Uniform Martin's conjecture, locally

Vittorio Bard

arxiv: 1907.10766 · v1 · pith:AQFXWDYOnew · submitted 2019-07-24 · 🧮 math.LO

Uniform Martin's conjecture, locally

Vittorio Bard This is my paper

Pith reviewed 2026-05-24 16:17 UTC · model grok-4.3

classification 🧮 math.LO

keywords uniform Martin's conjectureTuring degreesTuring determinacyBorel reducibilitycomputable reducibilityequivalence relations

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The pith

Part I of uniform Martin's conjecture follows from a local ordering property of Turing invariant functions.

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

The paper shows that part I of uniform Martin's conjecture follows from the local fact that any non-constant Turing invariant function taking a Turing degree x to a Turing degree y must satisfy x ≤_T y. This local condition is equivalent to Turing determinacy and therefore settles the global part I of the conjecture. The same local analysis yields that Turing reducibility is Borel reducible to computable reducibility on equivalence relations over the natural numbers. The paper ends by asking whether part II of the conjecture likewise reduces to local phenomena.

Core claim

If a non-constant Turing invariant function maps Turing degree x to Turing degree y then x ≤_T y. This local phenomenon implies part I of uniform Martin's conjecture, which is equivalent to Turing determinacy, and shows that ≤_T is Borel reducible to ≤_c on equivalence relations on ℕ.

What carries the argument

The local phenomenon that non-constant Turing invariant functions from degree x to y satisfy x ≤_T y.

If this is right

Part I of uniform Martin's conjecture holds.
Turing reducibility is Borel reducible to computable reducibility on equivalence relations over the naturals.
The structure of computable reducibility is at least as complex as that of Turing reducibility.

Where Pith is reading between the lines

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

Similar local conditions might be sought for part II of the conjecture.
The reduction between reducibility notions may apply to other degree structures or equivalence relations.

Load-bearing premise

The functions under consideration are Turing invariant and the setting includes the relevant notions of Borel reducibility and determinacy.

What would settle it

A non-constant Turing invariant function f with f(x) = y but x not Turing reducible to y.

read the original abstract

We show that part I of uniform Martin's conjecture follows from a local phenomenon, namely that if a non-constant Turing invariant function goes from the Turing degree $\boldsymbol x$ to the Turing degree $\boldsymbol y$, then $\boldsymbol x \le_T \boldsymbol y$. Besides improving our knowledge about part I of uniform Martin's conjecture (which turns out to be equivalent to Turing determinacy), the discovery of such local phenomenon also leads to new results that did not look strictly related to Martin's conjecture before. In particular, we get that computable reducibility $\le_c$ on equivalence relations on $\mathbb N$ has a very complicated structure, as $\le_T$ is Borel reducible to it. We conclude raising the question "Is part II of uniform Martin's conjecture implied by local phenomena, too?" and briefly indicating a possible direction.

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 derives part I of uniform Martin's conjecture from a local property of Turing-invariant functions, equates it to Turing determinacy, and obtains a Borel reduction of ≤_T into ≤_c on equivalence relations.

read the letter

The core claim is that part I of uniform Martin's conjecture follows from the local fact that a non-constant Turing-invariant function sending degree x to degree y must have x ≤_T y. From this the paper extracts an equivalence between that part of the conjecture and Turing determinacy, plus the concrete result that Turing reducibility is Borel reducible to computable reducibility on equivalence relations on the naturals. Both the equivalence and the reduction are presented as new consequences of the local observation. The local-to-global move and the application to equivalence relations are the parts that actually move the literature forward. The reduction in particular gives an independent lower bound on the complexity of ≤_c that does not obviously require the full strength of Martin's conjecture. The argument uses only the standard definition of Turing invariance and the usual notions of Borel reducibility and determinacy; there are no free parameters or self-referential constructions visible in the abstract. The main limitation is that the abstract supplies no proof details, so the steps that lift the local property to the global statement and that build the Borel reduction cannot be inspected for gaps. If those derivations are correct and self-contained, the paper is solid. If they rely on unstated choice principles or hidden determinacy assumptions, the claims would need tightening. This work is aimed at people who already follow Martin's conjecture or the theory of Borel reducibility on equivalence relations. A reader working on degree structures or on the fine structure of computable reducibility would get direct value from the equivalence and the reduction. It is worth sending to a serious referee because the statements are specific, the local premise is cleanly stated, and the consequences are checkable once the proofs are examined.

Referee Report

0 major / 1 minor

Summary. The manuscript shows that part I of uniform Martin's conjecture follows from a local property of non-constant Turing-invariant functions: if such a function maps Turing degree x to y then x ≤_T y. It establishes the equivalence of this part to Turing determinacy, derives that Turing reducibility ≤_T is Borel reducible to computable reducibility ≤_c on equivalence relations on ℕ (yielding a new result on the complexity of ≤_c), and poses an open question about whether part II is similarly implied by local phenomena.

Significance. If the local-to-global transfer and the Borel reduction hold, the result advances the study of Martin's conjecture by isolating a local condition equivalent to the global statement under determinacy, and it connects the conjecture to the structure theory of equivalence relations via Borel reducibility. The equivalence to Turing determinacy and the new reducibility result are notable contributions in descriptive set theory and computability.

minor comments (1)

The abstract invokes Turing invariance, Borel reducibility, and determinacy as background; a short paragraph in the introduction clarifying the precise ambient theory (e.g., ZF + AD or ZFC + PD) would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and their recommendation to accept. The report accurately reflects the main contributions: the local condition on non-constant Turing-invariant functions implying part I of uniform Martin's conjecture (and its equivalence to Turing determinacy), the resulting Borel reduction of ≤_T to ≤_c, and the open question posed about part II.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives part I of uniform Martin's conjecture from the stated local property of non-constant Turing-invariant functions (if f maps degree x to y then x ≤_T y). This local phenomenon is presented as an independent premise in the abstract, not as a restatement or fit of the conjecture itself. No equations, definitions, or self-citations in the provided text reduce the claimed implication or the noted equivalence to Turing determinacy back to the inputs by construction. The derivation chain remains self-contained against external benchmarks such as the background theory of Borel reducibility and determinacy.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background definitions of Turing degrees, Turing invariance, Borel reducibility, and determinacy from prior literature in computability and descriptive set theory; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard axioms of ZFC (or ZF+DC) sufficient to define Turing degrees and Borel sets
Invoked implicitly for the notions of Turing reducibility, invariance, and Borel reducibility used throughout the statements.

pith-pipeline@v0.9.0 · 5651 in / 1367 out tokens · 22999 ms · 2026-05-24T16:17:51.837597+00:00 · methodology

Uniform Martin's conjecture, locally

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)