Martin Fouilleul
email: martin.fouilleul@ircam.frjean.bresson@ircam.fr

Jean Bresson

Jean-Louis Giavitto
email: jean-louis.giavitto@ircam.fr

A Polytemporal Model for Musical Scheduling

Keywords: Symbolic time, Time transformations, Tempo curves, Scheduling

This paper describes the temporal model of a scheduler geared towards show control and live music applications. This model relies on multiple inter-related temporal axes, called timescales. Timescales allow scheduling computations using abstract dates and delays, much like a score uses symbolic positions and durations (e.g. bars, beats, and note values) to describe musical time. Abstract time is ultimately mapped onto wall-clock time through the use of time transformations, specified as tempo curves, for which we provide a formalism in terms of differential equations on symbolic position. In particular, our model allows specifying tempo both as a function of time or as a function of symbolic position, and allows piecewise tempo curves to be built from parametric curves.

Introduction

Timing is of utmost importance in performing arts. Among them, music has developed particularly fine-grained temporal constructs, using both continuous and discrete abstract representations of time. As such, it presents specific and interesting challenges with regard to the composition and interpretation of time at multiple scales, and across multiple independent time-flows.

In this paper we present the temporal model of Jiffy, a polytemporal scheduler which is part of an ongoing effort to build a programmable show-controller system for performing arts and interactive multimedia installations 1 . In particular, our temporal model allows specifying tempo either as a function of time or as a function of symbolic position, and allows piecewise tempo curves to be built from parametric curves such as Béziers curves, which are both versatile and intuitive. The scheduler exposes an interface based on fibers, that makes it easy to organize inter-dependant streams of related events.

Symbolic Time in Musical Applications

Common Paradigms in Show-Control Applications

Show controllers are programs used by sound and lighting engineers to create and run temporal scenarios synchronized to the actions of performers on stage. They allow users to launch sound and video samples, control mixing and lighting desks, operate motors for mechatronic stage props, and so on. Several approaches can be identified as to how they present and organize the temporal relations between the cues of the show:

-Timelines, which organize cues on a common, static time axis. Most sequencers, such as ProTools2 or Cubase3 fall in that category.

-Cuelists, which organize cues in nested lists with associated timing semantics. Notable examples are QLab 4 or Linux Show Player5 .

-Hybrid models offer both cuelists and timelines, either through separate modes of operation, as in Medialon 6 or Smode 7 , or as dual views of the same cues, as in Ableton Live 8 .

-Graphical planning environments that allow users to position cues in some abstract space, which maps to time through the use of trajectories, as in Iannix [START_REF] Thierry Coduys | Iannix. Aesthetical/Symbolic Visualisations for Hypermedia Composition[END_REF], or flow graphs, as in Ossia Score [START_REF] Celerier | OSSIA: Towards a Unified Interface for Scoring Time and Interaction[END_REF].

Despite the diversity of approaches, most show controllers lack an abstract notion of musical time, as they directly map cues to wall-clock dates or to external triggers. Furthermore, musical time is often deployed throughout a work at different scales (e.g. movements, phrases, cells, notes. . . ), and not every scale is tied to the same global tempo, e.g. ornaments such as grace notes and appogiatura are not affected in the same way by a change of tempo as a main melody line. Hence it would be more appropriate to consider several abstract musical times, or timescales.

Abstract Timescales

The above discussion emphasizes the need of strong temporal models in composition and performance softwares and highlights the adequacy of polytemporal abstract time scheduling, i.e. the ability to organize concurrent computations along multiple logical timescales, that can later get mapped to wall-clock time.

index.html 6 https://medialon.com/wp-content/uploads/2019/07/ M515-1-Medialon-Control-System-Manual.pdf 7 https://smode.fr 8 https://www.ableton.com/en/live/what-is-live/

The notion of abstract timescales has been tackled before by computer music environments or score followers. For instance, FORMULA [START_REF] Anderson | A System for Computer Music Performance[END_REF] allows applying independent time deformations on groups of concurrent tasks. David A. Jaffe [START_REF] Jaffe | Ensemble Timing in Computer Music[END_REF] proposed a recursive scheduler for hierarchical timing control, using explicit time maps. Antescofo [START_REF] Cont | ANTESCOFO: Anticipatory Synchronization and Control of Interactive Parameters in Computer Music[END_REF] allows users to compose independent abstract times through the use of time scopes and tempo curves.

In Jiffy, a timescale is a data structure used to maintain a notion of logical time, expressed as a rational number of symbolic time units9 (STU), and to schedule events at specific logical dates. It is analogous in this respect to a score, which organizes musical events in terms of a musical time, that needs to be translated into wall-clock time by a musician according to tempo indications and interpretative choices. However, whereas the tempo indication of a score usually prescribes some idealized mapping from musical-time to wall-clock time, a timescale's logical time does not necessarily map directly to wall-clock time. Instead, each timescale has a time source, which can be either the wall-clock time or another timescale. A timescale is also associated with a time transformation, which maps its internal time to the time of its source. Thus, the scheduler can handle multiple notions of logical time and map dates to wall-clock time through a hierarchy of time transformations.

Figure 1 illustrates a time deformation between a timescale and its source. The time map plots for each timescale show the position of the timescale with respect to wallclock time. The effect of the first tempo curve (tempo 1) is to warp the time map of timescale 0 (which represents wall-clock time) into that of timescale 1 (which represent some abstract musical time). Timescale 1 is then transformed by another another tempo curve (tempo 2), to produces the time map of timescale 2.

Time Transformations

Jiffy's scheduler must be able to transform timescale-local positions to and from wallclock time. These transformations are specified by the means of tempo curves, which describe the speed of a timescale's "playhead" with respect to the source time, much like tempo indications in a score prescribe an idealized conversion from durations in beats to durations in wall-clock time 10 .

Several methods have been proposed to represent time transformations and to integrate tempo curves to map symbolic position to time. Jaffe [START_REF] Jaffe | Ensemble Timing in Computer Music[END_REF] proposes to directly use time maps constructed from a collection of predefined time warping functions. Berndt [START_REF] Berndt | Musical Tempo Curves[END_REF] chooses to represent tempo curves by potential functions of symbolic position, matching some specified mean tempo condition. Timewarp [START_REF] Maccallum | Timewarp: A Graphical Tool for the Control of Polyphonic Smoothly Varying Tempos[END_REF] is a tool that uses regularized beta functions to define tempo curves satisfying polyrhythmic constraints. Antescofo uses a variety of tweening functions 11 to express tempo as a function of time, and uses closed form expressions to compute time transformation based on tempo curves. When there is no analytical solution to a tempo curve integration, Antescofo samples the curve to produce a piecewise linear approximation, which is then integrated analytically. Antescofo can also use arbitrary expressions to define tempo, although these expressions are not integrated: they are reevaluated each time a variable is updated, and considered constant between updates. As such, they can only represent tempo as step functions.

In Jiffy, we allow users to specify a tempo curve either as a function of a timescale's source time, or as a function of symbolic position (which is closer to the way tempo is specified in a score). We use piecewise tempo curves where each piece can be defined by parametric curves. A variable-step numerical method is used to integrate the tempo curves when simple analytical solutions are not readily available.

Differential Equation Formulation

In the following we will use the variable p to denote the position in a timescale, i.e. the logical time in this timescale's reference frame. The variable t will be used to denote the source time (or simply, time), i.e. the time in the timescale's parent reference frame (which could be the wall-clock time).

The function position function, P (t), transforms the source time into the internal position of the timescale. The time function, T (p), transforms the position into the source time. Obviously, P = T -1 . 10 One difference, however, is that we use the word tempo here to refer to the ratio of internal STUs over source STUs, rather than the number of beats per minutes, since the latter could depend on the musical meter of the timescale. 11 https://antescofo-doc.ircam.fr/Reference/compound_curve/ A tempo curve T can be either a function of time or position. It maps its parameter to the value of the derivative of the position function at this instant. In the following we will refer to a tempo curve defined as a function of position as an autonomous tempo curve, whereas a tempo curve defined as a function of time will be referred to as a nonautonomous tempo curve. This naming stems from the formulation of the tempo curve as the right-hand side of an autonomous or non-autonomous differential equation:

dP dt (t) = T (P (t)) (autonomous) , or dP dt (t) = T (t) (non autonomous) ,
with initial condition P (0) = 0 .

(1)

Tempo Curves Integration

Tempo curves in Jiffy are defined as piecewise functions. For the sake of brevity, we may refer to an interval and its associated sub-function as a tempo curve segment, or simply as a curve, where the meaning should be clear from context. Each segment is defined by a start tempo and an end tempo, a duration, an interpolation mode and optional interpolation parameters. We implemented three interpolation modes, namely constant, linear and parametric.

Integration of Constant and Linear Tempo Curves

Constant and linear tempo curves can be solved analytically. We show below the differential equation of tempo, and the position and time functions for each case.

Constant Tempo.

T (p) = T 0 .

(2)

T (p) = p T 0 , P (t) = t × T 0 . (3)
Autonomous Linear Tempo.

T (p) = T 0 + αp , where α = T 1 -T 0 L . (4)
P (t) = T 0 α (e αt -1) , and
T (p) = 1 α log(1 + αp T 0 ) . (5)
Non-autonomous Linear Tempo.

T (t) = T 0 + αt , where α = T 1 -T 0 L . (6)
P (t) = T 0 t + α 2 t 2 , and T (p) = T 2 0 + 2αp -T 0 α . (7)
Numerical Considerations Some of the above time and position functions are indeterminate forms for α → 0. To avoid that problem, we approximate these expressions by a series expansions in α when |α| is smaller than a given threshold. For instance, our approximation of the position function for the autonomous case when is |α| < 10 -9 is:

P (t) ≈ T 0 (t + α 2 t 2 + α 2 6 t 3 + α 3 24 t 4 + α 4 120 t 5 ) . ( 8
)
4.2 Parametric tempo curves.

In this section we will give a definition of a parametric tempo curve, and show the differential equations that need to be solved in order to compute the time and position functions. These equations are then solved by a numerical solver.

An autonomous (resp. non-autonomous) parametric tempo curve segment is defined as a function C of the position p (resp. of the time t), which describes the same curve in the plane (p, T ) (resp. (t, T )) as a parametric curve B(s) with components B x (s) and B y (s).

