P5
generators, relations, and Schwarzschild as the asymmetric realisation
A companion paper exhibited the Schwarzschild–de Sitter family as a single rigid geometry charted from several observer vantages, with the admissible vantages organised into a groupoid whose one invariant is the geometry [JanzenSlicing]. That paper established the rigidity and the discrete generation of the morphisms, and left open the relational content of the generated structure. The present paper takes the groupoid as established and develops that content.
We give the groupoid as a category, name its generators in coordinate-independent form, and derive the relations they satisfy: the root-exchange involution σ of order two and the sky-angle periodicity τ of order three, with (στ)2= id, generating D3≅S3—and shown complete, which closes the generation question the slicing paper left open.
Two structural results follow. First, the cosmological configuration is forced by the group structure rather than chosen: generic vantages fall into σ-pairs, while the Nariai member is the unique fixed point of σ, lying in a degenerate orbit of its own, so the reassignment that promotes a null direction to the fundamental timelike congruence—forbidden by the two-cycle structure at every generic vantage—selects it uniquely. The uniqueness is therefore a structural fact about the groupoid and not a fact about dynamics or fine-tuning, and it converges with the independent empirical forcing of a cosmic time from the redshift-isotropy floor [JanzenModernParallax]: two routes, sharing no premise, selecting one configuration.
Second, we classify the discrete symmetry of the solution space. The same-α between-member morphisms are the monodromy of the horizon cubic's three-sheeted cover branched at the Nariai points, with monodromy group S3—which is equally the Galois group of that cubic, one S3 worn as monodromy, Weyl and Galois symmetry alike. Adjoining the mass-reflection 2M↦-2M gives the full discrete symmetry Aut(A2)=S3×Z2≅D6, and the action between distinct α is the continuous homothety, under which the discrete structure is invariant. That mass-reflection is shown to be the de Sitter↔Schwarzschild correspondence itself rather than a structure standing outside the group: it acts on the metric function by parity, f(r)=(1-r2/α2)+(-2M/r) splitting into an -even piece, the invariant de Sitter geometry, and an -odd piece, the Schwarzschild mass the vantage carries and the swap reverses.
The construction accordingly carries three distinct discrete operations, which must not be conflated: σ (the Weyl, diagonal reflection—mass-invariant, Nariai-fixed, no branch-point crossing), (the diagram, anti-diagonal reflection—the vantage-swap, de Sitter-fixed, the r=0 crossing), and ξ (the partial involution at the throat seam). The first two generate D6; the third is the analytic continuation joining the Riemannian and Lorentzian pieces of the slicing curve, its invertibility securing the correspondence's exactness.
We read the Schwarzschild description as the canonical asymmetric realisation—a partial arc swept, by an exterior vantage that cannot use the manifold's own axis of symmetry, about a selected off-axis point—and mark the scope of that reading explicitly: the attribution of the parting curvature term to the forced pivot rather than to the mass is the perspectival reading's interpretive payoff, not a claim proven independently of it. We close with what remains open, and with one question the companion settles: whether the throat radius is the intrinsic gravitational mass in the standard senses—the standard definitions return the slicing-dependent , so α is the invariant curvature radius, a length rather than a mass.
The Schwarzschild and de Sitter geometries appear, in the standard reading, as two distinct exact solutions to the Einstein field equations. They share a one-parameter family, the Schwarzschild–de Sitter family, but are read in the standard treatment as different geometries on different manifolds. The companion paper [JanzenSlicing] showed that this reading is the wrong reading. The SdS family is exhibited there as a single radial curve r(l), with dr/dl=√|f(r)| and f(r)=1-2M/r-r2/α2, at fixed throat radius α=√3/Λ, of which the Schwarzschild and de Sitter forms are not two limits but two readings of one slicing, exchanged by an involution. The horizons of the family are the turning points of this curve. The curve is intrinsic to the underlying de Sitter manifold: moving the observer who charts the construction changes the image on the observer's celestial sphere but does not change the geometry. The SdS geometry is therefore rigid (no continuous moduli act on it), and the de Sitter and Schwarzschild forms are two descriptions of one slicing curve, related within a groupoid of admissible charting vantages whose single invariant is the geometry.
That paper established the structure of the groupoid at the level of generators. The Rigidity proposition of [JanzenSlicing] shows the geometry carries no continuous modulus under change of charting vantage. Its discrete-generation proposition shows the morphisms acting within a single geometry are discretely generated by a unique involution (the root-exchange of the horizon cubic, equivalently the reflection of the sky angle about the value at which the cubic's three roots collide) together with the periodicity of the sky angle. What that paper did not develop, and what it explicitly named as open in its closing section, is the relational content of the generated structure: the relations the generators satisfy, the global organisation of the discrete morphisms, the within-one-geometry reassignments at fixed α (in particular the overcritical slicings beyond the Nariai bound, reached at fixed α), and the action of vantages between distinct de Sitter representations (different α). It also left explicitly open whether the throat radius α=√3/Λ established there as the invariant of the construction is also the correct intrinsic gravitational mass in the standard quasi-local or asymptotic definitions—a question the present paper and the companion now settle (§10): the standard definitions return the slicing-dependent , so α is the invariant length and not the gravitational mass.
The present paper takes the groupoid as established and develops the relational content within a single geometry. Four results structure the paper. First, we give the groupoid as a category, identify its generators in coordinate-independent form, and derive the relations they satisfy: the root-exchange involution σ has order two and the sky-angle periodicity τ has order three, and the relation that organises them, (στ)2= id, makes the generated group the dihedral group D3≅S3—shown moreover complete (Proposition 5), which closes the generation question the slicing paper left open. Second, we characterise the partial involution at the equatorial seam—the analytic continuation θ↦π/2+iψ that the slicing paper identified as joining the Riemannian piece of the slicing curve to the Lorentzian piece, with the metric signature flipping automatically because dθ=i dψ—as a vantage-change distinct from the within-single-geometry morphisms. The signature flip is a property of the continuation, not of the descriptions. And the continuation's invertibility, on which the correspondence's exactness rests, is a property one can name. Written in the substrate's own eigenvalue λ=α2/(α2-u) with u= x2 [JanzenGeometricCore], the map from position to signature is Möbius, (a,b,c,d)=(0,α2,-1,α2) with ad-bc=α2≠0, and its inverse u=α2-α2/λ is Möbius again. So ξ is invertible because a Möbius map is a bijection of the Riemann sphere, and the two signature regions are two arcs of that sphere joined through the point at infinity: λgt;0 Riemannian below the seam, λlt;0 Lorentzian above it, and λ=∞ at the seam itself. The map's zero sits at u=∞ and not at the seam, which is the same statement as the metric never degenerating thereL7L19. Third, we read the Schwarzschild description as the canonical asymmetric realisation: a partial arc of the slicing curve swept, by an exterior vantage that cannot use the manifold's own axis of symmetry, about a selected off-axis point of the manifold—the critical point r=0. This forced off-axis pivot is the geometric origin of the horizon-versus-singularity asymmetry that the companion circle paper [JanzenCircle] identified algebraically as two metric singularities of identical analytic type. The asymmetric features that distinguish the Schwarzschild description from the de Sitter description are signatures of the forced pivot, not features of the underlying geometry—an attribution that is the perspectival reading's interpretive payoff, and which we mark as such rather than as a claim proven independently of that reading (Proposition 10).
