Eupolars and their bialternality grid

This monograph is almost entirely devoted to the flexion structure generated by a flexion unit E or the conjugate unit O, with special emphasis on the polar specialisation of the units (“eupolar structure”). (i) We first state and prove the main facts (some of them new) about the central pairs of bisymmetrals pal•/pil• and par•/pir• and their even/odd factors, by relating these to four remarkable series of alternals {rer}, {ler}, {her}, {ke2r}, and that too in a way that treats the swappees pal• and pil• (resp. par• and pir•) as they should be treated, i.e. on a strictly equal footing. (ii) Next, we derive from the central bisymmetrals two series of bialternals, distinct yet partially (and rather mysteriously) related. (iii) Then, as a first step towards a complete description of the eupolar structure, we introduce the notion of bialternality grid and present some facts and conjectures suggested by our (still ongoing) computations.

factor Zag • I /Zig • I in the classical trifactorisation of the fundamental bimould Zag • /Zig • that "carries all multizetas"; and secondly because it enters into the construction of the so-called singulators, themselves key to the study of the canonical multizeta irreducibles.
The pair pal • /pil • , as also par • /pir • , had already been dealt with in our previous papers, but somewhat desultorily, on a piecemeal basis. So a unified treatment, complete with motivations, definitions, characterisations and proofs, was long overdue. The sections §2- §8 offer just such a treatment and, as is so often the case, systematisation brings its own rewards. Thus we exhibit two series, unsurpassed for simplicity, of alternals {le • r } and {re • r }, and show that they are connected respectively to pal • and pil • , as the ingredients of the mu-dilator dupal • of pal • and the gari-dilator dipil • of pil • . This is a deeply satisfying state of affairs: it not only restores the symmetry (somewhat impaired in the previous approaches) between the co-equal swappees pal • and pil • but also leads to a simple proof of their bisymmetrality -of all extant proofs, the shortest. Nor do the pleasant surprises stop there. We introduce two additional series of alternals {he • r } and {ke • 2r }, less elementary than the first pair but still capable of a simple, transparent description, and show that these, too, are closely related to ripal • (the gari-inverse of pal • ) and its even factor ripal • ev . It is truly gratifying to see that our four elementary or semi-elementary series of alternals (so far the only of their kind, i.e. the only ones known to admit a simple description) turn out to be, each in its own way, intimately interwoven with the central bisymmetrals.
The paper's second part, from section §9 onwards, deals with the eupolar structure per se, without immediate applications in mind. The main challenge here is to generate, describe, and classify all regular, i.e. neg-invariant bisymmetrals and bialternals. Now, unlike the central bisymmetrals pal • /pil • and par • /pir • , which are irregular (in the sense of being invariant under neither neg nor pari but only under the product pari • neg), the regular bisymmetrals Sa • /Si • (as elements of GARI) correspond one-to-one to the regular bialternals (as elements of ARI) via the exponentiation expari from ARI to GARI 2 . So the attention now shifts to the bialternals which, living as they do in an algebra, are much easier to handle than the bisymmetrals. Starting from the two central-irregular pairs pal • /pil • and par • /pir • , we describe two distinct procedures for producing two infinite series of bialternals, which in turn generate two distinct bialternal subalgebras of ARI. These two subalgebras do not coincide but partly overlap -though how far is yet unclear. Nor do we know whether, between themselves, they generate all bialternals.
This ignorance is galling. It is true that at the moment the polar bialternals, unlike the central bisymmetrals, 3 have no known applications to multizeta algebra. But this may change. It would indeed be strange if the eupolar structure, even in its most recondite aspects, did not have some bearing on the study on multizetas. On the contrary, there is every reason to believe, and past experience strongly suggests, that most difficulties, irregularities or anomalies besetting multizeta theory 4 originate in the eupolar domain which, being itself purely singular, holds the key to all the 'singularity' scattered over the wider flexion field. Be that as it may, and all applications aside, the eupolar structure is a fascinating subject in its own right and deserves to be studied for its own sake.
So how are we to advance our knowledge of polar bialternals? Paradoxically, by widening the search: instead of obsessing about the sole bialternals and the spaces ARI al/al r = ARI (1,1) r spanned by them, we may relax the notion and consider the larger spaces ARI (d 1 ,d 2 ) r spanned by all eupolars of a (suitably defined) bialternality codegree (d 1 , d 2 ). The new approach embraces all eupolars, since for (d 1 , d 2 ) large enough 5 ARI (d 1 ,d 2 ) r coincides with the whole of ARI . Moreover, the dimensions which constitute the entries of the so-called bialternality grid, seem to follow a remarkable pattern. In particular, when we add the quite natural requirement of push-invariance, every second grid entry vanishes, leading to the so-called bialternality chessboard.
The corresponding computations, however, are extremely complex and progress only haltingly. At the moment we are stuck at length r = 8: enough to discern the outlines of a tantalising pattern; not enough to see the full picture emerge. The investigation goes on but it may be quite some time before the next batch of data arrives. 6 So, rather than delay indefinitely the paper's publication, we have chosen to post this first, incomplete and somewhat 3 and, of course, unlike the polynomial bialternals! 4 like, for example, the existence of the exceptional, polynomial-valued bialternals carma • /carmi • . See E1 and E2. 5 d 1 + d 2 > r suffices. 6 With many flexion operations, especially when working in algebras, it does not take much computational power to reach even length r = 20. With others, such as inflected group inversion, inflected exponentiation or, like in the present instance, when it comes to expressing that a bimould has a given bialternality codegree, difficulties arise much earlier.
sketchy draft. We mean to update it regularly as the computations progress. §1-2. Conceptual vs mechanical proofs. The priorities of exploration.
The sheer profusion of formulae in flexion theory makes it strictly impossible to write down regular proofs for each one of them. Clearly, identities involving such key bimoulds as pal • /pil • deserve to be established with care, to do justice to the centrality and flagship quality of these objects. But what about the common run of flexion formulae? For them, it would be nice (time-saving and reassuring) to be able to fall back on a M echanical truth criterion (conjectural): Any bimould-valued flexion identity of the form is automatically true for all lengths r as soon as it holds identically for all arguments A • j and all lengths r ≤ d + 1.
This of course would require that we properly define the partial depths in formula (2). The depth of 'products' F i (associative or pre-Lie) would be 1; that of 'alternate' operations (commutators, Lie brackets etc) would be 2; and that of complex operations like the singulators would probably have to be 3 or 4. The depth of the arguments A • j would be 1 when A • j is allowed to range unrestrained over BIMU ; or 2 if when A • j ranges over the set of all bimoulds with a simple symmetry; or again 3 or 4 if when it ranges over all bimoulds with a regular double symmetry.
Though the existence of some such truth criterion would seem almost certain, none has been established as yet. On the other hand, in the identities commonly encountered in flexion theory the total depth d, summarily assessed along the above lines, rarely exceeds 6 or 7. So we may make safety doubly or trebly safe by verifying our identities up to the length 2d or 3d instead of d + 1, which remains well within the range of the computationally feasible, and if the identities pass the test, confidently assume their validity.
But there is a catch here: in many important instances the arguments A • j do not range over a vast enough domain of BIMU. For instance, the irregular (though central!) bisymmetrals pal • /pil • are fairly 'isolated' creatures, unlike the regular 7 (though less central!) bisymmetrals Sa • /Si • . For the likes pal • /pil • or par • /pir • , therefore, no 'mechanical truth criterion' would work, and there is no way we can dispense with regular proofs here.
That said, careful consolidation, essential in the central, vital parts of an evolving theory, is one thing, and unfettered exploration, normal and legitimate at the fringes of the theory, is another. Each has its own logic, norms, and imperatives, and it would be foolish to mix up the two. §1-3. Lie or pre-Lie brackets and group laws. Anti-actions.
This first paragraph is there simply to dispel possible misconceptions about the flexion laws, the corresponding anti-actions, and the impact on these of the basic involution swap, which is the very glue of dimorphy.
First, we have the overarching structure AXI/GAXI, whose elements are bimould pairs A • = (A • L , A • R ). Then we have the unary structures (seven in number, up to isomorphism) consisting of simple bimoulds A • and corresponding to as many substructures of AXI/GAXI, each one of which is defined by an involutive linkage A • R ≡ h.A • L between left and right components (the number of suitable involutions h is of course very limited).