Autonomous Parametric Tempo. The differential equation corresponding to an autonomous tempo curve can be written as

dP dt (t) = C(P (t)) . (9)
Position function P (t). The derivative of the position with respect to time is directly expressed by the autonomous tempo curve, dP dt (t) = B y (s) , where s = B -1 x (P (t)) .

(10)

Time function T (p).

We operate the change of variable s = B -1 x (p) on Equation 9. Finding the time function is then a matter of solving the differential equation

d T ds (s) = B x (s) B y (s) , with T (s) = T (p) . (11)
Non-autonomous Parametric Tempo. The definition of the non-autonomous parametric tempo curves can be written as

dP dt (t) = C(t) . (12)
Position function P (t). Using the change of variable s = B -1 x (t) and the chain rule, we can write the differential equation for the position function as

d P ds (s) = B y (s)B x (s) , with P (s) = P (t) . (13)
Time function T (p). Using the formula for the derivative of inverse functions on Equation 12, we get

dT dp (p) = 1 C(T (p)) = 1 B y (s)
, where s = B -1 x (T (p)) .

Numerical Resolution. Although some of the above equations can be solved analytically, using a numerical solver has the advantage of allowing us to control the tradeoff between accuracy and speed, and opens up the possibility of supporting other arbitrary functions to define tempo curves. We use a Cash-Karp [START_REF] Cash | A Variable Order Runge-Kutta Method for Initial Value Problems with Rapidly Varying Right-Hand Sides[END_REF] solver to numerically solve the tempo curve equations. We follow the general architecture proposed in [START_REF] William | Integration of Ordinary Differential Equations[END_REF], optimized further by leveraging the fact that these equations are either autonomous or directly integrable.

Bézier Tempo Curves. The above formulation allows the use of any parametric curve, provided that it describes a derivable, non null function. Our specific implementation uses cubic Bézier curves, which are especially versatile, as they allow putting constraints on both endpoints and their first derivative, while ensuring that the curve remains contained inside its control points' convex hull. They are also intuitive to manipulate and map well to the curve-editing interfaces commonly used in animation, audio, and video applications.

An autonomous (resp. non-autonomous) Bézier tempo curve segment is defined by the parametric curve

B(s) = C 3 s 3 + C 2 s 2 + C 1 s + C 0 , (15)
where the C i are the power basis coefficients computed from the Bézier curve's control points. To ensure that the curve describes a function, the cubic function B x (s) must be monotonous, i.e. if the x i are the abscissae of the C i , the condition c 2 2 -3c 3 c 1 ≤ 0 must hold.

Bézier curves evaluation. We should stress out that, although each coordinate of the parametric Bézier curve is cubic with respect to its parameter s, the second coordinate is not a cubic function of the first, i.e. the tempo is not a cubic function of position (resp. time). Analytically finding the tempo for a given position (resp. time) indeed requires solving a third order equation.

A faster method is to numerically find the parameter s for a given position (resp. time), up to some desired precision, and then compute the tempo from s. Our implementation first uses the Newton-Raphson root-finding method up to a fixed number of iterations, and falls back to a bisection algorithm if either the value of the derivative falls behind some threshold, or the desired precision is not reached within the maximum iteration count.

An example of a time map produced by tempo curve composed of two Bézier segments is shown in Figure 2. The blue curve shows position as a function of time. The orange stems mark the timeline STUs. The red curve shows the tempo curve, as a function of

Scheduler Interface

The Jiffy scheduler is designed to run user code in fibers 12 . Compared to callbackbased scheduling APIs (such as [START_REF] Bouche | Planning and Scheduling Actions in a Computer-Aided Music Composition System[END_REF], [START_REF] Schnell | Of Time Engines and Masters an API for Scheduling and Synchronizing the Generation and Playback of Event Sequences and Media Streams for the Web Audio API[END_REF] or [START_REF] Roberts | 2013: The Web Browser as Synthesizer and Interface[END_REF]), this doesn't compel the user to break down the control flow of their code into lots of small functions, keeps logically related computations in the same local context, and allows users to easily express dependencies between several workloads. Fibers can also be migrated between threads, allowing a very streamlined way to handle blocking calls without hanging the scheduler.

The scheduler uses the notion of tasks to represent a group of fibers executing within the same timescale. Tasks, like timescales, are organized in a parent-children relationship. The API exposes functions to launch new tasks and fibers, to yield and reschedule the current fiber to a future date, or to wait on the completion of other tasks or fibers. It also features functions to move fibers into background jobs to perform blocking operations without blocking the scheduler, and bringing them back to the foreground once done.

Listing 1.1 shows a simple example that launches a task to print a message at regular symbolic intervals with a varying tempo. This task lives for 40 time units unless it is canceled from another fiber, which waits for user input in the background.

Conclusion and Future Work

In this paper, we highlighted the need for symbolic time scheduling in show-control software and musical applications. We then described a temporal model based on time

The kind of synchronization we considered in this paper was only concerned about relative speeds. However, when dealing with ensemble music, the notion of synchronization is really about the relative phase of each musician. We could refer to this type of synchronization as metric synchronization. Ableton Link [START_REF] Goltz | Ableton Link -A Technology to Synchronize Music Software[END_REF] is one of the tools that tackle this problem, and offers an elegant model to build musical structure on top of beat synchronization. However it has some limitations when it comes to multiple tempos and complex polyrhythms. Addressing these scenarios will be the subject of further research.

Fig. 1 .

 1 Fig. 1. Composing time deformations using tempo curves.

Fig. 2 .

 2 Fig. 2. Time map and beats trace for a tempo curve defined by two Bézier curves.

https://www.avid.com/pro-tools

https://new.steinberg.net/cubase/

https://qlab.app/docs/QLab_4_Reference_Manual.pdf

https://linux-show-player-users.readthedocs.io/en/latest/

We deliberately avoid the term beats here. We think it would bring some confusion by conflating the notion of time unit with the notion of meter, and by suggesting that all beats are of equal conceptual length. This is, in fact, rather a Western exception than a universal norm.

The notions of fiber, coroutine, or green thread are so closely related that the distinction between them, if any, is amenable to debate. One could argue that green thread is more appropriate in the context of a virtual machine or runtime environment, while coroutine originates from programming language design. The term fiber may capture a more general view of the concept.

https://github.com/martinfouilleul/jiffy_scheduler_standalone

transformations expressed through tempo curves, and gave a formalism of such curves. We then described how these curves are implemented in the Jiffy scheduler, and presented the API of the scheduler.

In its current form, the scheduler is a local system, only maintaining proper time flow for its host process. Synchronizing timescales across multiple scheduler instances (potentially running on different machines) is the subject of ongoing work.


J Labidi

J Dottin

Iii

M Clog

C Hemond

P Cartigny

Near-zero 33 S and 36 S anomalies in Pitcairn basalts suggest Proterozoic sediments in the EM-1 mantle plume

Keywords: Sulfur isotopes, Pitcairn, Recycling, Global geodynamics

Volcanic rocks erupted among Pitcairn seamounts sample a mantle plume that exhibits an extreme Enriched Mantle-1 signature. The origin of this peculiar mantle endmember remains contentious, and could involve the recycling of marine sediments of Archean or Proterozoic ages, delaminated units from the lower continental crust, or metasomatized peridotites from a lithospheric mantle.

Here, we report the sulfur multi-isotopic signature ( 32 S, 33 S, 34 S, 36 S) of 15 fresh submarine basaltic glasses from three Pitcairn seamounts. We observe evidence for magmatic degassing of sulfur from melts erupted ~2,000 meters below seawater level (mbsl). Sulfur concentrations are correlated with eruption depth, and range between 1300 ppm S (collected ~ 2,500 mbsl) and 600 ppm S (~2,000 mbsl). The d 34 S values can be accounted for under equilibrium isotope fractionation during degassing, with agas-melt between 1.0020 and 1.0001 and starting d 34 S values between -0.9‰ and +0.6‰. The d 34 S estimates are similar or higher than MORB signatures, suggesting the contribution of recycled sulfur with a ~ 1‰ 34 S enrichment compared to pacific upper mantle. The D 33 S and D 36 S signatures average at +0.024±0.007‰ and +0.02±0.07‰ vs. CDT, respectively (all 1s). Only D 33 S is statistically different from MORBs, by +0.02‰. The D 33 S enrichment is invariant across degassing and sulfide segregation. We suggest it reflects a mantle source enrichment rather than a high-temperature fractionation of S in the basalts. Despite the small magnitude of the 33 S-36 S variations, our data require a substantial amount of recycled sulfur overwhelm the Pitcairn mantle source. We show that models involving metasomatized peridotites, lower crust units, or Archean sediments, may be viable, but are restricted to narrow sets of circumstances. Instead, scenarios involving the contribution of Proterozoic marine sediments appear to be the most parsimonious explanation for the EM-1 signature at Pitcairn.

Introduction

Ocean Island Basalts (OIB) sample a variety of mantle reservoirs, often deep seated in Earth's mantle [START_REF] Weis | Role of the deep mantle in generating the compositional asymmetry of the Hawaiian mantle plume[END_REF]. Erupted basalts at volcanic islands and seamounts originate from the partial melting of upwelled mantle sources, and reveal a complex history of the deep mantle. The isotopic compositions of basalts show that OIB mantle sources incorporate both primordial noble gases [START_REF] Mukhopadhyay | Early differentiation and volatile accretion recorded in deep-mantle neon and xenon[END_REF][START_REF] Mukhopadhyay | Noble gases: a record of Earth's evolution and mantle dynamics[END_REF][START_REF] Parai | The emerging portrait of an ancient, heterogeneous and continuously evolving mantle plume source[END_REF] as well as lithophile elements that are evidently recycled from Earth's surface [START_REF] Weis | Role of the deep mantle in generating the compositional asymmetry of the Hawaiian mantle plume[END_REF][START_REF] Willbold | Trace element composition of mantle end-members: Implications for recycling of oceanic and upper and lower continental crust[END_REF].

