Octonionic polynomials provide a promising approach to the understanding of zero energy ontology (ZEO) and causal diamonds (CDs) defined as intersections of future and past directed lightcones: CD makes sense both in octonionic (8D) and quaternionic (4D) context. Lightlike boundary of CD as also lightcone emerge naturally as zeros of octonionic polynomials. This does not yet give CDs and ZEO: one should have intersection of future and past directed lightcones. The intuitive picture is that one has a hierarchy of CDs and that also the spacetime surfaces inside different CDs an interact. It turns out that CDs and thus also ZEO emerge naturally both at the level of M^{8} and M^{4} .
Remark: In the sequel RE(o) and IM(o) refer to real and imaginary parts of octonions in quaternionic sense: one has o= RE(o)+IM(o)I_{4}, where RE(o) and IM(o) are quaternions.
1. General view about solutions to RE(P)=0 and IM(P)=0 conditions
The first challenge is to understand at general level the nature of RE(P)=0 and IM(P)=0 conditions expected to give 4D spacetime surfaces as zero loci. Appendix shows explicitly for P(o)=o^{2} that Minkowski signature gives rise to unexpected phenomena. In the following these phenomena are shown to be completely general but not quite what one obtains for P(o)=o^{2} having double root at origin.
 Consider first the octonionic polynomials P(o) satisfying P(0)=0 restricted to the lightlike boundary δ M^{8}_{+} assignable to 8D CD, where the octonionic norm of o vanishes.
 P(o) reduces along each lightray of δ M^{8}_{+} to the same real valued polynomial P(t) of a real variable t apart from a multiplicative unit E= (1+in)/2 satisfying E^{2}=E. Here n is purely octonionimaginary unit vector defining the direction of the lightray.
IM(P)=0 corresponds to quaterniocity. If the E^{4} (M^{8}= M^{4}× E^{4}) projection is vanishing, there is no additional condition. 4D lightcones M^{4}_{+/} are obtained as solutions of IM(P)=0. Note that M^{4}_{+/} can correspond to any quaternionic subspace.
If the lightlike ray has a nonvanishing projection to E^{4}, one must have P(t)=0. The solutions form a collection of 6spheres labelled by the roots t_{n} of P(t)=0. 6spheres are not associative.
 RE(PE)=0 corresponding to coquaternionicity leads to P(t)=0 always and gives a collection of 6spheres.
 Suppose now that P(t) is shifted to P_{1}(t)=P(t)c, c a real number. Also now M^{4}_{+/} is obtained as solutions to IM(P)=0. For RE(P)=0 one obtains two conditions P(t)=0 and P(tc)=0. The common roots define a subset of 6spheres which for special values of c is not empty.
The above discussion was limited to δ M^{8}_{+} and lightlikeness of its points played a central role. What about the interior of 8D CD?
 The natural expectation is that in the interior of CD one obtains a 4D variety X^{4}. For IM(P)=0 the outcome would be union of X^{4} with M^{4}_{+} and the set of 6spheres for IM(P)=0. 4D variety would intersect M^{4}_{+} in a discrete set of points and the 6spheres along 2D varieties X^{2}. The higher the degree of P, the larger the number of 6spheres and these 2varieties.
 For RE(P)=0 X^{4} would intersect the union of 6spheres along 2D varieties. What comes in mind that these 2varieties correspond in H to partonic 2surfaces defining lightlike 3surfaces at which the induced metric is degenerate.
 One can consider also the situation in the complement of 8D CD which corresponds to the complement of 4D CD. One expects that RE(P)=0 condition is replaced with IM(P)=0 condition in the complement and RE(P)= IM(P)=0 holds true at the boundary of 4D CD.
6spheres and 4D empty lightcones are special solutions of the conditions and clearly analogs of branes. Should one make the (reluctanttome) conclusion that they might be relevant for TGD at the level of M^{8}.
 Could M^{4}_{+} (or CDs as 4D objects) and 6spheres integrate the spacetime varieties inside different 4D CDs to single connected structure with spacetime varieties glued to the 6spheres along 2surfaces X^{2} perhaps identifiable as preimages of partonic 2surfaces and maybe string world sheets? Could the interactions between spacetime varieties X^{4}_{i} assignable with different CDs be describable by regarding 6spheres as bridges between X^{4}_{i} having only a discrete set of common points. Could one say that X^{2}_{i} interact via the 6sphere somehow. Note however that 6spheres are not dynamical.