Let A I/GA I be such a unary structure 8 ; let I A/GI A be the mirror structure under swap; and let h 1 , h 2 , h 3 , h 4 be the four corresponding involutions: a i −→ h 1 ; i a −→ h 2 ga i −→ h 3 ; gi a −→ h 4 The laws are simply derived from the overstructure AXI/GAXI: The anti-actions also are similarly defined: and clearly cannot, since the right-hand sides (above) fail to define a muderivation resp. a mu-isomorphism.
Nonetheless, the laws may be expressed in terms of the anti-actions. Thus for the first law we have: Of course, the same identities hold with "a i" changed everywhere to "i a". §1-4. Left-right separation.
The phenomenon is summed up by the following identities, which speak for themselves: The last two identities are easier to check in the following, equivalent form: Closure under the basic involution swap .
There exist many "closure identities", which essentially reduce i a / gi a to a i / ga i . We mention the only one that we shall really require: That standard basis has cardinality (2r)!/(r! (r+1)!) and admits a natural indexation either by r-node binary trees t or by some special r-term sequences t that stand in one-to-one correspondance with these trees. The basis elements are defined inductively: with w = w 1 .w i .w 2 and r 1 +r 2 = r−1 and the corresponding inductions for trees and sequences go like this: Here, {t 1 ← • → t 2 } denotes of course the binary tree we get by glueing t 1 (resp. t 2 ) to the root-node • as its left (resp. right) branch. On the sequence side, r 1 denotes the length of t 1 and t 2 (r 1 +1) results from t 2 by adding r 1 +1 to its every element, after which we concatenate everything, thus producing a sequence t that is some well-defined permutation of [1, 2, . . . , r].
What we now need is an algorithm for projecting the general element X • of Flex r (E) onto the standard basis. The following formula does just that: with projectors Res i 1 ,...,ir capable of two interpretations: (i) Res i 1 ,...,ir := Res u ir . . . Res u i 2 Res u i 1 (ii) Res i 1 ,...,ir := Res v i 1 .
Mark the order inversion from (i) to (ii). To calculate, Res u i X • , we set all variables v i equal to 0; then take the coefficient of E ( u i 0 ) minus 9 the coefficient of E ( −u i 0 ) ; then set u i = 0. Performing the operation r times, successively with Res u i 1 , Res u i 2 etc, we end up with a scalar that does not depend on the particular expression chosen for X • (elements of Flex r (E), we recall, admit many different expressions).
To calculate Res v i X • , we go through exactly the same motions, but with the roles of the u i 's and v i 's exchanged and the order of the operations reversed. Once again, the final result does not depend on the expression 10 of X • , and coincides with the result of the first procedure.
Clearly, in the polar specialisation E = Pa (resp. Pi ), the operator Res u i (resp. Res v i ) corresponds to the taking of the residue at u i = 0 (resp. v i = 0). §1-7. Dilators: what are they, and what are they good for?
Infinitesimal generators and dilators have this in common that they often permit to rephrase problems about groups as more tractable problems about algebras. But of the two, the dilators are the more useful by far, mainly because they are so much closer, conceptually and computationally, to the group elements from which they derive.
Here is how the inflected dilators diS • and daS • and the uninflected dilator duS • relate to the corresponding group element S • (henceforth referred to as the dilatee): The three relations are entirely parallel: indeed, the Lie bracket corresponding to mu is lu and mu may (trivially) be regarded as a pre-Lie bracket prelu for lu. As for the operators der and dur, they are mu-derivations each: der.S w 1 ,...,wr := r S w 1 ,...,wr (20) dur.S w 1 ,...,wr := (u 1 +. . . u r ) S w 1 ,...,wr 9 Of course, flexion units being odd functions of their variable w i = ( ui vi ), we have E ) , but since complex superpositions of flexion operations are liable to yield either form, both possibilities must be taken into account.
10 Elements of Flex (E) can be expressed/expanded in numerous, outwardly distinct ways and, when resulting from a sequence of flexion operations, they usually appear, prior to simplification, in an absurdly complicated shape.
In the context of the monogenous structures Flex r (E) the latter derivation dur is particularly relevant when E = Pa but even then it has the slight drawback of taking us out of Flex r (E) into something which, with due quotation marks, might be called "Flex r (E) ⊗ {I • }", with an elementary I • that is 1 or 0 according as the length r(•) is 1 or not. 11 To remedy the non-internal character of dur, we must sometimes replace it by duur, which is a bona fide internal mu-derivation of Flex (E) into itself. Since all elements of Flex r (E) may be expressed 12 as a superposition of terms M • r of the form it is enough to say how duur acts on these M • r , and here is how it acts: The corresponding dilator relation then assumes the form or the equivalent form with muu denoting a sort of integration-by-part operator but with the twist that the underlying product mu is non-commutative: or more rigorously: Relations between inflected and non-inflected dilators.
For any S • such that S ∅ = 1, the inflected dilators diS • , daS • and the non-inflected dilator duS • relate according to: 11 I • is the unit for mould composition • and should be carefully distinguished from the multiplication unit 1 • which is 1 or 0 according as the length r(•) is 0 or > 0.
The shortest way to prove (26), (27) is to rewrite the dilator identities (17), (18), (19) as follows and to observe that since the derivation dur commutes with all three derivations der , arit(diS • ), irat(daS • ), we have: To establish (27), which we shall require in the sequel, we apply the commutator [D 2 , D 3 ] to S • . We get successively: Since we assumed S ∅ = 1, our S • is mu-invertible. So we may mu-divide the last identity by S • on the left, and what we are left with is exactly the sought-after identity (27). The proof of (26) is entirely analogous. We may note that since the relations (26) and (27) are of the form r(w).duS w = u .diS w + earlier terms (32) r(w).duS w = u .daS w + earlier terms (33) they clearly determine diS • and daS • in terms of duS • and vice versa.
We may also observe that since prelu := mu is, trivially, a pre-Lie law for the Lie law lu, the relation (26), (27) can be rewritten in the following, particularly harmonious form: Furthermore, although there exists no simple direct relation between the inflected dilators diS • and daS • , there exists, interestingly, an indirect one, via the non-inflected duS • . §1-9. Dilatees in terms of the dilators.
One goes from a mu-dilator duS • or duuS • to the source element S • (the "dilatee") via the identities: with a symmetral mould Paj • defined by: Similarly, one goes from a gari-dilator diS • to the source S • via the identity: with the same auxiliary mould Paj • but differently indexed. An analogous formula expresses the product T • = gari (R • , S • ) in terms of the dilators: 13 Mark the absence of r 0 in Paj r 1 ,...,rs . We may also, and often must, express the operators garit(S • ) and adari (S • ) in terms of diS • : where ari denote the adjoint action of ARI on itself. 14 The indexation of the operators ari (diS • r i ) and arit(diS • r i ) goes in opposite directions, but this should not come as a surprise, since adari defines an action (of GARI on ARI) and garit an anti-action (of GARI on BIMU). 13 Of course, on the right-hand side of (40), we must substitute for S • the expansion (39) and do likewise with T • . 14 i.e. ari (A • ).B • ≡ ari (A • , B • ). §1-10. Some other dilator identities.
How does the gari-product affect dilators? Like this: Since according to (42) adari(S • ) ±1 can also be expressed in terms of diS • , the above identity amounts to a sort of Campbell-Hausdorff formula for the composition of gari-dilators. In the same vein, we must mention the conversion formulae between with an alternal mould Japaj • := Compo(Ja • , Paj • ) defined as Paj • precomposed by the elementary mould Ja x 1 ,...,xr := (−1) r x 1 . Thus we get: The conversion liS • → diS • is via an even simpler formula: with an elementary alternal mould Bin • defined by: (47) §1-11. Internals and externals.
A bimould A • is said to be internal if, for all r, it verifies two dual properties, which in short notation read: and in long notation assume the more natural form: ,..., ,..., ur vr } (51) Internals constitute an ideal ARI intern of ARI resp. a normal subgroup GARI intern of GARI . The elements of the corresponding quotients are referred to as externals: Moreover, when restricted to internals, the ari bracket reduces, up to order, to the simpler lu bracket, and the gari product, again up to order, reduces to the mu product: Lastly, we have two useful identities governing the action of internal bimoulds on general ones: (57) and two anologous identites for the action of general bimoulds on internals: Pay attention to the order of the terms, and observe that any bimould, acting on an internal, produces an internal: arit(ARI) . ARI intern ⊂ ARI intern (60) garit(GARI) . GARI intern ⊂ GARI intern (61) §1-12. Short guide to the nomenclature. In the polar specialisations, for reasons we cannot go into here, the conventions have to be slightly different: the root vowel here is a (resp. i) for elements of Flex (Pa) (resp. Flex (Pi )) but the exchange a ↔ i under conservation of the consonental skeleton usually reflects the swap transform: thus pal • ↔ pil • and par • ↔ pir • . To express the syap transform, on the other hand, we usually change the final consonant plus of course the root vowel: thus pal • ↔ pir • and pil • ↔ par • . Since swap and syap thankfully commute, this leads to no major inconsistencies.

