Jean-Baptiste Joinet

COLLUSIONS AND QUOTIENTS: GENERALIZING EQUIVALENCE RELATIONS AND DEFINITIONS BY ABSTRACTION

§1. Introduction. During the XIXth century, the increasing diffusion in mathematical practice of what the Peano school coined "definitions by abstraction" 1 (i.e. Definitions which "allow us to talk about certain entities by defining what their equality means even if we are not in a position to provide an explicit definition of the terms flanking the equality", as P. Mancosu says) is attested through various instances (see [1]). Among numerous examples, one may cite cardinal numbers abstracted from bijectability between sets, a.k.a. Hume's principle 2 ; directions abstracted from parallelism relation between lines; shapes abstracted from topological homologies; von Helmholtz's weights, brightness, pitch of tones etc.

1.1. Historical overview on the theory of definitions by abstraction. The Peano's School (Peano, Burali-Forti, Vailati, Padoa, Pieri and others) seems to have provided the first systematic focus on the theory of such definitions (within a programme targeting a typology of mathematical definitions), including their characterization as statements of the form: f R (x) = f R (y) ⇔ xRy where f R is a newly introduced operator and xRy is a binary predicate satisfying the properties of what has been progressively called equivalence relations. Their debates were partly definitional (determination of the relevant properties of the relation R, discussions about the possible codomains for f R ), epistemological (discussions about the status for such statements as definitions, compared in particular to nominal, stipulatory definitions and to implicit, axiomatic definitions) and ontological (how to relate the status for the abstracta to the status of the objects with respect to which the later are induced). In sensibly the same period, notably from his Grundlagen der Arithmetik (see Research supported by the french-brazilian Capes-Cofecub Action Sh-873 17, Philosophy and Computing Theory: Contemporary Interactions. The results of this paper have been presented for the first time during the second occurrence of Jornadas de Mirantão organized within Action Sh-873 17 by Luiz Carlos Pereira in August 2017, in Brazil (but theorems 20 and 21 presented during the third occurrence of the Jornadas, in August 2018). The author would like to thank Thomas Seiller, Paulo Veloso, Marco Panza and Paolo Pistone for their availability for fruitful discussions.

1 Following P. Mancosu, the term definitions by abstraction appeared in print in 1894, but it was used earlier by members of the Peano school at least from 1888. 2 In §63 of his Grundlagen, Frege cites Hume as an ancestor of this idea (meanwhile developed by Cantor). The terminology Hume's principle comes from [START_REF] Boolos | The consistency of Freges Foundations, On Being and Saying: Essays in Honor of Richard Cartwright[END_REF]. 1 [START_REF]Platonism in Russell's Early Philosophy and the Principle of Abstraction, Bertrand Russell. A Collection of Critical Essays[END_REF], section 6), Frege also investigated this kind of "definitions", focusing specifically on higher order abstraction, as tools for his logicist programme. Early, some problematic epistemological aspects of "definitions by abstraction" as definitions have been pointed out: statements of the form presented above do not guarantee unicity nor existence for the definiendum f R , and they are generally not eliminable. From 1902, invoking those defects, Russell pleads for abandoning the terminology "definitions by abstraction" when describing statements of the form presented above 3 . He instead qualifies them as "Abstraction principles", bringing out the role of the introduced operators f R as classifiers mapping any element of the domain of R to its equivalence class, and advocating for limiting abstraction principles to that interpretation 4 . The treatment proposed by Russell in those years is centered on the nowadays standard remark that an equivalence relation R over a set a induces a classification of all the elements a into non empty disjoint subsets of a (or, as we would say nowadays, that the set of equivalence classes induced by R forms a partition of a) 5 .

Even if Russell's treatment reducing abstracta to equivalence classes clarified the epistemological status of definitions by abstraction as well as the status of abstracta henceforth set theoretically conceived (to such an extent that, later on, Russell himself will concede that Abstraction principles are Occam's razors which "dispense with abstraction", see [START_REF] Francisco | The Mathematical Philosophy of Bertrand Russell: Origins and Development[END_REF], p. 326), this did not lead the debate at its end. Let us mention a first line, developed in particular by P. Lorenzen (see [START_REF][END_REF], [START_REF] Mancosu | Abstraction and Infinity[END_REF] and [START_REF] Russell | Logical Atomism, Logic and knowledge[END_REF]), pointing out that the subtraction which accompanies abstraction (entities "disappear" through the equalization induced by the quotientation) propagates to predicates initially definable over the vanishing individuals: some of those predicates cease afterwards to be distinguishable, in other terms they also are equalized, and it is with respect to the "surviving" ones that the abstraction need to be justified. In particular, inasmuch equivalence classes are grasped through particular representatives, the means required to define those ones may be submitted to constructivity demands. A second line, in particular in J. Vuillemin, stresses that, as the definiens of equivalence classes includes definite descriptions (abstraction binders) not belonging to the original primitive language, their eliminability has in any case to be justified (cf. J. Vuillemin [START_REF] Frege | Eine logisch mathematische Untersuchung ber den Begriff der Zahl[END_REF] and [START_REF]The Foundations of Arithmetic[END_REF]), especially as equivalence classes cannot always be defined by separation (indeed, in bad cases, they are not sets, but proper classes defined by comprehension, as in the case of the bijectability relation).

A third line, has been extensively studied by the so called neo-fregean trend initiated by C. Wright in [START_REF]The Principles of Mathematics[END_REF], properly giving a second life to the investigations on Abstraction principles (see [START_REF] Van Der Waerden | Abstrakte Algebra[END_REF], [START_REF] Vuillemin | L'élimination des définitions par abstraction chez Frege[END_REF]). Those works focus on the fact that higher order principles (typically when the domain of the operator f R is the powerset of the co-domain of f R ) may induce an injection from the domain to the codomain, hence contradicting Cantor's theorem. The canonical example of this problem is Frege's Basic Law V, Ext(x) = Ext(y) ⇔ ∀z (z ∈ x ↔ z ∈ y) (the second abstraction principle introduced in the Grungesetze with fatal consequences, beside to Hume's principle), which assigns objects to concepts and is based on the equi-membership relation. If the domain of concepts is the power-set of the firstorder codomain, then, as the operator Ext (for Extension) is then an injection of concepts into objects, this enters in contradiction with Cantor's theorem 6 .

1.2. Aim and structure of the article. The aim of our paper is to introduce and investigate a generalization of definitions by abstraction 7 . In the § 1, we start introducing a generalization of equivalence relations -collusion relations -able to characterize the partitioning discipline. Indeed, to be an equivalence relation is a sufficient but not necessary condition with that respect (the fact that one can go from whatever equivalence relation to the partition it induces, and then back from that partition to the original equivalence relation, sometimes misleads into believing that the properties defining equivalence relations form a condition not only sufficient but also necessary for partitioning -which is not the case). The § 2 is devoted to show that the quotients induced by collusions (collusional quotients) may be recovered as standard equi-quotients induced by equivalence relations defined in terms of the original collusions. The § 3 focuses on the less familiar collusions, namely the non equivalence ones, and investigates in particular whether any partition may be induced by a non equivalence collusion. The § 4, concentrates on how to extend to collusions the theory of abstraction proposed by the Peano's school for equivalence relations. This requires a double shift: a mathematical one and a philosophical one. Mathematically, a wider version of "Abstraction principles" has to be formulated: indeed, under penalty of inconsistency, introducing a classifier f R such that f R (x) = f R (y) ⇔ xRy requires that R is an equivalence relation, so that one cannot just replace equivalence by collusion in the formulation of "Abstraction Principles". Philosophically, the "sameness interpretation" of equivalence relations (a commonplace from Peano's school to Frege-Russell and further) has to be complementarily replaced by another interpretation relavant for general (non equivalence) collusions. The point is not so much to renounce to the conception that a definition by abstraction captures some form of similarity (by abstracting, subtracting differences) and favours "a change of scale" (the classes are higher order notions), than to abandon the idea that the relation R itself expresses that sameness. To replace the "sameness interpretation", we will progressively present, along the various sections, an atypic 6 A short and good introduction to those problematics can be found at www.bbk.ac.uk/philosophy/our-research/ppp/ConferenceAntonelli.pdf 7 In particular, we will leave for further works an evaluation of the impact of that generalization of "Abstraction principles" on the debates of the so called neo-fregean trend initiated by [START_REF]The Principles of Mathematics[END_REF], which gave a second life to the investigations on Abstraction principles (see [START_REF] Van Der Waerden | Abstrakte Algebra[END_REF], [START_REF] Vuillemin | L'élimination des définitions par abstraction chez Frege[END_REF]).

interpretation of quotientation and classes in terms of "coalitions": the agonal interpretation. §2. Collusions and equivalences: partitioning.

Notations, terminology, conventions. By default, R will always denote a binary relation on a non empty set a. When one has xRy and x Ry (for some x, x , y ∈ a), one says that x and x do converge (on y). Whenever x ∈ a, the subset of a, noted [x] R and defined by

[x] R = def {y ∈ a ; xRy} is called the class of x (for R). But, in the special case R is an equivalence relation, [x] R will be sometimes called the equivalence class of x for R, as usual. If b ⊆ a is such that b = [x] R , for some x ∈ a, one says that b is a class (for R).
Let us recall that a set P of subsets of a forms a partition of a, whenever the three following conditions are satisfied:

• for any b, c ∈ P,

(b = c ⇒ b ∩ c = ∅) (pairwise disjointness) • for any b ∈ P , b = ∅ (non emptiness)
• a ⊆ P (exhaustivity). Reformulated with the notations and terminology just introduced, the standard observation recalled in our introduction and upon which Peano's, Frege's and Russell's approach of abstraction is based, is thus that in case R is an equivalence relation, the set [x] R x∈a of classes form a partition of a.

We will now see that though sufficient to get a partition, this condition is however not necessary. We now will see (see the three lemmas below) that collusivity, surjectivity and totality (which, by the way, are completely independent properties) correspond exactly and respectively to the three conditions which define partitions: pairwise disjointness, exhaustivity, non emptiness. Consequently, the conjunction of those three properties (which defines collusions) is a sufficient and necessary condition to produce a partition of the set on which the relation is defined.

Theorem 2. [x] R x∈a is a partition of a ⇔ R is a collusion
Proof. By the lemmas 3, 4 and 5 which follow.

Lemma 3. ∅ / ∈ [x] R x∈a ⇔ R is total Proof. ⇒ If R is not total, then some x ∈ a has no image for R. But then [x] R = ∅. ⇐ Let b ∈ [x] R x∈a . Then b = [x] R for some x ∈ a. If R is total, there exists y ∈ a, such that xRy. Thus y ∈ [x] R , hence [x] R = ∅ Lemma 4. a ⊆ x∈a [x] R ⇔ R is surjective Proof. ⇒ If R is not surjective, then some y ∈ a has no antecedent for R. But then y belongs to no b in [x] R x∈a . So again [x] R x∈a is not a partition of a. ⇐ Let y ∈ a. As R is surjective, xRy for some x ∈ a, so that y ∈ [x] R , i.e. y
belongs to at least one class belonging to a/R.

Lemma 5. ∀b, b ∈ [x] R x∈a (b ∩ b = ∅ ⇒ b = b ) ⇔ R is collusive
Proof. ⇒ If R is not collusive, there exists x, x , y, z ∈ a such that: 1.

x and x do converge on y, which implies that

[x] R ∩ [x ] R = ∅ 2. xRz but x Rz, which implies that [x] R = [x ] R ⇐ Let b, b ∈ [x] R x∈a such that b ∩ b = ∅. Let y ∈ b ∩ b . By definition of [x] R x∈a , we know that b = [x] R for some x ∈ a and that b = [x ] R for some x ∈ a. As y ∈ [x] R ,
one has xRy and as y ∈ [x ] R , one has x Ry. So that x and x do converge on y. Now, R being collusive, we know that x and x do converge on the same elements, so that

[x] R = [x ] R , i.e. b = b .

