arXiv:2607.11408v1

The combinatorics of sector renormalization

Willie Rush Lim

math.DSmath.NT37F2011A5511J7037F5037E05

Paper metadata

Original submission: July 13, 2026
Claimed MSC: 37F20, 11A55, 11J70, 37F50, 37E05
arXiv categories: math.DS, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 34 pages, 5 figures

Abstract

The goal of this note is to systematically develop the fundamental arithmetic and combinatorial properties of the sector renormalization operation on rigid rotations. We employ the specific framework of modified continued fractions appropriate for sector renormalization and analyze their properties. By allowing infinite first return times, this framework yields a dynamical compactification characterized by a universal property. We also discuss the corresponding natural extension and introduce the notion of a time (semi-)group. For example, we demonstrate how a bi-infinite tower of sector renormalizations of irrational rotations can be packaged within a single dynamical plane as a cascade of translations. This note will serve as a foundational combinatorial tool for studying the geometric properties of sector renormalizations of holomorphic maps with irrationally indifferent fixed points, particularly neutral quadratic polynomials.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsCorrect

The modified continued-fraction coding, the two universal characterizations of the compactification, and the natural-extension cascade statements are supported. A missing pair of absolute-value bars in one quantitative convergent estimate is a uniquely determined typo and does not affect these conclusions.

Theorem A / Theorem 3.4Correct

Modified continued fractions conjugate sector renormalization to the shift

Pages 2 and 12–15 · Theorem A, Lemma 3.2, Proposition 3.3, and Theorem 3.4 · arXiv:2607.11408v1

The determinant identity $p^{[n]}q^{[n-1]}-p^{[n-1]}q^{[n]}=\varepsilon_n$ and the uniform ratio bound on successive denominators give convergence of the modified convergents and shrinking symbolic cylinders. The branch formula $g(\theta)=-1/\theta+\varepsilon_1b_1$ removes the first symbol. Expansion by at least $4$ makes each itinerary fiber a singleton, while the cylinder intervals provide continuity in both directions. Thus $X$ is a homeomorphism and $X\circ s=g\circ X$ .

Theorem B / Theorems 4.7–4.8Correct

The segment and dynamical embeddings determine the same minimal compactification

Pages 4 and 18–24 · Propositions 4.4–4.5 and Theorems 4.6–4.8 · arXiv:2607.11408v1

The general orbit-closure construction in $Y^{\mathbb N}$ has the stated universal factor property. The segment orbit distinguishes the two possible one-sided rational enrichments at every shift, making the induced factor from the symbolic compactification injective. The extended cyclic-order map is itself an embedding, so its orbit factor is also injective. Compactness and the Hausdorff target then upgrade both continuous bijections to homeomorphisms and yield Corollary 4.9.

Section 5Correct

The natural extension, ordered time group, and translational cascade agree

Pages 26–33 · Propositions 5.2–5.13 · arXiv:2607.11408v1

The slow return times satisfy the usual continuant recurrence, which gives the chronological and left-right positive cones. Mapping each formal return-time generator $Q^{[n]}$ to translation by $-\varepsilon_{n+1}l^{[n]}$ respects all relations; injectivity follows because an integral relation between two consecutive generators would force an irrational $|\theta_{n+1}|$ to be rational. The nested half-strips then project by $z\mapsto e^{-2\pi iz/l^{[n-1]}}$ to the successive sector-renormalization domains, establishing the cascade and triangulation claims.

02Proofs3 reported findingsCorrect

The central arithmetic, compactification, and cascade proofs are correct and complete after immediate-consequence closure. The report records two mechanically determined notation corrections as typos; neither changes a theorem, proof strategy, or overall status.

Proposition 3.3Typo

Absolute-value bars are missing from the convergent-error estimate

Page 14 · Proposition 3.3 and its proof · arXiv:2607.11408v1

The displayed estimate must read $\frac{\theta_{\rm gm}}{q^{[n]}q^{[n+1]}}\leq\left|X(\sigma)-\frac{p^{[n]}}{q^{[n]}}\right|\leq\frac{2-\theta_{\rm gm}}{q^{[n]}q^{[n+1]}}.$ Without the bars it is false when $\varepsilon_{n+1}=-1$ : for example, $\varepsilon_1=1$ , $\varepsilon_2=\varepsilon_3=-1$ , and $\bar a_1=\bar a_2=2$ give consecutive convergents $1/2$ and $3/7$ , so the error after the first convergent has negative sign. Lemma 3.2(5) makes the remaining tail smaller than its first term, while the proof already estimates its magnitude. Inserting the bars in the statement and the two corresponding proof lines is therefore the unique intended correction, and the convergence, irrationality, and cylinder-shrinking consequences remain valid.

Full paper, version 1 ↗

Theorems 3.4 and 4.7; Proposition 5.11Typo

Several local symbols are mechanically mismatched

Page 15 · proof of Theorem 3.4; page 23 · proof of Theorem 4.7; page 32 · Proposition 5.11 · arXiv:2607.11408v1

In Theorem 3.4, the undefined symbols $\vartheta_1,\vartheta_n$ must be $\varepsilon_1,\varepsilon_n$ . In Theorem 4.7, the initial unequal pair and the second cases must compare the primed sequence with the unprimed one; the surrounding equations uniquely determine the missing primes. In Proposition 5.11(1), the union of the half-strips is the closed lower half-plane $-\mathbb H$ , not $\mathbb R$ , exactly as stated immediately before the display. These are literal notation mismatches with unique local corrections and no mathematical downstream effect.

Full paper, version 1 ↗

Sections 2–5Correct and complete

Central proof chain

Pages 7–33 · Sections 2–5 · arXiv:2607.11408v1

The first-return recurrences, best-approximation estimates, orbit-closure universal property, positive-cone constructions, and exponential projection of the cascade are mutually consistent. Endpoint cases with infinite symbols are handled by the one-point compactification of the alphabet, and the order arguments introduce a new free generator exactly at each parabolic level. No missing nontrivial case remains after the two notation corrections above.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.11408v1
Authors listed: Willie Rush Lim
Audit date: August 15, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. 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P(delta)

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P(delta)

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requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

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The combinatorics of sector renormalization

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Modified continued fractions conjugate sector renormalization to the shift

The segment and dynamical embeddings determine the same minimal compactification

The natural extension, ordered time group, and translational cascade agree

Absolute-value bars are missing from the convergent-error estimate

Several local symbols are mechanically mismatched

Central proof chain

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