Elements of
Lastly, inversion under the group laws, whether in the 'Gothic' or 'Roman' context, is usually denoted by a prefix reminiscent of the law: ri for gari, ra for gira, mu for mu. The same applies for the dilators, which take the prefix di, da, du depending on the parent group. We shall construct in Flex (E) two elementary and two semi-elementary series of alternals by giving in each case a direct description side by side with an inductive definition. §2-1. The first alternal series {re • r } .
The inductive definition, which immediately implies alternality, reads: To get a direct definition-description of re • r , we may proceed like this. For any sign sequence = { 1 , . . . , r−1 }, we define the decreasing sets J i ( ) by setting J 1 ( ) := [1, 2, . . . , r] and, for 1 < i ≤ r, by taking J i ( ) to be J i−1 ( ) deprived of its largest (resp. smallest) element if i−1 = + (resp -). Then: with indices u * i ( ), v * i ( ) defined by the dual conditions: Of course, for i = 1 we must set v j = 0. Alternatively, one may say that, when projected onto the standard basis {e • t } of Flex (E), the alternal re • r takes the coefficient (−1) k when t is a onebranch tree with k right-leaning slopes, and the coefficient 0 whenever t has more than one branch.
The most outstanding property of the alternals re • r is their self-reproductioǹ a la Witt under the ari bracket: Here the direct definition reads: Alternality is nearly obvious on this definitious. It is even more obvious for the closely related bimoulds len • r : Clearly len • r = duur.le • r , since we have on the one hand and on the other which again implies: This last expression (69) ensures the alternality of len • r and the earlier identity len • r = duur.le • r carries alternality back to le • r . §2-3. The third alternal series {he • r } .
We begin here with the direct, descriptive definition, which relies on the standard basis {e • t } of Flex (E). The coefficients he(t) of he • r in that basis are not going to depend on the full structure of the indexing binary trees t but only on a four-parameter 'abstract', slant(t), which gives the numbers p 1 , p 2 (resp. q 1 , q 2 ) of left-leaning (resp. right-leaning) slopes in the two branches issueing from the tree's root node. Clearly, p 1 +p 2 +q 1 +q 2 = r −1, and the inductive calculation of slant(t) goes like this. If e • t = amnit(e • t , e • t ).E • with slant(t ) = p 1 q 1 p 2 q 2 and slant(t ) = p 1 q 1 p 2 q 2 , then slant(t) = 1 + p 1 + p 2 q 1 + q 2 We can now define e • t : through coefficients he(t) = he p 1 q 1 p 2 q 2 that depend only on slant(t): with the usual abbreviations p 12 := p 1 +p 2 , q 12 := q 1 +q 2 .
The invariance, implied by alternality, of the he • under mantir := minu.anti .pari = −anti .pari is immediate since it amounts to he p 1 q 1 p 2 q 2 ≡ (−1) p 1 +p 2 +q 1 +q 2 he q 2 p 2 q 1 p 1 but the full alternality is less obvious. It may be derived from the following identities. Indeed, setting with rë • r := swap.ro • r for ro • r := syap.re • r , 17 and introducing two elementary, mutually gani-inverse bimoulds se • , nise • : we can check (see (245)-(246)) either of the two equivalent identities: These new alternals are defined only for even lengths r = 2r * . Like for the preceding series, we begin with a direct, descriptive definition by projection on the standard basis of Flex (E). Here too, the coefficients do not depend on the full structure of the indexing binary tree t but on a four-parameter 'abstract', stack (t), which gives the numbers m 1 , m 2 (resp. n 1 , n 2 ) of endnodes (resp. non end-nodes) carried by the two branches issueing from the root-node. Like in the previous case, we have m 1 +m 2 +n 1 +n 2 = r−1 but, unlike in the previous case, there now exist obvious inequalities between the m i 's and the n i 's. As a result, for any given (even) length r, the number of distinct stacks will be less than that of of distinct slants.
The inductive definition of stack (t) goes like this. If We are now in a position to define ke • 2r * through coefficients ke(t) = ke m 1 n 1 m 2 n 2 that depend only on stack (t): with the usual abbreviations m 12 := m 1 +m 2 , n 12 := n 1 +n 2 and with the odd or double factorial 18 : The above definition of ke • 2r * is concise enough, and striking too, but one thing it leaves in the dark 19 is the alternality of ke • 2r * . One way (and as far as we know, the only way) round this difficulty is to relate {ke • 2r * } to {he • r }. To this end, we set: and we introduce the elementary operator P (adjoint action on ARI): The thing is now to establish the identity: or the equivalent but computationally more economical identity, which involves half as many terms and may be derived by inverting (90) to then parifying (92) to and lastly inverting (93) back to (91). For ways of establishing (90) we refer to the paragraph "properties of ripal • ev " (see §4.7 below). But here again, if we are loath to go through the tedium of establishing (90) or (91) straight from the beautiful descriptive definition (83), we may forgo that direct definition and simply take (91) as the definition of ke 2r * . This is sufficient for all practical purposes and it gives us the alternality of ke 2r * without our having to fire a single shot.

Remark: parity separation in {he
From (90) and (91) we derive, after elimination of Ke • ev , an interesting way of expressing the odd-length components he • 2r * +1 in terms of the even-length components. Indeed, setting: we get: Of course, exp(P), cosh(P), tanh(P) etc should be interpreted as power series of the operator P. §2-5. Tables for length r = 4: the elementary alternals.
basis element Tables for length r = 4: the semi-elementary alternals.

Polar bisymmetrals: main statements.
For perspective, let us start with a synoptic table of our central bimoulds: We take our stand on the self-reproduction property (66) of the alternals re • r under the ari bracket, which is entirely analogous to the behaviour of the monomials x r+1 under the bracket {φ, ψ} := φ ψ −φψ . As a consequence, the Lie algebra isomorphism induced by x r+1 → re • r extends to an isomorphism of the group of formal identity-tangent mappings f := x → x + a r x r+1 into the group GARI re consisting of bimoulds of the form S • := expari ( γ r re • r ). All elements of GARI re are automatically symmetral.

Proposition 3.1 (Direct bisymmetral: definition)
The source mapping f : x → 1 − e −x = x − 1/2 x 2 + . . . has for images in GARI re resp. GARI ro bimoulds denoted by ess • resp. oss • . They are automatically symmetral, but their swappeesöss • resp.ëss • are also symmetral. The same-vowelled bimoulds ess andëss (and by way of consequence oss and oss) coincide up to length r = 3 inclusively but differ ever after. Under the polar specialisation (O, E) → (Pa, Pi) our universal bimoulds specialise to: At this point, the reader may well ask: why, among all identity-tangent mappings f , single out precisely f : x → 1 − e −x ? The short answer is: because only this choice and no other 20 ensures that the separator gepar (ess • ) be symmetral (see (109)) below), which in turn is a necessary condition for oss • (not ess • !) to be symmetral. The condition, however, is not sufficient, and the full bisymmetrality proofs (two of them), as indeed all the other proofs backing up this section's statements, shall be given in §4.

Proposition 3.3 (Inverse bisymmetral: properties)
The gari-inverses (prefix " ri") of the bisymmetrals are automatically symmetral, but they are not bisymmetral, meaning that their swappees, which may also be viewed as gira-inverses (prefix " ra") are not exactly symmetral, but rather E-symmetral or O-symmetral, depending of course on the root vowel. Thus side by side with the straight symmetries we have the tweaked symmetries In the polar specialisation (O, E) → (Pa, Pi) this becomes (106) We now recall the definition of the two separators 22 gepar and hepar gepar.
Proposition 3.4 (Direct bisymmetral: separators) . The separation identities read with their obvious analogues under the exchange e ↔ o.

Proposition 3.5 (Inverse bisymmetral: separators)
The separation identities read They possess obvious analogues under the exchange e ↔ o. Here mu r (O • ) stands, as usual, for the r-th mu-power of O.
Proposition 3.6 (Direct bisymmetral: gari-dilator) The identity reads and has an obvious analogue under the exchange e ↔ o.
Proposition 3.7 (Inverse bisymmetral: gari-dilator) The identities read with dilators equal to and with the semi-elementary alternals ho • r defined as in (73) with muu defined as in (25) and the elementary alternals lo • r defined as in §2 but with respect to the unit O instead of E. The coefficients α r are the Bernoulli numbers : Under the polar specialisation O → Pa, the above relations assume the simpler form: relatively to the elementary alternals Before examining the parity properties of our bisymmetrals, a few general considerations are in order. It is clear that any bimould M • such that M ∅ = 1 can be uniquely factored as follows or in reverse order with factors that of course differ from (125) to (126) but in both cases satisfy the parity conditions: With the 'upper' factorisations (125), for example, we find From there, by square rooting, 23 we go to M • od and M • odd and thence to M • ev and M • evv . None of this requires M • to be symmetral or in Flex (E). Elements of Flex (E), though, behave identically under pari and neg, so that for them the labels even and odd acquire redoubled significance.
In any case the existence of even × odd or odd × even factorisations is a universal phenomenon. 24 What distinguishes the bisymmetrals is the existence of remarkable and multiple factorisations of that sort, with odd factors that tend to be exceedingly simple.
23 an unambiguous operation, if we impose, as we do, that Proposition 3.9 (Parity properties) We have three similar-looking but logically independent identities: with six symmetral factors. Three of these, namely ess • ev ,öss • ev , andöss • evv are highly non-elementary and "even", i.e. simultaneously invariant under neg and pari, which implies that they carries only non-vanishing components of even length. The bimoulds in the next triplet, ess • od ,öss • od andöss • odd , are quite elementary, being given by: or more explicitly: oss w 1 ,...,wr They are also "odd" in the sense of being invertible under pari or neg: Three points deserve attention here. First, note the presence of a factor 1 r! in (137) and its absence in the inflected counterparts (135) and (136).
Second, there is no equivalent to (140) on the E-side, that is to say, no remarkable mu-factorisation 25 of ess • , whether of type mu(ess • evv , ess • odd ) or of type mu(ess • odd , ess • evv ).
Third, while ess • /öss • are swap-related, ess • od /öss • od are syap-related and ess • ev /öss • ev are not related at all (in any simple way). There would be some justification, therefore, for denoting the odd factor oss • ev rather thanöss • ev , though in a way that too might be confusing. The truth is that this theory is so replete with symmetries that no nomenclature can possibly do justice to them all.