 One can also have Poincare transforms of 8D CDs. Could the description of their interactions involve 4D intersections of corresponding 6spheres?
 6spheres in IM(P)=0 case do not have image under M^{8}H correspondence. This does not seem to be possible for RE(P)=0 either: it is not possible to map the 2D normal space to a unique CP_{2} point since there is 2D continuum of quaternionic subspaces containing it.
2. Some general observations about CDs
CD defines the basic geometric object in ZEO. It is good to list some basic features of CDS, which appear as both 4D and 8D variants.
 There are both 4D and 8D CDs defined as intersections of future and past directed lightcones with tips at say origin 0 at real point T at quaternionic or octonionic time axis. CDs can be contained inside each other. CDs form a fractal hierarchy with CDs within CDs: one can add smaller CDs with given CD in all possible manners and repeat the process for the subCDs. One can also allow overlapping CDs and one can ask whether CDs define the analog of covering of O so that one would have something analogous to a manifold.
 The boundaries of two CDs (both 4D and 8D) can intersect along lightlike ray. For 4D CD the image of this ray in H is lightlike ray in M^{4} at boundary of CD. For 8D CD the image is in general curved line and the question is whether the lightlike curves representing fermion orbits at the orbits of partonic 2surfaces could be images of these lines.
 The 3surfaces at the boundaries of the two 4D CDs are expected to have a discrete intersection since 4 + 4 conditions must be satisfied (say RE(P_{i}^{k}))=0 for i=1,2, k=1,4. Along line octonionic coordinate reduces effectively to real coordinate since one has E^{2}=E for E=(1+in)/2, n octonionic unit. The origins of CDs are shifted by a lightlike vector kE so that the lightlike coordinates differ by a shift: t_{2}= t_{1}k. Therefore one has common zero for real polynomials RE(P_{1}^{k}(t)) and RE(P_{2}^{k}(tk)).
Are these intersection points somehow special physically? Could they correspond to the ends of fermionic lines? Could it happen that the intersection is 1D in some special cases? The example of o^{2} suggest that this might be the case. Does 1D intersection of 3surfaces at boundaries of 8D CDs make possible interaction between spacetime surfaces assignable to separate CDs as suggested by the proposed TGD based twistorial construction of scattering amplitudes?
 Both tips of CD define naturally an origin of quaternionic coordinates for D=4 and the origin of octonionic coordinates for D=8. Real analyticity requires that the octonionic polynomials have real coefficients. This forces the origin of octonionic coordinates to be along the real line (time axis) connecting the tips of CD. Only the translations in this specified direction are symmetries preserving the commutativity and associativity of the polynomial algebra.
 One expects that also Lorentz boosts of 4D CDs are relevant. Lorentz boosts leave second boundary of CD invariant and Lorentz transforms the other one. Same applies to 8D CDs. Lorentz boosts define nonequivalent octonionic and quaternionic structures and it seems that one assume moduli spaces of them.
One can of course ask whether the still somewhat ad hoc notion of CD general enough. Should one generalize it to the analog of the polygonal diagram with lightlike geodesic lines as its edges appearing in the twistor Grassmannian approach to scattering diagrams? Octonionic approach gives naturally the lightlike boundaries assignable to CDs but leaves open the question whether more complex structures with lightlike boundaries are possible. How do the spacetime surfaces associated with different quaternionic structures of M^{8} and with different positions of tips of CD interact?
3. The emergence of CDs
CDs are a key notion of zero energy ontology (ZEO). Could the emergence of CDs be understood in terms of singularities of octonion polynomials located at the lightlike boundaries of CDs? In Minkowskian case the complex norm qq^{c}_{i} is present in P (^{c} is conjugation changing the sign of quaternionic unit but not that of the commuting imaginary unit i). Could this allow to blow up the singular point to a 3D boundary of lightcone and allow to understand the emergence of causal diamonds (CDs) crucial in ZEO.
The study of the special properties for zero loci of general polynomial P(o) at lightrays of O indeed demonstrated that both 8D land 4D lightcones and their complements emerge naturally, and that the M^{4} projections of these lightcones and even of their boundaries are 4D future  or past directed lightcones. What one should understand is how CDs as their intersections, and therefore ZEO, emerge.
 One manner to obtain CDs naturally is that the polynomials are sums P(t)= ∑_{k} P_{k}(o) of products of form P_{k}(o) =P_{1,k}(o)P_{2,k}(oT), where T is real octonion defining the time coordinate. Single product of this kind gives two disjoint 4varieties inside future and past directed lightcones M^{4}_{+}(0) and M^{4}_{}(T) for either RE(P)=0 (or IM(P)=0) condition. The complements of these cones correspond to IM(P)=0 (or RE(P)=0) condition.
