arxiv: 2604.10492 · v2 · submitted 2026-04-12 · 💱 q-fin.MF

Recognition: unknown

Aharanov-Bohm Type Arbitrage and Homological Obstructions in Financial Markets

Takanori Adachi

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:07 UTC · model grok-4.3

classification 💱 q-fin.MF

keywords arbitrageholonomyfiltrationconditional expectationAharonov-Bohm arbitrageself-financing strategycohomologymarket consistency

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The pith

Non-trivial holonomy in market filtrations converts into a predictable self-financing trading strategy under admissibility conditions.

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

The paper models filtrations in financial markets as contravariant functors from a time category to probability spaces. It defines a multiplicative distortion induced by the conditional expectation functor that measures how constant functions fail to stay constant across non-measure-preserving transitions. Holonomy of this distortion along loops in the time category captures global inconsistencies invisible to local checks. When the holonomy is non-trivial, the paper shows it can be realized as an economically admissible trading strategy. This shifts the view of arbitrage from pointwise price mismatches to cohomological features of the information structure.

Core claim

Given a filtration modeled as a contravariant functor F : T^op → Prob, the composition with the conditional expectation functor induces a canonical multiplicative distortion dF(i) := (E ∘ F)(i)(1). The holonomy of dF along loops in T is defined and interpreted as Aharonov-Bohm type arbitrage, representing a global inconsistency not detectable at individual transitions. Under suitable admissibility conditions, non-trivial holonomy yields a predictable self-financing trading strategy, establishing a direct link between cohomological structures and realizable arbitrage in financial markets.

What carries the argument

The multiplicative distortion dF induced by the conditional expectation functor, whose holonomy along loops in the time category T converts global inconsistencies into trading strategies.

Load-bearing premise

The multiplicative distortion dF induced by the conditional expectation functor has direct economic meaning as arbitrage, and holonomy along loops in the time category corresponds to a realizable trading opportunity.

What would settle it

An explicit market filtration containing a loop with non-unit holonomy for which no admissible predictable self-financing strategy produces a positive gain with no risk of loss.

read the original abstract

We introduce a new perspective on arbitrage based on global loop effects in filtered market systems, providing a conceptual extension of classical arbitrage theory beyond local consistency conditions. Given a filtration modeled as a contravariant functor $F : \mathcal{T}^{op} \to \mathrm{Prob}$, we consider the associated conditional expectation functor $\mathcal{E} \circ F$ and show that it induces a canonical multiplicative distortion $dF(i) := (\mathcal{E} \circ F)(i)(1)$, which measures the failure of constant functions to be preserved under non-measure-preserving transitions. We define the holonomy of $dF$ along loops in $\mathcal{T}$ and interpret non-trivial holonomy as a global inconsistency that is invisible at the level of individual transitions. This leads to a notion of Aharonov--Bohm (AB) arbitrage, in which arbitrage arises from loop effects rather than local price discrepancies. We further show that, under suitable admissibility conditions, non-trivial holonomy can be converted into a predictable self-financing trading strategy. This establishes a connection between cohomological structures and economically realizable arbitrage, highlighting the role of global invariants in the structure of financial markets.

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 sketches a categorical take on global arbitrage via holonomy of a filtration functor but does not deliver the promised link to an explicit self-financing strategy.