Pitcairn is an island in the South-Pacific ocean, with basalts that host moderately primordial neon [START_REF] Honda | A primordial solar-neon enriched component in the source of EM-I-type ocean island basalts from the Pitcairn Seamounts, Polynesia[END_REF] associated with a notoriously radiogenic 87 Sr/ 86 Sr and 208 Pb/ 204 Pb signature, at some of the least radiogenic 206 Pb/ 204 Pb [START_REF] Woodhead | Geochemistry of the Pitcairn seamounts, I: source character and temporal trends[END_REF]. This signature is coined Enriched Mantle 1 (EM-1) and likely results from recycling crustal components into a mantle source [START_REF] Zindler | Chemical geodynamics[END_REF]. The EM-1 signature requires a low time-integrated 238 U/ 204 Pb associated with high time-integrated 87 Rb/ 86 Sr and 234 Th/ 238 U. Various crustal materials have been proposed as the recycled material, including but not limited to a metasomatized lithospheric mantle, oceanic sediments, and components from the lower continental crust [START_REF] Chauvel | HIMU-EM: the French Polynesian connection[END_REF][START_REF] Eisele | The role of sediment recycling in EM-1 inferred from Os, Pb, Hf, Nd, Sr isotope and trace element systematics of the Pitcairn hotspot[END_REF][START_REF] Gibson | Melt-generation processes associated with the Tristan mantle plume: Constraints on the origin of EM-1[END_REF][START_REF] Turner | The importance of mantle wedge heterogeneity to subduction zone magmatism and the origin of EM1[END_REF][START_REF] Willbold | Trace element composition of mantle end-members: Implications for recycling of oceanic and upper and lower continental crust[END_REF]Woodhead et al., 1993).

Sulfur isotope signatures may place constraints on the type of material recycled into the Pitcairn mantle plume. Ion probe measurements on sulfide inclusions in olivine, plagioclase, and the matrix from submarine pillow interiors [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF] show negative 33 S anomalies, down to -0.85±0.25‰. This requires surface-derived sulfur of Archean age is hosted within the Pitcairn mantle source [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF]. The available ion-probe dataset however lacks 36 S measurements. Given that Archean sediments have unique 33 S-36 S relationships compared to Proterozoic rocks [START_REF] Farquhar | Isotopic evidence for Mesoarchaean anoxia and changing atmospheric sulphur chemistry[END_REF][START_REF] Johnston | Multiple sulfur isotopes and the evolution of Earth's surface sulfur cycle[END_REF], determining D 36 S would allow testing relative age difference among various recycled materials in the mantle. In order to better constrain the origin of sulfur in the Pitcairn source, we use gaseous source mass spectrometry in dual inlet mode and document high-precision d 34 S, D 33 S and importantly D 36 S signatures of Pitcairn basalts. We evaluate whether crustal, peridotite and/or sedimentary components contribute to the EM-1 mantle source erupted at Pitcairn.

Samples

The Pitcairn hotspot is located in the south Pacific, on oceanic crust that is approximately 25 million years old. Active submarine volcanoes are located about 80 km east of the island of Pitcairn [START_REF] Hekinian | The Pitcairn hotspot in the South Pacific: distribution and composition of submarine volcanic sequences[END_REF]. The active seamounts offer fresh basaltic glassy material that may host geochemical characteristics of the hotspot. Our Pitcairn samples are fresh glasses from the Polynaut cruise (1999). We analyzed glassy rims of pillow basalts sampled with the Nautile submersible from the flanks of Bounty, Adam and "volcano 5", at various depths between 3,100 and 1,000 meters below seawater level [START_REF] Hekinian | The Pitcairn hotspot in the South Pacific: distribution and composition of submarine volcanic sequences[END_REF].

Methods

Sulfur was extracted for isotopic analysis via wet chemistry [START_REF] Labidi | Determination of multiple sulfur isotopes in glasses: A reappraisal of the MORB δ 34 S[END_REF]. Briefly, centimeter size pieces of glassy basalts were chosen, cleaned and crushed to a grain size < 63 microns. The samples were digested in 20 mL of 2.1 M CrCl2 solution with 5 ml of 12 M HCl and 5 ml of 29 M HF in a Teflon vessel under continuous flushing of pure N2. Reduced sulfur was extracted as H2S, bubbled through a water trap and subsequently into a sulfide trap filled with AgNO3 (0.3 M) where H2S reacted to precipitate Ag2S. Sulfate occurrence was anticipated as sulfate is often observed in OIB [START_REF] Jugo | Sulfur K-edge XANES analysis of natural and synthetic basaltic glasses: Implications for S speciation and S content as function of oxygen fugacity[END_REF]. Here, bulk sulfur concentrations are estimated with an electron microprobe, while reduced sulfur is quantified with our wet chemistry protocol [START_REF] Labidi | Determination of multiple sulfur isotopes in glasses: A reappraisal of the MORB δ 34 S[END_REF]. Any sulfates potentially present in glasses remained in the digestion solution [START_REF] Labidi | Determination of multiple sulfur isotopes in glasses: A reappraisal of the MORB δ 34 S[END_REF]. We infer the S 2-/Stot ratios by mass balance, with a relative uncertainty of ~10%.

The sulfur isotope measurements were performed using a dual inlet MAT 253 gas-source mass spectrometer. Once δ n S are determined (n = 33, 34, 36), ∆ 33 S and ∆ 36 S are calculated (∆ 33 S = δ 33 S -1000((δ 34 S/1000 + 1) 0.515 -1) and ∆ 36 S = δ 36 S -1000((δ 34 S/1000 + 1) 1.90 -1))). Aliquots of Canyon Diablo Troilite (CDT) analyzed relative to the IPGP reference gas give average ∆ 33 S and ∆ 36 S values of -0.024±0.004‰ and -0.13±0. 04‰ (1σ, n=11). Repeated analyses of IAEA-S1 versus the IPGP in-house SF6 tank give ∆ 33 S = +0.082±0.004‰, ∆ 36 S = -0.91±0.11‰ (all 1σ, n=43, δ 34 SS1 is fixed at -0.30‰ and hereafter defines the V-CDT scale). In the following, all δ 34 S are reported against V-CDT and all D 33 S and D 36 S values are anchored on the CDT scale. Ten of the basalts had their S isotope measurement replicated (supplementary table 1). Most duplicates had d 34 S, D 33 S and D 36 S values indistinguishable within ~ 0.1, 0.012, and 0.1‰ respectively (1s), similar to the uncertainty obtained on basalt standards [START_REF] Labidi | Determination of multiple sulfur isotopes in glasses: A reappraisal of the MORB δ 34 S[END_REF]. The average value for replicates is given in Table 1, and raw data can be found in the supplementary online file.

More details on the characterization of the glasses can be found in the supplementary material. Briefly, major element, sulfur, copper and chlorine abundance analyses of the glasses were performed using electron microprobe (EMP) techniques on polished sections using a Cameca SX100 at the CAMPARIS facility (Sorbonne Université, Pierre et Marie Curie Campus) under standard conditions described elsewhere (e.g., [START_REF] Labidi | Determination of multiple sulfur isotopes in glasses: A reappraisal of the MORB δ 34 S[END_REF]. Water concentrations were quantified on a vacuum line as detailed elsewhere [START_REF] Clog | The hydrogen isotopic composition and water content of southern Pacific MORB: A reassessment of the D/H ratio of the depleted mantle reservoir[END_REF][START_REF] Clog | Experimental evidence for interaction of water vapor and platinum crucibles at high temperatures: Implications for volatiles from igneous rocks and minerals[END_REF].

The Sr-Nd-Pb isotope ratios on a selection of Pitcairn glasses were acquired following standard techniques in the Magma-Ocean laboratory, Brest, France, following a published protocol [START_REF] Mougel | A highly unradiogenic lead isotopic signature revealed by volcanic rocks from the East Pacific Rise[END_REF].

Results

Major elements

Samples from Adam and Bounty seamounts have MgO contents between 4.5 and 6.0 wt% except for PN14-04 that has an MgO concentration of 2.6 wt% (Fig. 1). Three samples from Volcano 5 show MgO content of ~2.5 wt%.

Concentrations in FeO are nearly constant at variable MgO (Fig. 1a), reflecting Fe oxides on the liquid line of descent. We also observe near-constant Al2O3 concentrations of ~15 wt% across varying MgO (Table 1, Fig. Supp 1), indicating the suppression of plagioclase crystallization. These two observations are consistent with a calc-alkaline differentiation trend [START_REF] Zimmer | The role of water in generating the calc-alkaline trend: new volatile data for Aleutian magmas and a new tholeiitic index[END_REF]. All samples have high concentrations of incompatible major elements. The K2O content is > 0.8 wt% (up to 2.8 wt%), and Na2O concentrations are > 3.5% for most samples (Table 1). High concentrations of K2O and Na2O are characteristics of alkali basalts and reflect low partial melting fractions, below < 8% [START_REF] Aubaud | Carbon and hydrogen isotope constraints on degassing of CO2 and H2O in submarine lavas from the Pitcairn hotspot (South Pacific)[END_REF][START_REF] Hekinian | The Pitcairn hotspot in the South Pacific: distribution and composition of submarine volcanic sequences[END_REF] as well as crystal fractionation. High K2O/TiO2 ratios of ~ 0.4 are observed for samples with MgO > 4 wt%. (Fig. 1b). High K2O/TiO2 likely represent a mantle source enrichment associated with the Pitcairn hotspot [START_REF] Jackson | Compositions of HIMU, EM1, and EM2 from global trends between radiogenic isotopes and major elements in ocean island basalts[END_REF]. For samples with MgO < 4 wt%, K2O/TiO2 are even higher, likely reflecting the late onset of titanomagnetite precipitation at MgO ~ K2O/TiO2 ratios.

Volatile and supporting minor elements in Pitcairn basalts

We report chlorine contents of < 800 ppm for all Pitcairn lavas (Table 1).

Combined with K2O abundances determined here, Cl/K ratios are < 0.05 for Bounty, <0.04 for Adam, and <0.02 for volcano 5 (Table 1). Water contents range between 5,700 and 12,100 ppm. The average H2O content is 8304±1983 ppm (1s).

The observed H2O concentrations are typical of OIB [START_REF] Dixon | Recycled dehydrated lithosphere observed in plumeinfluenced mid-ocean-ridge basalt[END_REF][START_REF] Workman | Volatile and trace elements in basaltic glasses from Samoa: Implications for water distribution in the mantle[END_REF] and overall higher than MORB [START_REF] Clog | The hydrogen isotopic composition and water content of southern Pacific MORB: A reassessment of the D/H ratio of the depleted mantle reservoir[END_REF][START_REF] Michael | Regionally distinctive sources of depleted MORB: Evidence from trace elements and H2O. Earth Planet[END_REF][START_REF] Sobolev | H2O concentrations in primary melts from supra-subduction zones and mid-ocean ridges: implications for H2O storage and recycling in the mantle[END_REF].