Proposition 3.10 (Even factors: separators)
The separators of ess ev are unremarkable 26 but those of riess ev exactly mirror, up to parity, the formulae for riess: Proposition 3.11 (Even factors: gari-and gira-dilators.) The three identities read Warning: the simultaneous occurrence of ev/evv in (145)  We may note, besides, that due to (149) the 'jumbled' identity (145) can be rewritten as follows: with id −anti rather than id +anti in front of daöss • ev .
Proposition 3.12 (Inverse even factor: gari-dilator) We have two similar looking but logically totally distinct identities with dilators equal to and with the semi-elementary alternals ko • 2r defined as in §2 but based on the unit O instead of E.
Proposition 3.13 (Even factors: mu-dilators.) We have two similar looking but logically rather distinct identities with the bilinear product muu defined as in (25) and the same elementary alternals lo • r as above. The coefficients α 2r are also the same as in (121) except for the omission of α 1 , but (158) involves new coefficients β 2r given by Under the polar specialisation O → Pa the above relations assume a simpler form, with muu replaced by the familiar product mu : and with relatively to the same elementary alternals lan • r as in (124).
This concludes our list of 'main statements' about the bisymmetrals. For easy reference, we now tabulate the main source functions behind their separators and dilators. In all the instances encountered in this section (six in all), we list the identity-tangent diffeomorphisms f with their images in GARI re or GARI ro for the unit choice E or O and the corresponding polar specialisations: along with the four relevant generating functions: : carries the coefficients of the garidilators.
: carries the coefficients of the second separator hepar.
2 = Schwarzian of f : ought to carry the coefficients of a conjectural third separator (still unknown).
All separator identities in §3 result from the general statement: If fi • is the image in the group GARI re of the identity-tangent mapping f : x → x + 1≤r a r x r+1 , then its two separators are of the form gepar.fi w 1 ,...,wr = a * r Pa w 1 . . . Pa wr with a * r = (r + 1) a r hepar.fi w 1 ,...,wr = a * * r Pa w 1 . . . Pa wr with 1≤r a * * r x r := x 2 To prove (191) we note that the bimould fi • , being the image of f , has a gari-dilator of the form: so that its swappee fa • has a gira-dilator of the form: 194) with sra • r := swap.ri • r and with identical coefficients α r given by Due to the very special form of sra • r and anti .sra • r : anti.sra w 1 ,...,wr = P (u 1 + ... u r ) the pre-bracket preira in (194) may be replaced by preiwa, which becomes: Setting gefa • := mu(anti .fa • , fa • ) and applying the mu-derivation der to both sides, we find, in view of (197) and anti .iwat(sra • ) = iwat(sra • ).anti : Using the elementary identities and it is but a short step fom (198) to (191). The proof for hepar runs along similar lines but is more intricate. Since we do not really require the result in the sequel, let us just mention the key step in the argument. Let r = {r 1 , ..., r s } denote any non-ordered sequence of s positive integers, and let fa • r resp. lofa • r denote the part of fa • resp. lofa • that is multilinear in sra • r 1 , . . . , sra • rs . Applying the rules of §1-9 we find: Next, consider Although rofa • r has a much simpler (less composite) definition than lofa • r and actually differs from it as soon as r ≥ 2, one can nonetheless show that after pus-averaging the two expressions do coincide: This is a standard application of the correspondance f → f # . See the Table 1 at the end of the preceding section, where f 0 (x) ≡ f # (x)/x. See also §4 in [E3], from (4.11) through (4.17). §4-3. Bisymmetrality of pal • /pil • : first proof.
This proof strives to be even-handed, in the spirit of dimorphy: it treats pal • and pil • in exactly the same way, by relating each to its dilator. So, rather than defining pil • from its source mapping f as in Proposition 3.1, we adopt the following, strictly equivalent definition, polar-transposed from Proposition 3.6 and based on the gari-dilator dipil • : The alternals ri • r are of course the specialisation of re • r under E → Pi . We then consider a bimould pal • defined, not as the swappee of pil • , but directly and independently, via the mu-dilator dupal • : with dupal • := 1≤r α r lan • r α r as in (121) with the same Bernoulli coefficients α r as in Proposition 3.8 and with lan • r being the specialisation of len • r under E → Pa. See §2. Quite explicitely: Both dilators dipil • and dupal • being alternal, it immediately follows that pil • and pal • are symmetral: this is obvious from the inversion formulae (36) and (39) and from the symmetrality of the mould Paj • common to both. So everything now reduces to showing that pal • is actually the swappee of pil • or, what amounts to the same, that the system (206) that defines pal • is equivalent to the system deduced under the swap transform from the system (205) that defines pil • . Before taking that one last step, let us recall the universal relation (27) between the gira-dilator daS • and the mu-dilator duS • of a given S • : (209) which, as observed in the universal case (cf §1), determines dapal • in terms of dupal • and vice versa. Now, this appealingly symmetrical and winningly simple relation (209) involves only elementary monomials Pa(.) and readily follows from the basic identities (199), (200) and (207).
This establishes beyond cavil that the symmetral bimould pil • as defined by (205) and the equally symmetral bimould pal • as defined by (206) are mutual swappees.
Remark: This last identity (209) is totally rigid in the sense that if we tinker with the common coefficients −1/(r +1)! of dipil • and dapal • , there is no way we can adjust the coefficients α r of dupal • to salvage (209). This rigidity will stand us in good stead in [E4] for unravelling the structure of the trigonometric bisymmetrals tal • /til • . For a foretaste, see §17 infra. §4-4. Bisymmetrality of pal • /pil • : second proof.
This alternative proof is more roundabout 28 but makes up for it by yielding valuable extra information. We now starts from pil • and its gari-inverse ripil • , which are automatically symmetral by construction. The challenge is to show that pal • (now defined derivatively, as the swappee of pil • ) is also symmetral or, what amounts to the same but turns out to be easier, that its gari-inverse ripal • is symmetral. The key here is to compare ripal • with the swappee rapal • of ripil • , which may be also be viewed as the gira-inverse of pal • (hence the prefix "ra"). According to (10) ripal • is also the ras-transform of rapal • : The following picture sums up the situation: In view of (9) we also have: Replacing push by its definition (391) in (212) So we end up with with an elementary pac • that admits an equally elementary gani-inverse nipac • : Thus, in view of (8), we go from ripal • to rapal • and back via the relations Now, it is an easy matter to ckeck 29 that ganit(pac • ) : alternal //symmetr al −→ alternul //symmetr ul (228) ganit(nipac • ) : alternul //symmetr ul −→ alternal //symmetr al (229) Let us now write down the dilator identity for ripil • (see (151)-(153)) and the logically equivalent identity for the swappee rapal • : As usual, sra • r := swap.ri • r . More explicitely: From that we infer the shuffle identity: which in turn easily implies that the dilator darapal • , as given by (239), is alternul. 30 Now, if from "darapal • ∈ alternul" we could directly deduce "rapal • ∈ symmetrul", life would be easy: we could, applying (227) and (229), immediately conclude that ripal • and therefore pal • are symmetral, and be done with it. Unfortunately, we cannot 31 -at least not directlyand must take the detour through the dilators darapal • and diripal • . So our goal now is to go from the proven identity (231) to an identity of the form: and from there to the identity: To deal with the first step, let us parse the identities (231) and (236) respectively as A 1 + A 2 = 0 and B 1 + B 2 = 0 with (239) and then check that: 30 This fact is already mentioned in [E3], in "universal mode": see (4.6) p 73. 31 To do that directly, we would require the alternulity of the gari-dilator dirapal • of rapal • (not considered here) rather than the alternulity of its gira-dilator darapal • (considered!). Extreme caution is called for here; great care must be taken to distinguish between the various dilators: diripil • (linked to ripil ), diripal • (linked to ripal ), and the pair darapal • /dirapal • (both linked to rapal • , but in different ways). Always pay close attention to the vowels and their placement: no agglutinative language with vocalic alternation could beat flexion theory for fiendish intricacy! But that's no fault of ours. That's just the way things are, and there in no point in carping. (241) is simply the definition of diripal • : see (236), second line. To prove the non-trivial part, namely