read the letter

The main new element is the setup that treats a filtration as a contravariant functor F from a time category to probability spaces, then pulls out a multiplicative distortion dF(i) from the composite conditional-expectation functor and defines its holonomy around loops. That construction is not in the usual arbitrage literature the abstract cites, so the analogy to Aharonov-Bohm phases is genuinely fresh on the formal side. It gives a clean way to talk about inconsistencies that are invisible locally but appear when you go around a closed path in time. That framing is worth having on the table for people who already work with categorical probability or global invariants in stochastic processes. The paper does a reasonable job stating the objects and the interpretation it wants. The soft spot is exactly where the stress-test note flags it: the abstract asserts that non-trivial holonomy yields a predictable self-financing strategy under suitable admissibility conditions, yet supplies no derivation, no explicit integrand H, and no check that the resulting gains process meets the standard semimartingale requirements for arbitrage. The admissibility conditions themselves are not written down in terms of integrability or lower bounds. Without those steps the economic claim stays formal. The distortion is defined directly from the functor, so there is also a risk that the arbitrage reading is circular unless an independent verification is added. This is the sort of paper that belongs in a reading group for people already comfortable with category theory in finance; it is not yet ready for someone who just wants a new no-arbitrage theorem they can apply tomorrow. I would send it to peer review. The idea is coherent on its own terms and the authors appear to be thinking carefully about the setup, but the referee will need to see the missing construction and at least one worked example before the central claim can be assessed.

Referee Report

2 major / 2 minor

Summary. The manuscript models a filtration as a contravariant functor F : T^op → Prob and considers the composite conditional-expectation functor E ∘ F. It defines a multiplicative distortion dF(i) := (E ∘ F)(i)(1) that quantifies the failure of constant functions to be preserved under non-measure-preserving transitions, introduces the holonomy of dF along loops in the time category T, and interprets non-trivial holonomy as Aharonov-Bohm-type arbitrage. The central claim is that, under suitable admissibility conditions, this holonomy can be realized as a predictable self-financing trading strategy.

Significance. If the missing derivation were supplied, the work would constitute a genuine conceptual extension of arbitrage theory by exhibiting global cohomological invariants that are invisible to local no-arbitrage conditions. The attempt to import categorical language and holonomy into mathematical finance is novel and, if made rigorous, could stimulate further research on topological obstructions in stochastic models.

major comments (2)

[Abstract] Abstract: the claim that 'under suitable admissibility conditions, non-trivial holonomy can be converted into a predictable self-financing trading strategy' is asserted without any derivation, without an explicit construction of an adapted integrand H such that the stochastic integral ∫ H dS is self-financing and predictable, and without verification that the terminal value satisfies the standard arbitrage definition (non-negative a.s., strictly positive with positive probability). This step is load-bearing for the entire contribution.
[Abstract] Abstract: the multiplicative distortion is defined directly as dF(i) := (E ∘ F)(i)(1); the economic reading of this quantity as 'arbitrage' therefore reduces to the definitional observation that conditional expectations do not preserve constants, creating a circularity that is not resolved by any independent market-theoretic argument.

minor comments (2)

The title misspells the effect as 'Aharanov-Bohm'; the standard spelling is 'Aharonov-Bohm'.
[Abstract] The notation for the time category T, the functor F, and the precise meaning of 'loop' should be introduced with a short diagram or explicit list of objects and morphisms before the distortion is defined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. The two major comments identify key gaps in the rigor of our claims. We will revise the manuscript accordingly to include the missing derivation and to strengthen the economic motivation for the multiplicative distortion. Below we respond point by point.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that 'under suitable admissibility conditions, non-trivial holonomy can be converted into a predictable self-financing trading strategy' is asserted without any derivation, without an explicit construction of an adapted integrand H such that the stochastic integral ∫ H dS is self-financing and predictable, and without verification that the terminal value satisfies the standard arbitrage definition (non-negative a.s., strictly positive with positive probability). This step is load-bearing for the entire contribution.

Authors: We accept this criticism. The current manuscript introduces the holonomy concept but does not include the full step-by-step construction of the integrand H or the verification of the arbitrage properties. In the revised manuscript, we will add a new section (Section 4) that explicitly constructs H as the predictable process derived from the holonomy cocycle, demonstrates that the stochastic integral is self-financing by showing that the value process satisfies the integral equation for self-financing strategies, and proves that the terminal wealth is non-negative almost surely and positive with positive probability when the holonomy is non-trivial. This addresses the load-bearing claim directly. revision: yes
Referee: [Abstract] Abstract: the multiplicative distortion is defined directly as dF(i) := (E ∘ F)(i)(1); the economic reading of this quantity as 'arbitrage' therefore reduces to the definitional observation that conditional expectations do not preserve constants, creating a circularity that is not resolved by any independent market-theoretic argument.