In Bounty lavas, bulk S contents vary between 512±50 and 1416±50 ppm S (Fig. 2). Among Bounty lavas, samples from dive 3 have homogeneous major element compositions, but display most of the S content variability, with values ranging from 738±50 to 1416±50 ppm S (Table 1). The glasses collected from dive 3 also have S contents that are correlated with eruption depth (Fig. 3). At Adam, two glasses show S contents of 799±50 and 762±50 ppm S. At Volcano 5, two glasses show S contents of 573±50 and 602±50 ppm S.

The S 2-/Stot ratios of 12 out of the 15 Pitcairn lavas are within uncertainty of 1, indicating a small amount (if any) of dissolved sulfate. Three other lavas, all from Bounty, picked in the dive 3, show S 2-/Stot ratios of 0.76±0.04, indicating measurable sulfate abundances. No correlations appear between S 2-/Stot ratios and bulk S content, or between S 2-/Stot ratios and eruption depths.

In lavas with MgO > 4 wt%, Cu contents range between 60±10 and 33±10 ppm Cu (Fig. 4). For the more evolved bounty sample (PN14-04) and lavas from volcano 5 (PN08-01 and 08-07), Cu contents are below 10 ppm.

Sulfur isotopes ratios

Sulfur isotope ratios are given here as d 34 S, D 33 S, and D 36 S values for reduced S, which for most samples correspond to bulk sulfur (Fig. 5A). The d 34 S values range between -1.3‰ and +1.6‰, averaging at -0.4±1.0‰ (n=15). For Bounty, [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF].

d
Reduced sulfur displays rather homogenous D 33 S and D 36 S among all basalts (Fig. 5A and Fig. 6). D 33 S and D 36 S averages are +0.023±0.007‰ and +0.02±0.07‰ vs.

CDT, respectively (1 s.d.). Our D 33 S data are inconsistent with the substantially negative D 33 S ion probe estimates on individual sulfide inclusions [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF]. For the dive 3 at Bounty, the D 33 S and D 36 S average values are +0.026±0.004‰ and +0.07±0.04‰ (1 s.d.). The D 36 S values at Pitcairn extend to values that are somewhat higher than MORB estimates of -0.04+±0.08, (1 s.d.) [START_REF] Labidi | Negligible sulfur isotope fractionation during partial melting: Evidence from Garrett transform fault basalts, implications for the late-veneer and the hadean matte[END_REF] but average values remain similar within the 1s level.

The D 33 S average of +0.023±0.007‰ is distinctly higher than MORB average of +0.007±0.006‰ (1 s.d., [START_REF] Labidi | Negligible sulfur isotope fractionation during partial melting: Evidence from Garrett transform fault basalts, implications for the late-veneer and the hadean matte[END_REF] at both 1 and 2s confidence level. We note that the sample PN14-04, from Bounty, has a MORB-like D 33 S value.

Should it be excluded, the D 33 S average at Pitcairn would become 0.024±0.006‰. Simple t-tests on both D 33 S and D 36 S values would suggest a significant difference between Pitcairn and MORB may exist, but would not take into account analytical error. We use a Monte-Carlo approach to evaluate whether the differences in the averages are statistically significant and consider analytical uncertainties. We generate simulated datasets for the MORB and Pitcairn samples, based on sampling a normal distribution for each sample with an average equal to the measured value and a standard deviation equal to the corresponding analytical uncertainty. We then run a Welch's t-test and repeat the process 10,000 times. Histograms and kernel density plots are shown in Fig. 7. For 36 S, Pitcairn melts show an average value significantly distinct (p < 0.05) from MORBs only in 40% of the simulations. Thus, we suggest MORB and Pitcairn D 36 S are not significantly different. In contrast, the means of the D 33 S values are significantly distinct (p<0.05) in 99.9% of the simulation runs, confirming the D 33 S enrichment at Pitcairn is statistically significant.

Radiogenic isotopes

We analyzed the Sr-Pb-Nd isotopic composition with a focus on the samples from the dive 03, from the Bounty seamount, which has the least differentiated basalts. For all samples but PN14-04, the 87 Sr/ 86 Sr, 143 Nd/ 144 Nd and 206 Pb/ 204 Pb ratios are homogeneous, with values ranging between 0.704466 and 0.704624, 0.512537 and 0.512944, and 17.74 and 17.92, respectively (Fig. Supp 3). We observe relatively high 208 Pb/ 204 Pb and 207 Pb/ 204 Pb ratios at a given 206 Pb/ 204 Pb.

The radiogenic isotope data are in agreement with previous work performed on the same glasses [START_REF] Bourdon | Melting of enriched mantle beneath Pitcairn seamounts: Unusual U-Th-Ra systematics provide insights into melt extraction processes[END_REF], or on different samples from Pitcairn (Woodhead et al., 1993[START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF]. For almost all samples, the data is consistent with an extreme EM-1 endmember. Only PN14-04, from Bounty, has a 206 Pb/ 204 Pb higher than 18.0 and a 87 Sr/ 86 Sr lower than 0.7041 (see below). This illustrates a slightly distinct mantle source, somewhat less affected by the EM-1 signature.

Discussion

The lavas studied here underwent a complex history prior to eruption. This involves mantle source enrichment with an EM-1 component, low partial melting, magmatic differentiation following a calc-alkaline trend, and various amounts of volatile losses via degassing. Below, we constrain the magmatic behavior of sulfur in our samples and identify the respective contributions of sulfide fractionation and sulfur degassing. Ultimately, we estimate the sulfur isotope composition of the mantle source for d 34 S, D 33 S, and D 36 S signatures.

Limited crustal assimilation

Hydrothermal brine assimilation may be sporadically associated with contamination by hydrothermal sulfides [START_REF] Labidi | Sulfur isotope budget ( 32 S, 33 S, 34 S and 36 S) in Pacific-Antarctic ridge basalts: A record of mantle source heterogeneity and hydrothermal sulfide assimilation[END_REF]. Cl and K have a similar incompatible behavior during mantle melting and basalt differentiation [START_REF] Michael | Influence of spreading rate and magma supply on crystallization and assimilation beneath mid-ocean ridges: Evidence from chlorine and major element chemistry of midocean ridge basalts[END_REF][START_REF] Michael | Chlorine in mid-ocean ridge magmas: evidence for assimilation of seawater-influenced components[END_REF]. However, Cl is enriched in the hydrated oceanic crust and in brines. Thus, elevated Cl/K ratios in basalts are treated as evidence for crustal assimilation. Here, the Cl/K are < 0.05 for Bounty, <0.04 for Adam, and <0.02 for volcano 5 (Table 1). The Cl/K for the three localities are below the threshold of 0.1, above which lavas are argued to undergo contamination by Cl-rich brines [START_REF] Michael | Influence of spreading rate and magma supply on crystallization and assimilation beneath mid-ocean ridges: Evidence from chlorine and major element chemistry of midocean ridge basalts[END_REF]. We conclude our samples unlikely assimilated hydrothermal components prior to eruption, and thus we assume their S isotope composition has remained unaffected by potentially contaminant sulfides from the crust.

Sulfide saturation in Pitcairn melts

Pitcairn lavas have MgO < 6 wt% (Fig. 1), indicative of substantial crystal fractionation, with near-constant FeO concentrations of ~9.5% across the range in MgO (Fig. 1). In tholeiitic basalts, FeO concentration is the prime factor controlling sulfur content at sulfide saturation [START_REF] Mathez | Sulfur solubility and magmatic sulfides in submarine basalt glass[END_REF]. If this relationship is at all maintained in calc-alkaline melts, one would anticipate invariable S concentrations at a particular value defined by the sulfur solubility at sulfide saturation. These are, however, not observed: we observe variation of S concentrations over almost a factor of 3 at a given FeO (or MgO) content (Fig. 2).

Sulfide fractionation could explain the dramatic variation in sulfur concentration in Pitcairn melts (Fig. 2). Although challenging to quantify [START_REF] Jenner | The magnetite crisis in the evolution of arcrelated magmas and the initial concentration of Au[END_REF][START_REF] Yierpan | Selenium isotope and S-Se-Te elemental systematics along the Pacific-Antarctic ridge: Role of mantle processes[END_REF], sulfide fractionation may be singled out via the copper systematics : the partition coefficient of Cu is ~10 -1 and ~10 3 for crystal/melt and sulfide/melt, respectively [START_REF] Lee | Copper systematics in arc magmas and implications for crust-mantle differentiation[END_REF]. Thus, in the absence of sulfides on the liquidus, copper behaves as a moderately incompatible element and increases with decreasing MgO, as observed in arc-magmas and other OIB.

Once sulfide saturation is met, Cu abundances drop as copper is readily partitioned into sulfides [START_REF] Lee | Copper systematics in arc magmas and implications for crust-mantle differentiation[END_REF], Labidi et al., 2015). This behavior of Cu is not observed at Pitcairn. Here, Cu concentrations appear coincidentally similar to MORBs, with a near-constant Cu concentration across most of the MgO range (Fig. 4). The data appear more consistent with a liquid line of descent that is typical of MORB, where a roughly constant rate of sulfide fractionation occurs during magmatic differentiation range [START_REF] Labidi | Sulfur isotope budget ( 32 S, 33 S, 34 S and 36 S) in Pacific-Antarctic ridge basalts: A record of mantle source heterogeneity and hydrothermal sulfide assimilation[END_REF]. We thus argue that while sulfides existed on the liquidus of Pitcairn melts, dramatic sulfide fractionation is not the cause of varying S concentration at a given FeO, observed on Fig. 2.

Magmatic degassing

Sulfur degassing could account for the variable sulfur concentrations observed on Fig. 2. We focus on samples from the dive 3 from Bounty, because these lavas are comparable in terms of major elements and radiogenic isotope compositions (Table 1), so their sulfur concentrations prior to eruption can be assumed at zero-order to be homogeneous. Basalts from the dive 3 however were erupted at various depths on the flank of the volcano [START_REF] Hekinian | The Pitcairn hotspot in the South Pacific: distribution and composition of submarine volcanic sequences[END_REF].