Terminology and notation

Because of theorem 2, in case R is a collusion, we will continue to use the standard terminology (quotient) and notation (a/R) for calling and noting [x] R x∈a . However, in situations where we wish to underline that the collusion generating such or such quotient under consideration is an equivalence relation, we will call it an equi-quotient. §3. Collusional quotients as equi-quotients. As they introduce an equality, "Abstraction principles" underpin the traditional philosophical interpretation of equivalence relations in terms of sameness: following such a reading, the fact that two elements are in such a relation suggests their similarity from a point of view expressed by R, a similarity which becomes salient when one forms the quotient, i.e. once one forgets all other points of view with respect to which those elements elsewhere differ. Clearly, as some collusions are not equivalence (and in particular not reflexive), such a reading cannot apply anymore ; and to substitute to the sameness interpretation an adequate interpretation would be welcome. A rather intuitive and actually fruitful candidate is what we call the agonal interpretation (from agôn i.e. fight in ancient greek), an interpretation which will be clearer once the results ending the current section will be presented. To give meanwhile a hint, let us stress that the word "collusion" has been chosen in the line of that interpretation. If one reads xRy as x targets y (or as y is targeted by x as well) or maybe, using a gorier but mnemonically efficient declination of the same idea, as x is a predator for y (or y is a prey for x as well), the collusivity property corresponds to situations where any two individuals having a common target did collude: from then on, all the targets of one of them are targets of the other one. As we will see, which such an interpretation in mind, the status of "abstracta" induced by collusions, i.e. the classes belonging to the quotients induced by them, receive a clear meaning: they are coalitions of individuals having the same targets.

To start with, let us prove the following crucial property of collusions. Lemma 6. Collusions are closed by converse: R collusion ⇒ R -1 collusion.

Proof.

• R collusive ⇒ R -1 collusive: Let x, x ∈ a converging on some y ∈ a for R -1 . Let now z ∈ a such that xR -1 z. As xR -1 y and xR -1 z, one has yRx and zRx, so that y and z do converge (on x). So, if R is collusive, y and z have exactly the same images; so, knowing that yRx , one must also have zRx . Hence x R -1 z. Hence R -1 is collusive. • R total and surjective ⇒ R -1 surjective and total:

As the switching from R to R -1 inverts the role of images and antecedents, it maps surjectivity on totality and conversely.

Collusions being closed by converse, any collusion R over a actually induces two partitions: a/R and a/R -1 . They will be called the (two) collusional quotients generated by R.

We will now see how the two collusional quotients generated by a collusion R can be recovered as equi-quotients generated from two equivalence relations defined in terms of R. ∼ R on a are defined as follows:

• x tr ∼ R x ⇔ def ∀y ∈ a (xRy ⇔ x Ry)
When x tr ∼ R x , one says that x, x are equi-targeters in R (or, when stressing the agonal interpretation, that they are coalized for attack );

• x td ∼ R x ⇔ def ∀y ∈ a (yRx ⇔ yRx ) When x td ∼ R
x , one says that x, x are equi-targeted in R (or that they are coalized for defence).

∼ R x ⇔ [x] R = [x ] R and x td ∼ R x ⇔ [x] R -1 = [x ] R -1 . Lemma 7. td ∼ R = tr ∼ R -1 and tr ∼ R = td ∼ R -1 .
Proof. For the first statement, the converse operation on binary relations switches the first notion to the second one. The second one then follows from the first, as the converse operation is involutive.

Lemma 8. Let R be a collusive relation on a. If xRy, then:

1. [x] R = [y] td ∼ R 2. [y] R -1 = [x] tr ∼ R Proof. Let us assume that xRy. 1. [x] R = [y] td ∼ R : • Let y ∈ [x] R .
This entails that xRy . As we have also xRy, this means that y and y do converge on x for R -1 . Now, if R is collusive, so is R -1 (by lemma 6). The fact that y and y do converge thus implies that y tr ∼ R -1 y or, equivalently, that

y td ∼ R y . So that y ∈ [y] td ∼ R . We thus proved that [x] R ⊆ [y] td ∼ R . • Let y ∈ [y] td ∼ R
. One thus has y td ∼ y and, as xRy, one must have xRy , so that y ∈ [x] R . We thus proved that [y] td ∼ R ⊆ [x] R (an inclusion by the way valid even when R is not a collusion).

[y]

R -1 = [x] tr ∼ R
As xRy, one has yR -1 x. As R is collusive, so is R -1 (by lemma 6), so that item 1. applies, which means

[y] R -1 = [x] td ∼ R -1 . So [y] R -1 = [x] tr ∼ R Proposition 9. Let R be a relation over a. {[x] R } x∈a = a/ td ∼ R ⇔ R is a collusion Proof. ⇒ As td ∼ R is an equivalence, a/ td ∼ R is a partition of a. So, if a/ td ∼ R = {[x] R } x∈a , then {[x] R } x∈a is also a partition of a. By theorem 2, this implies that R is a collusion. ⇐ -a/ td ∼ R ⊆ a/R. Let b ∈ a/ td ∼ R . By definition of a/ td ∼ R , b = [y] td ∼ R
for some y ∈ a. If R is a collusion, it is surjective and there exists x ∈ a such that xRy. Then by lemma 8, one has

[y] td ∼ R = [x] R . So b ∈ a/R. -a/R ⊆ a/ td ∼ R . Let b ∈ a/R. By definition of a/R, one has b = [x] R for some x ∈ a. If R is a collusion, it
is total and one has xRy for some y ∈ a. Then by lemma 8, one has

[x] R = [y] td ∼ R . So b ∈ a/ td ∼ R .
Note that the "⇐" direction of this proposition gives us a second proof for the "⇐" direction of our theorem 2, which states that {[x] R } x∈a is a partition of a, inasmuch the relation R on a is a collusion.

Proposition 10. Let R be a relation over a.

{[x] R -1 } x∈a = a/ tr ∼ R ⇔ R is a collusion Proof. R is a collusion ⇔ R -1 is a collusion (by lemma 6) ⇔ {[x] R -1 } x∈a = a/ td ∼ R -1 (by proposition 9). ⇔ {[x] R -1 } x∈a = a/ tr ∼ R (by lemma 7). Theorem 11. {[x] R } x∈a = a/ td ∼ R ⇔ R is a collusion ⇔ {[x] R -1 } x∈a = a/ tr ∼ R
Proof. By propositions 9 and 10.

To make this result more salient, we could rephrase it by abusively using the generalized quotientation notation (a priori potentially inappropriate for the first ⇒ and the second ⇐, as the notation a/R is well defined only for collusionseven if it appears a posteriori non problematic):

Theorem 11. (reformulated ): a/R = a/ td ∼ R ⇔ R is a collusion ⇔ a/R -1 = a/ tr ∼ R
To conclude this section, let us go back to the agonal interpretation and introduce a bit of terminology: when a subset b of a is made of individuals pairwise colluded for Defence (resp. for Attack), we will say that b is a D-coalition (resp. an Acoalition). In the same manner an equivalence class is made of pairwise equivalent elements and is maximal for that property w.r.t. inclusion (a remark whose justification is not given here, for sake of brevity), the status to the classes of the collusional quotients can be described through the agonal interpretation in simple terms: they are maximal coalitions. §4. Focusing on non equivalence collusions. Among collusions, the non reflexive ones (which thus are not equivalence relations) are likely unfamiliar to most readers, and this section is devoted to make some observations on them. We will in particular focus on irreflexive collusions8 , i.e. the ones such that ∀x ∈ a x Rx.

If R is an irreflexive collusion, the picture for classes belonging to the induced collusional quotient is orthogonal to the one for classes induced by equivalence relations. Indeed, as shows the second lemma below, in the irreflexive case, no one is related in R with someone inside a class (in agonal terms: the coalitions are peaceful -nobody targets a member of its own coalition). Proof. One gets the contraposed implication by particularizing to x the universally quantified x in lemma 12. In the subsections below, we will now examine some peculiarities of non reflexive collusions with respect to the partitioning discipline. It is well known that for any partition P of a, there exists an equivalence relation R on a and a unique one, such that P = a/R. As we will see now, the picture is very different (for existence and for unicity as well) when one considers non reflexive collusions, instead of equivalence relations: (a) some (actually rare and special) partitions cease to be "reachable", namely the degenerated ones, i.e. the ones with a unique class (actually, these ones are the only non reachable ones); (b) all other partitions are reachable, but a reachable partition appears to be reachable from different collusions. Proof. Let us assume that a/R = {a}. As a = ∅ (by a convention set in section 2), there is some x ∈ a. And as R is total, there exists x ∈ a such that xRx . Applying the lemma 12 to (the class) a, we deduce that ∀x, x ∈ a xRx . In particular, ∀x ∈ a xRx. So, when they are non reflexive, and a fortiori irreflexive, collusions are so to speak "incomplete w.r.t. partitions": they cannot reach the degenerated one 9 . We will now show that the degenerated partition is however the only missed one.

Irreflexive collusions are complete for non degenerated partitions.

In this subsection, whenever P is a partition of a and x ∈ a, we will note [x] P (by a slight notational abuse) the unique b ∈ P such that x ∈ b. Definition 4.1. Let P a partition of a and σ an application from P to P . The binary relation R σ over a is defined by:

xR σ x ⇔ def σ([x] P ) = [x ] P Lemma 15.
Let P a partition of a and σ a permutation of a (i.e. a bijection from P to P ). Then R σ is a collusion on a. Moreover, if σ has no fix-points (i.e. is such that for any b ∈ P , σ(b) = b), the collusion R σ is irreflexive.

Proof.

• R σ is total and surjective. Indeed, let x ∈ a. As σ is total (resp. surjective), there exists b ∈ P such that σ( Theorem 20. Let P a non degenerated partition of a, i.e. with P = {a}.

There exists an irreflexive collusion such that the collusional quotient it induces coincides with P10 .

Proof. Because of proposition 19, we just need to find a permutation σ of P without fix points. While P is finite or denumerable, any "successor like" relation with no endpoints (see below) would do the job. However, this being not suitable for the non denumerable case, we treat the infinite case through another strategy.

• in case P is finite, to get a permutation of P without fix points, we choose for σ any circular permutation with a unique cycle including all the elements of a, in other terms such that any element of P has any element of P in its orbit (note that the no fix points condition is satisfiable if and only if the cardinal of partition P is strictly greater than 1, i.e. when the partition is non degenerated, as assumed); • in case P is denumerable, σ could be the binary relation (on P ) which is the image of the successor relation on Z through any bijection from Z to P . • in case P is not denumerable (but that treatment applies also for a denumerable P ), let us consider any P 1 , P 2 ⊆ P such that {P 1 , P 2 } is a partition of P and (P 1 ) = (P 2 ). The existence of such a partition of P is for instance proved in [START_REF] Wright | Freges Conception of Numbers as Objects[END_REF] (p. 184 and 287-288; Axiom of Choice used). As (P 1 ) = (P 2 ), there exists a bijection β from P 1 to P 2 . We then define the binary relation σ on P as β ∪ β -1 . As {P 1 , P 2 } is a partition of P , it is immediate, by construction, that σ is a permutation of P with no fix points.

Let us close this subsection by stressing an intentional refinement introduced by non reflexive collusions.