 If one has nontrivial sum over the products, one obtains a connected 4variety due the interaction terms. One has also as special solutions M^{4}_{+/} and the 6spheres associated with the zeros P(t) or equivalently P_{1}(t_{1})== P(t), t_{1}=Tt vanishing at the upper tip of CD. The causal diamond M^{4}_{+}(0)∩ M^{4}_{}(T) belongs to the intersection.
Remark: Also the union M^{4}_{}(0)∪ M^{4}_{+}(T) past and future directed lightcones belongs to the intersection but the latter is not considered in the proposed physical interpretation.
 The time values defined by the roots t_{n} of P(t) define a sequence of 6spheres intersecting 4D CD along 3balls at times t_{n}. These time slices of CD must be physically somehow special. Spacetime variety intersects 6spheres along 2varieties X^{2}_{n} at times t_{n}. The varieties X^{2}_{n} are perhaps identifiable as 2D interaction vertices, preimages of corresponding vertices in H at which the lightlike orbits of partonic 2surfaces arriving from the opposite boundaries of CD meet.
The expectation is that in H one as generalized Feynman diagram with interaction vertices at times t_{n}. The higher the evolutionary level in algebraic sense is, the higher the degree of the polynomial P(t), the number of t_{n}, and more complex the algebraic numbers t_{n}. P(t) would be coded by the values of interaction times t_{n}. If their number is measurable, it would provide important information about the extension of rationals defining the evolutionary level. One can also hope of measuring t_{n} with some accuracy! Octonionic dynamics would solve the roots of a polynomial! This would give a direct connection with adelic physics.
Remark: Could corresponding construction for higher algebras obtained by CayleyDickson construction solve the "roots" of polynomials with larger number of variables? Or could Cartesian product of octonionic spaces perhaps needed to describe interactions of CDs with arbitrary positions of tips lead to this?
 Above I have considered only the interiors of lightcones. Also their complements are possible. The natural possibility is that varieties with RE(P)=0 and IM(P)=0 are glued at the boundary of CD, where RE(P)=IM(P)=0 is satisfied. The complement should contain the external (free) particles, and the natural expectation is that in this region the associativity/coassociativity conditions can be satisfied.
 The 4varieties representing external particles would be glued at boundaries of CD to the interacting nonassociative solution in the complement of CD. The interaction terms should be nonvanishing only inside CD so that in the exterior one would have just product P(o)=P_{1,k0}(o)P_{2,k0}(oT) giving rise to a disjoint union of associative varieties representing external particles. In the interior one could have interaction terms proportional to say t^{2}(Tt)^{2} vanishing at the boundaries of CD in accordance with the idea that the interactions are switched one slowly. These terms would spoil the associativity.
Remark: One can also consider sums of the products ∏_{k} P_{k}(oT_{k}) of n polynomials and this gives a sequence CDs intersecting at their tips. It seems that something else is required to make the picture physical.
4. How could the spacetime varieties associated with different CDs interact?
The interaction of spacetime surfaces inside given CD is welldefined. Sitation is not so clear for different CDs for which the choice of octonionic coordinate origin is in general different and polynomial bases for different CDs do not commute nor associate.
The intuitive expectation is that 4D/8D CDs can be located everywhere in M^{4}/M^{8}. The polynomials with different origins neither commute nor are associative. Their sum is a polynomial whose coefficients are not real. How could one avoid losing the extremely beautiful associative and commutative algebra?
It seems that one cannot form their products and sums and must form the Cartesian product of M^{8}:s with different origins and formulate the interaction at M^{8} level in this framework. Note that CayleyDickson hierarchy does not seem to be relevant since the dimension are powers of 2 rather than multiples of 8.
Should one give up associativity and allow products (but not sums since one should give up the assumption that the coefficients of polynomials are real) of polynomials associated with different CDs as an analog for the formation of free manyparticle states. One can still have separate vanishing of the polynomials in separate CDs but how could one describe their interaction?
If one does not give up associativity and commutativity, how can one describe the interactions between spacetime surfaces inside different CDs at the level of M^{8}?
 Could the intersection of spacetime varieties with zero loci for RE(P_{i}) and IM(P_{i}) define the loci of interaction. As already found, the 6D spheres S^{6} with radii t_{n} given by the zeros of P(t) are universal and have interpretation as t=t_{n} snapshots of 7D spherical light front.