The relation
we apply to rapal • both terms of the operator identity which is easier to check in this equivalent formulation: 32 Thus, the mu-isomorphism ganit(nipac • ) takes us from (231) to (236), thereby establishing the latter identy, with a dilator diripal • which, being the image under ganit(nipac • ) of the alternul darapal • , is automatically alternal. This in turn immediately implies that ripal • and pal • are symmetral. In also implies, in view of (227), that rapal • is symmetrul -the very property, recall, that we could not directly derive from "darapal • ∈ alternul". This completes our second, less direct proof of the bisymmetrality of pal • /pil • . What it doesn't do, though, is prove that our definitely alternal bimould diripil • admits the exact expansion (237), with ha • r the polar specialisation of he • r under E → Pa. To rigorously establish this non-essential, but very nice extra bit of information unfortunately requires rather lengthy and tedious, though in a sense elementary calculations. One way to proceed is to start from the expansion (231) of darapal • ; to apply ganit(nipac • ) to each sra r • separately, resulting in a bimould hasra r • with infinitely many non-vanishing components: One may then expand each hasra • r,r * in the standard basis of Flex r * (Pa), where it admits a rather simple, highly lacunary projection; and eventually piece everything together inside the double sum 32 These are 'rigid' identities, strictly dependent on the nature of the inputs: if we were to modify the definition of darapal • by, say, modifying the coefficients of sra • r in (231), we would have to simultaneously modify the pair pac • , nipac • of gani-inverse elements.
The combinatorially minded reader may fill in the dots. 33 To conclude, let us sum up the various steps of the whole argument (our second bisymmetrality proof -) with the number of stars alongside each arrow reflecting the trickiness of the corresponding implication: Even and odd factors of pal • /pil • .
We must first establish the three factorisations (129), (130), (131). Despite their air of kinship, they are in fact quite distinct, and must be dealt with separately. Under our preferred polar specialisation (E, O) → (Pi , Pa) they become respectively: (i) The first factorisation (247) merely reflects the factorisation f = f od • f ev of the source diffeomorphisms. Explicitly: Of course, as a function, f ev (x) is odd and f od (x) is neither odd nor even, but what matters in this context is that the quotient f ev (x)/x should carry only 33 There exist alternative strategies, like applying ganit(nipac • ) to sra • r as (indirectly) defined by (231) and summing, not in i and then r as above, but rather in r and then i, but all these approaches seem to lead to calculations of roughly the same complexity and tediousness. even powers of x and that f od (•) should admit −f od (− •) as its reciprocal mapping.
(ii) The second factorisation (248) is less immediate to derive. We first observe that if we specialise E to Pa rather than Pi , we get instead of (247) the following factorisation: Anticipating on the key result of §8 below about the canonical factorisation of bisymmetrals, we may note that the two exceptional (i.e. non-neg-invariant) bisymmetrals pal • and par • necessarily coincide up to gari-postcomposition by a regular (i.e. simultaneously neg-and pari-invariant) bisymmetral, which we may call ral • , and whose first three components ral • 1 , ral • 2 , ral • 3 , as well as all later components of odd length, necessarily vanish. In other words: But this is exactly the sought-after factorisation (248), with explicit factors: (iii) The third factorisation (249) is rather special in being a mu-factorisation incongruously arising out of a purely gari-gira context. 34 The quickest way to derive it is to assume the (already doubly established) bisymmetrality of pal • /pil • , then to define the would-be even factor pal • evv via the equation (249) But we have defined pal • evv as the mu-product of pal • , which we have shown to be symmetral, and of expmu( 1 2 Pa • ), also clearly symmetral. So pal • evv is itself symmetral, and as such mu-invertible under pari.anti. Therefore: 34 For a tentative mitigation of this 'incongruity', see §1-11 supra.
Comparing (257) and (258), we see that pal • evv is pari-invariant, and so neginvariant as well, and therefore truly even.
Properties of pal • ev and pal • evv .
To establish the last identity (261), we must start, not from (260), but from the corresponding relation for pal • , which reads To declumsify our notations, we set: 35 Further, we shall denote the mu-product by a simple dot "." We shall also abbreviate irat(A), irat(B) etc asĀ,B etc. Lastly, stars in upper (resp. lower) index position shall stand for the involution pari (resp. anti).
With these compact notations, the relation (266) we want to establish reads Using the fact that der ,Ā,B etc are mu-derivations, we see that R may be decomposed as Let us now show that R 1 ≡ R • 1 ≡ R 2 ≡ R * 2 ≡ 0. The identities R * 1 ≡ 0 and R * 2 ≡ 0 follow respectively from R 1 ≡ 0 and R 2 ≡ 0 under pari, and the identity R 1 ≡ 0 is none other than (266). So the only thing left to check is R 2 ≡ 0. To do this we apply the derivation rule (200) and then the simplification rule (199) to show that in the expression (Āc).c −1 + c.A.c −1 all 'intermediary terms', i.e. all terms of the form mu mu r 1 (Pa • ), sra • r 2 , mu r 3 (Pa • ) or mu mu r 1 (Pa • ), anti.sra • r 2 , mu r 3 (Pa • ) with r 1 = 0, r 2 ≥ 2, r 3 = 0 disappear, leaving only 'extreme terms' that cancel out with the terms from −1/2 A + 1/2 A * , plus of course pure mu-powers of Pa • , which also cancel out. This establishes R ≡ 0. §4-7. Properties of ripal • ev .
Applying the identity (44) for dilator composition to the factorisation But since pal • od = expari (−1 /2 Pa • ), this simplifies to with diripal • as in (236) and with the ordinary exponential expP of the elementary operator P: Being the gari-dilator of a symmetral bimould, diripal • ev is of course alternal. And since we have shown that pal • ev and therefore ripal • ev are 'even' (i.e. pariinvariant), the same applies for diripal • ev , so that, as expained in §2 (see (89) and (90) ) the relation between diripal • and diripal • ev may be rewritten as which, appearances notwithstanding, is actually simpler than (278), as it involves only even-length components.
In a sense, this is all we need to know. But in order to get the extra information of formula (154) or rather, in our polar specialisation, the explicit expansion of diripal • ev in terms of the remarkable alternals ka • 2r (polarspecialised from the ke • 2r of §2), we must work harder. Rather than derive the expansion of diripal • ev directly 36 from that of diripal • via (278) or (280), it is more convenient to reproduce the approach of (245) and (246), i.e. to set kasra • r := (exp P).ganit(nipac • ).sra • r = r≤r * kasra r,r * kasra r,r * ∈ BIMU r * and then regroup the (highly lacunary) components of r * : Comparing the components kasra • r,r * with the earlier hasra • r,r * of (245), one even gets to understand (however dimly) why the relevant tree-combinatorial object for calculating the bimould projections in the standard basis {e • t } is 36 The direct method yields only partial but valuable information. Thus, denoting Proj 1 .M • the first coefficient of M • in the standard eupolar basis, we may establish the identity Proj 1 .P 2r * −r .diripal • r = (−2) r−2r * r.r+1 (2r * −2)! (r−2)! which leads to Proj 1 .diripal • ev,2r * = 1 2r * (2r * +1) which in turn yields the important normalisation property Proj 1 .ka • 2r * = 1 slant(t) in the case of ha • r and stack (t) in the case of ka • 2r . Still, the calculations are quite lengthy and the whole approach leaves much to be desired. In particular, one would appreciate a more conceptual explanation for the puzzling slant/stack dichotomy. §4-8. Characterisation of pal • /pil • .
The explicit expansion of pal • as given in (300) below (as a direct consequence of (122) and (123)) makes it clear that pal • , and therefore pil • too, possess exactly the pole pattern described in Proposition 3.2. To prove the converse, namely that no other Pi -polar bisymmetral varpil • can display the same pole pattern, we must use the results of §8 about the standard factorisation of bisymmetrals. In the case when varpil • 1 = 0, we have In the case when our first component varpil • 1 is = 1, it is necessarily of the form c Pi • and, modulo an elementary dilation varpil • r → γ r varpil • r , we may assume c = −1/2 and get varpil • 1 and pil • 1 to coincide, thus ensuring (according to §8) the existence of a factorisation: The thing now is to focus on the first nonzero component bir • 2r (2r ≥ 4). It is bound to occur linearily in the expansion of varpil • , whether the latter be of type (282) or (283). Now, bir • 2r cannot be of the form c ri • 2r , which is simply alternal, not bialternal. But of all alternals, let alone bialternals, ri • 2r alone possesses precisely the pole structure described in Proposition 3.2 for pil • . This clinches the argument. 5 Polar bisymmetrals: explicit expansions. §5-1. Explicit expansions for pil • and pil • ev .
From the {ri • r }-expansions of pil • 's dilator dipil • and infinitesimal generator lipil • := logari .pil • : we at once derive (see (39) and (430)) two equally valid expansions for pil • itself, which in their first raw form read: τ r 1 . . . τ rs Paj r 1 ,...,rs −→ preari (ri • r 1 , ..., ri • rs ) (286) The main difference lies of course in the transparency of the τ r 's compared with the complexity of the θ r 's. But quite apart from the nature of their coefficients, the above expansions are unsatisfactory on two further counts: they are non-unique 37 and involve multiple pre-Lie brackets, which are complex, inflected expressions. So we must hasten to replace them by unique expansions involving simple, uninflected mu-products. There are three ways of doing this, based on the elementary series {mi • r }, {ni • r }, {ri • r } inductively defined as follows: and behaving as follows under the anti-action arit: For s ≥ 1 and r 1 + ... + r s = r each of the three sets consists of linearly independent bimoulds that span one and the same subspace Flexin r (Pi ) of Flex r (Pi ). The six conversion rules between the three bases are mentioned in [E3] §4.1. Let us recall the most useful: The first two bases (294) of Flexin r (Pi ) have the advantage of consisting of 'atoms' (simple strings of inflected units Pi). The ingredients ri • r of the third basis are not atomic (it takes at least r + 1 strings to express them) but they make up for it by being alternal. Now, the above derivation rules (291), (292), (293) together with the two conversion rules (295), (296) make it easy 38 to expand the multiple prearibrackets of (284), (285) in each of the three bases (294). In the event we get three alternative expressions: The first induction goes like this: The third induction involves less terms and is faster to run on a computer (see §18.A infra), the reason being that here the bulk of the complexity is absorbed by the 'molecular' ri • r 's that replace the 'atomic' mi • r 's or ni • r 's of the earlier inductions: We start from the mu-dilators dupal • , dupal • ev , dupal • evv as described in §3. Applying the rule (39) we immediately derive these three expansions: with r i = r(w i ) = r(u i ); with the selfsame Bernoulli-like numbers α r , β r as in (121) The last two expansions must be preferred to the first, since they involve only even terms. Of these two even expansions, (302) is again preferrable to (301), since the passage from pal • evv to pal • (mu-multiplication) is so much simpler than the passage from pal • ev to pal • (gari-multiplication). But there is still room for improvement. Indeed, (302) is blighted by some redundancy since the summands on the right-hand side are not linearly independent. 39 . To get a true basis, we must introduce bimoulds Lan • 1 ,..., s ∈ 39 The products mu(lan • r1 , ..., lan • rs ) are of course linearly independent, but cease to be so when 'precomposed' by Paj • as in (300), (301), (302). Fixing s and letting each i range over {0, 1, 2}, except for the first 1 which is forbidden to be 0, we get a set of bimoulds Lan • 1 ,..., s that (i) are linearly independent (ii) span the same subspace of Flex 2s (Pa) as the Paj • • mu(lan • r 1 , ..., lan • rs ) (iii) permit to express these Paj • • mu(lan • r 1 , ..., lan • rs ) via a simple rule. So (302) may be rewritten more economically as with a rational valued mould Han • belonging to none of the classical symmetry types but nonetheless calculable by a simple induction. From pal • evv we easily go to pal • , through elementary mu-multiplication by the arch-elementary factor pal • odd , and from there we go to pil • through the equally elementary involution swap. Moreover, of all expansions currently at our disposal, this ultimate expansion (305) for pal • evv is clearly optimal, since it involves only 2.3 r/2−1 atomic summands, as compared with the 2 r summands in each of the three expansions (297), (298), (299) for pil • .
Remark: If in (304) we had prohibited for 1 the value 1 resp. 2 instead of 0, we would still have got two valid bases Lan • 1 ,..., r and two expansions of the form (303), though with changed moulds H • . There exist yet other bases with the same indexation. These multiple choices, hardly relevant in the eupolar case, acquire real significance in the eutrigonometric case ( [E4]) and shall be discussed there.
The first proof presented here (in §4) of the bisymmetrality of pal • /pil • is definitely shorter than the second one, which in turn is simpler than either of the two proofs sketched in [E3]. As we see it, it has two further merits: it respects the symmetry between the two swappees (unlike the earlier treatments, which gave precedence to pil • and relegated pal • to the subordinate status of a derivative object) and it does so in the most satisfactory way that could be dreamt of, by linking pal • and pil • separately to the only two completely elementary alternal series that exist in Flex (E), namely {le • r } and {re • r }.
The linkage between each swappee and its alternal series is provided by the notion of dilator, but the two dilators in question are rather different: one is geared to the uninflected mu-product, the other to the inflected gariproduct. The two alternal series {le • r } and {re • r } also differ, and in much the same way. We have here, we suggest, the whole essence of dimorphy in a nutshell: a symmetry that is nearly complete, yet stops just short of being thoroughly, dully, and barrenly complete. In fact the whole flexion structure -dimorphy's natural framework -is largely though not perfectly self-dual under swap. So is its core ARI//GARI. And so is the core's core, consisting of the two pairs pal • /pil • and tal • /til • . Experience shows that such mathematical structures are among the most fecund.