Remark 4. Extensionally, there exists a unique equivalence relation generating a given partition P (i.e., P being given, there is a unique reflexive collusion, i.e. a unique equivalence relation, R on a such that P = a/R). Such an "extensional collapse" does not happen when one also considers non reflexive collusions. Then, indeed, a given partition P of a may generally well be generated through different collusions on a (see the relations R 1 and R 2 over a = {α, β, γ, ρ} pictured below as sagittal diagrams, where the partition induced by R i is noted under

each diagram). ' & $ % R1 α G G β Ð Ð γ y y G G ρ y y |αρ|β|γ| ' & $ % R2 α G G β Ð Ð γ y y G G ρ o o |αρ|β|γ| §5
. Generalizing Abstraction principles. In this final section, we come back to "Abstraction Principles":

f R (x) = f R (y) ⇔ xRy
As introducing such a statement requires that R is an equivalence, one needs to modify their formulation, inasmuch one wants to cover the (more general case of) collusion relations. Let R be any binary relation over a set a. We consider the two following "Generalized Abstraction Principles", where R is a collusion relation:

1. d-Generalized Abstraction Principle: f R (x) = f R (y) ⇔ x td ∼ R y 2. a-Generalized-Abstraction principle: f R (x) = f R (y) ⇔ x tr ∼ R y
Let us note that the word "generalization" may seem odd in the current situation. Indeed, as for any relation R, the relations td ∼ R and tr ∼ R are equivalence relations, one could prefer to say that "Regular Abstraction principles" can be narrowed to the particular case where the equivalence relation is of the form indicated in these "Generalized" abstraction principles ( td ∼ R or tr ∼ R ). Because of theorem 11, the word generalization is nevertheless relevant: inasmuch R is a collusion, one can extend the solution proposed by Russell for determining the co-domain of f R (i.e. f R (x) then denotes the class of x in the corresponding collusional quotient). Let us finally observe that, because of the proposition below, in the particular case R is an equivalence relation, both those Generalized Abstraction Principles collapse to the usual, regular "Abstraction principles", i.e. they both are equivalent to: The generalization of equivalence relations that we introduced under the name collusions, characterizes the condition needed for the quotientation of a set from a binary relation. It can thus be seen as generalizing one of the standard tool for the theory of classification. The renewed theory of classification which is at hand, appears no more based on a primitive notion of similarity (as in the case of equivalence relations), but on a (non primitive) notion of similarity induced by the primitive notion of interaction with the context (agonality) 11 . As pointed out by Thomas Seiller (currently collaborating with me on those topics), the crucial fact that collusions are closed by converse invites to reformulate the resulting classification theory in terms of orthogonality (or equivalently duality), this notion being here used with the precise meaning it has in the proofs-as-programs paradigm, where it is employed to define the notion of type (i.e. sets of processes closed by bi-orthogonal, for a dynamic dependent notion of orthogonality between processes). In that framework, the words "reaction to the context" receive a proper sense, linked with the dynamics of computation. But as Seiller advocates, this notion of orthogonality may well be studied at an abstract, time-independent level, considering directly axiomatic properties of the relation with respect to which the orthogonality is defined Independently of this general promising research programme, it would be useful to uncover and investigate "concrete" models of non equivalence collusions. The typical kind of applications we have in mind is algorithms separation 12 . A last (but not least) kind of investigations for the future would be to reinvest, through the notion of collusion, the philosophical and mathematical debates about definitions by abstraction, in particular those based on higher order abstraction principles.

Definition 3 . 1 .

 31 (equi-targeting relations) Let R any binary relation on a. The two binary relations td ∼ R and tr

∼

  R are (evidently) equivalence relations. By the way, they could have been defined in terms of equality of classes as x tr

Lemma 12 .

 12 Let R over a collusive, b a class for R and x ∈ b. ∃x ∈ b xRx ⇒ ∀x ∈ b xRx Proof. Let y ∈ a such that b = [y] R and let x ∈ b. One thus has yRx . So, if xRx , then x and y do converge (on x ). So, as R is collusive, x has exactly the same targets/images for R than y. Hence ∀x ∈ b xRx . Lemma 13. Let R be collusive relation on a. If R is irreflexive, then for any targeting class b for R and any x, x ∈ b, one has x Rx .

4. 1 .

 1 Incompleteness of non reflexive collusions w.r.t. partitions. Lemma 14. Let R a collusion. If a/R = {a}, then R is reflexive.

Lemma 17 .

 17 [x] P ) = b (resp σ(b) = [x] P ). As P is a partition, b is non empty, i.e. b = [x ] P for some x ∈ b. As σ([x] P ) = [x ] P (resp σ([x ] P ) = [x] P ), by σ's definition, we have that xR σ x (resp. x R σ x). Hence, R σ is total (resp. surjective). • R σ is collusive. Indeed, let x, y, x , y ∈ a, such that xR σ x , yR σ x and xR σ y . By R σ 's definition, one thus has σ([x] P ) = [x ] P , σ([y] P ) = [x ] P and σ([x] P ) = [y ] P . So [x] P = [y] P (as σ is injective) and [x ] P = [y ] P (as σ is functional). So, as σ([x] P ) = [x ] P , one has σ([y] P ) = [y ] P . Thus yR σ y . • R σ is irreflexive. Indeed, one never has xR σ x, since for any b ∈ P , σ(b) = b Lemma 16. Let σ any application from P (a partition of a) to itself. For any y∈ σ([x] P ), one has [x] Rσ = [y] P . Proof. Let y ∈ σ([x] P ). As [x] Rσ = z ∈ a ; xR σ z = z ∈ a ; σ([x] P ) = [z] P , it amounts to show that z ∈ a ; σ([x] P ) = [z] P = [y] P • For the ⊆ direction, let z ∈ a such that σ([x] P ) = [z] P . As we assumed that y ∈ σ([x] P ), we have that y ∈ [z] P . Hence [z] P = [y] P . So z ∈ [y] P . • For the ⊇ direction, let z ∈ [y] P . This entails [y] P = [z] P . But as we assumed that y ∈ σ([x] P ), one has σ([x] P ) = [y] P . Hence σ([x] P ) = [z] P . For any b, b ∈ P : σ(b) = b ⇔ ∀x ∈ b ∀x ∈ b xR σ x Proof. ⇒ Let us assume that σ(b) = b , and let any x ∈ b and any x ∈ b . As b = [x] P and b = [x ] P , one has σ([x] P ) = [x ] P , hence xR σ x . ⇐ Let us assume that ∀x ∈ b ∀x ∈ b xR σ x , i.e. ∀x ∈ b ∀x ∈ b σ([x] P ) = [x ] P . As b, b = ∅, one has for some x, x ∈ a that [x] P = b and [x ] P = b . We thus have σ([x] P ) = [x ] P , i.e. σ(b) = b Lemma 18. Let P a partition of a and σ an application from P to P . Then σ([x] P ) = [x] Rσ Proof. • σ([x] P ) ⊆ [x] Rσ . Indeed, let y ∈ σ([x] P ). This implies that σ([x] P ) = [y] P . By lemma 17, one has xR σ y. So y ∈ [x] Rσ . • [x] Rσ ⊆ σ([x] P ). Indeed, let y ∈ [x] Rσ . By definition of classes, one has xR σ y. By definition of R σ , one has σ([x] P ) = [z] P . Hence z ∈ σ([x] P ). Proposition 19. Let P a partition of a and σ a bijection from P to P with no fix-points. Then a/R σ = P . Proof. 1. a/R σ ⊆ P . Indeed, let b ∈ a/R σ . This means that, for some x ∈ a, one has b = [x] Rσ . As b belongs to a/R σ (which, by lemma 15 and theorem 2, forms a partition of a), it is non empty. Let then y ∈ b. As y ∈ [x] Rσ , by definition of targeting classes, one has xR σ y, which, by definition of R σ means that σ([x] P ) = [y] P . One thus has y ∈ σ([x] P ). Hence, by the lemma 16, [x] Rσ = [y] P . Hence b ∈ P . 2. P ⊆ a/R σ . Indeed, let b ∈ P . Classes in a partition being non empty, there exists x ∈ a such that b = [x ] P . As σ is a surjective, there exists b ∈ P such that σ(b) = b . Again, there exists x ∈ a such that b = [x] P . Finally, one has b = σ(b) = σ([x] P ) = [x] Rσ (by lemma 18). Finally, one thus has b ∈ a/R σ .

  f R (x) = f R (y) ⇔ xR y Proposition 21. R is an equivalence relation on a ⇔ R = tr ∼ R ⇔ R = td ∼ R Proof. The ⇒ directions amount to say that, when R is an equivalence relation, xRy ⇔ [x] R = [y] R (a standard observation, already made by Burali-Forti) and xRy ⇔ [x] R -1 = [y] R -1 . The ⇐ directions are immediate as td ∼ R and tr ∼ R are equivalence relations, whatever the relation R. §6. Conclusion and Future work.

  So if R is transitive, as x Ry, one has x Rx, and as xRy , one has x Ry . ⇐ • Let us first show that any collusive and reflexive relation, is symmet-Let us now show that any collusive and reflexive relation, is transitive. Let x, y, z ∈ a, such that xRy and yRz. If R is reflexive, yRy. So that x and y do converge (on y). Hence, if R is collusive, they must have exactly the same images by R. So, as yRz, one must have xRz. Remark 2. As any reflexive relation is evidently total and surjective, any equivalence relation is a collusion. But in general, collusions are not equivalence relations (actually, some non reflexive collusions are not symmetrical or not transitive; some of them have none of these properties).

Definition 2.1. The relation R over a is: • total, if ∀x ∈ a ∃y ∈ a xRy • surjective, if ∀y ∈ a ∃x ∈ a xRy • collusive, if ∀x ∈ a ∀x ∈ a ∃y ∈ a (xRy ∧ x Ry) ⇒ ∀y ∈ a (xRy ⇒ x Ry) • a collusion relation, if it is total, surjective and collusive Remark 1. The meaning of the collusivity property is more salient ("convergence somewhere entails convergence everywhere") when one expresses it in the following redondant form: ∀x ∈ a ∀x ∈ a ∃y ∈ a (xRy∧x Ry) ⇒ ∀y ∈ a (xRy ⇔ x Ry) . Theorem 1. Let R be a binary relation on a. R is an equivalence relation ⇔ R is collusive and reflexive Proof. ⇒ Let us show that any symmetrical and transitive relation is collusive. Let x, x ∈ a. Let us assume that xRy and x Ry for some y ∈ a and that xRy for some y ∈ a. If R is symmetrical, as xRy, one has yRx. rical. Let x, y ∈ a, such that xRy. If R is reflexive, yRy, so that x and y do converge (on y). If R is collusive, x and y must have exactly the same images by R. As xRx (R reflexive being assumed), one thus must also have yRx. •

Cf. [5], §.110, p.114-115 and [19].

Following F. Consuegra, p.132 of[START_REF]The Arché Papers on the Mathematics of Abstraction[END_REF], that view was already sensible or explicit in the Peano School, with a priority accorded to Pieri by Peano himself. For Mancosu[1], the first explicit reading of equivalence classes as being the defined "entities" -rather than what we would call today a specific representant of the class or maybe a proper, independent entity -is also made early by Frege, influenced in that matter by observations of Grassmann from 1844.

Following Mancosu, even if a systematic and complete exposition of the partitioning/quotienting discipline seems to occur only in the late twenties (maybe for the first time in van der Waerdens Abstrakte Algebra[START_REF] Lorenzen | Einführung in die operative Logik und Mathematik[END_REF], in the § 5, entitled "Klasseneinteilung. Äquivalenzrelation"), "the technical details were already clear in the 1910s" (see[1], p. 20)

Through the agonal reading, there is nothing odd in considering auto-targeting elements, i.e. elements x such that xRx: after all, logic knows about paradoxes (and life about suicide). The important point is to observe that such situations generate special effects which do have an agonal meaning: when the relation is a collusion, the two maximal coalitions (for Attack and for Defence) to which such an auto-targeting element belongs are necessarily degenerated coalitions, i.e. coalitions in which everybody targets everybody.

As a corollary, observe that if a is a singleton, no irreflexive collusion exists over a. In terms of the targeting interpretation: in a non suicidal world, one needs to be at least two to fight.

A similar proof has been given independently by Paulo Veloso -private communication, unpublished.

As an illustration of the usefulness of the agonal reading of collusions, let us observe that it suggests for instance that, a given irreflexive collusion R being given, one could extend R toward a larger collusion, in such a way that a given maximal coalition for R (i.e. a class in the collusional quotient a/R) so to speak absorbs the ennemies of the ennemies of its members (under the condition, of course, that the later ones are not themselves ennemies of members of the initial coalition) -a processus rather frequent in the political life.

Far from this computational direction, let us mention that Paulo Veloso presented some concrete arithmetical models of collusions during the third occurrence of Jornadas de Mirantão in August 2018.


