiv: 2605.22624 · v1 · pith:TEDW7FRTnew · submitted 2026-05-21 · 🧮 math.CO · math.RA

On the self-similarity of rational power series with matrix coefficients

Pierre-Emmanuel Caprace , Justin Vast This is my paper

Pith reviewed 2026-05-22 04:09 UTC · model grok-4.3

classification 🧮 math.CO math.RA
keywords self-similarityrational power seriesmatrix coefficientstilingssubstitutionsfinite fieldsbinomial coefficients
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The pith

Rational power series with matrix coefficients over finite fields generate self-similar colored tilings of n-dimensional space.

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

The paper establishes that for polynomials P and Q with coefficients in d by d matrices over the field with p elements, where Q is invertible as a power series, the coefficients of the expansion of P Q inverse define a map M from non-negative integer n-tuples to matrices. This map corresponds to coloring the unit cubes of real n-space, and the resulting colored tiling is invariant under substitutions. A sympathetic reader would care because the construction extends the familiar fractal self-similarity of binomial coefficients modulo p to arbitrary dimensions and to matrix-valued colors, supplying an algebraic source for substitution-invariant patterns.

Core claim

The map M that sends each multi-index to the corresponding coefficient in the power series expansion of the rational fraction P Q inverse in the ring of matrix formal power series is self-similar when interpreted as a tiling of R^n by unit cubes colored by matrices; self-similarity takes the form of invariance under substitutions.

What carries the argument

Invariance under substitutions of the colored tiling whose colors are the matrix coefficients of the power series expansion of P Q inverse.

If this is right

  • The classical self-similarity of binomial coefficients modulo a prime is recovered exactly by setting d=1, n=2, P=1 and Q=1-x1-x2.
  • The same substitution invariance holds for any dimension n and any matrix size d.
  • Rational functions with non-commuting matrix coefficients supply an explicit algebraic recipe for producing self-similar colorings.

Where Pith is reading between the lines

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

  • The substitution rules furnished by the matrix construction could be used to generate infinite families of higher-dimensional substitution tilings by direct computation.
  • Similar coefficient extractions from other algebraic series might be checked for the same invariance property, extending the method beyond the rational case.
  • The colored tilings provide concrete examples of multi-dimensional automatic structures whose fractal geometry can be studied via linear algebra over finite fields.

Load-bearing premise

Q must be invertible in the ring of formal power series with matrix coefficients so that the rational fraction possesses a well-defined power-series expansion whose coefficients define the map M.

What would settle it

Choose small concrete values of p, d, n, P and Q satisfying the invertibility condition, compute the matrix coefficients of P Q inverse up to a fixed multi-degree, assemble the corresponding finite colored tiling, and verify whether it satisfies the substitution-invariance relation claimed for the infinite tiling.

Figures

Figures reproduced from arXiv: 2605.22624 by Justin Vast, Pierre-Emmanuel Caprace.

Figure 1
Figure 1. Figure 1: Two illustrations of Corollary 1.6 with d = 2 and p = 2. The pictures represent a portion of size 1024 of the first octant; the origin is placed on the top left corner and the y-axis is oriented downward. Each color represents an element of A; the zero matrix corresponds to the red color. On the left: a = b = [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Two illustrations of Theorem 1.3 with d = 1 and p = 2. The pictures represent a portion of size 1024 of the first octant; the origin is placed on the top left corner and the y-axis is oriented downward. Each color represents an element of A = F2; the zero element corresponds to the red color. On the left: P = 1 and Q = 1 + x 2 1 + x 2 2 + x1x2 + x 2 1x 2 2 , On the right: P = 1 and Q = 1 + x 2 1x2 + x1x 2 … view at source ↗
Figure 3
Figure 3. Figure 3: Other illustrations of Corollary 1.6 with d = 2. In the left column p = 3; in the right column p = 5. On the top line: a = b = [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Q = x 2 y 2 + xy + x 2 + y 2 + 1, p = 2, 3, 5, 7 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Q = x 2 y 2 + xy + x + y + 1, p = 2, 3, 5, 7 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: p = 2, Q = x 3 y 3 + x 2 + y 2 + x + y + 1 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: p = 2, Q = x 3 y 3 + xy + x + y + 1 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: p = 2, Q = x 3 + y 3 + x 2 y 2 + x 2 y + xy2 + 1 [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: p = 3, Q = −x 2 y 2 − x 2 y − xy2 − xy + 1 (compare [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: p = 3, Q = x 2 y 2 − x 2 − y 2 + y + 1 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: p = 3, Q = x 3 y 3 + x 2 + y 2 + x + y + 1 (compare [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
read the original abstract

Let $p$ be a prime, let $d \geq 1$ be an integer and $A$ be the algebra of square matrices of size $d$ over the field of order $p$. Let $P, Q \in A[x_1, \dots x_n]$ be polynomials in $n$ indeterminates with coefficients in $A$, such that $Q$ is invertible in $ A[\![x_1, \dots, x_n]\!]$. Let also $\mathcal M \colon \mathbf Z^n \to A$ be the map associating to the $n$-tuple of integers $(\alpha_1, \dots, \alpha_n)$ the coefficient of the monomial $x_1^{\alpha_1} \dots x_n^{\alpha_n}$ in the development of the rational fraction $PQ^{-1}$ as a power series (the support of $\mathcal M$ is contained in $\mathbf N^n$). Our main result ensures that the map $\mathcal M$, viewed as a tiling of $\mathbf R^n$ by unit cubes with color set $A$, is self-similar. The self-similarity is expressed in terms of invariance under substitutions. By specializing to $d=1$, $n=2$, $P=1$ and $Q =1-x_1-x_2$, we recover the well-known self-similarity feature of the binomial coefficients modulo $p$.

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.

Circularity Check

0 steps flagged

No significant circularity; derivation is algebraic and self-contained

full rationale

The central result establishes self-similarity of the coefficient map M of the rational series PQ^{-1} by deriving a recurrence on coefficients from the invertibility of Q in the power-series ring, then verifying that this recurrence is preserved under the base-p digit substitution operators. The argument proceeds by direct verification that the substitution operators commute with the matrix multiplication implicit in the recurrence relation, reducing to the classical binomial case when d=1. No step equates a derived quantity to a fitted parameter, renames a known pattern as a new result, or relies on a load-bearing self-citation whose content is itself unverified. The special-case recovery of binomial self-similarity is presented as an instance of the general construction rather than its justification. The derivation therefore rests on explicit algebraic identities rather than circular re-labeling or imported uniqueness.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result depends on the algebraic setup of formal power series rings and matrix algebras over finite fields; the key non-standard assumption is the invertibility of Q.

axioms (1)
  • domain assumption Q is invertible in the formal power series ring A[[x1, …, xn]]
    Required so that PQ^{-1} admits a power-series expansion whose coefficients define the map M.

pith-pipeline@v0.9.0 · 5786 in / 1210 out tokens · 50421 ms · 2026-05-22T04:09:55.931884+00:00 · methodology

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