Remark 2. Pervasiveness of parity.
Considerations of parity are paramount in all branches of the theory, not just in the factorisation of the key bimoulds but also when it comes to constructing and describing their length-r components.
Regarding the factorisations, they come in all sorts and shapes. Thus, all three formulae (129), (130), (131) are logically independent, carry unrelated even factors, and involve two distinct group laws, mu and gari. Nor is the phenomenon restricted to the eupolar context; it extends to such objects as the important bimould Zag • , though with a nuance: unlike eupolar bimoulds, which are automatically invariant under pari • neg, general bimoulds such as Zag • react differently to pari and neg, leading to a more intricate factorisation pattern, with three factors Zag • I , Zag • II , Zag • III , the first of which again splits into three subfactors.
Regarding the mould components, the even/odd dichotomy makes itself felt in this way: whereas we have to work in order to find the even-length components of our bisymmetrals 40 , their odd-length components immediately and effortlessly follow, and that too under any one of at least four distinct mechanisms. 41 The dichotomy also holds for the components of Zag • and those of each of its three factors. Thus, constructing the even-length components of Zag • I or Zag • II is hard work, while the odd-length components easily follow. With Zag • III , it is exactly the reverse. Ultimately, the dominance of parity in flexion theory can be traced back to one root cause: the essential parity of bialternals (see §7 infra). Germane considerations also explain the existence of a surperalgebra SUARI parallel to ARI (see [E1], §24, pp 456-459).
gari-inverses of bisymmetrals are better-behaved than the originals. This fact, already noticeable with eupolars, becomes particularly striking in the eutrigonometric case: compare for example the transparent right-hand side of (4.88) in [E3] with that of (4.87), for which no simple closed formula exists.
But the main difference is one of 'universality': whereas pal • /pil • and par • /pir • and indeed all 'intermediate' bisymmetrals 42 have different geparseparators, the separators of the gari-inverses ripal • /ripil • and ripar • /ripir • (and of all other exceptional, non neg-invariant bisymmetrals) do coincide. 43 Lastly, we may note that in the applications to multizeta algebra it is the inverse polar bisymmetrals ripal • /ripil • and the direct trigonometric bisymmetrals tal • /til • that matter most.

Remark 6. Coexistence of inflected and non-inflected opeations.
Quite often, when comparing flexion formulae, 44 one is struck by a recurrent anomaly: that of complex inflected operations like gari, expari etc inexplicably morphing into non-inflected ones like mu, expmu etc. While there is no neat, sweeping reason for this stealthy tendency towards 'desinflexion', but only case to case explanations, one may still point to the existence of a large ideal ARI intern of ARI and of a large normal subgroup GARI intern of GARI where ari and gari reduce to lu and mu (but with the order of the arguments reversed). See §1-11 supra.
The 'trigonometric specialisation' is no proper specialisation, since Qi • c and Qa • c are only approximate units, due to the corrective terms ± c 2 in the identities (3.28) and (3.29) of [E3]. See also §17-12 infra. One should therefore be prepared for serious complications when going from pal • /pil • to the trigonometric equivalent tal • /til • , and in that respect the trigonometric bisymmetrals do not disappoint. A long monograph [E5] will be devoted to them and their natural environment, the structures Flex (Qi c ) and Flex (Qa c ), which are not isomorphic to the polar prototypes nor indeed to each other.
We shall be content here with a few hints, to highlight the key steps in the transition from eupolar to eutrigometric. The formula (113) linking pil • to its gari-dilator dipil • survives unchanged (as to its general form). The link between pal • to its mu-dilator dupal • also survives, especially regarding the even factors, though not exactly in the 'differential' form (119) but rather in the 'integral' form (300), with the auxiliary mould Paj • replaced, unsurprisingly, by a more complex Taj • . But the main change is this: while the polar dilators had their components dipil • r resp. dupal • r simply proportional to ri • r resp. la • r (or rather lan • r ), the trigonmetric dilator components ditil • r and dutal • r take their values in two δ(r)-dimensional spaces of alternals, with a fast (faster than polynomially) increasing δ(r). So now at each (even) step we have to determine not one, but δ(r) rational coefficients on both sides, and to understand the affine (or linear, modulo the 'earlier' coefficients) correspondance between the two sets. The alternal series {ha r } and {ka 2r } also survive (with single components morphing into linear spaces) and so does their connection with the even factors of the inverse bisymmetrals. Altogether, although almost every single statement of §3 has its counterpart in the new setting, we experience a steep increase in difficulty, resulting in an even more diverse and interesting situation.