Maarten Lupker
email: maarten.lupker@erdw.ethz.ch

Christian France-Lanord

Valier Galy

Jérôme Lavé

Hermann Kudrass

Increasing chemical weathering in the Himalayan system since the Last Glacial Maximum

Continental chemical weathering is central in

Earth's surface biogeochemical cycles as it redistributes elements across reservoirs such as the crust and the oceans. However the evolution of weathering through time and its response to external forcing such as changes in climate remain poorly constrained. In this work, a composite sediment record from the Bay of Bengal is used to document the evolution of chemical weathering in the Himalayan system (Himalayan range and Indo-Gangetic floodplain), the world largest sediment conveyor to the oceans, since the Last Glacial Maximum (LGM). The degree of weathering of the sediments is documented using mobile to immobile ratios such as K/Si and H2O + /Si as well as detrital calcite abundance. Robust weathering proxies are derived by correcting the chemical composition of sediment for sorting effects that occur during transport and deposition. The Bay of Bengal record is also further compared to the chemical composition of modern river sediments from the Ganga & Brahmaputra basin. Weathering proxies all indicate that the sediments exported by the Ganga & Brahmaputra Rivers became increasingly weathered over the past ~21 kyr, whereas, Sr, Nd and major elements suggest a constant sediment provenance in the system over the last 21 kyr. These changes in the degree of weathering of the sediments show that the weathering flux exported by the system to the Indian Ocean during the LGM was significantly lower than at present and demonstrate that chemical weathering in continental scale basins such as the Ganga & Brahmaputra responds to Late Quaternary climate changes.

Introduction:

Chemical weathering of continental rocks is the main source of soluble elements to the oceans and therefore exerts a first order control on oceanic bio-geochemical cycles. Over geologic time scales, silicate weathering and subsequent carbonate precipitation also draws down atmospheric CO2 thereby stabilizing the earth's climate [START_REF] Walker | A negative feedback mechanism for the longterm stabilization of the Earth's surface temperature[END_REF].

Continental weathering fluxes are a function of continent lithology, tectonic activity and climate but the mechanisms and feedbacks linking silicate weathering to the earth's climate remain debated (France-Lanord and Derry, 1997;[START_REF] Willenbring | Long-term stability of global erosion rates and weathering during late-Cenozoic cooling[END_REF].

A better understanding of these complex climate-weathering couplings can be gained by studying the response of continents to climatic transitions in the past.

The role of the Himalayan orogeny in the long-term forcing of oceanic cycles and the global climate has retained much attention (France-Lanord and Derry, 1997;[START_REF] Raymo | Influence of late Cenozoic mountain building on ocean geochemical cycles[END_REF]. However, the temporal variability of and controls on the evolution of weathering fluxes exported by large basins remains poorly constrained and is often difficult to reconstruct [START_REF] Limmer | Geochemical record of Holocene to Recent sedimentation on the Western Indus continental shelf, Arabian Sea[END_REF]. In particular, direct characterization of the response of chemical weathering to abrupt climate perturbations has heretofore remained a challenge. We present a record of chemical weathering intensity in the Himalayan system since the LGM. The Himalayan system is defined here as the Himalayan range and the downstream Indo-Gangetic floodplain drained by Ganga and Brahmaputra Rivers. No major change in tectonic forcing of the Himalayan orogeny has been documented so far over the last glacial-interglacial period, which enables us to discuss variations in sediment weathering as a response to climate change.

Setting and methods:

The central part of the Himalayan range is drained by two major rivers, the Ganga and Brahmaputra (Figure 1). Currently, the Ganga & Brahmaputra transfers ca. 1.10 9 Mt of sediments to the Bay of Bengal, of which 90% is transported during the monsoon [START_REF] Rsp | River Survey Project, Flood Action Plan (FAP) final report[END_REF]. River sediments were sampled mainly during monsoon seasons, in 2002,2004,2005,2007,2008 and 2010 across the Indo-Gangetic floodplain. Sampling locations mainly included the front of the Himalayan range for the main trans-Himalayan Rivers of the Ganga & Brahmaputra system, the outlet of the Ganga and the Brahmaputra as well as their confluence -the lower Meghna -which integrates the entire sediment load exported from the Ganga & Brahmaputra basin (Figure 1). When possible, sediments were sampled along vertical depth profiles through the water column in order to capture the full variability of the sediment load. These profiles were complemented by bedload samples dredged from the active channel of the rivers and freshly exposed bar sediments. Details on the sampling procedure are described in [START_REF] Lupker | A Rouse-based method to integrate the chemical composition of river sediments: Application to the Ganga basin[END_REF].

Sediment cores were retrieved during RV Sonne cruise SO93 in 1994 from the shelf (105KL and 107KL) and the active channel levee system of the middle fan (117KL, 118KL and 120KL) [START_REF] Ittekkot | The Bay of Bengal--An Introduction[END_REF]. The chronology of the cores is provided by re-calibrated [START_REF] Reimer | Intcal09 and Marine09 Radiocarbon Age Calibration Curves, 0-50,000 Years[END_REF] 14 C ages on planktonic foraminifers [START_REF] Weber | Active growth of the bengal Fan during sea-level rise and highstand[END_REF] as well as 137 Cs, 210 Pb and 228 Ra gamma spectrometry [START_REF] Michels | The submarine delta of the Ganges-Brahmaputra: cyclone-dominated sedimentation patterns[END_REF]. Scarce hemipelagic layers were excluded from the study in order to focus on turbiditic sediments representing the Ganga & Brahmaputra detrital signal [START_REF] Pierson-Wickmann | Os-Sr-Nd results from sediments in the Bay of Bengal: Implications for sediment transport and the marine Os record[END_REF][START_REF] Weber | Active growth of the bengal Fan during sea-level rise and highstand[END_REF]. We interpret our data by constructing a composite Log from the shelf and levees core following the approach detailed in [START_REF] Galy | C4 plants decline in the Himalayan basin since the Last Glacial Maximum[END_REF].

River suspended river sediments were filtered through 0.22 µm PES membrane filters within 24 h of sampling and dried in the lab. Major element concentration was determined by ICP-OES after LiBO2 fusion at SARM, Nancy -France [START_REF] Carignan | Routine analyses of trace elements in geological samples using flow injection and low pressure on-line liquid chromatography coupled to ICP-MS: A study of geochemical reference materials BR, DR-N, UB-N, AN-G and GH[END_REF]. Core samples were repeatedly rinsed with water in order to remove salt contributions. Nevertheless, Na was not considered in this study due to seawater contamination even after these rinsing steps.

Carbonate contents were measured manometrically from the CO2 released after reaction with H3PO4 and the evolved CO2 was subsequently analyzed for C and O isotopic composition [START_REF] Galy | The strontium isotopic budget of Himalayan Rivers in Nepal and Bangladesh[END_REF]. Sr and Nd isotopic compositions were measured on the acid leached, silicate fraction of the river and core sediments by TIMS after classical separation and purification by ion exchange resins. Sediment hydration [H2O + ] and isotopic composition dD-SMOW was measured by EA-IRMS after degassing the samples to remove adsorbed freewater (Lupker et al., 2012b). The long-term reproducibility is better than 0.1 H2O + wt % and 2 ‰ dD-SMOW.

Sediment sorting effects on the chemical composition of detrital sediments:

Hydrodynamic control of sediment sorting

The chemical composition of river sediments has to be interpreted carefully. Sediment vertical depth profiles recovered from the water column of large rivers show a strong heterogeneity in terms of concentration and chemical composition. These heterogeneities are the result of the hydrodynamic sorting of sediments, based on grain-size, shape and density, which induce mineralogical and thus chemical differentiation of sediments within the water column (Bouchez et al., 2011a;Bouchez et al., 2011b;[START_REF] Garzanti | Mineralogical and chemical variability of fluvial sediments 2. Suspended-load silt (Ganga-Brahmaputra, Bangladesh)[END_REF][START_REF] Lupker | A Rouse-based method to integrate the chemical composition of river sediments: Application to the Ganga basin[END_REF]Singh and France-Lanord, 2002a). The mineralogy of sediments from the Himalayan system is dominated by quartz, micas and feldspar, with occurrences of other phyllosilicates, clay assemblages and hydroxides in the finer fraction [START_REF] Garzanti | Mineralogical and chemical variability of fluvial sediments 2. Suspended-load silt (Ganga-Brahmaputra, Bangladesh)[END_REF][START_REF] Garzanti | Mineralogical and chemical variability of fluvial sediments 1. Bedload sand (Ganga-Brahmaputra, Bangladesh)[END_REF]. These minerals are segregated during sediment transport: coarse-grained quartz is enriched in the bedload and at the bottom of the water column while phyllosilicates such as micas and clays are comparatively enriched in the shallow surface waters of the river, leading to a strong contrast in chemical composition between samples of a single reach. This is best illustrated using the immobile element ratios Fe/Si and Al/Si (Figure 2). Fe, Al and Si are, to a first order, immobile elements during Himalayan erosion and are thus not notably affected by chemical weathering [START_REF] Galy | Higher erosion rates in the Himalaya: Geochemical constraints on riverine fluxes[END_REF]. The sediments sampled across the Indo-Gangetic plain form a unique mixing trend (to the exception of a few bedload samples that are enriched in Fe due to heavy mineral placer effects) between quartz rich (low Al/Si and Fe/Si) bottom sediments and clay rich (high Al/Si and Fe/Si) surface sediments. Al/Si and Fe/Si are directly correlated to the bulk sediment grain-size and are a proxy of mineral sorting (Bouchez et al., 2011a;[START_REF] Lupker | A Rouse-based method to integrate the chemical composition of river sediments: Application to the Ganga basin[END_REF]. A single sediment sample from a given river reach can therefore not be considered, a priori, as representative of the whole sediment load, its chemical composition being dependent on river hydrodynamics at the moment of sampling.

Weathering proxies in modern river sediments

The degree of chemical weathering experienced by sediments in the Himalayan system can be traced from detrital sediments by using the mobile to immobile elemental ratios of river sediments such as K/Si and H2O + /Si. The structural hydroxyl content of sediments, H2O + , is a proxy of chemical weathering that traces mineral hydrolysis and secondary neoformations (Lupker et al., 2012b). Sediment sorting effects also apply to mobile elements such as K and H2O + . The K/Si and H2O + /Si ratios are dominantly controlled by the grain size of the sediments, i.e. clay to quartz mixing (Lupker et al., 2012b) and are, to a first order, linearly correlated to their Al/Si ratio as shown by Figure 3, where these ratios are plotted for sediments from the modern Himalayan system. By using a sediment depth profile sampling approach on modern rivers, we characterize these sorting effects as this method allows sampling a full set of sediments covering a wide range of grain sizes. The degree of chemical weathering can then be assessed based on the trend formed by the chemical composition of the sediments in diagrams such as Figure 3, this trend being representative of the entire sediment load exported by the considered river.

In addition to silicate weathering proxies, we determined the calcite content and the Ca/Si ratio in Bay of Bengal sediments that reflects the carbonate weathering intensity in the Himalayan system. Carbonates, including calcite, account for over 80 % of the bulk Ca and Ca/Si therefore mainly reflects carbonate content. Both carbonates and Ca content are not grain size dependent [START_REF] Lupker | A Rouse-based method to integrate the chemical composition of river sediments: Application to the Ganga basin[END_REF] and are therefore not corrected for sediment sorting.