Fourth, two structural results follow from the discrete classification and are worth stating at the outset, since the corpus leans on both. The cosmological configuration is forced by the group structure: the generic vantages fall into σ-pairs, while the Nariai member is the unique fixed point of σ, lying in a degenerate orbit of its own, so the reassignment promoting a null direction to the fundamental timelike congruence—forbidden by the two-cycle structure at every generic vantage—selects it uniquely (Proposition 8). The uniqueness of the cosmological configuration is therefore a structural fact about the groupoid, not a fact about dynamics or fine-tuning, and it converges with the independent empirical forcing of a cosmic time by the redshift-isotropy floor [JanzenModernParallax]. And the mass-reflection —the diagram automorphism of Aut(A2)—is the de Sitter↔Schwarzschild correspondence itself, acting on the metric function by parity, f(r)=(1-r2/α2)+(-2M/r), whose -even piece is the invariant de Sitter geometry and whose -odd piece is the Schwarzschild mass the vantage carries (Remark 13)Corpus convention: is the orientation/mass-reflection parity—the A2 diagram automorphism, the orientation-reversing element R∈O(5,1)∖SO0(5,1), acting on a cut spinor as γ5 (Section 9). The corpus reserves “” for the areal spatial parity r↦-r (Clifford generator γ1γ2γ3, anticommuting with γ5), which does not appear in this paper; is the time reflection.}. The construction accordingly carries three distinct discrete operations—σ (Weyl, mass-invariant, Nariai-fixed, no branch-point crossing), (diagram, the vantage-swap, the r=0 crossing), and ξ (the throat seam at X=α)—which share a continuation mechanism but do not coincide as maps, and must not be conflated.
The ontological setting carries forward from the slicing paper: the smooth de Sitter manifold is taken as fundamental and the charts built upon it as perspectival constructions, and the throat radius α, which the present paper holds invariant under the groupoid, is that manifold's own scale. The fundamental object is therefore the de Sitter manifold, smooth and with no privileged points. A charting vantage is a fixed projection axis together with the chart it induces. Features standardly attributed to a Schwarzschild spacetime—the horizon, the curvature singularity at r=0—are properties of the perspectival construction, of where the chart degenerates, and not of the fundamental manifold. This is not offered as one interpretation among others; it is the reading in which the paper is written, and the constructions below proceed accordingly. The empirical footing is in place before any of the algebra below: the redshift-isotropy floor forces a cosmic foliation of the existing universe, and with it the necessary and sufficient augmentation of general relativity, on evidence and independently of any structure laid over it [JanzenModernParallax]. The present paper supplies algebraic content for a structure already grounded, not a structure awaiting a ground.
We work in geometrised units. Where convenient we set α=1 as a gauge; this is a choice of units, and we are explicit about it. The mass parameter is, throughout, related to the slicing parameter by 2M=r0-r03 in this gauge; the throat radius α is invariant.
We begin by specifying the groupoid as a category in the standard sense. The objects are charting vantages on the de Sitter manifold; the morphisms are vantage-changes that preserve the SdS geometry; the invariant of the groupoid is the geometry itself.
A charting vantage is a fixed point on the de Sitter hyperboloid together with the gnomonic projection induced by that point onto the orthogonal hyperplane. The slicing paper showed (Section 3.2 of [JanzenSlicing]) that obtaining a planar chart of the slicing curve forces the gnomonic projection: the hole's image lies on the observer's celestial sphere, and the planar chart that maps lines through the projection point to lines in the plane is forced to be the gnomonic one (the orthographic projection being excluded as it does not satisfy the line-mapping requirement). The vantage is therefore specified by the projection direction in the embedding space.
We collect the admissible vantages into a set V. Each v∈V determines a chart: a labelling of the slicing curve by a sky angle on the projection plane, with a genuine geometric angle (the angle the slicing curve subtends on the observer's celestial sphere). The same intrinsic curve is therefore charted by different elements of V as different sky-angle functions wv: slicing curve→R/2πZ.
A vantage-change is a map v→v' that takes one chart of the slicing curve to another while leaving the underlying de Sitter manifold invariant. The sky angle throughout is the distinguished observer's—the labelling for which the horizon relation takes the clean triple-angle form 2M=(2/3√3) sin 3w (the horizon-relation (sky-angle) proposition of [JanzenSlicing]; every other charting observer reads the same curve with a residual harmonic). In that labelling a vantage-change is a transformation w→w' under which the horizon relation keeps its triple-angle form.
There are two kinds of vantage-change. The first kind acts within a single geometry: it permutes the sky-angle labelling while leaving the value of 2M fixed. These are the morphisms the slicing paper proved discretely generated (the Discrete-generation proposition of [JanzenSlicing]). The second kind moves between geometries: it changes the value of 2M, equivalently the offset r0, and so charts a different member of the SdS family. The slicing paper carries the first kind and hands the classification of the discrete group here [JanzenSlicing].