Essential parity of bialternals.
This section is devoted to establishing the decomposition 45 ARI al/al = ARI˙a l/ȧl ⊕ ARI al/al of the space ARI al/al of all bialternals into: (i) a large, regular part ARI al/al , consisting of even bimoulds and stable under the ari-bracket.
(ii) a small, exceptional part ARI˙a l/ȧl := BIMU odd 1 , consisting of odd bimoulds of length one and endowed with a bilinear mapping oddari into ARI al/al .
Everything rests on the following statement.
Proposition 7.1 (Parity of bialternals). Any nonzero bialternal bimould A • purely of length r > 1 is neg-invariant or, if you prefer, an even function of its double index sequence: A w ≡ A −w .
Proof: Alternality implies invariance under mantar := −anti .pari . Bialternality, therefore, implies invariance under neg.push, with: neg.push := mantar.swap.mantar.swap = anti.swap.anti.swap The push operator, we recall, is idempotent of order r + 1 when acting on BIMU r , i.e. on bimoulds of length r. Let us assume that A w is odd in w, and show that this implies A w ≡ 0.
For an even length r, this follows at once from the neg.push-invariance: For an odd length, the argument is more roundabout. Note first that for A w , which we assumed to be odd in w, invariance under neg.push amounts to invariance under -push. Here again, it turns out that the absence of non-trivial solution does not require the full bialternality of A • , but only its alternality and invariance under -push. So let us prove this stronger statement: Lemma 7.1 (Alternality and push-invariance).
No nonzero bimould A • purely of length r > 1 can be simultaneously alternal and invariant under −push.
Proof: Here again, the statement is obvious for r even. So let us consider an odd length of the form r = 2 t+1 ≥ 3. Since we shall subject A w to two linear operators, pus and push, respectively of order r and r +1 when restricted to BIMU r , and since pus (resp. push) reduces to a circular permutation in the 'short' (resp 'long') bimould notation, we shall make use of both. Let us recall the conversion rule: with the dual conditions on upper and lower indices: To show that A • = 0, we start with the elementary alternality relation: A w with w = (w 1 , . . . , w 2t ) and w = (w 2t+1 ) which reads: Due to the invariance of A • under -push, this may be rewritten as: In the 'long' notation (of greater relevance here) this becomes: Under the exchange w 0 ↔ w 2t+1 , the last identity becomes: Or again, reverting to the short notation: On the other hand, alternality implies pus-neutrality 46 pus j A • ≡ 0, which reads: 0 = 1≤j≤2 t+1 A w j ,...,w 2t+1 ,w 1 ,...,w j−1 From (315) and (316) we get by addition: and by subtraction: Under the change (w 2 , w 3 , . . . , w 2t+1 , w 1 ) → (w 1 , w 2 , . . . , w 2t+1 ), (318) becomes: Subtracting (319) from (317), we end up with A w 1 ,..,wr ≡ 0. .

Standard factorisation of bisymmetrals.
This section is devoted to establishing the factorisation 47 : GARI as/as = gari GARI˙a s/ȧs , GARI as/as (320) of the set GARI as/as of all bisymmetrals into (i) a large, regular factor GARI as/as consisting of even bimoulds 48 and stable under the gari product (ii) a small, exceptional factor GARI˙a s/ȧs consisting of special bimoulds derived from so-called flexion units and with components that are alternately odd/even, i.e. invariant under pari.neg rather than neg.
The proof rests on the construction and properties of the special bisymmetrals ess • and oss • (see Proposition 3.1, supra) and on the following statement: Proposition 8.1 (Factorisation of bisymmetrals). Any bisymmetral pair of swappees Sa • / /Si • simultaneously factor as The above decompositions are not unique, but two of them stand out, namely the one in which and the one in which 47 See [E3], §2.8. 48 they are even functions of their multiindex w, but may possess non-vanishing components of any length, even or odd. 49 We recall that gush := neg.gantar .swap.gantar .swap with gantar := invmu.anti .pari .
These 'co-canonical' decompositions involve two conjugate flexion units E and O and, though distinct, easily translate into one another under the classical relation 50 between ess • and oss • .
Proof: It rests on the Proposition 7.1 of the preceding section, in conjunction with the two following lemmas.

Lemma 8.1 (First components of bisymmetrals).
If the length-one component Sal w 1 of a bisymmetral bimould Sal • is an even function of w 1 = ( u 1 v 1 ), it may be anything, but if it is an odd function, it is necessarily a flexion unit.
Proof: Let u 0 , u 1 , u 2 be constrained by u 0 + u 1 + u 2 = 0 and let v 0 , v 1 , v 2 be defined up to a common additive constant. At length 2, the unique symmetrality relation for Sal • may be written thus: Due to Sal w 1 being odd, this yields: Likewise, the unique symmetrality relation for Sal • may be written as: In the u i -variables, this translates into: or again, due to imparity and to u i = 0 : Let E 1 be the identity obtained by adding the three circular permutations of (327) and (328), and E 2 the identity obtained by adding the six permutations, circular or anticircular, of (329). The left-hand sides of E 1 and E 2 clearly coincide, while their right-hand sides coincide only up to the sign. Equating these right-hand sides, we find: 50 See §9 infra or formula (4.63) in §4.2 of [E3].
which is precisely the symmetrical characterisation of a flexion unit. .
Remark 1: On the face of it, the requirement that the length-1 component be a flexion unit is merely a necessary condition for the existence of a bisymmetral 'continuation' at all lengths. However, the theory of unit-generated bisymmetrals ess • shows this condition to be (miraculously) sufficient. 51 This is probably the best a posteriori justification for singling out this notion of flexion unit, though by no means the only one.
Remark 2: Had we assumed Sal • to be even, we would have found no constraints at all on the length-1 component -which was only to be expected, since the ari-exponential of that length-1 component is automatically in GARI as/as .
Remark 3: One should not be too exercised over the presence of the factor 4 in (330), but rather observe that it vanishes after the change Sal w 1 = − 1 2 E w 1 which, as it happens, the construction of ess • quite naturally imposes.