The 2D intersections X^{2} of 4D spacetime variety X^{4} with S^{6} would define natural candidates for the intersections and might allow interpretation as preimages of partonic 2surfaces. X^{2} would be the contact of X^{4} with S^{6} associated with second 8D CD. Together with SH this gives hopes about an elegant description of interactions in terms of connected spacetime varieties.
 The following picture is suggestive. Consider two spacetime varieties X^{4}_{i}, i=1,2 associated with CDs with different origins and connected by a connected sum contact, which at the level of H corresponds to a wormhole contact connecting spacetime sheets with different octonionic coordinates. The partonic 2varieties X^{2}_{i}= X^{4}_{i}∩ S^{6}_{i} are labelled by time values t=t_{i,ni}.
Assume that there is tubelike 3surface X^{3}_{1,2} connecting X^{2}_{1} and X^{2}_{2}. The union X^{2}_{1}∩ X^{2}_{2} of partonic 2surfaces must be homologically trivial in order to define a boundary of 3surface X^{3}_{1,2}. The surfaces X^{2}_{i} must therefore have opposite homology charges. X^{3}_{1,2} would be preimage of a wormhole contact connecting different spacetime sheets to which the CDs are assigned.
The 6spheres S^{6}_{i} intersect along 4D surface X^{4}_{1,2}= S^{6}_{1}∩ S^{6}_{2} in M^{8}. One should have X^{3}_{1,2}⊂ X^{4}_{1,2} and X^{3}_{1,2} should be noncritical but associative and therefore 3D. This surface should allow a realization as a zero locus of RE(P_{1,2}(u)) or IM(P_{1,2}(u)) and belong to X^{4}_{1,2}. One would not have manifoldtopology. Rather, one could speak of two 4D branes X^{4}_{i} (3branes) connected by a 3D brane X^{3}_{1,2} (2brane). Two 2 parallel 4planes joint by a 1D curve is the lowerdimensional analogy. The interaction would be instantaneous inside X^{4}_{i}.
 The polynomials associated with different 8D CDs do not commute nor associate. Should one allow their products so that one would still effectively have a Cartesian product of commutative and associative algebras?
Or should one introduce Cartesian powers of O and CD:s inside these powers to describe the interaction? This would be analogous to what one does in condensed matter physics. What seems clear is that M^{8}H correspondence should map all the factors of (M^{8})^{n} to the same M^{4}× CP_{2} by a kind of diagonal projection.
 Partonic 2surfaces define wormhole throats and appear in pairs if they carry monopole charges. Could one think that the above mentioned 2surfaces are intersections of X^{1}_{i} with S^{k}_{i+1} for the pair of spacetime sheets assignable to different CDs? Could the image in H of the structure formed by {X^{2}_{1},X^{2}_{2}, S^{6}_{1}, S^{6}_{2}} under M^{8}H correspondences be wormhole contact.
5. Summary
All big pieces of quantum TGD are now tightly interlinked.
 The notion of causal diamond (CD) and therefore also ZEO can be now regarded as a consequence of the number theoretic vision and M^{8}H correspondence, which is also understood physically.
 The hierarchy of algebraic extensions of rationals defining evolutionary hierarchy corresponds to the hierarchy of octonionic polynomials.
 Associative varieties for which the dynamics is critical are mapped to minimal surfaces with universal dynamics without any dependence on coupling constants as predicted by twistor lift of TGD. The 3D associative boundaries of nonassociative 4varieties are mapped to initial values of spacetime surfaces inside CDs for which there is coupling between Kähler action and volume term.
 Free many particle states as algebraic 4varieties correspond to product polynomials in the complement of CD and are associative. Inside CD the addition of interaction terms vanishing at its boundaries spoils associativity and makes these varieties connected.
 The basic building bricks of topological scattering diagrams identified as spacetime surfaces having as vertices partonic 2surfaces emerge from the special features of the octonionic algebraic geometry predicting sequence of 3balls as intersections of hyperplanes t= t_{n} with CD. One can say that octonionic dynamics solves roots of the polynomial P(t) whose octonionic extension defines spacetime surfaces as zero loci. Furthermore, the generic prediction is the existence of 6spheres inside octonionic CDs having 2D partonic 2variety as intersection with spacetime surface inside CD and interpreted as a vertex of generalized scattering diagram.
See the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the articles Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry?: part II.
