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arxiv: 2606.00396 · v1 · pith:MFWAGKYEnew · submitted 2026-05-29 · 🧮 math.AG · math.RT

Properties of deformed mass and phase functions

Daniel Halpern-Leistner , Antonios-Alexandros Robotis This is my paper

Pith reviewed 2026-06-28 19:43 UTC · model grok-4.3

classification 🧮 math.AG math.RT
keywords stability conditionsdeformed massdeformed phasecontinuitytriangle inequalitytruncationsalgebraic geometry
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The pith

Deformed mass and phase functions on stability conditions are continuous, yielding a homeomorphic embedding into a product of finite measure spaces.

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

This paper proves that deformed mass and phase functions defined on the space of stability conditions are continuous. The continuity allows embedding the space homeomorphically into a product of spaces of finite measures. It also establishes the triangle inequality for these mass functions and gives estimates for the deformed mass of truncations relative to a slicing. These results provide a way to topologically control the space of stability conditions using measure theory.

Core claim

We establish basic properties of the deformed mass and phase functions on the space of stability conditions. We prove that these functions are continuous and deduce that the space of stability conditions admits a homeomorphic embedding into a product space of finite measures. Subsequently, we give a proof of the triangle inequality for deformed mass functions and provide estimates for the deformed mass of truncations of objects with respect to a slicing.

What carries the argument

The deformed mass and phase functions, which assign measures and phases to stability conditions and support the continuity and embedding arguments.

If this is right

  • The space of stability conditions admits a homeomorphic embedding into a product space of finite measures.
  • The triangle inequality holds for the deformed mass functions.
  • Estimates hold for the deformed mass of truncations of objects with respect to a slicing.

Where Pith is reading between the lines

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

  • The embedding may let researchers transfer metric or convergence properties from finite measures back to stability conditions.
  • Similar continuity proofs could extend to related functions like central charges on the same space.
  • The results may help analyze the topology of connected components within the space of stability conditions.

Load-bearing premise

The deformed mass and phase functions are well-defined and satisfy the structural properties needed for continuity arguments on the space of stability conditions.

What would settle it

An explicit stability condition at which the deformed mass function or phase function fails to be continuous.

Figures

Figures reproduced from arXiv: 2606.00396 by Antonios-Alexandros Robotis, Daniel Halpern-Leistner.

Figure 1
Figure 1. Figure 1: An example of the kind of curve to which Lemma 5.1 applies. Conditions (1) and (2) of the lemma can be interpreted as follows: If γz is the piecewise linear curve connecting 0, Σz1, Σz2, . . . , Σzn, and γw is the analogous curve for w•, then the section of the curve γz over which ℑ(−) is increasing (resp. decreasing) must lie to the left (resp. right) of the section of γw where ℑ(−) is increasing (resp. d… view at source ↗
Figure 2
Figure 2. Figure 2: This illustrates the polygons appearing in the proof of Lemma 6.2. The blue horizontal line is the curve ℑ(z) = h, the green dashed line is the curve γE, the black line is the HN curve of E, the purple line is Z(G)−(HN curve of G), and the gray dashed line is γG. Z(im(δ)) is constrained to lie in the polygon P bounded by the blue, black, and purple curves. For the purposes of our bound, though, we maximize… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the base case for the induction in the proof of Lemma 5.1, when m = 1. The fact that ϕ(ui+1) ≤ ϕ(zi) < ϕ(ui+1) + 1 and ui = ui+1 +zi implies the triangle inequality gt(ui) ≤ gt(ui+1) + gt(zi). The hypotheses of the lemma imply that ϕ(zi+1) ≤ ϕ(Σzi) ≤ ϕ(zi+1) + 1 for i = 1, . . . , r, and that each square-bracketed expression also satisfies the hypotheses of the lemma. See [PITH_FULL_IMAGE:… view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the inductive argument used in the proof of Lemma 5.2. The curve (z1, . . . , z7) begins with a section on which ℑ(z) is monotone increasing (blue), followed by a section on which ℑ(z) is monotone decreasing (purple), whereas ℑ(z) is monotone increasing on the curve (w•) (orange). This will always be the case under the hypotheses of Lemma 5.1. Note that a vertex Σz4 has been added where the… view at source ↗
read the original abstract

We establish basic properties of the deformed mass and phase functions on the space of stability conditions. We prove that these functions are continuous and deduce that the space of stability conditions admits a homeomorphic embedding into a product space of finite measures. Subsequently, we give a proof of the triangle inequality for deformed mass functions and provide estimates for the deformed mass of truncations of objects with respect to a slicing.

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.

Referee Report

0 major / 0 minor

Summary. The paper establishes basic properties of the deformed mass and phase functions on the space of stability conditions. It proves that these functions are continuous and deduces that the space of stability conditions admits a homeomorphic embedding into a product space of finite measures. It subsequently proves the triangle inequality for deformed mass functions and provides estimates for the deformed mass of truncations of objects with respect to a slicing.

Significance. If the proofs hold, the results supply foundational continuity and embedding properties for deformed functions on stability spaces. This strengthens the toolkit for analyzing the topology of spaces of stability conditions and their applications to moduli problems and wall-crossing in algebraic geometry and derived categories. The direct proofs from definitions, without ad-hoc parameters, are a positive feature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance, and recommendation to accept.

Circularity Check

0 steps flagged

No circularity detected; derivation is self-contained

full rationale

The paper proves continuity of the deformed mass and phase functions directly from their definitions on the space of stability conditions and deduces the homeomorphic embedding into a product of finite measures as a consequence. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the triangle inequality and truncation estimates are likewise established via explicit arguments from the structural properties. The derivation chain is independent of the target results and relies on standard mathematical reasoning rather than renaming or smuggling prior assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters, invented entities, or non-standard axioms are mentioned. The work relies on the standard mathematical framework of stability conditions in algebraic geometry and representation theory.

axioms (1)
  • standard math Standard axioms of category theory and triangulated categories underlying the definition of stability conditions.
    Invoked implicitly as the setting for stability conditions and the deformed functions.

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