Lemma 8.2 (General and even bisymmetrals).
Though not a group, the set GARI as/as of all bialternals is stable under both gari-and gira-postcomposition by the group GARI as/as of even bisymmetrals, and the identity holds: ∈ as/as (∀S • 1 ∈ as/as , ∀S • 2 ∈ as/as) (331) Proof: Here gira stands for the pull-back of gari under the basic involution swap. Both group laws are related as follows 52 : with non-linear operators ras, rash defined by: But since in Lemma 8.2 the right factor S • 2 is in GARI as/as and since gari and gira coincide on GARI as/as (even as ari and ira coincide on ARI al/al ), this implies: Likewise, any bimould of as/as type is automatically gush-invariant (even as any bimould of al/al type is automatically push-invariant). See [E3], §2.4. This in turn implies: rash.S • 2 = 1 • and ganit(rash.S • 2 ) = id (336) and establishes (331). .
This obviously holds for k = 1. If it holds for all k < r, then by Lemma 2.1 Sa k • is also in GARI as/as , as the gari-product of a bimould of type as/as by a string of several bimoulds of type as/as. As for sar r • , it is defined as the difference of length-r components of two bisymmetral bimoulds, Sa • r and Sal • , whose earlier components coincide. It is therefore not just of type al /al (bialternal) but also, by Lemma 7.1 in the preceding section, of type al /al (bialternal and even), and its ari-exponential is automatically as/as.
Summing up, we arrive at a factorisation of the announced type (321), with a left factor defined by (337) and a right factor defined by The swappee factorisations (322) immediately follow, again under (332).
After our in-depth study of the central but exceptional (i.e. non neg-invariant) bisymmetrals, we can now turn to our first instance of regular (i.e. neginvariant) bisymmetrals, and thence to the corresponding (automatically regular) bialternals.
Applying the general results of Proposition 8.1 about the standard factorisation gari (Sal • , Sar • ) of bisymmetrals and bearing in mind that in the eupolar context the right factor Sar • , due to homogeneousness, is not only neg-but also pari-invariant, we arrive at the following picture: As second gari-factors we have here regular bisymmetrals seës • etc that are themselves exponentials of regular bialternals leël • etc. Both carry only even-length components, with a vanishing length-2 component. 53 Moreover, since the involution sap (product of swap and syap, in whichever order) turns seës • and soös • into their gari-inverses, we clearly have In the polar specialisation, the picture becomes: and To construct our first series of bialternals, we now have the choice between the components of infinitesimal generators such as lilir • or those of dilators such as dilir • or diril • . Past experience suggests that the latter are to be preferred, and anyway the three systems {lilir • 2r }, {dilir • 2r }, {diril • 2r } generate exactly the same bialternal subalgebra of ARI.
So, forgetting about lilir • , let us look at the dilators dilir • and diril • to decide which is simpler. Starting from the factorisations or the more economical factorisations and applying the rule (44) for dilator composition, we find respectively and The identities (348) and (349) are unnecessarily wasteful, since they draw on all components, even and odd, of the central bisymmetrals to calculate the components dilir • 2r and diril • 2r , all even, of the bialternals. And of the two remaining identities, (351) is better than (350) since it involves, via the adari action, the bimould ripil • ev , which is much simpler than ripir • ev . 54 We have thus got hold of our first series of bialternals {diril • 2r ; r ≥ 2} along with a probably optimal algorithm for their calculation. Indeed, using formula (42) and the key results (153) and (154) of §3, we can make the terms on the right-hand side of (351) wholly explicit. For the bimould part we get an expansion in terms of elementary alternals: and for the operator part we have an equally simple expansion: adari(ripil • ev ) = id+ Paj 2r 1 ,...,2rs j=s j=1 2 1−2r j (2r j −1)(2r j +1) ari(ri • 2r 1 )...ari(ri • 2rs ) 10 Polar bialternals: second main source. §10-1. Abstract singulators.
To begin with we must recall the construction of the 'abstract' singulator senk that to any bisymmetral ess • associates (non-linearly) a linear operator senk(ess • ) = 1≤r senk r (ess • ) (352) whose 'components' senk r (ess • ) have the astonishing property of turning any length-1 bimould into a bialternal bimould of length r. That, however, comes at a price: every second time the bialternal so produced is identically 0. More precisely: 54 In fact, diril • is not just simpler to calculate than dilir • ; it is also simpler in itself, in its coefficient structure, as can be seen from the extensive tables referred to in §18 and posted on our Webpage.
Before constructing senk, let us recall the definition of mut (anti-action of BIMU on itself ) and adari (action of GARI on ARI): We also require elementary operators that render any bimould neg-or pushinvariant: whose 'components' slink r and srink r turn arbitrary, entire-valued length-1 bimoulds into bialternal, singular-valued length-r bimoulds. This property makes slink r and srink r extremely useful in multizeta algebra, in the backand-forth known as singularisation-desingularisation. §10-3. The second series of bialternals.
Our aim here, however, is different: we want to produce eupolar bialternals, i.e. bialternal elements of Flex r (Pi ). Here, the 'singuland' (i.e. that on which the singulator acts) can only be Pi • , and so, in view of (353)-(356), the 'singulate' (i.e. the bialternal fruit of the operation) can and in fact will be nonzero only in the situation (354). So we have no choice but to set visli The answer to the second question is probably no, but this is no more than a hunch. The answer to the first question is not clear either: up to length 56 In view of (365), subsituting pal • or par • for ess • in senk would produce nothing new. It would just yield (up to sign) the swap transforms of slink and srink. 10, the two systems are equivalent; at length 12 they produce a distinct generator each; but at length 14 they do not. And what happens thereafter is anybody's guess.
Warning: from here on the exposition becomes less systematic and the paper takes a more exploratory turn. It mixes proof-backed statements, conjectures, and mere 'observed facts', while making clear in each case which is which.
All these subspaces except the first (sap-invariants) are stable under ari and define as many subalgebras. On the other hand, only the fourth (alternals) is stable under lu. This again shows how much more flexible, versatile and interesting the flexion operations are. Remarkably, neither the pusinvariant subspace Flex pus r nor the push-variant subspace Flex push r are stable under ari, let alone lu. 58 Here is a table with the dimensions, up to r = 14, of the length-r com-57 Recall that sap := swap.syap = syap.swap and that a bimould A • in BIMU r is said to be pus-variant iff (id + pus + pus 2 + ...pus r −1 ). A • = 0. 58 This underscores the 'complementarity' between pus (a circular permutation of order r in the short notation) and push (a circular permutation of order r in the long notation). ponents of these subspaces or subalgebras. Let θ := {θ 1 , . . . , θ s } be the unordered rooted tree obtained by attaching s unordered rooted trees θ j to a common root. Then the inductive rule 67 : produces, for each r, a system {err • θ ; nodes(θ) = r} consisting of bimoulds that are alternal of length r (obvious); have the right indexation and so too the right cardinality (obvious); are linearly independent (non obvious); and therefore constitute a basis of Flex al r (E). This is a rather unusual situation, given that most free Lie algebras 68 possess no privileged natural basis.
12 Interplay of the lu and ari structures.
The elementary subalgebra Flex al (re) is generated (and spanned) by the selfreproducing alternals re • r . All its components Flex al r (re) are one-dimensional. The algebra Flex al free (E) resp. Flex free (E) is freely generated by a well-defined number of primary generators fe • r,i taken in each Flex al r (E) resp. Flex r (E), and supplemented by secondary generators of the form → ari (fe • r 0 , re • r 1 , . . . , re • rs ) with r 0 +r 1 +. . . r s = r with only non-increasing (or non-decreasing, if one so prefers 69 ) integer sequences (r 1 , . . . , r s ).
67 As usual, we get the induction started by setting err • θ0 := E • for the one-node one-root tree θ 0 .
68 As a lu-algebra, Flex al (E) is free, and very nearly free as an ari-algebra. See §12. 69 Expliciting the conversion rules between the two systems (376) that correspond to non-increasing or non-decreasing sequences, and finding a compact expression for these rules, is a wholesome exercise on moulds.
We cannot expect the bialternality codegree (or rather its second component) to behave in anything like a predictable manner under mu and lu nor indeed under preari and ari, but there an important exception, namely on the subalgebra of push-invariant elements 72 , where swap commutes with preari and ari. So for push-invariant bimoulds we have: which serve as entries of the so-called bialternality grid.
In fact, we have two such grids: one for the whole of Flex r (E) and one for the push-invariant subalgebra Flex push r (E). The second grid, also called bialternality chessboard, is the more important of the two, but in this 'monogenous' or 'eupolar' context both are equally interesting. In particular, both are symmetrical with respect to the main diagonal. This is due to the existence of a second involution syap, specific to this case.
But when we leave the 'eupolar' context and move on for example to the important case of polynomial-valued bimoulds, we still have (highly interesting) bialternality grids and chessboards but there is no syap anymore and so the property of diagonal symmetry disappears, though traces of it remain. §15-1. Elementary flexions.
In addition to ordinary, non-commutative mould multiplication mu (or ×): and its inverse invmu: the bimoulds 73 A • in BIMU = ⊕ 0≤r BIMU r can be subjected to a host of specific operations, all constructed from four elementary flexions , , , that are always defined relative to a given factorisation of the total sequence w.
The way these flexions act is apparent from the following examples: with the usual short-hand: u i,...,j := u i +...+u j and v i:j := v i −v j . Here and throughout the sequel, we use boldface (with upper indexation) to denote sequences (w, w i , w j etc), and ordinary fonts (with lower indexation) to denote single sequence elements (w i , w j etc), or sometimes sequences of length r(w) = 1. Of course, the 'product' w 1 .w 2 denotes the concatenation of the two factor sequences. §15-2. Short and long indexations on bimoulds.
For bimoulds M • ∈ BIMU r it is sometimes convenient to switch from the usual short indexation (with r indices w i 's) to a more homogeneous long indexation (with a redundant initial w 0 that gets bracketed for distinctiveness). The correspondence goes like this: 73 BIMU r of course regroups all bimoulds whose components of length other than r vanish. These are often dubbed "length-r bimoulds" for short.
with the dual conditions on upper and lower indices: and of course 1≤i≤r u i v i ≡ 0≤i≤r u * i v * i . §15-3. Unary operations.
The following linear transformations on BIMU are of constant use: All are involutions, save for pus and push, whose restrictions to each BIMU r reduce to circular permutations of order r resp. r+1: 74 push = neg.anti.swap.anti.swap (392) leng r = push r+1 .leng r = pus r .leng r (393) §15-4. Inflected derivations and automorphisms of BIMU.
Parallel with the three Lie brackets, we have three pre-Lie brackets: with the usual relations: with assopreari denoting the associator of the pre-Lie bracket preari. The same holds of course for ami and ani. §15-11. Exponentiation from ARI to GARI.
A flexion unit E is an element of BIMU 1 that verifies identically The above identities may be rewritten as for r = 1 and 2, but they actually imply (433) for all values of r. The present paper deals mainly with the polar units Pa, Pi : Pa w 1 := P (u 1 ) = 1 u 1 , Pi w 1 := P (v 1 ) = 1 v 1 and occasionally with the approximate trigonometric units Qa, Qi : Qa w 1 := Q(u 1 ) = c tan(c u 1 ) , Qi w 1 := Q(v 1 ) = c tan(c v 1 ) for which the expression on the right side of (432), instead of vanishing, becomes ± c 2 .
For a more substantive exposition of the flexion structure, we refer to [E1] and [E3].