Accounting for sediment sorting effects in detrital records

For detrital records such as the Bay of Bengal cores, changes in sediment transport and deposition patterns induce changes in the grain size of deposited sediments that, in turn, affect the chemical composition of the record. Hence, sediment sorting also has to be taken into account in order to compare the degree of weathering of samples with variable grain-sizes (variable Al/Si ratios). However, unlike for modern rivers, it is not possible for a given age interval of the record to recover a set of sediment samples spanning different grain sizes that would fully describe the sediment weathering trend. By considering that each sample of the Bay of Bengal record is the result of the mixing between a coarse-grained end-member, similar to bedload material and a fine-grained, clay rich sediment, it is nevertheless possible to normalize mobile to immobile ratios (X/Si) to a common Al/Si composition as illustrated in Figure 4 and equation 1: (1) where A and B are respectively the slope and intercept of the linear regression through the sample and coarse grained end-member composition (Figure 4). Such normalized ratios allow comparing the chemical composition of sediments that have contrasted grain sizes, while properly accounting for sorting effects. Doing so, we assume that a unique coarse-grained end-member composition can be assigned to the low Al/Si mixing end-member and that the weathering signature is mainly carried by the small grain size, clay fraction of the sediment load. This is a reasonable approximation as: 1) contrary to the fine grained suspended load, no systematic evolution of K/Si and H2O + /Si ratios of modern bedloads in the Ganga basin is detected from up to downstream (Lupker et al., 2012b), suggesting that the effect of chemical weathering during transfer in the floodplain is negligible for the coarse fraction of the transported sediments, and 2) the transfer time of bedload material is thought to be an order of magnitude longer than that of fine-grained suspended load [START_REF] Granet | Useries disequilibria in suspended river sediments and implication for sediment transfer time in alluvial plains: The case of the Himalayan rivers[END_REF], which makes it less sensitive than the suspended load to short-term upstream variations.

All corrected ratios presented here were normalized to a common Al/Si value of 0.23. They are referred to as X/Si* and are listed in the supplementary information along with the composition of the coarse grained end-member used in the calculations. The Al/Si value of 0.23 was chosen as the average composition of the sediments exported by the modern Ganga, which was determined by modeling the chemical composition of sediments as a function of the river's hydrological conditions and integrated over a 25-year average hydrograph [START_REF] Lupker | A Rouse-based method to integrate the chemical composition of river sediments: Application to the Ganga basin[END_REF]. This composition is also close to the composition of the Himalayan crust eroded in the Ganga basin [START_REF] Lupker | A Rouse-based method to integrate the chemical composition of river sediments: Application to the Ganga basin[END_REF]. It should be noted that the choice of the Al/Si ratio used for normalizing K/Si and H2O + /Si ratios only determines the absolute values of K/Si*

X Si * = Al Si A + B
and H2O + /Si* but has no influence on the general trends observed for Bay of Bengal sediments.

Before normalization, the raw K/Si and H2O + /Si ratios of sediments from the Bay of Bengal cores are correlated to Al/Si (Figure 5), which supports that K/Si and H2O + /Si are dominantly controlled by grain size and sorting effects. However the normalized K/Si* and H2O + /Si* ratios show no correlation with Al/Si, which suggests that our normalization properly accounts for sediment sorting.

Comparison with modern river sediments

In order to compare the weathering degree of the Bay of Bengal sediment record to modern river sediments we applied a similar normalization to the river sediments. The average compositions of modern river sediments for elements prone to sorting effects (K/Si*, H2O + /Si*), reported in Figure 5, were determined by evaluating the regression through all available data of each river reach and a coarse-grained end-member at Al/Si = 0.23. The Himalayan front end-member was reconstructed from the river sediments sampled at the Himalayan front in the Ganga basin and the Brahmaputra basin separately assuming that ca.

60% of the total sediment flux exported to the Bay of Bengal come from the Brahmaputra and the remaining 40 % come from the Ganga basin [START_REF] Rsp | River Survey Project, Flood Action Plan (FAP) final report[END_REF]. For compositions that are not affected by sorting effects (calcite, Ca/Si, 87 Sr/ 86 Sr, εNd, d 18 O and d 13 C), a simple average for the Ganga, Brahmaputra and lower Meghna and a weighted mean for the Himalayan front were considered (details can be found in supplementary information and Table S2).

Weathering in the Himalayan system:

Weathering in the modern system

Chemical weathering is marked by a change in the mobile to immobile ratio at constant Al/Si ratio. This mobile element depletion is highest for fine-grained (high Al/Si) sediments (Figure 3). Modern Himalayan rivers sediments, sampled at the front of the range show high K/Si ratios and low H2O + /Si ratios compared to the sediments sampled at the outlet of the Ganga & Brahmaputra basins and their confluence, the lower Meghna (Figure 3). The loss of K and gain in hydration during floodplain transfer marks the weathering of the sediments within the floodplain. Even if part of this weathering signal may stem from the sediment contribution of the southern (Ganga) and eastern (Brahmaputra) tributaries, their sediment fluxes remain small compared to that of Himalayan tributaries (Lupker et al., 2012b;Singh and France-Lanord, 2002a). Calcite weathering in the modern floodplain is significant and leads to almost complete dissolution in Brahmaputra sediments in Bangladesh (Singh and France-Lanord, 2002a;[START_REF] Singh | Chemical erosion in the eastern Himalaya: Major ion composition of the Brahmaputra and delta C-13 of dissolved inorganic carbon[END_REF] while modern Ganga sediments in Bangladesh display a low calcite content of ca. 3 wt% (Lupker et al., 2012b).

It has been shown that in the modern Ganga basin, weathering within the floodplain outweighs weathering within the Himalayan range (Lupker et al., 2012b;[START_REF] West | Small-catchment perspective on Himalayan weathering fluxes[END_REF].

Significant K depletion and H2O + gain is only recorded when sediments transit through the floodplain (Lupker et al., 2012b), which is attributed to the longer residence time of sediments in the floodplain than in the Himalayan range. Overall, these observations emphasize the strong floodplain control on weathering in the Himalayan system.

Weathering record of the Bay of Bengal:

The normalized weathering indexes (K/Si* and H2O + /Si*) are not biased by sorting effects and can thus be applied to the Bay of Bengal record to reconstruct temporal variations in chemical weathering. Composite logs of K/Si* and H2O + /Si* (Figure 6) show a progressive evolution towards lower K/Si* and higher H2O + /Si* ratios from the LGM to present. Both weathering proxies suggest that during the LGM, sediments exported to the Bay of Bengal were less weathered than recent Bay of Bengal and modern lower-Meghna sediments. It should also be noted that the chemical composition of sediments from the most recent part of the Bay of Bengal record agrees well with that of the sediments currently exported by the lower Meghna (Figure 6).

Calcite content and Ca/Si ratios also decrease from LGM to present (Figure 6). Calcite in sediments from the Bay of Bengal is of detrital origin as demonstrated by their isotopic composition (d 18 OPDB ranging from -15 to -6 ‰, see electronic appendix), much lower than marine biogenic calcite [START_REF] Hoehndorf | Transfer of the Sr isotopic signature of the Himalayas to the Bay of Bengal[END_REF] and within the range of composition of detrital Himalayan river calcite (Lupker et al., 2012b). The decrease in calcite concentration and Ca/Si ratios since the last glacial maximum supplement the K/Si* and H2O + /Si* records, further reinforcing the conclusion that sediment weathering increased during the studied period.

Estuarine and early digenetic reactions such as weathering and reverse weathering may affect the composition of deposited sediments [START_REF] Jones | An experimental study of the interaction of basaltic riverine particulate material and seawater[END_REF][START_REF] Michalopoulos | Rapid Clay Mineral Formation in Amazon Delta Sediments: Reverse Weathering and Oceanic Elemental Cycles[END_REF][START_REF] Michalopoulos | Early diagenesis of biogenic silica in the Amazon delta: alteration, authigenic clay formation, and storage[END_REF][START_REF] Wallmann | Silicate weathering in anoxic marine sediments[END_REF]. The close agreement between the chemical composition of modern lower Meghna sediments and the recent part of the record, both from the shelf and the levee, suggests that estuarine processes do not alter significantly the composition of river sediments delivered to the Bay of Bengal. The chemical weathering of sediments after deposition cannot explain the trends observed in the Bay of Bengal record either. Indeed, post-depositional weathering would lead to down-core increase in the weathering intensity which is opposite to the observed trend. Reverse weathering leads to an uptake of K and to an overall decrease in structural OH radicals of the sediments [START_REF] Michalopoulos | Rapid Clay Mineral Formation in Amazon Delta Sediments: Reverse Weathering and Oceanic Elemental Cycles[END_REF] but cannot explain the observed down-core increase in detrital carbonate content. Furthermore, during reverse weathering reactions, the newly formed minerals will form in equilibrium with seawater so that the D/H isotopic composition of their hydroxyls groups should become heavier as reverse weathering reactions proceed.

However no systematic down-core increase in dDH2O+ is observed (supplementary information, Figure S1).

Sediment provenance since the LGM:

Potential sediment source effects on the chemical composition of Bay of Bengal sediments since the LGM must be taken into account as changes in precipitation and erosion patterns within and across the Himalayan range may have affected the sedimentary record.

The Sr and Nd isotopic signatures of river sediments in the Himalayan range are a first order indicator of the erosion balance between the Lesser Himalaya (LH) and High Himalaya (HHC) (and Transhimalayan formations -THB -for Brahmaputra) as these units have contrasted isotopic compositions [START_REF] France-Lanord | Evolution of the Himalaya since Miocene time: isotopic and sedimentologic evidence from the Bengal Fan[END_REF]Singh and France-Lanord, 2002a). Sediments from the Ganga and Brahmaputra basins also show distinctive Sr and Nd isotopic compositions so that changes in the balance of sediment supply from both basins would be detected (Figure 7).

Previous studies have shown contrasting changes in the balance between sediments from the LH and HHC in the Indus basin [START_REF] Clift | Holocene erosion of the Lesser Himalaya triggered by intensified summer monsoon[END_REF] and the western part of the Ganga basin [START_REF] Rahaman | Climate control on erosion distribution over the Himalaya during the past ~100 ka[END_REF][START_REF] Rahaman | Climate control on erosion distribution over the Himalaya during the past similar to 100 ka: REPLY[END_REF][START_REF] Srivastava | Climate control on erosion distribution over the Himalaya during past similar to 100 ka: COMMENT[END_REF] since deglaciation. However, both Sr and Nd isotopic compositions of Bay of Bengal sediments appear constant, implying that there is no evidence for major changes in the distribution of erosion within the Himalayan range at this scale (Figure 7). The Sr and Nd record also suggests that the balance between the Brahmaputra and Ganga has remained roughly constant over the last 20 kyrs. A significant change in the sediment balance between Ganga and Brahmaputra would furthermore result in a correlated evolution of K/Si* and H2O + /Si*, whereas these ratios are actually anti-correlated since the LGM (Figure 6), demonstrating that the weathering signal overprints any potential change in sediment source.

An increase in sediments from the Siwalik or from southern tributaries through time is also very unlikely. Siwalik rocks and sediments are characterized by high carbonate content and low d 13 C compositions (Lupker et al., 2012b;[START_REF] Sanyal | Chemical diagenesis of Siwalik sandstone: Isotopic and mineralogical proxies from Surai Khola section, Nepal[END_REF]. Thus, the concomitant decrease of both carbonate content and d 13 C values observed in the Bay of Bengal sediments is not compatible with an increase of sediment supply from the Siwaliks. In addition, sediments from the Chambal River in the Ganga basin and the Eastern tributaries of the Brahmaputra basin (Lohit and Dibang) have distinctively higher Fe/Si ratios and unradiogenic Sr and Nd signatures (Lupker et al., 2012b;Singh and France-Lanord, 2002a;[START_REF] Singh | Sr and Nd isotopes in river sediments from the Ganga Basin: Sediment provenance and spatial variability in physical erosion[END_REF] that are not identified in our record (see supplementary information, Figure S1).

Altogether, the observed changes in Bay of Bengal sediment mobile elements composition since the LGM can thus be primarily attributed to changes in the chemical weathering intensity in the Ganga & Brahmaputra basin.

