arXiv:2606.28014v1

The Natural Extension for the Triangle Map (a Multi-dimensional Continued Fraction) with An Internal Symmetry from Young Conjugation

Joe Fox, Thomas Garrity

math.NTmath.DS11J7011K5037A44

Paper metadata

Original submission: June 26, 2026
Claimed MSC: 11J70, 11K50, 37A44
arXiv categories: math.NT, math.DS
Paper license: CC BY 4.0 ↗

Abstract

The natural extension of the triangle map (a type of multi-dimensional continued fraction algorithm) is completely described in all possible dimensions. The motivation and inspiration for this natural extension stems from the triangle map's recent link to the classical study of integer partitions. Inspired by Young conjugation for integer partitions, we show that the natural extension has an internal symmetry and allows the natural extension to be subdivided into four natural subdomains. This appears to be new even for the classical case of the natural extension for continued fractions, namely for both the classical Gauss map and the classical Farey map.

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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 findingsContains unsupported statements

The formulas for Young conjugation and the invariant densities are supported, but the four central natural-extension assertions are not able to be verified under the paper's own definition: the proofs establish invertible branch dynamics but do not establish the required measurable generation or universal factorization properties.

Theorems 5.4, 6.5, 9.4, and 10.3Not able to verify

The natural-extension assertions are not fully established

Pages 2, 11, 14, 27, and 30 · Definition in Section 2.1 and the four natural-extension theorems · arXiv:2606.28014v1

Section 2.1 defines a natural extension using measure-preserving spaces, an invertible extension, an onto factor map, commutation, generation of the extension sigma-algebra by the iterated factor sigma-algebras, and a universal factorization property. The cited cylinder identities prove explicit invertibility and commutation for the proposed domains, but the manuscript does not specify and verify all measure spaces modulo null sets, prove the sigma-algebra generation condition, or prove universal factorization. These missing properties are part of the stated definition and are not immediate consequences of the coordinate formulas. The proposed systems may satisfy them, but the supplied arguments do not permit that conclusion.

Full paper, version 1 ↗

Propositions 8.2 and 8.4–8.6Correct

Young conjugation and the four-domain symmetry

Pages 19–24 · Section 8 · arXiv:2606.28014v1

After correcting the local index typos recorded under Proofs, the homogeneous Young map is an involution, preserves the ordering and positivity inequalities defining the simplex cone, and exchanges the indicated branch coordinates. Its affine projection has the stated inverse relation. The four regions are exactly the sign choices induced by comparing the relevant first and last coordinates, so the claimed internal symmetry and domain subdivision follow from the displayed algebra.

Section 12Correct

Invariant densities from the affine fibers

Pages 31–36 · Theorem 12.3 and Sections 12.1–12.2 · arXiv:2606.28014v1

The homogeneous branch transformations have unit Jacobian, and integrating Lebesgue measure over the stated fibers gives the affine densities $\frac{1}{m!\,t_1\cdots t_m}$ and $\frac{1}{m!\,t_1\cdots t_{m-1}(1+t_m)}.$ Direct substitution in the branch formulas preserves the resulting measures. This verifies the density calculations, although it does not supply the separate sigma-algebra and universality arguments required by the paper's definition of natural extension.

02Proofs5 reported findingsContains incorrect or incomplete proofs

The canonical inverse-limit map in the preliminary model has the wrong orientation, and the central natural-extension proofs omit conditions contained in the stated definition. Several displayed Young-conjugation and fast-map formulas also contain mechanically repairable notation errors.

Section 2.1Incorrect as written

The canonical inverse-limit map does not commute with the factor map

Page 2 · canonical natural extension in Section 2.1 · arXiv:2606.28014v1

The manuscript sets $X=\{(y_0,y_1,\ldots):S(y_{n+1})=y_n\},\qquad T(y_0,y_1,\ldots)=(y_1,y_2,\ldots),\qquad \psi(y)=y_0.$ Then $\psi(Ty)=y_1$ , whereas $S(\psi(y))=S(y_0)$ , so the required identity $\psi\circ T=S\circ\psi$ fails in general. Repair classification: Verified repair. Use the forward inverse-limit map $T(y_0,y_1,\ldots)=(S(y_0),y_0,y_1,\ldots),$ whose inverse is the printed left shift. This restores commutation and invertibility without changing the inverse-limit space.