Authors: We disagree that the interpretation is circular. The definition is the natural mathematical consequence of the functor, but the arbitrage content follows from the fact that non-trivial holonomy produces a global inconsistency across the filtration that cannot be arbitraged away by local trading at single times. We will revise the abstract, introduction, and add a new subsection with an independent market-theoretic argument: in a discrete two-period example, the product of distortions around a loop yields a strictly positive terminal wealth for a self-financing strategy with zero initial capital, which satisfies the classical arbitrage definition independently of the categorical language. This grounds the economic reading in standard no-arbitrage concepts. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation introduces definitions and claims a construction without reduction to inputs

full rationale

The paper models the filtration as a contravariant functor F, defines the conditional expectation functor E ∘ F, and explicitly constructs the multiplicative distortion via the equation dF(i) := (E ∘ F)(i)(1). It then defines holonomy of this dF along loops in T and interprets non-trivial values as AB arbitrage before claiming a conversion to a self-financing strategy under admissibility conditions. These steps consist of fresh categorical definitions followed by a purported existence result; no equation equates the final trading strategy or arbitrage profit to the input distortion by construction, no parameters are fitted and relabeled as predictions, and no load-bearing self-citation or uniqueness theorem is invoked. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The framework rests on modeling assumptions about filtrations and functors plus newly introduced entities whose economic content is asserted rather than derived from market primitives.

axioms (2)

domain assumption A filtration is modeled as a contravariant functor F : T^op → Prob
Invoked at the outset to represent time-dependent market information; no justification for why this categorical structure captures real filtrations is given.
domain assumption The composite E ∘ F induces a canonical multiplicative distortion dF(i) := (E ∘ F)(i)(1)
Central definitional step that turns the functor into a measurable quantity; treated as automatic rather than derived from economic primitives.

invented entities (2)

Aharonov-Bohm arbitrage no independent evidence
purpose: To name arbitrage opportunities arising from global loop effects rather than local price discrepancies
New interpretive label for non-trivial holonomy; no independent market evidence or falsifiable prediction supplied.
holonomy of dF no independent evidence
purpose: To quantify global inconsistency invisible at individual transitions
Defined mathematically from the distortion but lacks external validation or mapping to observable trading data.

pith-pipeline@v0.9.0 · 5505 in / 1762 out tokens · 46168 ms · 2026-05-10T16:07:09.983587+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Martingale Cohomology, Holonomy, and Homological Arbitrage
q-fin.MF 2026-05 unverdicted novelty 7.0

Martingales arise as 0-cocycles in a simplicial cochain complex from categorical filtrations, with the first cohomology group defining homological arbitrage as consistent gains unreachable by any price process after b...
Martingale Cohomology, Holonomy, and Homological Arbitrage
q-fin.MF 2026-05 unverdicted novelty 7.0

A transport cohomology framework on categorical filtrations produces holonomy operators and homological arbitrage as global effects from probabilistic distortions along closed simplicial loops.

Reference graph

Works this paper leans on

3 extracted references · 2 canonical work pages · cited by 1 Pith paper

[1]

Adachi, T. (2025). A geometric model of synthetic filtrations via context-dependent time. arXiv:2509.12919 [math.PR]

work page arXiv 2025
[2]

Adachi, T., Nakajima, K., and Ryu, Y. (2020). Generalized filtrations and its application to binomial asset pricing models. arXiv:2011.08531 [q-fin.MF]

work page arXiv 2020
[3]

and Ryu, Y

Adachi, T. and Ryu, Y. (2019). A category of probability spaces. J. Math. Sci. Univ. Tokyo , 26(2):201--221

2019