Discussion:

Climatic forcing of Himalayan chemical weathering

During the last 21 kyrs, climate has been the main external forcing of Himalayan erosion that could have affected the Bay of Bengal weathering signal. Paleo-temperature reconstructions over the Himalayan system are scarce but indicate a ca. 2 °C colder LGM sea surface temperature (SST) in the Bay of Bengal [START_REF] Kudrass | Modulation and amplification of climatic changes in the Northern Hemisphere by the Indian summer monsoon during the past 80 k.y[END_REF]. Hydrological changes were significant with strongly reduced monsoon and therefore a lower continental runoff during the last glacial [START_REF] Cullen | Microfossil evidence for changing salinity patterns in the Bay of Bengal over the last 20000 years[END_REF][START_REF] Duplessy | Glacial to interglacial contrasts in the northern Indian Ocean[END_REF][START_REF] Herzschuh | Palaeo-moisture evolution in monsoonal Central Asia during the last 50,000 years[END_REF][START_REF] Kudrass | Modulation and amplification of climatic changes in the Northern Hemisphere by the Indian summer monsoon during the past 80 k.y[END_REF], which reduced the sediment supply to the Bay of Bengal [START_REF] Goodbred | Response of the Ganges dispersal system to climate change: a source-to-sink view since the last interstade[END_REF][START_REF] Weber | Active growth of the bengal Fan during sea-level rise and highstand[END_REF]. Precipitation and temperature directly control the weathering rate of minerals [START_REF] White | Effects of climate on chemical weathering in watersheds[END_REF], and the conditions during the LGM where therefore less favorable for intense weathering in the floodplain. These changes in floodplain weathering are confirmed by the evolution of clay assemblages in the southern part of the basin that suggests an increase in weathering intensity from at least 10 kyr to present [START_REF] Heroy | Mineralogy of the Ganges and Brahmaputra rivers: Implications for river switching and Late Quaternary climate change[END_REF].

Additional, independent processes should not be excluded [START_REF] Blum | Fluvial responses to climate and sea-level change: a review and look forward[END_REF]. The lower base level and a reduced sediment supply during the LGM could have lead to channel incision and a narrower active floodplain [START_REF] Goodbred | Response of the Ganges dispersal system to climate change: a source-to-sink view since the last interstade[END_REF][START_REF] Heroy | Mineralogy of the Ganges and Brahmaputra rivers: Implications for river switching and Late Quaternary climate change[END_REF], which would reduce channel -floodplain interactions and limit the supply of weathered floodplain sediments to the Bay of Bengal. A complete abandonment of the floodplain during the LGM is nevertheless unlikely as floodplain derived organic material is persistent in the Bay of Bengal cores throughout the deglaciation [START_REF] Galy | C4 plants decline in the Himalayan basin since the Last Glacial Maximum[END_REF] but changes in the sediment locus and volume being mobilized should not be excluded [START_REF] Dosseto | Climatic and vegetation control on sediment dynamics during the last glacial cycle[END_REF].

The weathering degree of sediments does, however, not show any clear change in response to the decrease of monsoon intensity after the Holocene Climatic Optimum (HCO) [START_REF] Kudrass | Modulation and amplification of climatic changes in the Northern Hemisphere by the Indian summer monsoon during the past 80 k.y[END_REF]. The sediment fluxes during the HCO were up to double modern fluxes as indicated by sediment volume reconstructions [START_REF] Goodbred | Response of the Ganges dispersal system to climate change: a source-to-sink view since the last interstade[END_REF]. These high sediment fluxes may have obliterated the intense weathering conditions prevailing at that time. It can also not be ruled out that modern sediments reflect the export of material weathered during the HCO owing to a long-sediment transfer time so that the post-HCO decrease in precipitation is not yet recorded in the Bay of Bengal. The impact of anthropogenic activity in the Indo-Gangetic floodplain should also be considered as intense land use changes, certainly significant more than 3 centuries ago [START_REF] Ramankutty | Estimating historical changes in global land cover: Croplands from 1700 to 1992[END_REF], may overprint the climatic forcing of the weathering degree in the recent part of the record if already significant thousands of years ago [START_REF] Bayon | Intensifying Weathering and Land Use in Iron Age Central Africa[END_REF].

The sediment transfer time through the Himalayan system has to be considered as well, as a long transfer time would induce a temporal offset between the suggested climatic forcing of weathering in the Ganga & Brahmaputra basin and its recording in the sediment record of the Bay of Bengal. The sediment transfer time in large continental basins remains difficult to constrain. U-series disequilibrium studies on Ganga sediments suggest that the transfer time of coarse sediments may reach several 100 kyrs while fine grained suspended material travel in less than 25 kyrs [START_REF] Chabaux | Determination of source and transfer-time of river sediments in alluvial plain from Useries nuclides: Evidence from the Ganges River System[END_REF][START_REF] Chabaux | U-series in Himalayan rivers: Timescale of sedimentary transfer[END_REF][START_REF] Granet | Useries disequilibria in suspended river sediments and implication for sediment transfer time in alluvial plains: The case of the Himalayan rivers[END_REF][START_REF] Granet | Time-scales of sedimentary transfer and weathering processes from U-series nuclides: Clues from the Himalayan rivers[END_REF]. However, simple mass balance considerations based on the volume of the active floodplain and grain size along with modern sediment fluxes suggest shorter transfer time scales of 1 kyrs for fine (<100 µm) sediments along the Ganga floodplain (Lupker et al., 2012a) and most probably even shorter transfer time along the narrower Brahmaputra floodplain. In addition, short term variability of the chemical composition of Ganga sediments observed in Bangladesh and attributed to the variable input of a > 1000 km distant, upstream tributary suggests that for fine-grained suspended sediment the transfer time can be of the order of the season (Lupker et al., 2012b). Finally, a relatively short transfer time is also supported by the organic carbon Bay of Bengal record, which shows a change in vegetation photosynthetic type over the Ganga & Brahmaputra basin from the last glacial to present [START_REF] Galy | C4 plants decline in the Himalayan basin since the Last Glacial Maximum[END_REF]. This vegetation change follows the model predictions and would not be recorded if the sediment transfer time exceeded glacial -interglacial time scales.

Although the transfer time of sediments is probably significant and remains difficult to quantify, a network of evidences suggests that it is lower than glacial-interglacial time scales

for fine sediments such as those composing the Bay of Bengal record (high Al/Si ratios).

Thus, the observed weathering trends most likely results from the direct or indirect climatic forcing of weathering in the Ganga & Brahmaputra basin.

Towards a quantitative weathering budget

Changes in the net weathering flux or dissolved flux (φX) can be estimated by considering that, at steady state, the weathering flux exported as solutes to the oceans is the product of the mobile element depletion of sediments (ΔX) and the sediment flux (Fsed) (Lupker et al., 2012b):

(

Where is the average mobile to immobile ratio of the source and sink sediments and the average Si concentration of the sediments. A high-resolution reconstruction of weathering fluxes is pointless, as the steady state hypothesis is likely not fully satisfied over short time-scales because of late Holocene climatic variations and anthropogenic imprints (possibly acting over timescales shorter than the sediment residence time in the floodplain). A first order estimate can nevertheless be made by comparing the loss of K, and gain in H2O + , relative to Himalayan front sediments for the LGM (19 -22 kyr) and the modern (0 -2 kyr)

part of the record.

The average and ratios of the exported sediment flux can be estimated based on the average Al/Si ratio of the sediments, as Al/Si is relatively insensitive to chemical weathering. The average Al/Si ratio of sediments was estimated to be 0.23 for the Ganga in Bangladesh [START_REF] Lupker | A Rouse-based method to integrate the chemical composition of river sediments: Application to the Ganga basin[END_REF]. Hence, and are well

X = F sed X = F sed X Si Source Si [ ] Source X Si Sink Si [ ] Sink X / Si Si [ ] K / Si H 2 O + / Si K Si H 2 O + / Si
approximated by and , as for these ratios the composition of the sediment is normalized to an Al/Si ratio of 0.23.

Because the effect of weathering in the Himalayan range is not detected on the K/Si and H2O + /Si composition of modern river sediments sampled at the front of the Himalayan range (Lupker et al., 2012b), we conclude that the composition of these sediments is a good approximation of the average composition of the sediment source. This observation also excludes the possibility of a significantly lower degree of sediment weathering within the Himalayan range during the LGM, as weathering for K and H2O + is already limited at present.

We therefore consider that the Himalayan front end-member (as constrained from the mixture of 60 % of sediments from the Himalayan front in the Brahmaputra basin and 40 % in the Ganga basin) is a good first order approximation of the sediment source composition, both in the modern day system and during the LGM. The modern dissolved flux of Si is negligible compared to the solid Si fluxes [START_REF] Galy | Higher erosion rates in the Himalaya: Geochemical constraints on riverine fluxes[END_REF]) and this was most likely also the case during the LGM as the chemical weathering intensity was even lower than at present.

Furthermore the Si concentration of the exported sediments is only slightly affected by sequestration of Si-rich bedload sediments in the modern floodplain (Lupker et al., 2012b) and we make the assumption that this has been true during the last 21 kyrs. Hence, we consider constant during the time span of our record.

The change in the weathering flux (RφX) of mobile element X from LGM to present relative to the modern fluxes can therefore be written as:

(3)

K / Si * H 2 O + / Si * Si [ ] R X = X Modern X LGM X Modern = F Sed Modern X Si * HimalFront Modern X Si * BoB Modern F Sed LGM X Si * HimalFront LGM X Si * BoB LGM F Sed X Si * HimalFront Modern X Si * BoB Modern Modern
In order to estimate RφX from eq. 3, the sediment flux (Fsed) has to be estimated at present and during the LGM. Based on extensive gauging data the combined modern flux of the Ganga & Brahmaputra in Bangladesh is estimated to be ca. 1.10 9 t.yr -1 (RSP, 1996). The past sediment flux exported by the Ganga & Brahmaputra is largely unknown. However, qualitatively, the sediment flux was most probably lower during the LGM than at present. A weakened monsoon during the LGM [START_REF] Kudrass | Modulation and amplification of climatic changes in the Northern Hemisphere by the Indian summer monsoon during the past 80 k.y[END_REF][START_REF] Sharma | Lateglacial and Holocene environmental changes in Ganga plain, Northern India[END_REF] may have limited physical erosion in the Himalayan range and lowered the transport capacity of the rivers, leading to a lower sediment flux to the Bay of Bengal [START_REF] Weber | Active growth of the bengal Fan during sea-level rise and highstand[END_REF][START_REF] Goodbred | Response of the Ganges dispersal system to climate change: a source-to-sink view since the last interstade[END_REF].

A very conservative estimate of the minimum value of RφX can therefore be made assuming a constant sediment flux over the period. By doing so we also isolate the effect of the change in the chemical composition of sediments only.

Rφ was computed using eq. 3 for K and H2O + and is summarized in Table 1.

Uncertainties on the K/Si* and H2O + /Si* ratios of individual samples are ca 3% (supplementary information S3). This is of the same order as the variability of K/Si* and H2O + /Si* ratios over the 0 -2 and 19-22 kyr intervals or of the uncertainties on the regression through the data for the Himalayan front sediments. These uncertainties were propagated through the calculations of RφX and reported in Table 1. The sensitivity of the weathering budget to the average Al/Si ratio of sediments is lower than the uncertainty propagated from the variability of K/Si* and H2O + /Si* ratios (see supplementary information S3). This first order estimation shows that the weathering flux during the LGM was at least 67 (±28) and 89 (±36) % lower (for K and H2O + , respectively) than for the modern period (0-2 kyrs) for a constant sediment flux. The main uncertainty on these estimates comes from the uncertainty on the sediment Ganga and Brahmaputra sediment flux during the LGM. A sediment flux during the LGM of half the present day sediment flux would induce a 83 and 95 % lower weathering flux during the LGM compared to present for K and H2O + , respectively.

The loss of K in the Himalayan system is mainly attributed to the weathering of biotite while the gain in hydration also depends on clay secondary mineral formation (Lupker et al., 2012). Because the reaction mechanisms in both cases are different, their sensitivity to changes in precipitation and temperature is also most probably different. Despite our estimates for both proxies being similar within uncertainty, it would not be surprising that the relative changes in weathering fluxes recorded by K and H2O + could be different. It is not possible to estimate the relative change in calcite weathering using a similar approach, because for calcite the Himalayan source composition cannot be reliably estimated. Indeed, significant carbonate dissolution has been documented within the Himalayan range [START_REF] Galy | The strontium isotopic budget of Himalayan Rivers in Nepal and Bangladesh[END_REF], hence, Himalayan front river sediments cannot accurately represent the eroded source. In addition, no direct estimation of the average calcite content of Himalayan rocks is available.