Both kinds are settled below. The within-geometry group — the first kind — is D3≅S3 (Proposition 5). The second kind is the reflection R:2M↦-2M, which adjoined to the Weyl S3 gives the full discrete symmetry of the solution space, Aut(A2)≅D6 (Proposition 13); and between distinct α the action is a continuous homothety leaving that structure invariant (Section 9), so no further discrete morphism remains to classify. That homothety is a symmetry of the causal structure and not of the metric, and the conservation law reads the distinction exactly: a homothetic Killing field ξ gives d(ξ·p)/dτ=c p ·p along a geodesic, so its charge is conserved on the null cone, where p ·p=0, and nowhere else—the failure off the cone going as -m2, so rest mass is what spoils it. A symmetry that is not a symmetry of the action carries no Noether charge, which is why α is the quantity no dynamics fixes: it is moved by a transformation the causal structure has and the metric does notL20. The two statements are reconciled by naming what the null cone is to the flow: papa is itself conserved along an affinely parametrised geodesic, so the cone is an invariant submanifold, and on it ξ·p is a first integral of the restricted system—{ξ·p,p ·p}=2 p ·p, weakly zero on the constraint surface and not identically zero, which is the same “and nowhere else” read as a bracket. It therefore counts toward the null subsystem's tally of independent integrals and toward no other, the massive case admitting it as no integral at allP5R12L14.
The condition states that the value of 2M read off by the chart at vantage g(v) agrees with the value read off at vantage , since 2M=(2/3√3) sin 3w in either chart. Vantage-changes within a single geometry therefore preserve the horizon relation as an identity of the chart on the same intrinsic curve. The continuity requirement is not an extra hypothesis but a property of vantage-changes as defined: a vantage is a point on the de Sitter hyperboloid with its induced gnomonic projection (Definition 1), and a vantage-change is the relabelling of the sky angle induced by moving that point, a smooth reparametrisation of the same intrinsic curve. We record continuity explicitly because it is load-bearing in the completeness proof below: without it the condition sin 3wg(v)= sin 3wv, which is satisfied by every permutation of a sky-angle orbit, would not single out the dihedral action.
A point here calls for care, since the objects V form a continuum while the within-single-geometry morphisms characterised below are discrete. The two are reconciled by the rigidity of the geometry under change of charting vantage (the Rigidity and Discrete-generation propositions of [JanzenSlicing]): a continuous change of charting vantage either fails to act on the slicing curve—contributing only the identity to G—or re-images it onto an off-distinguished observer, whose reading carries a residual harmonic and so is not a transformation of the distinguished sky angle preserving the clean triple-angle form (Definition 2), hence not a morphism of G; the vantage-changes that act non-trivially within a single geometry are therefore discrete. This is why the within-geometry morphism group is discrete though V is a continuum, and it is what the completeness proof (Proposition 5) makes precise: those non-trivial morphisms are exactly D3.
The objects V and morphisms {g} together form a category. Identity morphisms exist (the identity vantage-change at each ); composition is associative (vantage-changes compose by composing the maps V→V); and every morphism has an inverse (a vantage-change can always be reversed by changing vantage back the other way). The category is therefore a groupoid.
The groupoid is the structural object the present paper analyses. The two generators of its morphisms within a single geometry are identified in the next section.
The slicing paper identified the two generators of the within-single-geometry morphisms (the Discrete-generation proposition of [JanzenSlicing]): the involution σ exchanging roots of the horizon cubic, and the periodicity of the sky angle . We state them here in coordinate-independent form and derive the relations they satisfy.
The horizon cubic of the SdS family is r3-α2r+2Mα2=0, which in the gauge α=1 becomes r3-r+2M=0. When the slicing parameter is taken as one of its own roots, r0, the cubic factors as (r-r0)(r2+rr0+r02-1)=0, with the quadratic factor carrying the other two roots. The root-exchange involution proposition of [JanzenSlicing] shows that the parameter map exchanging which root is designated r0,
is an involution: σ(σ(r0))=r0. Its fixed point is at r0=1/√3, the Nariai configuration at which the two positive horizons of the cubic coincide. The involution's endpoints are the de Sitter configuration r0=0 and the throat-tangent configuration r0=1, which it exchanges (the root-exchange involution proposition of [JanzenSlicing]). Separately, at r0=0 the single w=0 curve carries both the de Sitter and the Schwarzschild readings (the r0=0 proposition and the two-readings discussion of [JanzenSlicing]): that de Sitter↔Schwarzschild correspondence is the vantage-swap between two readings of one curve at fixed α, a between-description move distinct from the root-exchange σ itself.
In the sky angle , with r0=(2/√3) sin w, the involution acts as the reflection w↔π/3 - w (the root-exchange involution proposition of [JanzenSlicing]). This is its coordinate-independent statement: the involution is the reflection of the sky angle about w=π/6, the midpoint of one fundamental domain. We denote this generator σ.
The horizon relation 2M=(2/3√3) sin 3w has period 2π/3 in , since sin 3w has period 2π/3 in . The natural fundamental domain for modulo the periodicity is [0,2π/3), and the sky-angle periodicity acts as
Composed twice it gives w↦w+4π/3; composed three times it gives w↦w+2π, which is the trivial periodicity of any angle. The sky-angle periodicity is therefore of order three on the fundamental domain [0,2π/3) if we identify the endpoints, with τ3= id as the appropriate relation.
We denote this generator τ and note its order.
With σ of order two and τ of order three, the group generated by σ and τ acts on the circle w∈R/2πZ. The relation between them is determined by computing the composition στ and identifying its order.
The composition στ first translates by 2π/3 and then reflects about π/6:
Composed with itself this gives
so στ is also of order two. The group generated by σ and τ with the relations σ2=τ3=(στ)2= id is the dihedral group of order six, D3≅S3.
By Proposition 5 the within-single-geometry morphisms of G are exactly the dihedral group D3. The six elements are the identity, the two non-trivial powers of τ, the involution σ, and the two products στ and στ2. They act on the sky-angle circle by permuting six labelled positions, corresponding to the six distinct sky-angle labels a single intrinsic curve can carry under the admissible vantage-changes within one geometry.
The slicing paper showed (the Conjugacy proposition of [JanzenSlicing]) that the cubic involution σ on and the chart involution g1 on the throat angle are conjugate by an explicit closed-form map χ: that is, the same involution appears in two coordinate systems related by sin u = (2/√3) sin w.
Within the present development, this conjugacy is the statement that the relations (5) hold in either coordinate system. The throat-angle coordinate carries the same D3 structure, with σ realised as the reflection of about its midpoint in one fundamental domain and τ realised as the periodicity of on the throat circle. The conjugacy is the change-of-coordinates within G; the group structure is invariant.
The slicing paper proved (the Rigidity proposition of [JanzenSlicing]) that the SdS geometry carries no continuous moduli under change of charting vantage. The geometric reason was identified in Section 7 of that paper: the throat radius α=√3/Λ is invariant under the reading-swap involution, across all slicings, and under the projection; only the mass parameter varies with the slicing, via the relation 2M=α ((r0/α)-(r0/α)3). That relation, and the degeneracy this section formalises, are already in the 2012 dissertation: it writes 2M=r0-r03 in the scale-invariant gauge and records that the three roots of the horizon equation form one triplet—“for a given value of any one unambiguously fixes those of the other two”—so that the geometry at one admissible r0 is the same geometry as at either of the other two, with the parameter ranges related by explicit formal equivalences [JanzenThesis, § ch. 4]. What the present section adds is the group-theoretic content: that this degeneracy is a groupoid, that its morphisms are generated by σ and the sky-angle periodicity, and that the collapse is dimensional. In the gauge α=1, the slicing parameter r0 moves continuously over its admissible range. Two distinct invariances are in play, and the dimensional collapse rests on the second. First, at fixed r0, a change of charting vantage leaves the whole SdS geometry—, , and all—invariant: this is the Rigidity proposition, and it is what makes the within-single-geometry groupoid G well defined. Second, as r0 varies, varies with it (so the SdS member changes), but the underlying de Sitter manifold, fixed by the throat radius α, does not: this is the invariance of Section 7 of [JanzenSlicing]. It is this second invariance that the dimensional collapse formalises—the continuous family of slicings collapses onto the single de Sitter manifold α they all chart, not onto a single SdS member.
The present section gives the algebraic content of this rigidity. We call the mechanism dimensional collapse: a continuous parameter space of slicings collapses onto a single geometric invariant—the de Sitter manifold α—with the discrete morphisms of G permuting the labels of a single member and the continuous variation of r0 producing no variation in the underlying manifold α (though it does move between members ).
The slicing parameter r0 continuously parametrises the vantages: choosing r0 chooses the point on the slicing curve at which the chart is centred, and the chart unfolds outward from r0. As r0 varies continuously over the admissible range, the chart's centre moves continuously along the slicing curve, and the labels the chart attaches to points on the curve change continuously with r0.
The geometry of the slicing curve does not change with r0. Different choices of r0 label the same intrinsic curve differently; the curve itself is fixed. The relation 2M=r0-r03 (in the gauge α=1) reads off the chart as a function of r0, but thus computed is the throat radius modulated by the chart's slicing profile, not a property of the geometry independent of the slicing.
The proposition formalises the α-invariance of Section 7 in algebraic terms. The continuous parameter r0 does not vary the underlying de Sitter manifold; it varies the slicing of it (and with it ), selecting different SdS members charted on the one manifold. This is distinct from the action of G: the discrete morphisms of G (the dihedral group D3) act at fixed , permuting the six sky-angle labels of one member while leaving that member's geometry fixed. The continuous variation of r0 over a fundamental domain of D3 sweeps once through the members of the family, all charted on the one de Sitter manifold α; the discrete morphisms relate the six such fundamental domains of the sky-angle circle to one another, matching member-by-member across them.
The geometric invariant of Vα is the throat radius α. The slicing paper established (Section 7 of [JanzenSlicing]) that α is invariant under the reading-swap involution, across all slicings, and under the projection of the celestial-sphere chart; in contrast is the throat radius modulated by the slicing profile, bounded by α and vanishing at r0=0. In the language of the present paper: α is the invariant of the whole slicing family Vα, fixed across every member; is the slicing-dependent quantity that varies between members as r0 varies. The morphisms of G do not move —they act at fixed , permuting the sky-angle labels of a single member (Section 4).
Consider a reassignment that promotes a null direction to the fundamental timelike congruence—the move a cosmological reading of the family requires. The group structure decides where it is available, and the answer is a single member. Acting on the slicing family Vα, the involution σ (w↦π/3-w) preserves each member's mass 2M and so acts within each member. At a generic member it exchanges the member's two vantages: the vantages come in σ-pairs, and a reassignment made at one is undone at its partner, so the two-cycle structure forbids it. At the single fixed point w=π/6 there is no partner to undo it—the member's one vantage is its own—and the reassignment is available there and nowhere else.
That fixed point is the Nariai member, and the requirement has a geometric reading: it is the unique member whose comoving fundamental congruence runs along a null generator of the de Sitter embedding without pivoting into a horizon, since only the direction at which the two positive horizons merge gives a comoving curve with no pivot. So the selection is a structural fact about the groupoid rather than a fact about dynamics or fine-tuning. A companion paper reaches the same configuration on independent grounds and carries the relation ΛG2M2/c4=1/9 as an equality [JanzenCRframework]; nothing in the argument below draws on that.
The involution σ is the reflection of the sky angle about w=π/6 (Proposition 2). The fixed point of this reflection is at w=π/6, where the involution leaves unchanged. At w=π/6, the slicing parameter is r0=(2/√3) sin (π/6)=1/√3, which is the Nariai value. The mass parameter at Nariai is 2M=(2/3√3) sin (3·π/6)=(2/3√3) sin (π/2)=2/(3√3), giving the Nariai relation 2M=2/(3√3) in the gauge α=1, equivalently ΛG2M2/c4=1/9 when units are restored.
the unique fixed point of the vantage involution σ is exactly the radius at which the areal acceleration changes sign. \end{quote} The two statements are of different kinds and neither entails the other here. This paper's is a fact about the groupoid: the vantage at which the two non-fixed roots collide and σ acts trivially. The other is a fact about a second derivative on a curve, computed from f' alone and holding generally in the mass. They meet only on the forced member, and they meet exactly. We record the coincidence without claiming a derivation of either from the other; what it does show is that the vantage the reassignment selects is not merely distinguished within the family of charts but is the locus at which the member's own expansion history turns.
The cosmological reassignment of de Sitter space (the CR causal reassignment of [JanzenCRframework]) promotes a null direction at the horizon to the fundamental timelike congruence. Within the groupoid G, this reassignment is structurally constrained: only the Nariai vantage admits a comoving curve along a null generator without pivoting into a horizon. Every other vantage in Vα produces a fundamental congruence that crosses the embedding's symmetry structure transversally, and that transverse crossing manufactures the horizon-like features of the corresponding black-hole reading.
The single-reassignment uniqueness is therefore the statement that the Nariai vantage is the unique vantage at which the reassignment produces a cosmology rather than a localised black hole. In groupoid terms: the Nariai vantage is the unique fixed point of σ, and the reassignment that selects a null generator without pivoting is the reassignment that selects the fixed-point vantage.
The single-reassignment uniqueness has a structural implication for the groupoid action. The generic vantages fall into σ-pairs (the involution acts on them by two-cycles); the Nariai vantage is the exception, the one fixed point of σ, lying in a degenerate orbit of its own rather than as a member of a generic six-element orbit. The cosmological reassignment selects that fixed point and is forbidden by the two-cycle structure at every generic vantage. The discrete symmetry of G therefore predicts the uniqueness of σ's fixed point as a structural fact about the groupoid, not a fact about dynamics or fine-tuning; that this fixed point is the cosmological configuration is the identification imported from [JanzenCRframework] (Proposition 8), so the uniqueness of the cosmological configuration follows from the groupoid structure together with that import.
This is the algebraic content of the CR cosmology's geometric forcing: the cosmology is the fixed point of the within-single-geometry involution, and the cosmological reassignment is the unique vantage-change in G at which the reassignment to a null generator is consistent with the fixed-point structure. Read against the cosmology's layered ontology, this locates the reassignment precisely. The vantages of G are charts—synchronizations of one rigid geometry, the shift-freedom of the framework's lapse–shift split—and the geometry is their single invariant; the reassignment that promotes the collapse horizon's null generator to the fundamental timelike congruence is the discrete morphism that installs the evolving layer whose rate is the lapse, carrying the collapsing interior across the branch point into the expanding cosmology [JanzenCRframework, JanzenCosmogenesis]. That it selects Nariai as σ's unique fixed point “not as a fact about dynamics or fine-tuning” converges with the empirical route—the redshift-isotropy floor forcing the same cosmic foliation from the data [JanzenModernParallax]: two independent routes, the algebraic here and the empirical there, selecting one configuration. And the eigenspace reading of the mass-reflection that the same discrete group carries—the invariant de Sitter geometry its -even part, the Schwarzschild mass its -odd perspectival artefact—is the shadow-reading partition in closed form [JanzenShadowExistence]: the same separation of the existent from the chart the layered rate rule runs on, here exhibited as an exact involution rather than inferred from a spread of perspectives.
The slicing paper showed (Section 5 of [JanzenSlicing]) that the seam—a turning point of the slicing curve—joins a Riemannian spherical piece to a Lorentzian de Sitter piece by the analytic continuation
and that the metric signature flips automatically across the seam because dθ=i dψ squares the continuation factor to -1. The signature flip is not imposed; it follows from the form of the continuation.
The present section reads this seam continuation as a vantage-change of a kind distinct from the within-single-geometry morphisms of Section 3. We call it a partial involution of the groupoid.
A vantage that charts the slicing curve from the Riemannian (spherical) side and a vantage that charts it from the Lorentzian (de Sitter) side are related by the continuation (6). The two vantages describe the same intrinsic slicing curve, but in different signature regimes: the spherical regime has ds2=dθ2+ sin2θdΩ2 (positive definite), and the de Sitter regime has ds2=-dψ2+ cosh2ψdΩ2 (Lorentzian). The continuation θ→π/2+iψ is therefore a vantage-change between two regimes of the same construction, with the signature flip a property of the continuation rather than of the descriptions.
This vantage-change is not a within-single-geometry morphism in the sense of Section 2. It does not act on the sky-angle labelling of a fixed slicing curve; it changes the analytic regime in which the slicing curve is read. We denote it ξ and characterise it as a partial involution.
We use the term partial because ξ does not act within a single sky-angle fundamental domain of G. It acts on the analytic continuation of the slicing curve into the complex parameter plane, where the spherical and de Sitter regimes are two real slices of the same analytic object. The within-single-geometry morphisms σ and τ permute the six labelled positions on the sky-angle circle; the seam continuation ξ moves between the two real slices of the analytic continuation. The two structures are complementary: σ and τ act within a regime, ξ acts across regimes.
The structural feature this captures is that the SdS construction unifies what the standard reading treats as distinct geometries (Schwarzschild, de Sitter, intermediate SdS family members) by exhibiting them as readings of one underlying intrinsic curve—one representation of one ontological layer, not autonomous geometries on distinct realities. The seam is the analytic structure that makes the unification work: at the seam, the Riemannian and Lorentzian descriptions meet, and the analytic continuation through the seam is the structural bridge that joins them.
The slicing paper noted (Section 5.4 of [JanzenSlicing]) that overcritical SdS (the regime 2M gt;2/(3√3) in the gauge α=1, where the cubic has only one real horizon) is the same continuation (6) applied to the horizon angle past the Nariai crest. The overcritical continuation extends the seam structure beyond the under-critical regime: the locus of horizons is an ellipse continuing analytically past the Nariai threshold, with the two coincident horizons at Nariai continuing as a conjugate pair on the complex extension.
Within the groupoid structure, the overcritical continuation is a further partial involution: it relates an under-critical vantage to an overcritical vantage of the same throat radius α via the analytic continuation past Nariai. The discrete generators σ and τ act within the under-critical regime (or within the overcritical regime considered separately); the partial involution ξ moves between Riemannian and Lorentzian pieces of the slicing curve; the overcritical partial involution moves between under-critical and overcritical regimes of the horizon cubic.
The vantages that move between distinct de Sitter representations (different values of α—distinct at the representational level) are not addressed by these partial involutions and remain in the open category of Section 10.
We now read the Schwarzschild description as one specific vantage in the groupoid G, and identify the geometric origin of the horizon-versus-singularity asymmetry as the forced off-axis pivot the Schwarzschild sweep is driven onto—the cascade of a single symmetry break, not a second break standing beside the first. The slicing paper articulated the sweep mechanism geometrically (Section 6.4 of [JanzenSlicing], “The horizon–singularity asymmetry is a sweep-pivot artefact”); the present section gives its algebraic content within the groupoid, and connects it to the pivoting/non-pivoting structure of Section 5.
The Schwarzschild description corresponds, in the slicing paper's framework, to the swing-zero (w=0) member of the SdS family at fixed throat radius α, read from the pivot vantage: the slicing curve there is the cycloid r(z)=M(1+ cos z), the equator taken diametrically, read as Schwarzschild from the swing-pivot looking down (the two-readings-at-the-throat discussion of [JanzenSlicing]). The relevant section of the slicing curve is the partial arc z∈[0,π], running from the horizon at z=0 (r=2M) to the standard “singularity” at z=π (r=0).
The Schwarzschild vantage is then the choice of charting that takes this partial arc and sweeps it about a fixed axis to produce the four-dimensional spacetime. The sweep is the conventional “rotational” construction that produces the Schwarzschild geometry from the cycloid: at each point of the cycloid, a two-sphere is attached, and the resulting four-dimensional manifold is parametrised by (z,θ,φ) together with a time-like direction.
The de Sitter description sweeps the complete arc of the slicing curve about the manifold's own axis of symmetry. The slicing curve, in the de Sitter reading, is the full half-arc r=α sin θ for θ∈[0,π], and the sweep through the rotational symmetry of the de Sitter hyperboloid produces the full de Sitter manifold. No symmetry is broken in this sweep that is not already broken by the choice of foliation; the de Sitter sweep is structurally clean.
The Schwarzschild description, viewing the hole from outside in the timelike orientation, cannot use that axis: the symmetric sweep about the manifold's own axis is the de Sitter, interior one just described, and it is unavailable to the exterior vantage. The sweep still requires a pivot; denied the axis, it is forced to take a selected point of the manifold as the centre it sweeps about. That point is r=0. It is a point of the manifold—the off-axis critical point at the back of the throat ring, where the two branches of the slicing curve, having conjugated around the ring, meet—and the circle paper [JanzenCircle] showed it to be a regular, non-degenerate critical point of the C∞ function r(z)=M(1+ cos z) on the smooth manifold z∈R/2πZ, no more special intrinsically than the horizon critical point at z=0. What the Schwarzschild sweep does is make a centre of this off-axis point: it pivots the whole partial-arc construction about a point that sits off the surface's axis of symmetry, as if it were the axis. The smooth structure contains the point; what it withholds is any warrant for treating it as a centre of symmetry.
This is not a second symmetry break standing beside a first. There is one break, cascading. The single break is the exterior vantage's locating of the hole—taking the hole, off-axis, as the reference of the chart (Section 6.4 of [JanzenSlicing]). Having located it that way, the vantage is committed: the restriction to the partial arc and the forced off-axis pivot onto r=0 both follow of necessity, not as further independent choices. Together they produce the asymmetric features of the Schwarzschild geometry—the horizon at one end of the swept arc, the apparent curvature singularity at the other—as the cascade of that one break.
This places the Schwarzschild vantage within the structure of Section 5. There the distinction was read at the level of the fundamental congruence: Nariai—the fixed point of σ—is the unique vantage whose comoving congruence runs along a null generator without pivoting, while every other vantage crosses the embedding's symmetry structure transversally. The forced off-axis pivot of the sweep is that same off-axis forcing, seen now at the level of the spatial construction rather than the congruence. Nariai alone sweeps about the manifold's axis and manufactures no asymmetry; the Schwarzschild description is a canonical pivoting vantage, and its horizon–singularity asymmetry is what the forced pivot leaves behind.
The circle paper [JanzenCircle] established that the two critical points of r(z)—at z=0 (r=2M, the horizon) and z=π (r=0, the standard “singularity”)—are non-degenerate critical points of identical analytic character, with d2r/dz2=±M alternately. As algebraic features of the cycloid, they are structurally identical. The standard reading classifies the horizon as a removable coordinate singularity and r=0 as a true curvature singularity, but this asymmetry is not forced by the analytic structure of r(z).
The present section's contribution is to identify the geometric mechanism that manufactures the asymmetric standard classification: the Schwarzschild sweep, forced to pivot on the off-axis manifold point r=0, makes that point the centre of radial infall, and the appearance of a singularity there is the shadow of the forced pivot—the Kretschmann scalar 48M2/r6 as a function on the parameter diverges at r=0 via the chain rule applied to the composition with the cycloid. The divergence tracks the chart's labelling of the pivot, not the underlying smooth manifold.
Under the perspectival reading in which this paper is written, the horizon–singularity asymmetry is read not as a feature of the geometry but as the signature of the forced pivot—the reading's interpretive payoff, the geometric account behind the algebraic asymmetry of Papers 1–2 [JanzenBHcausality, JanzenCircle], rather than a further proposition of the construction. So read, the result of Section 7 supplies a diagnostic for the groupoid programme: features of a chart that appear to be features of the geometry can be sweep artefacts. The diagnostic is: identify the sweep used to produce the chart; identify the point about which the sweep is pivoted; check whether that point is the manifold's own axis of symmetry or a selected point off it. If the sweep is forced onto an off-axis point, the asymmetric features arising in the swept chart are diagnostic of the forced pivot, not of the geometry.
The Schwarzschild description is the canonical case. Its horizon–singularity asymmetry is the canonical sweep artefact. Other applications of the diagnostic await development.
The scope of that attribution, and one computation often brought to bear on it. The two critical points are identical through first order and parted at second by the curvature term -M/r3; attributing that term to the forced pivot rather than to the mass is the perspectival reading's payoff and not a claim proven independently of it. A computation is often offered as though it settled the matter, and it is worth separating what it shows from what it does not. Along the cosmological branch the same perspectival invariant is free of the mass: with an amplitude carrying r∝M1/3 [JanzenCRframework] the M2 in K=48M2/r6 divides out exactly, leaving K=12α-4 sinh-4(3 τ/2α), a function of cosmic time alone—while along the interior cycloid, whose amplitude carries r∝M, the same invariant goes as M-4 [JanzenCircle]. What it shows is that the invariant parting the two critical points carries no memory of the mass along that branch, the cancellation being exact and structural. What it does not show is which attribution is right: the -dependence of follows entirely from the amplitude's r(M), a property of the trajectory swept and common to both readings.
The discriminating term has a second reading that the attribution question does not touch. On any member of the family d2r/d τ2=rKG [JanzenSlicing]P3R12, so the quantity parting the two critical points at second order is the comoving acceleration up to a factor of , and the -M/r3 carrying that parting is the mass's contribution to the deceleration. The parting is therefore one in dynamics and not only in curvature. This still does not settle the attribution—the term is -M/r3 on either reading, and giving it a dynamical name leaves the question exactly where it was.
The cosmological completion [JanzenCRframework] reads the within-groupoid uniqueness of the Nariai vantage as the structural identification of the horizon's null direction (selected asymptotically in collapse, per [JanzenBHcausality]) with the cosmic temporal direction of expansion. The present paper's contribution to this identification is the algebraic groundwork: the groupoid action establishes that the Nariai vantage is the unique fixed point of the within-single-geometry involution, and the cosmological reassignment is the unique reassignment that selects this fixed point.
The Null-Boundary Correspondence Theorem of [JanzenCRframework] reads, in groupoid terms, as the identification of the horizon's null direction in the collapse geometry with the cosmic temporal direction in the SdS-Nariai cosmology. The present paper supplies only the within-single-geometry component of this identification: that the Nariai member is the unique fixed point of the within-geometry involution σ, and that the cosmological reassignment is the unique reassignment selecting it (the preceding paragraph). The remaining component is cross-geometry—“collapse geometry” and “cosmology” label different members of the SdS family (different α), so relating them is a between-member move, not the same-member seam continuation ξ of Section 6 (which relates Riemannian and Lorentzian regimes of one slicing curve at fixed 2M). That between-member reassignment is in the open category of Section 10; the present paper does not establish it.
Sections 3–6 developed the morphisms one regime at a time: the within-single-geometry group D3≅S3 (Proposition 4), the seam involution ξ relating Riemannian and Lorentzian regimes at fixed 2M (Proposition 9), and the overcritical continuation relating under- and over-critical regimes at fixed α (Section 6). The slicing paper left the global organisation of the same-α between-member morphisms open, and named the action between distinct α as a further open direction. This section closes both. The same-α between-member morphisms are the monodromy of a branched cover, generated by the involution σ itself; the full discrete symmetry of the solution space is the automorphism group of the A2 root system the horizon cubic realises; and the action between distinct α is a continuous homothety under which that discrete symmetry is invariant.
In the gauge α=1 the horizon cubic is r3-r+2M=0, with discriminant Δ=4-27(2M)2 vanishing at the Nariai values 2M=±2/(3√3). Read 2M as a complex parameter: the three roots are the three sheets of a cover of the 2M-plane, branched at the two Nariai points where two roots collide.
The horizon cubic's roots, equivalently the sky-angle triple, realise the A2 root system [Humphreys1972] (the six roots arranged as a regular hexagon); the within-geometry group of Proposition 4 and the monodromy group of Proposition 12 are both the Weyl group W(A2)=S3. The coupled operations that generate this discrete symmetry—the root-exchange, the backward-radial reflection, their conjugacy, and their action across both regimes—are collected as the slicing's own in the companion slicing paper [JanzenSlicing]; the present section advances them into their group-theoretic form.
The morphisms between distinct α—the open direction named in the slicing paper and at the head of Section 10—are not a further discrete structure. The throat radius α=√3/Λ is the single scale of the construction (Section 4); the map between de Sitter representations of different α is the homothety r0↦λr0, α↦λα, a continuous one-parameter scaling, not a discrete morphism. That scaling has a projective identity worth recording, since it explains why it acts between representations rather than within one: a linear map preserves the substrate's absolute—the null cone η(X,X)=0, which is what fixes the causal structure—exactly when it is an element of O(5,1) composed with a dilation, and the dilation carries η(X,X)=α2 to η(X,X)=λ2α2P5R4. The homothety is that dilation. So the isometry group of a single de Sitter representation is O(5,1) while the group preserving the causal structure alone is O(5,1)×R+, and the one generator by which they differ is precisely the one that moves α: the causal structure fixes the geometry up to the scale, and the scale is what it does not fix. The word linear in that statement is not a restriction, which is worth saying because it looks like one: the conformal group of a pseudo-Euclidean space acts linearly on the null cone of one dimension higher, translations, dilations and the special conformal transformations alike, an SCT appearing non-linear only in an affine chart of the cone's projectivisationP3R24. So the count above is of the whole conformal group and not of a linear part of it. The same generator has a conservation-law face, and it is worth recording because it says which observers can and cannot detect the scale. A dilation is a homothety, Lξg=2cg with constant, and for any vector field the charge ξapa obeys d(ξapa)/ dτ= 12 papb(Lξg)ab along an affinely parametrised geodesic. For a Killing field the right-hand side vanishes identically; for a homothety it is c papa, which vanishes if and only if the geodesic is nullP5R11. So the dilation carries a conserved charge on the null cone and on nothing else—which is the statement above met from the conservation side, a symmetry conserving a charge exactly on the structure it preserves. The failure off the cone is proportional to papa=-m2, so it is rest mass that spoils it, and rest mass is the perspectival -odd quantity of this construction rather than an invariant of the substrate. We state the identity as the result and mark the gloss as a gloss. The discrete classification above is formulated on the dimensionless ratio r0/α= sin u (equivalently the sky angle ), so it is invariant under this homothety: the same Aut(A2)=D6 governs every α-leaf. There is accordingly no further between-α discrete morphism to classify; the action between distinct α is the homothety orbit, along which the discrete structure is constant.
The Nariai point of the cover—the configuration at which two designated horizons merge—is the locus at which the discrete structure of this paper meets the continuous structure of the algebroid completion [JanzenAlgebroid]: it is a stratum at which the algebroid connection vanishes and the substrate's isotropy enhances (the Nariai degeneracy at which the cubic's two designated horizons merge). That the monodromy group's Nariai point and the connection-vanishing stratum coincide is a structural meeting of the discrete (monodromy) and continuous (algebroid) faces of the one solution space. The relation can now be named, and the naming needs only the vocabulary of the field the two objects belong to. The algebroid the companion paper constructs is an action algebroid—its own notation, so(5,1) C, is that of the infinitesimal action of the substrate's isometry algebra on the space of cuts [JanzenAlgebroid]—and an action algebroid is always integrable, its integrating object the action groupoid SO(5,1) C C, whose objects are cuts and whose arrows are the isometries carrying one cut to another [Mackenzie2005]. That is not the groupoid G of this paper, whose objects are vantages on one geometry, and the two are separated exactly as the companion paper separates their operations, by dimension: the slicing is dimension-reducing and continuous, the reassignment dimension-preserving and discrete. What relates them is isotropy. In an action groupoid the arrows from an object to itself form its isotropy group; the coset grading is symmetric precisely when that isotropy makes a symmetric pair, so the companion paper's connection —the leak of [m,m] into m—measures the isotropy's failure to be of that kind, and vanishes exactly where it is. Its two vanishing strata carry isotropy of dimension ten and six against a generic fourP5R7. And G's Nariai point is where σ has its fixed point, which is where two designated roots merge—a vantage being a choice of designated root, and the choices collapsing exactly when the isotropy that permutes them grows. So the discrete face and the continuous face are not two structures that happen to degenerate at one locus: they are the isotropy of one action groupoid, read discretely and infinitesimally, and Nariai is where that isotropy jumps. And the jump is not a further coincidence to be added to the list the slicing paper's corollary already carries. That corollary locates Nariai three ways—the cubic's double root, the singular point of the reducible locus, the end of the conic's minor axis—and the framework and optics readings add the merged horizon and the photon sphere's contact with it; every one of those reduces to the single condition f(rN)=f'(rN)=0P5R9. The isotropy jump is what that condition is, read on the near-horizon geometry: a double root makes that geometry dS2×S2, whose isometry SO(2,1)×SO(3) is six-dimensional against a generic four, which is exactly the stratum the algebroid paper records [JanzenAlgebroid]. So there is one condition wearing several faces, and the isotropy is the face that explains why a discrete structure and a continuous one both mark the same locus.}
The groupoid G is the structural object the present paper develops. It is generated by the root-exchange involution σ (order two) and the sky-angle periodicity τ (order three), organised as the dihedral group D3≅S3, and within each SdS member it permutes the member's six labelled sky-angle positions while leaving that member's geometry invariant. The continuous slicing parameter r0 runs between members of the family Vα, all charted on the one de Sitter manifold of throat radius α; the dimensional collapse is the statement that this continuous family carries a single invariant, α. The cosmological reassignment selects the unique fixed point of σ on the family (the Nariai member), and the partial involution ξ at the seam moves between Riemannian and Lorentzian descriptions of the same intrinsic curve. The Schwarzschild description is the canonical asymmetric realisation, in which the sweep, forced off the manifold's axis of symmetry onto the off-axis point r=0, manufactures the horizon-versus-singularity asymmetry that the circle paper identified algebraically as two metric singularities of identical type.
The paper establishes the following:
The first seven results constitute the relational content of the groupoid within a single SdS member; the eighth extends it to the discrete symmetry of the full SdS solution space.
This paper's relational programme leaves no open direction of its own. A direction the slicing paper named—the action between distinct α—is resolved in Section 9: it is the continuous homothety r0↦λr0, α↦λα, not a further discrete morphism, and the discrete symmetry Aut(A2)≅D6 is invariant under it. One further direction is noted below, and it is an instance of the programme's named frontiers rather than a standalone open of this paper.
The mass question, settled. Whether the throat radius α established here as the invariant of G is also the correct intrinsic gravitational mass in the standard quasi-local (Brown–York, Misner–Sharp) and asymptotic definitions is now settled: it is not. The present paper sharpens the question by showing any fully invariant mass must be built from α; the companion slicing paper's mass section evaluates the standard definitions and finds they all return the slicing-dependent (Misner–Sharp M+r3/2α2, Komar M-r3/α2, asymptotically-de Sitter ), so the gravitational mass is the perspectival , α is the invariant curvature radius (a length, not a mass), and only the Nariai member locks an α-built mass, M=α/(3√3).
The first is an instance of the programme's named frontiers—the matter dynamics and the irreducible interiors—pursued in the framework and cosmology papers rather than here; the second is settled above. Neither is a standalone open of this paper.
The paper sits between the foundational results of [JanzenBHcausality], [JanzenCircle], and [JanzenSlicing], and the cosmological endpoint of [JanzenCRframework]. Its job is to fill the algebraic gap between the geometric construction of the slicing paper (which establishes the groupoid at the level of generators) and the cosmological reading in the completion [JanzenCRframework] (which uses the within-groupoid uniqueness of Nariai as the structural identification of horizon and cosmic temporal direction). With the gap filled, the programme's foundational structure is established at the scope of a single de Sitter geometry: the geometric construction is rigid, the groupoid of admissible vantages is algebraically characterised, the cosmological reassignment is identified as the unique selector of σ's fixed point—its cross-geometry component (Section 8) and the further directions of Section 10 remaining open—and the Schwarzschild description's apparent asymmetric features are diagnosed, under the perspectival reading, as sweep artefacts. Seen in the framework's unification, this discrete symmetry is the joint. The framework reads one maximally symmetric de Sitter substrate as at once general relativity's solution space, the discrete and charge structure, and the Standard Model's flavour and colour skeleton [JanzenCRframework]; the Aut(A2)=D6 established here is the discrete backbone shared across those readings—the monodromy group of the vantage-groupoid on the gravitational side, and, in its A2 root system carried analytically onto the substrate's conjugate real form, the skeleton the fermion content's discrete structure is borne on [JanzenMatter, JanzenBoundary, JanzenGeometricCore]. The matter sector separates the two indices that structure carries: the S3 among the vantages is the within-state one, while the generations' threeness is the turnaround's deck Z3 (ibid.). The one discrete symmetry joins the covariance of geometries to the matter skeleton.
We have given the groupoid G of Schwarzschild–de Sitter description vantages as a category, identified its two generators and the relations they satisfy (D3≅S3), and characterised the rigidity of the SdS geometry as a dimensional collapse of the slicing family onto its single invariant, the de Sitter manifold α, distinct from the within-member action of the groupoid. The cosmological reassignment uniquely selects the Nariai vantage as the unique fixed point of the involution. The seam between the Riemannian and Lorentzian pieces of the slicing curve is a partial involution of the groupoid, with the signature flip a property of the analytic continuation. The Schwarzschild description is the canonical asymmetric realisation, in which the sweep, forced off the manifold's axis of symmetry onto the off-axis point r=0, manufactures the horizon-versus-singularity asymmetry that the circle paper identified algebraically. The full discrete symmetry of the solution space is Aut(A2)=S3×Z2≅D6: the within-geometry and same-α between-member root permutations form the Weyl group S3—the latter realised as the monodromy group of the horizon cubic's cover branched at Nariai, with monodromy the root-exchange σ—the mass-reflection 2M↦-2M adjoins the diagram automorphism Z2, and the action between distinct α is the continuous homothety under which the structure is invariant. The asymmetry is a signature of the chart, not a feature of the geometry.
The relational content this paper establishes is the algebraic core that the cosmological completion reads as the structural identification of horizon's null direction with cosmic temporal direction, and that the foundation reads back as the geometric content of CR's ontological augmentation of general relativity [JanzenModernParallax]. That identification is the seam of the cosmogenesis: the reassignment by which collapsed matter re-founds its clock and continues as an expanding universe is this groupoid morphism, carrying the horizon's null direction to cosmic time, and the companion cosmogenesis paper carries it forward to the primordial light-element abundances the re-expansion produces [JanzenCosmogenesis].