Full paper, version 1 ↗

Proofs of Theorems 5.4, 6.5, 9.4, and 10.3Incomplete as written

The natural-extension verification stops before the defining measure-theoretic conditions

Pages 11, 14, 27, and 30 · proofs of the four natural-extension theorems · arXiv:2606.28014v1

Each proof refers to the preceding branch or cylinder identities and says that the remaining verification is the same. Those identities establish one-to-one branch dynamics and the factor relation, but they do not prove generation of the full sigma-algebra by the iterated factor sigma-algebras or the universal factorization property required in Section 2.1. The later Jacobian computation establishes preservation of a candidate measure only. Repair classification: No repair supplied; a complete proof must specify the measure spaces and null-set conventions and verify the two missing conditions for each proposed extension.

Full paper, version 1 ↗

Propositions 8.3 and 8.6Typo

Two Young-conjugation specializations contain index and factor typos

Pages 20 and 25 · Proposition 8.3 and the one-dimensional affine specialization · arXiv:2606.28014v1

In Proposition 8.3, the reconstructed coordinates are printed with $\lambda_0=k_0+\cdots+\lambda_m$ and $\lambda_1=k_0+\cdots+\lambda_{m-1}$ ; the last terms must be $k_m$ and $k_{m-1}$ . In the $m=1$ affine specialization, direct projection of the preceding general formula gives $Y_A(t,u)=\left(\frac{N-tu}{N-tu+u},\ (N-tu+u)(1-t)\right),$ so the extra factor involving $t_{m-1}-t_m$ in the printed second coordinate must be deleted. Both corrections are uniquely determined by the general formulas and leave the involution argument unchanged.

Full paper, version 1 ↗

Sections 9–10Typo

Fast-map formulas and theorem labels contain local notation errors

Pages 26–30 · definitions of the fast dual branches and Theorems 9.4 and 10.3 · arXiv:2606.28014v1

The homogeneous dual branch prints $\lambda_1-n(\lambda_0-\lambda_m)$ where composition with the preceding branch gives $\lambda_1-n(\lambda_0-\lambda_1)$ ; the domain inequalities immediately below already use this correction. Theorem 9.4 then calls the extension an extension of $T$ although the section concerns the fast map $G$ , and Theorem 10.3 similarly names $T$ and $\widetilde G$ where the affine fast maps $G^A$ and $\widetilde G^A$ are meant. Replacing these symbols is mechanically forced by the section definitions and does not alter the branch calculations.

Full paper, version 1 ↗

Propositions 5.3, 6.4, 9.3, and 10.2Correct and complete

Explicit branch inverses and cylinder identities

Pages 10–14 and 26–30 · slow and fast branch computations · arXiv:2606.28014v1

On each stated branch domain, the matrix formulas are mutual inverses and map the inequalities to the corresponding primal and dual cylinders. Boundary conventions are compatible up to the usual null sets. These calculations correctly establish the algebraic invertibility and factor identities used later; the adverse proof verdict concerns the additional measure-theoretic requirements, not these branch computations.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.28014v1
Authors listed: Joe Fox, Thomas Garrity
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. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. 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. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

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

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

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for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

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The Natural Extension for the Triangle Map (a Multi-dimensional Continued Fraction) with An Internal Symmetry from Young Conjugation

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The natural-extension assertions are not fully established

Young conjugation and the four-domain symmetry

Invariant densities from the affine fibers

The canonical inverse-limit map does not commute with the factor map

The natural-extension verification stops before the defining measure-theoretic conditions

Two Young-conjugation specializations contain index and factor typos

Fast-map formulas and theorem labels contain local notation errors

Explicit branch inverses and cylinder identities

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