Our quantification of the change in the net weathering flux from LGM to present should only be considered as a rough estimate. In particular, it should be kept in mind that these estimates somewhat depend upon the assumption of a transition between two steady state weathering stages. Nevertheless, our estimates suggest that changes in weathering fluxes exported by the Himalayan system have been significant over short time scales of ca. 20 kyrs.

The magnitude of the variations in weathering fluxes reconstructed here exceeds previous estimates [START_REF] Foster | Negligible glacial-interglacial variation in continental chemical weathering rates[END_REF]. This may be attributed in part to the very high weathering rates of glacial outwash during the very early stage of an interglacial, as proposed earlier [START_REF] Foster | Negligible glacial-interglacial variation in continental chemical weathering rates[END_REF][START_REF] Vance | Variable Quaternary chemical weathering fluxes and imbalances in marine geochemical budgets[END_REF] combined to the short time span of our record.

However, for the Himalayan system the weathering intensity in the modern floodplain (Lupker et al., 2012b;[START_REF] West | Small-catchment perspective on Himalayan weathering fluxes[END_REF] supports the dominant role of the floodplain in modulating glacial to interglacial weathering fluxes. This study further emphasizes that present day weathering rates are likely higher than long-term averages, which supports the idea that the imbalance of oceanic geochemical cycles may at least partially derive from the over-estimation of Quaternary riverine fluxes to the oceans [START_REF] Vance | Variable Quaternary chemical weathering fluxes and imbalances in marine geochemical budgets[END_REF].

Conclusions:

The geochemistry of detrital sediment records holds valuable information on the weathering regime these sediments have undergone during their transit from sediment producing areas to their deposition in sedimentary basins. The chemical composition of these sediments is primarily controlled by sediment sorting effects that occur during transport and deposition. Nevertheless weathering signals can be extracted from the sediment chemical composition if sorting is properly taken into account by comparing sediment records to modern river systems.

The detrital record presented here shows that the degree of chemical weathering experienced by sediments exported by the Himalayan system, drained by the Ganga and Brahmaputra Rivers, has changed significantly since the LGM. An overall decrease in potassium and detrital calcite content along with an increase in sediment hydration all point towards a steady increase in the weathering degree of sediments over the last 21 kyrs, while the sediment provenance has remained unchanged. These changes in the weathering regime are attributed to the direct climatic forcing of chemical weathering in the floodplain, as increasing precipitation and temperatures after the deglaciation favored mineral weathering kinetics. Additional hydrological changes in response to increasing runoff and rising base level might also have impacted weathering through enhanced channel-floodplain interactions.

A first order estimation of changes in the weathering flux delivered to the oceans can be made using the detrital record and conservatively assuming a steady sediment flux. It shows that weathering fluxes from the Himalayan system could have been reduced by 90 % during the LGM. This supports previous evidence that current dissolved fluxes are likely higher than average Quaternary estimates. These drastic changes in solute supply to the oceans on glacial-interglacial time scales have to be accounted for to reconstruct global oceanic biogeochemical cycles. passes through a unique coarse-grained end-member with a composition close to river bedloads (given in the supplementary information, Table S1). which is a proxy for grain size (Bouchez et al., 2011a;[START_REF] Lupker | A Rouse-based method to integrate the chemical composition of river sediments: Application to the Ganga basin[END_REF]. Correction for hydraulic mineral sorting yields unbiased weathering indicators (K/Si* and H2O + /Si*). (Pierson-Wickmann et al., 1998;Galy et al., 2008 and this study) along with the modern river sediments composition of the Himalayan front, the Ganga, the Brahmaputra and the lower Meghna (average and 1 st -3 rd quartile range). The isotopic composition of sediments from the modern Siang River (Singh and France-Lanord, 2002b) is also plotted. The Siang delivers ca. 50 % of the sediment load of the Brahmaputra (Singh and France-Lanord, 2002b), contributing disproportionally to the final isotopic signature of the modern Brahmaputra. 7). We therefore suspect the average Sr and Nd isotopic composition of the Brahmaputra to be closer to that of the Brahmaputra in Guwahati (higher for Nd and lower for Sr) than our whole dataset suggests.

The Sr and Nd isotopic composition of Brahmaputra sediments sampled at Guwahati, upstream of the confluence with the Teesta show a homogeneous composition of 0.7201 ± 0.0014 87 Sr/ 86 Sr and -13.3 ± 0.7 εNd (Figure 7, Singh and France-Lanord, 2002). These sediments are mainly derived from the erosion of the Namche Barwa zone by the Siang River that accounts for ca. 50 % of the Brahmaputra sediment load (Singh and France-Lanord, 2002). The Teesta River has a contrasted isotopic composition due to the dominant exposure of Lesser Himalayan rocks in the basin (Figure 7, Singh and France-Lanord, 2002) and its confluence with the Brahmaputra mainstream is only ca. 120 km upstream of our sampling station in Bangladesh. These considerations may also explain why the contribution of Brahmaputra sediments to the lower Meghna is overestimated based on a simple Sr and Nd isotopic budget of the Ganga, Brahmaputra and lower Meghna.

Increase in Southern or Eastern tributaries and Siwalik contribution since the LGM:

An increase in Siwalik derived sediments would generate a more weathered signature of the Ganga & Brahmaputra sediment load. Because Siwalik are Neogene floodplain deposits produced by Himalayan erosion and uplifted by the southern propagation of the deformation front, Siwalik sediments have a more weathered silicate signature but similar Sr and Nd isotopic compositions than trans-Himalayan river sediments (Lupker et al., 2012).

However, the high carbonate content with lower d 13 C compositions of Siwalik rocks and sediments (e.g. [START_REF] Sanyal | Chemical diagenesis of Siwalik sandstone: Isotopic and mineralogical proxies from Surai Khola section, Nepal[END_REF]Lupker et al., 2012) is not compatible with the concomitant decrease of both carbonate content and d 13 C found in the Bay of Bengal sediments (Figure S1).

Further changes in erosion patterns involving cratonic tributaries are also unlikely.

The Chambal River in the Ganga basin and the Eastern tributaries of the Brahmaputra basin (Lohit and Dibang) have distinctively higher Fe/Si* ratios and unradiogenic Sr and Nd (Singh et al., 2002;[START_REF] Singh | Sr and Nd isotopes in river sediments from the Ganga Basin: Sediment provenance and spatial variability in physical erosion[END_REF]Lupker et al., 2012) signature that are not identified in our record (Figure S1).

S3 -Uncertainties and sensitivity of the weathering budget to the average Al/Si ratio:

The analytical uncertainty from the major element and H2O + measurements is 2% on the Al/Si and Fe/Si ratios, 3% on the K/Si and H2O + /Si ratios [START_REF] Carignan | Routine analyses of trace elements in geological samples using flow injection and low pressure on-line liquid chromatography coupled to ICP-MS: A study of geochemical reference materials BR, DR-N, UB-N, AN-G and GH[END_REF]. The uncertainty on the definition of the coarse grained end-member (Table S1) has only a minor effect on the normalized K/Si* and H2O + /Si* ratios. For an average Bay of Bengal sediment sample with an Al/Si ratio of 0.38, the total uncertainty propagated from the analytical and normalization uncertainty is ca. 3% for the K/Si*, H2O + /Si* and Fe/Si* ratios on a 1σ confidence level.

The sensitivity of the weathering budget to the average Al/si ratio of the sediments is assessed by comparing Rφ determined for an Al/Si of 0.23 to Rφ obtained using an Al/Si of 0.21 and 0.25 (Table S4). This corresponds to ca. 10% of uncertainty on Al/Si and is higher than the ca. 5% uncertainty estimated by [START_REF] Lupker | A Rouse-based method to integrate the chemical composition of river sediments: Application to the Ganga basin[END_REF] for the average Al/Si composition of sediments of the Ganga River but it takes into account the fact that in this case the average Al/Si ratio of Brahmaputra sediments has not been determined directly. Table S4 shows that for the sensitivity of the weathering budget to the average Al/Si is limited for a reasonable range of Al/Si ratios. 806 Table S2: Averaged chemical composition of samples from the Himalayan system. The compositions are given separately for samples from the Ganga Himalayan front, the Brahmaputra Himalayan front, the average Himalayan front, the Ganga in Bangladesh, the Brahmaputra in Bangladesh, the lower Meghna (confluence of the Ganga & Brahmaputra) and southern and eastern tributaries. a The Himalayan front is reconstructed assuming that 60 % of the total sediment flux is generated by front rivers in the Brahmaputra basin and 40 % by the front rivers in the Ganga basin. b For element ratios affected by sediment sorting, the average and standard deviation of each end-member is constrained by evaluating the linear regression through the entire data set of each reach under consideration along with the coarse-grained end-member at Al/Si=0.23. The 1 st and 3 rd quartile, median and number of samples are also reported for all reaches. Detail on the chemical composition of samples used can be found in the electronic appendix.

Figures:Figure 1 :

 1 Figures:

Figure 2 :Figure 3 :

 23 Figure 2: Hydraulic sorting effect on the Si-Al-Fe chemical composition of sediments.

Figure 4 :

 4 Figure 4: Normalization of the chemical composition of sediments in of the Bay of Bengal

Figure 5 :

 5 Figure 5: K/Si and H2O + /Si in Bay of Bengal core sediments are strongly correlated to Al/Si,

Figure 6 :Figure 7 :Figure 8 :

 678 Figure 6: Evolution of K/Si*, H2O + /Si*, Ca/Si and calcite content of sediments in the

87 Sr/ 86 Sr

	Tables								εNd
		0 . 7 0	0 . 7 2	0 . 7 4	0 . 7 6	0 . 7 8	-18 LGM	-16	-14 Modern -12
	Sediment flux (t/yr)					1.10 9	1.10 9
	Himalayan front 1				0.0574 (±0.0025) 0.0574 (±0.0025) 0.1009 (±0.0068) 0.1009 (±0.0068)
	Age ky. (cal. BP)	5 9 13 17 Bay of Bengal HCO			RφK RφH2O+	0.0553 (±0.0005) 0.1034 (±0.0075) 67 (±28) % 0.0510 (±0.0017) 0.1236 (±0.0026) 89 (±36) %
		21 Cores	LGM Shelf Levees	Modern Rivers	Himal. front Ganga Siang	Brahmaputra Lower Meghna

Table 1 :

 1 Relative changes in the weathering flux delivered to the Bay of Bengal as dissolved load from the LGM to present (RφK and RφH2O+) assuming a constant sediment flux over the period. The weathering flux was estimated using eq. 3 for K and H2O + from the Himalayan front and Bay of Bengal end-members reported in the table. H 2 O + / Si * compositions. This sample choice introduces a bias towards samples impacted by the influx from an upstream right-bank tributary, the Teesta, which has high εNd and low 87 Sr/ 86 Sr values (Figure