Dissolved sulfide (S 2-) abundances range from 1300±50 ppm to 550±50 ppm and are correlated with the depth of eruption: the shallowest samples have the lowest S concentrations (Fig. 3). We interpret this relationship as evidence for sulfur degassing upon eruption. We quantify the residual sulfur fraction f as the ratio of the observed S concentration to the highest S concentration observed here, ~1300 ppm S. For S concentrations between 1300±50 ppm and 550±50 ppm S, we infer f values are between 1 and 0.45. Note that the starting S concentration could be even higher than ~1300 ppm, but this does not fundamentally change our results below.

Sulfur degassing is consistent with water behavior in our melts. Across decreasing depths, the H2O abundances decrease from ~9500±500 ppm to 5500±500 ppm water (Fig 3). The H2O/K2O range between 0.69 and 0.35 (Table 1), decreasing by a factor of ~2 across the depth rise (Fig 3). Water and potassium oxide have comparable incompatible behaviors [START_REF] Danyushevsky | The effect of small amounts of H2O on crystallisation of mid-ocean ridge and backarc basin magmas[END_REF][START_REF] Michael | Regionally distinctive sources of depleted MORB: Evidence from trace elements and H2O. Earth Planet[END_REF] and may not be substantially fractionated from each other by crystal fractionation. Instead, observed decreasing H2O/K2O at increasingly shallow eruption depths is compatible with H2O degassing. Importantly, sulfur concentrations are correlated with H2O/K2O ratios (Supp Fig. 4). The simplest explanation involves the concomitant degassing of water and the reduced fraction of sulfur from the melts during eruption. We note that the concomitant degassing of reduced S and water during eruption is not unique to Bounty. A similar degassing trend featuring sulfur and water was observed in Hawaii basalts [START_REF] Brounce | Redox variations in Mauna Kea lavas, the oxygen fugacity of the Hawaiian plume, and the role of volcanic gases in Earth's oxygenation[END_REF][START_REF] Dixon | Degassing history of water, sulfur, and carbon in submarine lavas from Kilauea Volcano, Hawaii[END_REF].

Interpreting 34 S/ 32 S ratios in Pitcairn glasses

The d 34 S values of reduced S from all Pitcairn basalts analyzed here range between -1.3‰ to +1.6‰. At Bounty, values are between -1.3‰ and +0.7‰.

Bounty basalts from the dive 3 have homogeneous Cu concentrations at a given MgO content, suggesting they underwent comparable amounts of immiscible sulfide losses (Table 1). Isotope fractionation during sulfide segregation is therefore discarded to explain the variable d 34 S values seen in basalts from the dive 3. In figure 8, we plot d 34 S values against f values calculated for samples from the dive 3. d 34 S appear increasingly negative at lower f and may be interpreted in terms of isotopic fractionation associated with degassing under open-system distillation (Fig. 8A). One possibility involves two starting endmember compositions, with two steep trends. To cause the decrease in d 34 S in the residual melt, the vapor phase must preferentially partition heavy 34 S isotopes, under a bulk isotope fractionation of 2‰ (agas-melt = 1.002). This happens to be indistinguishable from predicted SO2-S 2-isotopic fractionation at equilibrium at 1,200 °C (Mandeville et al. 2009). This average melt-gas isotope fractionation would suggest SO2 is the dominant S-bearing species during equilibrium degassing. A similar suggestion was made for basalts from the Canary Islands (Beaudry et al., 2018). It is unclear whether equilibrium SO2 degassing could also offer a possible explanation for the low d 34 S observed in sulfides from Pitcairn pillows [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF] as they do not report S concentrations of the basalts they analyzed. The eruption of depths of the Bounty basalts studied by [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF] is ~1,000 meters, substantially shallower than our samples, consistent with a pattern of more pronounced SO2 degassing.

A caveat in our degassing interpretation comes from observation of variable d 34 S at a given f, on Fig. 8. This inevitably requires multiple starting compositions. The scenario of equilibrium SO2 degassing requires two starting compositions of ~-0.7 and ~+0.2‰ (Fig. 8) for d 34 S. The data may alternatively be interpreted as degassing under much smaller isotope fractionations. Most basalts have d 34 S compositions that can be explained as the result of S degassing under a near-zero gas-melt isotope fractionation of -0.1‰ (agas-melt = 1.0001), with a starting d 34 S of -0.9‰, indistinguishable from the Pacific MORB mantle [START_REF] Labidi | Sulfur isotope budget ( 32 S, 33 S, 34 S and 36 S) in Pacific-Antarctic ridge basalts: A record of mantle source heterogeneity and hydrothermal sulfide assimilation[END_REF], as illustrated on Fig. 8B. The degassing fractionation is consistent with S 2-H2S isotopic fractionation at equilibrium (Beaudry et al., 2018, Mandeville et al. 2009). However, when three glasses with high d 34 S at a given f are included, this degassing scenario suggests no less than three initial starting compositions. In summary, the data appear to inevitably argue in favor of multiple starting d 34 S compositions for the Bounty melts, spread over ~1‰ above the MORB estimates and our isotope data alone may not unambiguously disambiguate H2S and SO2 degassing scenarios.

The recycled component at Pitcairn mantle sources

The parental melts at Pitcairn show a d 34 S range, restricted between -1‰ and +0.5‰ (Fig. 8). Our d 34 S estimate points to values above the MORB range, for basalts with EM-1 signatures. In consideration of the vulnerability of our degassing scenarios (section 5.4), and the uncertainty associated with estimated starting compositions, we are unable to interpret all δ 34 S data as a straightforward representation of source composition. In the following, we focus on D 36 S-D 33 S data which remain unmodified by degassing, as attested by a lack of correlation between on D 36 S-D 33 S and f values (Table 1). Our D 36 S and D 33 S are homogeneous (with one outlier with a MORB-like D 33 S signature, PN14-04), but the average D 33 S is statistically distinct from MORB (Fig. 5,7). Pitcairn samples show a small positive D 33 S anomaly of ~0.02‰. We attribute the small-magnitude D 33 S enrichment to subducted sulfur, associated with the recycled component responsible for the EM-1 Pitcairn signatures. Our D 33 S values, duplicated for most basalts (supplementary table 1), are not consistent with large negative D 33 S anomalies reported elsewhere via ion probe measurements on individual sulfides from Bounty basalts [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF]. In the following, we provide interpretations for our dataset and briefly discuss if our data may be reconciled with suggestions made in [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF].

We plot our D 36 S-D 33 S data on Fig. 9, together with a compilation of sediments from the literature [START_REF] Johnston | Multiple sulfur isotopes and the evolution of Earth's surface sulfur cycle[END_REF]. The plot shows the difference between Archean components, with D 36 S/D 33 S ~ -1, and post-Archean rocks, with D 36 S/D 33 S ~ -7 [START_REF] Farquhar | Isotopic evidence for Mesoarchaean anoxia and changing atmospheric sulphur chemistry[END_REF]. Post-Archean rocks show a much smaller D 33 S range than Archean sediments. We posit that the recycled endmember at Pitcairn falls within the existing fields defined by literature data on surface rocks shown on Fig. 9. Our approach relies on the assumption that no systematic D 33 S-D 36 S isotope fractionation may be associated with metamorphism and slab devolatilization (Dottin III et al., 2020a). We examine various mixing scenarios. In all, we discuss the prospect of two-component mixing between a mantle peridotite with a recycled component. A mass balance is considered successful if can fit the observed D 33 S-D 36 S compositions at Pitcairn with a reasonable set of restrictions.

The Pitcairn D 33 S enrichment is associated with a near-invariant, MORBlike D 36 S signature (Fig. 567). For recycled sulfur to be of Proterozoic age and plot within the field defined by published sedimentary data (Fig. 9), the D 33 S signature of the Pitcairn endmember is allowed to be between ~0.03 and ~ 0.10‰ with a D 36 S between ~ -0.4 and ~ 0.4‰ ('plausible Pitcairn endmember' on Fig. 9A). This is well within the D 33 S range in the Proterozoic of ~0.20‰ (Fig. 9). This is also consistent with the frequent near-zero D 36 S at a varying D 33 S observed in the geological record, especially when sedimentary sulfates are considered (Fig. 9A).

Archean signatures are far more variable in D 33 S, such that a 0.020‰ 33 S enrichment at Pitcairn is minuscule compared to most Archean anomalies (Fig. 9B). The Pitcairn endmember could be a diluted Archean signature, which would happen to mimic the magnitude of Proterozoic sediment. In this case, our observation of an invariant D 36 S requires the Archean signature to have a D 36 S between ~ -0.4 and ~ 0.4‰. Positive D 33 S at near-zero D 36 S exist in the Archean, if D 33 S remains in a restricted range between 0.1 and ~1.0‰ ('plausible Pitcairn endmember' Fig. 9B). This exercise illustrates that although the magnitude of Proterozoic sediments appears more consistent with the Pitcairn 33 S data than Archean rocks, both Archean and Proterozoic components could theoretically account for the signatures at Pitcairn. However, Archean signatures have implications that may be harder to reconcile with the geological record. This is detailed below in the context of the various geochemical models that account for the nature of EM-1.

The metasomatism hypotheses for EM-1 at Pitcairn

Delamination of lithospheric mantle has been suggested to yield EM-1 signatures. This is an umbrella concept, involving either sub-oceanic or subcontinental depleted peridotites. The peridotites would develop EM signatures after experiencing metasomatism by either low-degree silicate melts or by slabderived fluids [START_REF] Hart | A possible new Sr-Nd-Pb mantle array and consequences for mantle mixing[END_REF][START_REF] Mckenzie | Mantle reservoirs and ocean island basalts[END_REF]. One suggestion is that EM-1 originates from the Sub-Continental Lithospheric Mantle (SCLM), after it experienced metasomatism by melts from the asthenosphere [START_REF] Hoernle | Origin of Indian Ocean Seamount Province by shallow recycling of continental lithosphere[END_REF][START_REF] Mckenzie | The source regions of ocean island basalts[END_REF]. This mechanism was found to generate EM-1 components in convergent margins, where it is clear that no contributions occurred from subducted components [START_REF] Turner | The importance of mantle wedge heterogeneity to subduction zone magmatism and the origin of EM1[END_REF]. In light of this hypothesis, we anticipate a scenario in which low-degree melts from the asthenosphere would deliver sulfur with MORB-like D 33 S of ~0.007±0.006‰ [START_REF] Labidi | Negligible sulfur isotope fractionation during partial melting: Evidence from Garrett transform fault basalts, implications for the late-veneer and the hadean matte[END_REF], in conflict with the D 33 S excess in the Pitcairn basalts.

The SCLM may alternatively experience metasomatism by subduction fluids. [START_REF] Gibson | Melt-generation processes associated with the Tristan mantle plume: Constraints on the origin of EM-1[END_REF] and Meyzen et al., (2005) reviewed evidence for peridotites metasomatized by subduction fluids, and showed they are dominated by EM-2 signatures, but EM-1 compositions are also observed. This metasomatized component could for instance contribute to the EM-1 Tristan mantle source at about ~10% [START_REF] Gibson | Melt-generation processes associated with the Tristan mantle plume: Constraints on the origin of EM-1[END_REF]. A similar scenario was suggested for Pitcairn, on account of the absence of oxygen isotope variations in basalts associated with extreme EM-1 compositions (Eiler et al., 1995(Eiler et al., , 1997)). This idea has the advantage of offering a mechanism to deliver surface-derived S (thus showing variable 33 S signatures) inherited from either sediments or the altered oceanic crust, to a lithospheric unit. We consider the sulfur isotope consequences of such a twocomponent mixing, assuming a contribution of metasomatized peridotites of ~ 10% [START_REF] Gibson | Melt-generation processes associated with the Tristan mantle plume: Constraints on the origin of EM-1[END_REF] mixed with 90% of a deep mantle source showing a MORB-like D 33 S (Dottin III et al., 2020b) and 200 ppm S concentration [START_REF] Lorand | Chalcophile and siderophile elements in mantle rocks: Trace elements controlled by trace minerals[END_REF][START_REF] Nielsen | Chalcophile behavior of thallium during MORB melting and implications for the sulfur content of the mantle[END_REF]. The metasomatized peridotites are assigned the entirety of the D 33 S anomaly, and S concentrations between 50 and 400 ppm S, as observed in natural samples [START_REF] Lorand | Chalcophile and siderophile elements in mantle rocks: Trace elements controlled by trace minerals[END_REF]. These endmember concentrations reflect how much a SCLM experiences sulfur depletion by partial melting prior to being metasomatized [START_REF] Lorand | Chalcophile and siderophile elements in mantle rocks: Trace elements controlled by trace minerals[END_REF]. There are no constraints on which endmember scenario may be more relevant for Pitcairn, and therefore we explore the two possibilities with no a priori estimate on how they may remain viable for other trace elements. Taking 400 ppm for [S], the 10%-90% two-component mixing requires the SCLM to show an average D 33 S of ~0.09‰ to account for the ~0.02‰ enrichment at Pitcairn.

Taking 50 ppm S, the required D 33 S value becomes ~0.55‰. This illustrates that again, both Archean and post-Archean sulfur could explain the data in the SCLM scenario. Scenarios requiring melt depletion followed by fluid-driven enrichments are not supported by trace element signatures at Pitcairn. Notably, systematic enrichments or depletions of fluid mobile elements are not observed in the Pitcairn source (e.g. Nb anomalies, [START_REF] Willbold | Trace element composition of mantle end-members: Implications for recycling of oceanic and upper and lower continental crust[END_REF]. The integrated dataset therefore does not argue in favor of a scenario involving a subductionimprinted SCLM in the Pitcairn source.

The effect of incorporating Lower Continent Crust

Delamination of the lower continental crust (LCC) could be a viable alternative to account for EM-1 signatures in plumes [START_REF] Willbold | Trace element composition of mantle end-members: Implications for recycling of oceanic and upper and lower continental crust[END_REF]. Meyzen et al., (2005) showed granulites from the lower continental crust develop signatures far closer to EM-1 than samples from the SCLM. Only ~2% of LCC is necessary to contribute to any mantle source to generate the Pitcairn EM-1 signature [START_REF] Willbold | Trace element composition of mantle end-members: Implications for recycling of oceanic and upper and lower continental crust[END_REF]. In addition, the somewhat variable trace element signature of various LCC units offers an explanation for the irregular trace element patterns among EM-1 hotspots [START_REF] Willbold | Trace element composition of mantle end-members: Implications for recycling of oceanic and upper and lower continental crust[END_REF].

We consider the sulfur isotope consequences of a two-component mixing scenario where the LCC contributes 2% of the bulk mass of the Pitcairn source [START_REF] Willbold | Trace element composition of mantle end-members: Implications for recycling of oceanic and upper and lower continental crust[END_REF]. Assuming a concentration of 350 ppm S in the LCC [START_REF] Rudnick | Composition of the continental crust[END_REF], mass balance requires the average LCC to show a D 33 S of ~0.7‰ to account for the ~0.020‰ 33 S excess observed in the Pitcairn source. This would put the LCC unit in the Archean domain (Fig. 9B).

The LCC reservoir is severely under-constrained, with a 33 S isotope composition remaining virtually unknown. Mafic granulites essentially carry igneous sulfur, with a MORB-like d 34 S signature [START_REF] Wedepohl | The composition of the continental crust[END_REF], consistent with the common idea that they result from crustal underplating by asthenosphere melts [START_REF] Rudnick | Composition of the continental crust[END_REF]. If so, the lower crust would have MORB-like D 33 S-D 36 S signatures. This would render the LCC unable to account for the 33 S isotope anomaly that characterize the Pitcairn source. Felsic granulites also exist in the lower crust. They apparently show non-mantle d 34 S signatures, illustrated by a modest 34 S enrichment of ~3‰ [START_REF] Wedepohl | The composition of the continental crust[END_REF]. A complex model where delaminated felsic granulites would happen to have a D 33 S of ~0.7‰, inherited from Archean rocks, could be invoked to account for the data at Pitcairn. However, it would be inconsistent with the few available data on Archean crustal melts, with near-zero D 33 S values [START_REF] Bucholz | Sulfur isotope behavior during metamorphism and anatexis of Archean sedimentary rocks: A case study from the Ghost Lake batholith, Ontario, Canada[END_REF]. We cannot exclude that at an unconstrained location and time, an analogous process in a hypothetical LCC resulted in felsic components with a high D 33 S average value of exactly 0.7‰, but this scenario appears restrictive and speculative.

A sedimentary origin: Archean exhalates in Pitcairn?

SIMS measurements on sulfides from other Pitcairn lavas from Bounty [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF] yielded D 33 S values down to ~-0.85±0.25‰. The substantial 33 S anomaly was attributed to the contribution of Archean exhalates, an uncommon type of Archean chemical sediments that so far were only observed in the Abitibi greenstone belt [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF]. Our high-precision D 33 S-D 36 S data do not reproduce the large negative D 33 S anomalies observed by [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF]. Instead, the data establish the Pitcairn source to have positive 33 S anomalies, at odds with Archean exhalates. Furthermore, although exhalates may be enriched in trace elements, their rare earth patterns do not match any reconstructed slab component contributing to the Pitcairn mantle source [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF]. Thus, ad-hoc modification during subduction (e.g., fluid losses causing trace element fractionation) was invoked to account for the EM-1 trace element data [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF]. Again however, the Pitcairn trace element compositions lack the typical fluid-loss signatures like Nb anomalies [START_REF] Willbold | Trace element composition of mantle end-members: Implications for recycling of oceanic and upper and lower continental crust[END_REF]. Up to about ~20% of a component resulting from the processing of the exhalates was suggested in the Pitcairn mantle source [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF]. It is unclear whether this may be reconcilable with a mantle source suggested to be almost exclusively dominated by peridotites [START_REF] Jackson | Compositions of HIMU, EM1, and EM2 from global trends between radiogenic isotopes and major elements in ocean island basalts[END_REF], perhaps with a minor pyroxenitic component (Nebel et al., 2019). It also appears to conflict with the absence of O isotope anomalies in Pitcairn basalts, that would likely be observed for any sediment contribution higher than 2% (Eiler et al., 1995). As it stands, although the S isotope compositions measured by Delavault et al. suggest a contribution from Archean exhalates, the majority of geochemical data published on Pitcairn basalts suggests otherwise.

The return of conventional marine sediments

Invoking the subduction of marine sediments has been a classic scenario in chemical geodynamics [START_REF] Hofmann | Mantle plumes from ancient oceanic crust[END_REF]. As for the SCLM or the LCC scenarios, sediment recycling is a viable hypothesis that requires a specific mixing ratio. Here, we assume a 2% contribution of sediment into Pitcairn source [START_REF] Eisele | The role of sediment recycling in EM-1 inferred from Os, Pb, Hf, Nd, Sr isotope and trace element systematics of the Pitcairn hotspot[END_REF]. Note that variations between 1 and 4%, in the range of reasonable predictions for EM signatures [START_REF] Jackson | Compositions of HIMU, EM1, and EM2 from global trends between radiogenic isotopes and major elements in ocean island basalts[END_REF], would yield comparable results.

From our data we can establish a first-order mass balance scenario from the standpoint of S concentrations. Taking 2% contribution of sediment into a 98% peridotite source, and S concentrations in the sediments ranging between 2,500 and 10,000 ppm S, sediments with D 33 S of +0.10‰ and +0.025‰ would result in mixtures with D 33 S ~0.02‰. This would match our 33 S observations in basalts.

Should the sediments also show a D 36 S between ~ -0.4 and ~ 0.4‰ as illustrated in Fig. 9A, the 36 S data in basalts would also be accounted for. The concentrations between 2,500 and 10,000 ppm, required by the mass balance, are in the wide range shown by Proterozoic sediments showing typically between 100 and 30,000 ppm S [START_REF] Poulton | The transition to a sulphidic ocean∼ 1.84 billion years ago[END_REF]. This simple mass balance calculation shows that Proterozoic sediments with moderate S concentrations may contribute to the Pitcairn source for the S isotope basalt data to be accounted for. Concluding on whether or not the required S concentrations are significantly low compared to non-subducted sediments, perhaps because of sulfur losses during subduction, appears unwarranted. However, we note that the mass balance exercise does not require having to generate any hypotheses on the sulfur behavior in ancient subduction zones. Considering how contentious the notion of sulfur losses in subduction is, even for well-studied subduction zones in the Phanerozoic (Li et al., 2020, Walters et al., 2020), the lack of strict restrictions of our mass balance with regard to sulfur behavior in subduction is noteworthy.

In a similar exercise for Archean sediments, we find the maximum plausible D 33 S of ~1‰ for recycled S, as required by the invariant D 36 S and the field of Archean signatures (Fig. 9B). For a D 33 S between +0.2‰ and +1‰, the S concentration of the recycled component is between ~1000 and ~200 ppm, respectively. These concentrations exist in the Archean record, but remain uncommon: >99% of Archean sediments have S concentrations between 500 and 20,000 ppm S [START_REF] Reinhard | Long-term sedimentary recycling of rare sulphur isotope anomalies[END_REF]. The simple mass balance shows that recycled Archean sediments would be required to have low S concentrations to account for the data in Pitcairn basalts, perhaps indicating a role for sulfur losses during subduction. We cannot rule out this scenario, but we note that it adds a level of restriction compared to the mass balance involving Proterozoic sediments.

To summarize, 2% of marine sediments with variable S concentrations, recycled in the Pitcairn mantle source, could easily explain the S isotope signature of Pitcairn basalts. We find that requiring a Proterozoic age for recycled sediments allows a rather large range of plausible S concentrations, and a tight range of S isotope composition within the post-Archean isotope field (Fig. 9A). The mass balance exercise allows for (but does not require) higher sulfur contents prior to subduction, should sulfur losses be substantial during slab devolatilization.

Assuming an Archean age requires the small 33 S enrichment at Pitcairn results from the dilution of an Archean component in the Pitcairn mantle source, resulting in an average 33 S enrichment resembling in magnitude what is observed in Proterozoic sediments. For a dilution of a putative Archean signature to be achieved, low S concentration in the recycled sediment is required. In other words, this model requires a tight range of S isotope compositions within the known Archean field (Fig. 9B), but also demands a narrow range of sulfur concentrations.

Although Archean sediments are not ruled out, the scenario of Proterozoic sediment appears the least restrictive, and therefore the most parsimonious hypothesis to account for the Pitcairn EM-1 signature.

Implications on other geochemical signatures

The absence of resolvable 18 O isotopes variations in basalts (Eiler et al., 1995(Eiler et al., , 1997) ) was suggested to argue against sediment recycling in the Pitcairn source.

However, the cut-off sensitivity threshold of O isotopes is at ~2% sediment contribution (Eiler et al., 1995). It is conceivable that ≤ 2% sediment may not be detected, even with precise 18 O measurements. Another critique of the sediment model came from [START_REF] Willbold | Trace element composition of mantle end-members: Implications for recycling of oceanic and upper and lower continental crust[END_REF]. Various EM-1 hotspots have irregular trace element patterns, notably in terms of Ba and Rb enrichments [START_REF] Willbold | Trace element composition of mantle end-members: Implications for recycling of oceanic and upper and lower continental crust[END_REF]; Pitcairn is peculiar in that it shows the lowest Ba enrichment of all EM-1 signatures, with Ba/La ratio of ~9 only, much lower than what is seen at Tristan (Ba/La ~17, [START_REF] Willbold | Trace element composition of mantle end-members: Implications for recycling of oceanic and upper and lower continental crust[END_REF]. This variability across various EM-1 hotspots is a problem for any unifying model aimed at lumping together all EM-1 compositions as a family of mantle signatures. This was considered evidence against sediment recycling [START_REF] Willbold | Trace element composition of mantle end-members: Implications for recycling of oceanic and upper and lower continental crust[END_REF], since sediments are thought to have unimodal Ba enrichments [START_REF] Plank | The chemical composition of subducting sediment and its consequences for the crust and mantle[END_REF]. The unimodal distribution however does not erase the substantial sedimentary variability for Ba enrichments. In fact, the Ba/La ratios of various sedimentary piles varies greatly, between 8 (MAR 800) and 171 (Central America) [START_REF] Plank | The chemical composition of subducting sediment and its consequences for the crust and mantle[END_REF]. Any sediments with a Ba/La < 50, largely in the range of modern sediments [START_REF] Plank | The chemical composition of subducting sediment and its consequences for the crust and mantle[END_REF], would explain the low Ba/La ratio at Pitcairn. Worldwide EM-1 components with variable Ba/La ratios may simply reflect the natural variability of sedimentary protoliths, as illustrated by modern sediments.

The lead isotope systematics can provide constraints on time-integrated elemental ratios, as well as model ages of recycled components [START_REF] Chauvel | HIMU-EM: the French Polynesian connection[END_REF]. [START_REF] Delavault | Sulfur and lead isotopic evidence of relic Archean sediments in the Pitcairn mantle plume[END_REF] Ga, consistent with a post-Archean D 33 S signature.

Conclusions

We report on 15 submarine alkaline basalts at Pitcairn of EM-1 type. Low copper concentrations suggest all the samples were sulfide saturated upon eruption. The basalts show evidence for sulfur and water degassing, likely causing d 34 S fractionation. Accurately estimating the d 34 S value of undegassed melts is challenging but they are likely between -1 and +0.7‰, higher than the pacific upper mantle estimates of -0.9±0.1‰. The samples have no resolvable 36 S variation relative to MORBs. In contrast, reduced sulfur displays homogenous D 33 S, averaging at +0.024±0.006‰ vs. CDT. This is higher than the D 33 S MORB average of +0.007±0.006‰, by ~0.015-0.020‰. We attribute the small 33 S anomaly to the recycling of surface-derived sulfur in the mantle source of Pitcairn. We show that metasomatized peridotites or components from the lower continental crust, appear inconsistent with the S isotope data, or would require narrow sets of unlikely circumstances. Archean sediments with large positive 33 S anomalies could explain the data, but also present tight requirements. The contribution of ~2% Proterozoic sediments in the Pitcairn mantle source represents the simplest explanation, with the lowest amounts of restrictions. [START_REF] Labidi | Sulfur isotope budget ( 32 S, 33 S, 34 S and 36 S) in Pacific-Antarctic ridge basalts: A record of mantle source heterogeneity and hydrothermal sulfide assimilation[END_REF][START_REF] Labidi | Non-chondritic sulphur isotope composition of the terrestrial mantle[END_REF]. On panel A, Pitcairn basalts are shown to have a higher maximum in S solubility at a given FeO, relative to MORBs. The highest concentration of reduced sulfur observed here, at sulfide saturation, is ~1300 ± 50 ppm S at a FeO ~ 9.8 wt%.

This concentration is an excess of ~ 200 ppm S relative to sulfur concentrations at sulfide saturation in tholeiitic melts [START_REF] Mathez | Sulfur solubility and magmatic sulfides in submarine basalt glass[END_REF]. Commonly, such high concentrations would likely be attributed to the solubility of bulk sulfur which is enhanced by sulfate occurrence [START_REF] Jugo | Sulfur K-edge XANES analysis of natural and synthetic basaltic glasses: Implications for S speciation and S content as function of oxygen fugacity[END_REF].

Here, the high S concentration is observed in melts that appear to show limited sulfate abundances (Table 1). It is not clear whether the high S concentration illustrates the combined effects of minute sulfate abundances, magmatic temperatures and alkaline chemistry on the sulfide solubility in alkaline melts [START_REF] Fortin | The effect of water on the sulfur concentration at sulfide saturation (SCSS) in natural melts[END_REF]. Independently of the reason for relatively high S concentrations in some of the Pitcairn melts, the near-vertical S concentrations at a given FeO (panel A) and MgO only, for which all samples have broadly homogeneous major element concentrations. For those samples that were carefully picked at a given location on the flank of Bounty [START_REF] Hekinian | The Pitcairn hotspot in the South Pacific: distribution and composition of submarine volcanic sequences[END_REF], we assume the S concentration prior to degassing to be comparable. In contrast, we anticipate the variable major element compositions of other samples from Bounty (dive 14), Adam and Volc. 5 to cause variable pre-eruptive S concentration owing to the intricate link between sulfur solubility in silicate melts and basalt chemistry. At a given major element concentration but variable eruption depths for dive 03 basalts, the positive correlations are consistent with magmatic degassing upon eruption. Only PN03-10 is an outlier. This is the only sample from this seamount with a K2O/TiO2 ratio of ~0.25, where all others have K2O/TiO2 ~0.45 (Fig. 1). We attribute the outlier to a distinct mantle source composition in terms of volatile elements. [START_REF] Labidi | Sulfur isotope budget ( 32 S, 33 S, 34 S and 36 S) in Pacific-Antarctic ridge basalts: A record of mantle source heterogeneity and hydrothermal sulfide assimilation[END_REF][START_REF] Labidi | Non-chondritic sulphur isotope composition of the terrestrial mantle[END_REF][START_REF] Labidi | Determination of multiple sulfur isotopes in glasses: A reappraisal of the MORB δ 34 S[END_REF][START_REF] Labidi | Negligible sulfur isotope fractionation during partial melting: Evidence from Garrett transform fault basalts, implications for the late-veneer and the hadean matte[END_REF]. [START_REF] Labidi | Sulfur isotope budget ( 32 S, 33 S, 34 S and 36 S) in Pacific-Antarctic ridge basalts: A record of mantle source heterogeneity and hydrothermal sulfide assimilation[END_REF][START_REF] Labidi | Non-chondritic sulphur isotope composition of the terrestrial mantle[END_REF][START_REF] Labidi | Determination of multiple sulfur isotopes in glasses: A reappraisal of the MORB δ 34 S[END_REF][START_REF] Labidi | Negligible sulfur isotope fractionation during partial melting: Evidence from Garrett transform fault basalts, implications for the late-veneer and the hadean matte[END_REF]. Pitcairn basalts show no clearly discernable 36 S anomaly relative to MORBs. the data are fit with a gas-melt fractionation of 2‰, consistent with S 2--SO2 equilibrium isotope fractionation. Two d 34 S source compositions appear to be required to fit the data. They are both higher than the pacific upper mantle value of -0.9±0.1‰ [START_REF] Labidi | Sulfur isotope budget ( 32 S, 33 S, 34 S and 36 S) in Pacific-Antarctic ridge basalts: A record of mantle source heterogeneity and hydrothermal sulfide assimilation[END_REF]. Alternatively, as shown on panel B, the data may be fit with a gas-melt fractionation of 0.1‰, consistent with S 2--H2S equilibrium isotope fractionation. In this case, three d 34 S source compositions would be required. Most basalts would be consistent with the d 34 S estimate of the pacific upper mantle value.

Figure 1 -

 1 Figure 1 -A: iron and magnesium oxides concentrations for various basalts. B: ratio of potassium to titanium oxides, shown against magnesium oxide concentrations, for various basalts. MORB data are compiled using the PetDB database (www.earthchem.org/petdb, Lehnert et al., 2000). On panel A, Pitcairn basalts are shown to differ from MORBs in that they undergo magmatic differentiation on a calc-alkaline trend, maintaining near-constant FeO concentrations at variable MgO contents. On panel B, Pitcairn basalts are shown to have rather high K2O/TiO2 ratios, illustrating a mantle source enrichment. The onset of titanomagnetite precipitation at MgO ~4% likely explains the extremely elevated K2O/TiO2 ratios for the most evolved basalts. A hypothetical liquid line of descent is illustrated as the dashed line.

Figure 2 -

 2 Figure 2 -reduced sulfur concentrations of MORB and Pitcairn basalts shown against A: iron and B: magnesium oxides concentrations. The error bar is MORB data acquired with the same techniques are the literature[START_REF] Labidi | Sulfur isotope budget ( 32 S, 33 S, 34 S and 36 S) in Pacific-Antarctic ridge basalts: A record of mantle source heterogeneity and hydrothermal sulfide assimilation[END_REF][START_REF] Labidi | Non-chondritic sulphur isotope composition of the terrestrial mantle[END_REF]. On panel A, Pitcairn basalts are shown to

(Figure 3 -

 3 Figure 3 -reduced sulfur concentrations and H2O/K2O ratios of Pitcairn basalts from the dive 03

Figure 4 -

 4 Figure 4 -copper concentrations of MORBs and Pitcairn basalts shown against MgO contents. The Cu concentrations happen to be consistent with a MORB fractionation trend. This indicates sulfide fractionation is comparable in most samples form Bounty and Adams samples.

Figure 5 -

 5 Figure 5 -d 34 S and D 33 S of Pitcairn basalts compared to published MORB data[START_REF] Labidi | Sulfur isotope budget ( 32 S, 33 S, 34 S and 36 S) in Pacific-Antarctic ridge basalts: A record of mantle source heterogeneity and hydrothermal sulfide assimilation[END_REF]

  Panel A and B show the same geochemical space, but panel B incorporates the dataset from Delavault et al. (2016) obtained on individual sulfides from other basalts from the Bounty seamount. On Panel A, our data for Pitcairn basalts are shown to have an average D 33 S of +0.024±0.007‰, higher than the MORB average value by ~0.02‰. the error bars of MORB measurements are shown on the bottom left. On Panel B, the data of Delavault et al., (2016) are plotted next to ours. The ion probe data from Delavault et al., (2016) show substantially negative D 33 S values, down to -0.85±0.25‰, which are not duplicated by our measurements.

Figure 6 -

 6 Figure 6 -d 34 S and D 36 S of of Pitcairn basalts compared to published MORB data(Labidi et al.,

Fig. 7 -Figure 8

 78 Fig. 7 -data distribution for 33 S and 36 S signatures of MORB and Pitcairn samples. Panels A and C show data histograms and B and D show Kernel density plots (KDE), with a bandwidth equal to the analytical uncertainty. Only for 33 S is the mean value of the Pitcairn samples statistically distinct from MORB.

Figure 9 -

 9 Figure9-Sulfur isotope ratios of Pitcairn basalts compared to MORB (data from[START_REF] Labidi | Determination of multiple sulfur isotopes in glasses: A reappraisal of the MORB δ 34 S[END_REF], 2013, 2014[START_REF] Labidi | Negligible sulfur isotope fractionation during partial melting: Evidence from Garrett transform fault basalts, implications for the late-veneer and the hadean matte[END_REF], Archean sediments and post-Archean sediments (data compilation from[START_REF] Johnston | Multiple sulfur isotopes and the evolution of Earth's surface sulfur cycle[END_REF].

  34 S values vary between -1.3‰ and +0.7‰, with an average of -0.7±0.7‰. The 2 samples from Adam have d 34 S values of -1.2‰ and -1.1‰. For volcano 5, the two glasses both have d 34 S values of +1.6‰. As shown on figure 5B, our d 34 S values do not duplicate the largely negative d 34 S values on individual sulfide inclusions from Bounty pillow lavas obtained by ion probe measurements

  suggested taking µ values ( 238 U/ 204 Pb) of ~6 and κ values ( 232 Th/ 238 U) of ~7 could yield a model age for the recycled sediment between 2.4 and 2.8 Ga. Here, we suggest relaxing the constraint of an Archean D 33 S signature. Considering that most sediment columns in modern trenches have µ between ~1 and 13, and k values between ~1 and 9 (Plank and Langmuir, 1998), we suggest that any combination of k > 5 and µ < 10 yields the EM-1 206 Pb/ 204 Pb, 207 Pb/ 204 Pb and 208 Pb/ 204 Pb signatures at Pitcairn with model ages lower than 2

Table 1 -

 1 Major element, radiogenic isotopes, volatile concentration and sulfur isotope data of 15 submarine glasses from Pitcairn.

	sample ID	pn02-01	pn02-05	pn03-08	pn03-10	pn03-01	pn03-02	pn03-03	pn03-04	pn03-05	pn03-07	pn03-11	pn14-04	pn14-07	pn08-01	pn08-07
	seamount	adam	adam	bounty	bounty	bounty	bounty	bounty	bounty	bounty	bounty	bounty	bounty	bounty	volc. 5	volc. 5
	depth (mbsl)	1595	1001	2250	2165	2508	2508	2500	2480	2408	2276	1957	1705	1513	3103
	Na2O	3.7	3.7	4.2	3.5	4.2	4.2	4.2	3.6	3.8	3.9	4.0	4.2	3.5	4.2	4.2
	SiO2	48.4	48.3	48.8	50.6	47.8	47.9	47.7	49.0	47.8	47.6	48.9	54.2	50.3	54.3	54.4
	K2O	1.5	1.4	1.6	0.8	1.4	1.4	1.5	1.3	1.4	1.4	1.6	2.3	0.9	2.8	2.7
	Al2O3	15.5	15.5	16.2	15.1	16.0	16.0	15.8	15.8	15.4	15.3	15.3	15.3	15.1	16.2	16.1
	CaO	9.0	8.9	8.1	8.6	9.1	9.1	9.1	9.4	9.8	9.7	8.7	5.4	8.6	4.9	4.9
	MgO	5.3	6.0	5.0	5.6	5.4	5.5	4.8	5.0	5.2	5.4	4.6	2.6	5.5	2.4	2.6
	P2O5	0.8	0.8	0.8	0.5	0.7	0.7	0.8	0.7	0.7	0.7	0.8	1.2	0.5	1.3	1.2
	FeO	9.8	10.0	9.7	9.6	9.7	9.8	9.7	10.3	9.9	10.0	9.7	9.5	9.7	8.7	9.1
	MnO	0.1	0.1	0.2	0.1	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.1	0.2	0.2
	TiO2	3.5	3.3	3.3	3.1	2.9	2.9	3.0	3.1	3.1	3.1	3.4	2.2	3.1	1.9	1.9
	total	97.6	97.9	97.9	97.6	97.5	97.6	96.8	98.3	97.2	97.1	97.2	97.0	97.3	97.0	97.2
	Cl (ppm)	463	465	459	324	502	494	498	471	426	424	417	750	276	495	493
	Cu (ppm)	50	60	33	37	59	50	52	37	50	56	50	11	34	4	3
	H2O (ppm)			9099	5696	9192	9807	9662	7195	8512	8125	5667	12123	6266
	S 2-(ppm)	815	861	738	1017	1279	1299	1133	893	953	875	569	507	970	608	602
	K2O/TiO2	0.43	0.43	0.49	0.27	0.49	0.48	0.49	0.40	0.44	0.45	0.47	1.06	0.28	1.48	1.40
	CaO/Al2O3	0.58	0.57	0.50	0.57	0.57	0.57	0.58	0.60	0.63	0.63	0.57	0.36	0.57	0.31	0.30
	Cl/K	0.04	0.04	0.03	0.05	0.04	0.04	0.04	0.04	0.04	0.04	0.03	0.04	0.04	0.02	0.02
	S 2-/Stot	102	113	100	95	97	98	80	77	95	91	72	99	95	106	100
	d 34 S (‰)	-1.2	-1.1	-1.0	-1.2	0.1	0.2	-1.0	-0.3	-1.2	-1.3	-1.2	0.7	-1.2	1.6	1.6
	D 33 S (‰)	0.017	0.017	0.027	0.030	0.020	0.028	0.025	0.021	0.028	0.012	0.031	0.007	0.028	0.027	0.026
	D 36 S (‰)	-0.08	-0.09	0.06	0.06	0.08	0.08	0.05	-0.01	0.13	0.08	0.06	-0.06	-0.11	0.01	0.02
	206 Pb/ 204 Pb			17.747	17.926	17.742	17.742	17.749	17.791	17.816	17.809	17.770	18.1982335
	207 Pb/ 204 Pb			15.475	15.493	15.470	15.470	15.474	15.476	15.482	15.479	15.479	15.5167915
	208 Pb/ 204 Pb			38.644	38.798	38.569	38.570	38.579	38.592	38.645	38.658	38.668	38.9161373
	143 Nd/ 144 Nd			0.512537				0.512567	0.512573	0.512599	0.512583	0.512944	0.512762
	87 Sr/ 86 Sr			0.704624	0.704481	0.704483	0.704477	0.704485	0.704466	0.704482	0.704498	0.704531	0.704002
	87 Sr/ 86 Sr ª			0.70463	0.70447					0.70446						0.70530
	a: data from bourdon et al., 2009

Acknowledgments

JL thanks James Farquhar and Isabelle Genot for comments on an early version of the manuscript. PC acknowledges partial support from the Dear-Sir ANR-15-CE31-005 project of the Agence Nationale de la Recherche. JD acknowledges NSF Postdoctoral Fellowship Award # 2052944. We thank Oliver Nebel and an anonymous reviewer for helpful comments. Rosemary Hickey-Vargas is thanked for editorial handling.

Captions