	K / Si *
	H 2 O + / Si *
	K / Si *

Table S2 : 817

 S2

			Ganga Himalayan Front	Brahmaputra Himalayan Front	Himalayan Front Rivers a	Ganga	Brahm aputra	L. Meghna	S. and E. tributaries
		Mean b	0.063	0.054	0.057	0.052	0.048	0.050	0.036
		sd b	0.001	0.004	0.002	0.001	0.001	0.001	0.007
	K/Si*	1st Q Median	0.061 0.062	0.051 0.055	0.057 0.061	0.051 0.054	0.048 0.049	0.050 0.050	0.029 0.039
		3rd Q	0.065	0.057	0.065	0.055	0.051	0.051	0.060
		n	39	14	53	68	106	60	6
		Mean b	0.099	0.102	0.101	0.122	0.114	0.122	0.065
		sd b	0.003	0.011	0.007	0.005	0.003	0.002	0.004
	H2O + /	1st Q	0.093	0.082	0.090	0.118	0.103	0.117	0.083
	Si*	Median	0.099	0.089	0.098	0.127	0.112	0.119	0.118
		3rd Q	0.109	0.104	0.107	0.142	0.115	0.122	0.159
		n	32	12	44	26	26	21	4
		Mean b	0.052	0.055	0.054	0.056	0.060	0.059	0.082
		sd b	0.001	0.004	0.003	0.001	0.007	0.002	0.012
	Fe/Si*	1st Q Median	0.050 0.052	0.052 0.056	0.050 0.053	0.054 0.055	0.059 0.061	0.057 0.058	0.077 0.078
		3rd Q	0.054	0.061	0.055	0.058	0.062	0.061	0.096
		n	39	14	53	68	106	60	6
		Mean	6.4	0.9	3.1	3.4	-	0.6	5.1
		sd	6.0	1.1	2.5	1.3	-	0.8	3.1
	Calcite	1st Q	0.8	0.0	0.6	2.2	-	0.2	2.8
	(wt%)	Median	4.0	0.1	3.1	3.9	-	0.2	4.9
		3rd Q	12.1	2.0	11.1	4.5	-	0.8	6.8
		n	31	5	36	30	-	6	5
		Mean	-1.3	-0.3	-0.7	-1.6	-	-1.2	-0.7
	δ 13 CCalci te (‰ -PDB)	sd 1st Q Median	1.2 -2.1 -1.2	0.8 -0.6 0.0	0.7 -2.0 -1.2	0.4 -1.6 -1.5	---	0.1 -1.3 -1.2	1.6 -1.8 -1.5
		3rd Q	-0.5	0.1	-0.3	-1.4	-	-1.1	0.2
		n	29	3	32	27	-	3	5
		Mean	-12.3	-9.4	-10.6	-10.0	-	-6.2	-6.8
	δ 18 OCalc ite (‰ -PDB)	sd 1st Q Median	1.4 -13.2 -12.5	3.8 -11.6 -11.4	2.3 -13.2 -12.4	2.6 -11.6 -10.4	---	6.2 -8.4 -3.5	3.7 -9.8 -4.4
		3rd Q	-12.2	-8.2	-11.7	-10.0	-	-2.6	-4.4
		n	29	3	32	27	-	3	5
		Mean	0.115	0.027	0.062	0.058	0.042	0.043	0.115
		sd	0.075	0.021	0.002	0.017	0.018	0.015	0.038
	Ca/Si	1st Q Median	0.040 0.104	0.017 0.018	0.028 0.064	0.042 0.063	0.036 0.040	0.032 0.039	0.098 0.108
		3rd Q	0.170	0.022	0.153	0.072	0.042	0.050	0.137
		n	39	14	53	68	106	60	6
		Mean	0.7713	0.7660	0.76813	0.77491	0.73562	0.73694	0.70705
		sd	0.0101	0.0439	0.02665	0.01009	0.01271	0.00833	-
	87 Sr/ 86	1st Q	0.7647	0.7329	0.74925	0.77111	0.72560	0.72547	-
	Sr	Median	0.7726	0.7600	0.76440	0.77252	0.72779	0.73790	-
		3rd Q	0.7769	0.7870	0.77921	0.77825	0.74756	0.74157	-
		n	8	11	19	21	13	9	2
		Mean	-18.9	-16.6	-17.5	-18.0	-16.2	-15.8	-9.7
		sd	0.9	3.5	2.3	0.4	1.5	1.1	-
	εNd	1st Q Median	-19.3 -18.8	-18.9 -16.0	-19.3 -18.2	-18.3 -18.0	-18.0 -16.0	-16.5 -15.9	--
		3rd Q	-18.3	-14.4	-16.0	-17.8	-14.7	-15.0	-
		n	10	11	21	22	13	9	2

Table S3 : 833

 S3

	Core	Depth	Age	Al/Si	K/Si*	H2O/Si* Ca/Si Fe/Si*	LGM	Calcite			87 Sr Modern	eNd
		cm 77-86	yr BP 47 Sediment flux (t/yr) mol mol 0.346 0.050	mol 0.123	mol 0.032	mol 0.058	dD wt% d 13 C -87 1.10 9		d 18 O 1.10 9	Sil	Sil
		129	74 Himalayan front 0.270 0.054	0.123	0.041	0.064 0.0574 (±0.0025) 0.0574 (±0.0025) -84
	Al/Si = 0.23	7-17 241 50-60 279-285 80 29	98 138 147 161 213 313 Bay of Bengal 0.401 0.052 0.179 0.049 0.391 0.051 0.411 0.049 0.215 0.053 0.405 0.048	0.123 0.124 0.124 0.119 0.121	0.032 0.051 0.034 0.028 0.042 0.034 RφK	0.060 0.048 0.059 0.060 0.069 0.063 0.1009 (±0.0068) -82 -85 2.1 -78 1.4 -78 -86 1.4 0.0553 (±0.0005) 0.1034 (±0.0075) 67 (±28) % -1.3 -12.5 -0.9 -7.5 -1.6 -13.4 0.1009 (±0.0068) 0.74740 0.73600 0.74040 0.0510 (±0.0017) 0.1236 (±0.0026)	-15.7 -15.9 -14.3
		564-572 664-670	325 382	0.335 0.398	0.050 0.051	0.124 RφH2O+ 0.040 0.037	0.060 0.060	-87	89 (±36) %	0.74218	-15.7
		678-685	390 Himalayan front 0.321 0.050	0.123	0.046	0.059 0.0524 (±0.0020) 0.0524 (±0.0020) -86
		686-690	394	0.192	0.049		0.049	0.059		3.1	-1.3	-11.2	0.73437	-15.9
	Al/Si = 0.21	758-765 70-80 53 145-155 134-145 165-175	436 681 1184 1214 1302 1603 Bay of Bengal 0.392 0.050 0.385 0.052 0.447 0.052 0.389 0.051 0.393 0.053 0.400 0.053	0.124 0.124 0.131 0.122 0.125 0.124	0.032 0.025 0.024 0.032 0.032 0.035 RφK	0.060 0.060 0.058 0.057 0.060 0.059 0.0871 (±0.0045) -86 -80 -93 -77 -83 -80 1.2 0.0499 (±0.0004) 0.0865 (±0.0046) 57 (±28) % -1.4 -12.8 0.0871 (±0.0045) 0.74255 0.0465 (±0.0014) 0.0988 (±0.0016)	-15.4
		83	1854	0.412	0.052	0.027 RφH2O+	0.058		105 (±42) %
		210-220 220-230	2212 2317 Himalayan front 0.412 0.051 0.385 0.053	0.122 0.125	0.038 0.028	0.057 0.060 0.0625 (±0.0029) 0.0625 (±0.0029) -75 -79
	Al/Si = 0.25	265-275 255-265 138 147-157 291-301 340-350	2799 2904 3083 3396 3457 3717 Bay of Bengal 0.385 0.053 0.395 0.051 0.378 0.052 0.411 0.051 0.376 0.053 0.385 0.053	0.124 0.124 0.120 0.130 0.126 0.129	0.027 0.040 0.034 0.103 0.043 0.027 RφK	0.060 0.057 0.058 0.061 0.057 0.060 0.1148 (±0.0091) -80 -77 -81 -80 -77 2.1 -84 0.0606 (±0.0006) 0.1203 (±0.0104) 73 (±32) % -0.4 0.1148 (±0.0091) 0.74872 0.74879 -9.1 0.0555 (±0.0021) 0.1484 (±0.0036)	-16.1 -15.3
		335-345	4133	0.337	0.053	0.124 RφH2O+ 0.044	0.057	-82	2.0 84 (±33) % -0.7	-3.9
		376-385	4748	0.362	0.052	0.127	0.074	0.056	-80				0.74153	-16.1
		284	6345	0.304	0.056	0.125	0.059	0.057	-87	2.8	-1.0	-16.9
		565-575	6496	0.397	0.053	0.121	0.030	0.058	-81
		575	6496	0.390	0.054		0.047	0.057
		656	6719	0.407	0.051		0.033	0.055
		920	7418	0.298	0.055		0.061	0.059					0.75414	-17.3
		348	7775	0.461	0.050		0.049	0.062					0.74467	-14.5
		1144	7787	0.340	0.054		0.063	0.060
		411	9182	0.422	0.053	0.123	0.030	0.066	-84
		242-257	10070 0.372	0.054	0.117	0.082	0.059	-80
		452	10098 0.302	0.058	0.115	0.067	0.058	-84
		555	11255 0.412	0.053	0.123	0.070	0.061	-83
		579	11376 0.306	0.054		0.076	0.060		7.2	-0.7		-6.6	0.74723	-15.3
		632-642	11668 0.429	0.056	0.121	0.079	0.061	-76
		691-701	11940 0.429	0.055	0.119	0.078	0.062	-78	4.5	-0.4	-11.0	0.74899	-16.0
		701	11991 0.418	0.054	0.119	0.080	0.063	-80
		762	12298 0.410	0.052	0.119	0.051	0.060	-77
		787	12424 0.322	0.051		0.063	0.064					0.74129	-14.4
		848	12731 0.370	0.054	0.118	0.056	0.059	-77
		845-855	12736 0.364	0.054	0.120	0.059	0.059	-81
		863	12865 0.423	0.051	0.123	0.055	0.061	-77
		722-725	13712 0.401	0.054		0.077	0.061		5.9	-0.4		-8.3	0.74180	-16.0
		929-939	13744 0.410	0.054	0.121	0.066	0.065	-78
		959	14053 0.412	0.053	0.119	0.061	0.061	-77
		772-774	14096 0.400	0.054		0.075	0.060		6.4	-0.4	-10.2	0.72754	-15.7

Table S4 :

 S4 Relative change in weathering fluxes exported as dissolved load to the oceans obtained for K and H2O + using an average Al/Si ratio of 0.23, 0.21 and 0.25. The end members used for the calculation are also reported along with their uncertainty.

	K / Si *
	H 2 O + / Si *
	K / Si *
	H 2 O + / Si *
	K / Si *
	H 2 O

+ / Si * K / Si * H 2 O + / Si * K / Si * H 2 O + / Si * K / Si * H 2 O + / Si *

Acknowledgments:

This work was supported by French INSU program "Relief de la Terre" and ANR Calimero. V.G. was supported by the U.S. National Science Fundation (Grant OCE-0851015, OCE-0928582). We thank C. Guilmette for assistance in the lab. H. Schopka and two anonymous reviewers are also thanked for their constructive comments that helped to improve the manuscript. T.M. Harrison is acknowledged for the very efficient handling of the manuscript.

Supplementary information and data: S1 -Chemical weathering proxies and data:

The coarse grained end-members used to normalize the chemical composition of the Bay of Bengal sediments are reported in Table S1. This coarse-grained mixing end-member is determined as the intersection between the linear regression through the composition of Himalayan front sediments and the composition of sediments in Bangladesh (Ganga, Brahmaputra and lower Meghna) for K/Si and H2O + /Si. For Fe/Si (used in section S2), the common coarse-grained end-member composition is defined by the intersection between the regressions through Himalayan front sediments and southern and eastern tributary sediments (Chambal, Lohit and Dibang) as sediments from these rivers show a very contrasted Fe/Si composition. This method was preferred over averaging all available bedloads as placer effects tend to introduce a very high variability in the chemical composition of the bedloads, which is not found in suspended load samples and therefore does not accurately describe the sediments that are mixed in the water column.

A summary of the composition of modern river sediments is reported in Table S2 and the composition of Bay of Bengal sediments in Table S3. The full, detailed data set of modern sediments and Bay of Bengal sediments is provided as electronic appendix.

S2 -Sediment provenance:

Sr & Nd signature of Brahmaputra sediments:

It should be noted that the Sr and Nd isotopic data set of the Brahmaputra in Bangladesh has partly been acquired in order to test sediment source effects in the Brahmaputra basin by selecting samples that covered the largest span of chemical

Supplementary information figures:

Supplementary information tables: