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P8

Covariance of geometries over the de Sitter substrate

the slicing operator, the vacuum kernel, and matter as the bend of the cut

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Introduction and ontological setting

The slicing paper [JanzenSlicing] established the Schwarzschild–de Sitter family as the readings of a single intrinsic curve on the de Sitter manifold: a radial curve r(l) with dr/dl=√|f(r)|, f=1-2M/r-r²/α², whose turning points are the roots of the SdS horizon cubic, read as the Schwarzschild cycloid and the de Sitter arc at fixed throat radius α. Two further readings of that radius belong beside this one, and the second is dynamical. The companion slicing paper identifies it as the flat locus of the slicing surface's intrinsic curvature, and the epistemic companion gathers the descriptions as several shadows of one fact about the existent [JanzenSlicing, JanzenShadowExistence]. To those may be added that the areal acceleration of the marginal congruence is d²r/d τ²=-f'/2, which vanishes exactly at r³=Mα²: the same radius is the exact locus at which the areal radius stops decelerating and begins to accelerate, generally in the mass. So the handover this paper describes—from the sub-marginal bound orbits to the marginally-bound congruence—coincides with a change of sign in the acceleration and with a change of sign in the intrinsic curvature, three readings of one radius, of which only the last is dynamical. The de Sitter and Schwarzschild forms are two manifold-observer vantages on one slicing; the throat radius α=√3/Λ is the invariant and the slicing- and projection-dependent factor. That paper used the curve to classify the vacuum family. The present paper uses it to generate: it shows that the same curve, sent through the Einstein tensor, returns the entire static spherically symmetric sector of general relativity—the vacuum solutions as one distinguished locus, and the stress-energy of every other member as a functional of how the curve departs from that locus.

The reading this paper adopts of its own results is the programme's covariance-of-geometries reading, formalized as the substrate's action Lie algebroid in a companion [JanzenAlgebroid], stated here at the outset. Within Cosmological Relativity the de Sitter manifold is the fundamental representation in which the evolving three-dimensional universe { S_t}—the individuating primitive [JanzenCRframework]—is described; it is not a second ontological primitive. The present work lives entirely at the level of that representation: it concerns how slicings of the de Sitter manifold generate the representational geometries (the exact-solution line elements and their stress-energy), and it takes no position here on the worldtube-first individuation beyond inheriting it. At the representational level the structure is a covariance, and it is one level above the covariance of general relativity. General relativity holds a single geometry invariant under change of chart; the construction here holds the de Sitter substrate invariant under change of geometry, with the slicing curve as the gauge object. “Many line elements, one geometry” becomes “many geometries, one substrate.” We will state which parts of this are theorems and which is the reading the theorems ground. That held-invariant substrate is the geometric–ontological core [JanzenGeometricCore]; the present operator is where its cuts are realized as line elements and their stress-energy, matter the bend of the cut.

The discipline is the programme's, in the sense the foundation makes explicit [JanzenShadowExistence]: the results below ground the structural reading; they do not entail it as a theorem entails a corollary, and we do not claim more. The vacuum kernel, the bend-density identity, the lapse split, the cosmological geodesic and its collapse limit, and the embedding of the synchronous space in the second ruling are computed and verified. The covariance-of-geometries reading is adopted on their strength. And the scope is bounded: everything proven here is static and spherically symmetric, or its spherical-cosmological reassignment. Whether the operator reaches the rest of general relativity is posed here as the principal open problem (Section 10) and is answered in the companion range paper [JanzenRange]: the range is the symmetry-reducible sector of general relativity—rotation and the anisotropic vacuum classes included—leaving the radiation-filled, dynamical-matter regime beyond the wall of inhomogeneity as the piece that remains open.

The construction gauge and the matter functional

The slicing curve carries one function. In the static chart it sets the areal radius against proper radial distance, dr/dl=√|f|, and the line element it induces is

the same filling the temporal and spatial slots. We call (1) the construction gauge; the lock g_ttg_rr=-1 is the single-curve condition, and Section 5 is the analysis of what freeing it does.

The reading has no quarrel with a variational route. Because the reading here is directional—the cut primary, ρ the name of its bend—it is worth saying plainly that the direction is not a stance taken against the usual derivation. Varying an action returns a field equality, and an equality has no direction; which side is the existent and which the name of the other's bend is not a question the variation asks. The direction is fixed here by measurement rather than by formalism: the microwave-redshift isotropy floor forces uniform expansion to some three orders of magnitude and so rejects a rate tracking the lumpy content [JanzenModernParallax]. So the two are orthogonal, not in competition^P8R11—the same field equations either way, and the reading settled where the corpus settles it, on the sky.

What kind of map the operator is. The operator is defined on cuts, and the substrate's own isometries carry cuts to cuts, so it is worth recording what it does to those—the answer organises three results the companion papers state separately. An isometry taking one cut to another takes the geometry read off the first to the geometry read off the second, and does so compatibly with composition, so the slicing operator carries the arrows as well as the objects^P5R8.

Three properties follow, and each is a companion paper's subject. It is surjective onto its image by construction, and that image is what the range paper computes [JanzenRange]. It is not injective on arrows: three distinct cuts, the three roots of one horizon cubic, return the same 2M and hence one geometry, so isometries that move the cut may leave the geometry fixed—and that failure is exactly the vantage multiplicity the groupoid paper classifies [JanzenGroupoid]. Whether every isometry between two geometries in the image arises from a substrate isometry splits cleanly in two, and the algebroid paper's stratification settles the first half. At each stratum it names, the isotropy of the cut is not merely of the same dimension as the geometry's isometry group but is that group—SO(4,1) at Type O, SO(2,1)×SO(3) at the Nariai member [Nariai1951], and R_t×SO(3) at the generic Schwarzschild–de Sitter stratum [JanzenAlgebroid]^P8R9. So at fixed throat radius nothing is left over: every isometry of a geometry in the image is induced by a substrate isometry fixing its cut.

That equality is stated for the strata tabulated and not in general, and the reason is worth giving rather than leaving as a limit of the survey. An isometry of a cut's induced metric extends to an isometry of the substrate exactly when it also preserves the way the cut sits inside it—its second fundamental form—so the isotropy of a cut is the subgroup of the geometry's isometry group that preserves the embedding, and may in general be proper^P8R10. The strata for which the equality is exhibited are those whose symmetry is large enough to fix the extrinsic data as well as the intrinsic: at Type O the cut is totally geodesic and there is nothing to preserve, and at Nariai and the generic Schwarzschild–de Sitter class the product and static-spherical structures leave it invariant. Where the symmetry is smaller the two need not agree, which is why the classes the companion paper names without tabulating are exactly the ones the equality is not claimed for. The remaining half is the one the groupoid paper leaves open, the component relating distinct throat radii, and it cannot be closed the same way: the map between representations of different α is the homothety, and a homothety is not an isometry—it is the dilation by which the group preserving the causal structure exceeds the isometry group [JanzenGroupoid]. So the failure across radii is not a gap in the classification but the single scale appearing again: the leaves are not related by isometries because α is exactly what the causal structure does not fix.

Forming the Einstein tensor of (1) and writing the field equations with cosmological constant as G^μ_ν+Λδ^μ_ν=8π μν, the mixed stress-energy components are

The lock g_ttg_rr=-1 already shows in (2): tt= rr identically, i.e. p_r=-ρ for every . The single curve therefore reaches only the equation-of-state slice p_r=-ρ. We take the three results of the operator in turn—the kernel of (2), its general non-vanishing value, and the consequence of unlocking the gauge—and then read them geometrically.

The vacuum kernel: straight cuts are vacuum

Theorem 1 is the precise content of “straight cuts are vacuum.” The vacuum condition is not imposed and then solved by the SdS metric; it is the linear equation whose solution space is the SdS family, the integration constant. The Schwarzschild–de Sitter members are the offset cuts, the de Sitter form the M→0 centre, the Nariai case the degenerate-horizon member; all are the kernel of the matter functional (2)–(3). (Pure Schwarzschild is the α→∞ flat-space limit of the family.) The slicing paper's vacuum family is identified here as a kernel.

Matter as the bend of the cut

A general curve departs from (5) by promoting the integration constant to a function. Write

with m(r) a mass function. A straight cut has m'≡0 (the mass is a constant, “already there”); the bend is m'≠0.

The reading is exact: m'=0 is vacuum (Theorem 1); any m'≠0 is energy density, and equal to the rate at which the curve bends away from the 1/r vacuum profile. As a check on a non-vacuum member, the Reissner–Nordström–de Sitter curve f=1-2M/r+q²/r²-Λr²/3 is the bend 2m(r)=2M-q²/r, m'=q²/2r², returning ρ=q²/8πr⁴=p_t, the electromagnetic stress-energy read off the q²/r² term. The general leaf. The identity (7) is the spherical instance of a statement that holds for any spatial leaf. For a general leaf the energy density is the leaf's intrinsic-curvature departure from the de Sitter substrate,

which is the Hamiltonian constraint; ρ=m'(r)/4πr² is its spherically symmetric case. The reading of the bend is therefore not an artefact of the symmetry the rest of this paper works in: what the bend is extends to the general leaf, and the departure-from-substrate reading extends with it. Why a given leaf bends as it does is the dynamics, and is treated in [JanzenDynamics, JanzenCanonicalTime].

The stress-energy is not an independent posit fed into the geometry; it is the curvature of the cut.

The lapse split: the second function and the equation of state

The single curve carries one function and so reaches only p_r=-ρ (Section 2). General matter, with p_r≠-ρ, requires the temporal slot to be freed from the spatial one. Write the general static spherically symmetric metric with an independent lapse,

the spatial profile (the slicing curve) and the lapse, the time-stacking of the leaves. The construction gauge (1) is the lock A=f.

Proposition 2 separates the operator's data. The leaf—the spatial slicing curve—carries the three-geometry and, through its bend, the density: planar gives ρ=0, bent gives ρ=m'/4πr². The stacking—the lapse , how successive leaves are spaced in time—carries the radial pressure: locked to the leaf it gives the rigid vacuum equation of state p_r=-ρ, unlocked it gives general p_r. The slicing paper's curve is the shape of one leaf; the lapse is the second, independent datum, and Section 8 identifies it geometrically.

The separation carries a load elsewhere that is worth naming here, since it is the reason this proposition is cited outside its own paper. The companion cosmogenesis account has the causal reassignment act on the stacking—resetting it to the geometry-set rate—and not on the leaf; by Proposition 2 the density is a functional of the leaf alone, so it crosses the reassignment as inherited content rather than being remade by it [JanzenCosmogenesis, JanzenCRcosmology]. That is the structural transition law, and this proposition is its whole content: without the separation of the data there would be no sense in which a reassignment of the time-stacking could leave the matter untouched.

A second and independent support for the same conclusion is given, and the two should not be confused. The segment on which the reassignment is reached carries no cosmic time—the real part of the complex cosmic time does not advance along it—so a structure carried on the leaf has no interval in which to be altered, whatever the reassignment acts on [JanzenCanonicalTime]. The two differ in what they establish. This proposition says that nothing acts on the leaf, which is the stronger statement and requires the separation to hold; the zero-duration argument says only that nothing could, which is weaker but survives even if the reassignment were found to touch the leaf. The transition law is therefore better supported than either alone makes it look, and a reader checking it should know which is being leaned on where.

The geometric dictionary

The algebra of Sections 3–5 has a clean image on the de Sitter hyperboloid, which fixes “straight” and “bent” as statements about cuts of a fixed surface.

A planar section of the hyperboloid is a flat cut: its image is the slicing curve of a vacuum member. The central section, the plane through the axis (X₀=0 in the embedding, the equatorial throat circle continued onto its Lorentzian arc), is a geodesic of the hyperboloid and is de Sitter, M=0. An offset planar section—a parallel plane not through the centre—is not a geodesic, and is SdS with M≠0; the offset is the mass, through 2M=α ((r₀/α)-(r₀/α)³) [JanzenSlicing]. That identification says something about the asymptotic machinery, and it is the sharpest form of the point. An asymptotic mass charge is built to measure a property the spacetime possesses, read off at its boundary; but if the mass is where the cut sits, there is no such property to read—the quantity being sought is a placement of the section, not a content of the geometry. So the standing difficulty that no conserved charge is well defined in an asymptotically-de Sitter spacetime is not a gap in the definitions [JanzenCRframework]: there is nothing at infinity to measure, because the mass was never in the geometry to begin with—it is in the cut. This offset–mass relation is an odd cubic in r₀—it is -odd, 2M↦-2M under r₀↦-r₀—and its three roots, the A₂ weights of the horizon cubic, are read on a fermion sector as the three-generation multiplicity—built as three chiral zero-modes, forced within CR, in the matter-sector paper [JanzenMatter]; the continuous SU(3) that shares this A₂ root system, not an isometry of the Lorentzian substrate (su(3)⊄ so(5,1)), is carried on the substrate's conjugate real form—the compact S⁵= SO(6)/ SO(5) the global Wick rotation reaches—in the geometric-core paper [JanzenGeometricCore]. That the mass is the cut's offset is the geometric content of Newton's constant in the substrate's constant ledger: enters only as the offset-length GM/c², so it is not a coupling of independent matter to geometry but the identification of the mass-label with the cut's offset—the geometrisation of matter's being the bend, and the perspectival counterpart to the invariant null-ruling reading of [JanzenGeometricCore]. The thing the offset cut loses is geodesy, not planarity: both cuts are flat, both are vacuum, and geodesic-versus-not is the finer split inside the vacuum family. The non-geodesy is the geometric face of the off-axis sweep-pivot that the slicing paper identified as the origin of the mass.

A section that bends off every plane is matter: by Proposition 1 the bend off the planar profile is the density. So the dictionary is

planar cut ⇔ vacuum (ρ=0); central/geodesic ⇔ de Sitter; offset/non-geodesic ⇔ SdS, the offset the mass; non-planar cut ⇔ matter (ρ=m'/4πr²).

The cosmological sector

Under the timelike vantage the operator is the cosmology, and the same dictionary holds. The expanding face of the substrate is the family of marginally-bound radial geodesics.

Two readings of (13) belong here before the branching, because the operator's whole account turns on which parametrisation is being read.} Near the origin the Λ term is negligible and the law reduces to r=6^2/3M^1/3τ^2/3/2, in which α has cancelled between amplitude and argument: there the substrate's own scale does not enter, and Λ fixes amplitude and rate only away from it.

And the amplitude's mass-scaling makes a curvature invariant mass-free. Because r∝M^1/3 one has r⁶∝M², so the perspectival Kretschmann scalar is

free of the mass, since 6× 13=2 exactly. Two collapses of different mass reach the same curvature at the same proper interval from the origin. Along the interior cycloid, whose amplitude carries r∝M instead, the same invariant goes as M^-4 [JanzenCircle]. So the mass-dependence of the invariant tracks the slicing being read, not the geometry read through it—which is what the dictionary above requires, the mass being a datum of the cut rather than of the substrate.

One property of (13) is worth reading off, because the construction turns on continuing it and its branching has an order.} Write u=3τ/2α. Near u=0 the hyperbolic sine is times a function analytic and non-vanishing there, so the whole branch structure of sits in the factor u^2/3: one circuit of the branch point multiplies by e^4πi/3, a primitive cube root of unity. The cosmogenesis branch point therefore carries a Z₃ monodromy—one circuit does not close it, and three do^P8R12. Of the three determinations exactly two carry real : the real axis gives r gt;0, the expanding leg, and the line Imu=-π/2—which is the offset Im τ=-πα/3 the companion papers continue along—gives r=-(2Mα²)^1/3 cosh^2/3, the conjugate branch [JanzenCircle, JanzenSlicing]. So the two legs the construction uses are two of three sheets, and the third carries no real geometry to continue onto. The order is inherited from the mass term: near r→0 the 2M/r in (12) dominates and gives r^3/2∝τ, so the 3 here is that exponent's and not the horizon cubic's, whose degree comes from the r²/α² term instead. These are two threes of the same order and different origin, and the distinction is recorded rather than elided: this paper's programme has already had to withdraw one conflation of two threes [JanzenMatter].

Where that locus is, is worth stating, because it does not lie on this branch. The effective potential is itself, so V_eff=1 is the condition r³=-2Mα²: the marginal energy is fixed at the conjugate-branch radius r=-(2Mα²)^1/3—the comoving turning point of the lap—where the field vanishes and (5) reduces to Minkowski. The same locus fixes the sign, since t=τ holds there if and only if E=+1. So the marginal value is fixed by the field rather than chosen, and the radius that fixes it lies on the branch the cosmogenesis continues onto [JanzenCircle, JanzenCRframework].

The E=1 congruence is the Painlevé–Gullstrand frame [Painleve1921, Gullstrand1922]: the comoving observers whose constant-time slices are flat, with comoving speed v=√1-f=√2M/r+r²/α², which reaches unity (v→1, null) at the horizon f=0. This is why the cosmology is flat ΛCDM and not the closed cosh universe: E=1 is the flat slicing of the same SdS geometry; the closed cosh slicing is the orthogonal, synchronous one. The choice between them is the third datum below (Section 8). The single E=1 geodesic, threaded through the horizon-regular Painlevé–Gullstrand chart, is dust collapse read inward and dust cosmology read outward; the horizon is the seam the one congruence crosses, not a wall between two regimes.

The same two terms locate the boundary between local boundedness and the cosmic flow^P8R3, and tie it to the slicing paper's curvature reading. The E=1 congruence is the marginally-bound expansion—the background cosmology—and an overdense shell, carrying more mass than the background, can turn around and remain bound only within the Hubble–Eddington radius [Eddington1933] r_HE=(Mα²)^1/3, where its inward pull (the same 2M/r of (12)) balances the outward r²/α²; beyond r_HE the cosmological term wins and the shell is carried outward with the expansion [PavlidouTomaras2014]. This r_HE is the maximum of the structure function , its zero of f', and the slicing paper reads it as the flat locus of the existent slice's own Gaussian curvature—K_G=1/α²-M/r³=0 there, the local bend of the cut exactly cancelling the substrate's cosmological curvature [JanzenSlicing]. The two readings are not two facts that agree: K_G=-f'/2r identically, and the factor never vanishes for r gt;0, so the zero of the operator's derivative and the flat locus of the cut are one locus described twice [JanzenSlicing]. And the cancellation has a dynamical face: on the E=1 worldline d²r/ d τ²=rK_G, so the epoch at which the cut's local bend exactly cancels the substrate's cosmological curvature is the epoch at which the expansion stops decelerating and begins to accelerate—the bend's cancellation read as a cosmological event rather than only a geometric one^P3R12. So the local structure and the global expansion are one Schwarzschild–de Sitter geometry read at two energies: the marginally-bound E=1 slicing is the cosmology, the sub-marginal bound structures turn within their own r_HE, and the Hubble–Eddington radius—a per-structure boundary carried by every mass—is where a structure's own bend of the cut gives way to the substrate's Λ-set rate. Its standard use as a local test of Λ from the largest bound structures [PavlidouTomaras2014] is, in this picture, a measurement of that handover. This boundary is gathered, in the framework paper's first synthesis, with the other standing problems the layered distinction dissolves [JanzenCRframework]; its several descriptions—the flat locus of the slicing paper, this energy handover, the one scale read at two ranges—are resolved by the epistemic discipline into a single existent fact, one of that discipline's own assessments [JanzenShadowExistence].

The matter content reads off (13) exactly as in the static sector—as the bend. Writing the closed-form Friedmann readout of (13),

the identity ²=1+ csch² splits the right side into the “1,” which is Λ (pure de Sitter, the unbent expansion), and the csch², which is a pressureless dust, ρ∝a^-3, p=0. The dust is the bend of the expansion off pure de Sitter; its amplitude is fixed by Λ, not dialled. Equation (15) is read here leftward: the cut r(τ) is the primary object—a slicing of the substrate along the comoving worldlines—and ρ is the name of its bend, not its cause. The expansion rate is set by the geometry (which cut); the density is read off afterward. This is the framework paper's result [JanzenCRframework] that the expansion is governed by geometric structure and not by the stress-energy content of space.

Read rightward, the same equation is Einstein's 1917 proposition, and that is the point of contrast the phrase marks wherever it recurs below. His static solution fixes the cosmic scale from the content—with λ=κρ/2 the radius is R=1/√λ, so the density of the world-matter determines the size of the world [Einstein1917]—and Friedmann's expanding solutions carry that direction of determination forward, the scale factor evolving as the content sources it [Friedmann1922]. The leftward reading reverses exactly that: Λ is geometrically primary, the throat radius α=√3/Λ is the substrate's own single scale, and the content is carried on a rate it does not set. So the term Einstein introduced to hold the world at a fixed size against its matter is here the one thing that fixes the scale, and it fixes it as the geometry's rather than as a balance. This leftward reading is the level the cosmology papers run their observable expansion on—the foliation stacking rate, “L1” of the three-level scoping that keeps it apart from the leaf-level local dynamics (the self-gravitating collapse, radiation included) and the E=1 projection: radiation is excluded from it not by a mechanism but because content is read off the clock the geometry sets, never summed into it [JanzenCosmogenesis].

That the amplitude and rate are both fixed by Λ rests on the slicing framework, not on a free choice. The slicing paper [JanzenSlicing] shows that the overcritical curve—the line that misses the throat circle—leaves unconstrained, whereas the cosmological curve is forced to the Nariai tangency, the curve meeting the throat at its critical point (ΛM²=1/9, the double root at r=1/√3 in the gauge α=1). The tangency is the mass: it fixes 2M=2/3√3α and the amplitude to 2^1/3α/√3, by Λ alone. The framework paper's selection of Nariai as the unique non-pivoting cosmological member [JanzenCRframework] is, in the present language, the requirement that the cosmological cut be the tangent one.

The synchronous space is the second ruling

The lapse of Proposition 2 has a geometric identity, and it answers what the third datum—the choice of synchronous slicing—is on the substrate. The de Sitter hyperboloid is doubly ruled by null generators; the comoving congruence is built from one ruling and the synchronous space from the other.

In the five-dimensional Minkowski embedding -X₀²+X₁²+…+X₄²=α², the flat-slicing comoving worldline of pure de Sitter is the unit timelike geodesic

strung between two distinct null directions: it asymptotes to the future null generator as τ→+∞ and to the past one as τ→-∞. The future generators are the per-worldline cosmological horizons (the Null-Boundary Correspondence map p↦H⁺(p) of [JanzenCRframework]); the past asymptote is common to the whole congruence—every comoving worldline, at every spatial position, runs into the same as τ→-∞.

The flat synchronous slices are the level sets of η(X,B): a direct computation gives η(X,B)=1/2α²e^τ/αP8R6, independent of the transverse coordinates and so constant on each slice, so the constant-time surfaces are the horospheres normal to the second (past) ruling . The synchronous space is the second ruling. This identifies the lapse: the closed cosh slicing is synchronized to the timelike vertical (not a ruling), the flat slicing to the null ruling , and the choice between them—the lapse-shift of Proposition 2—is which structure the comoving frame synchronizes to. Synchronizing to is what makes the slices flat, hence flat ΛCDM rather than the closed cosh universe. The leaf, the stacking, and the second ruling are one set of data: the curve is the leaf, the lapse is the stacking, and the stacking is the second ruling realized as the synchronous space. This second ruling is the geometric origin of what the cosmology papers read observables through—the E=1 projection, the synchronous shadow (“L3” of the three-level scoping); the standing error the layered framework names is to take that synchronous appearance for the existent, computing on the shadow rather than carrying the physics on the layer and projecting [JanzenCRframework, JanzenShadowExistence, JanzenCosmogenesis].

This also fixes the nature of the cosmological singularity. As τ→-∞, η(X,B)→0: the horospheres pile onto the null plane through and the whole congruence converges on the single common asymptote—“a single point at t=-∞.” That point is the smooth null generator ; de Sitter has no curvature singularity there. In the SdS reading the same structure becomes the r=0 origin of (13), which the comoving reading reports as a Big Bang singularity. It is the same artifact dressed with mass: the fundamental metric is the closed-de Sitter sphere X(T)=α cosh (T/α), which is the substrate's defining constant α, and the r=0 singularity is the ill-definition of a derivative metric there [JanzenCRframework, Janzen2015]. The reading inherited from the synchronous slicing—that flat space extends from r=0 outward with matter ejected from a point singularity at the origin—was Lemaître's [Lemaitre1949]; it inverts the geometry, reading the smooth common asymptote as a source. The leftward reading of Section 7 is its correction: r=0 is not a point matter emerges from, it is a horizon the slicing is anchored to. The singularity taxonomy of the companion papers makes this exact. The cosmological beginning is a finite-curvature metric singularity—the smooth past null boundary , a horizon at which the comoving ruler collapses while the substrate curvature stays finite, the same species the event horizon was shown to be [JanzenBHcausality, JanzenCircle]—and not the infinite-curvature centre it is conventionally pictured as. The Big-Bang reading conflates the two species of the one genus, the cosmological face of precisely the doubled category error the circle paper identifies in the static hole; the slicing anchors to the finite-curvature horizon, never to the infinite-curvature centre.

The scope of this identification is worth drawing, because the property it turns on is exactly the one the collapse face lacks. The construction requires the constant-time surfaces to be normal to the second ruling—that is what the level sets of η(X,B) are, and the reading is not that a null ruling is available but that the slices are orthogonal to it. A collapse horizon's limiting causal direction is generically non-orthogonal to any spacelike slice [JanzenBHcausality], meeting its horizon tangentially—which is the condition that picks the merged double root, and the same condition that disfavours the synchronous identification there [JanzenCRframework]. So the identification above is a statement about the cosmological face and does not extend to the collapse face^P8R7. The two are separated by the horizon cubic's discriminant and not by a modelling choice: the massless member has discriminant 4α⁶ with three distinct roots and a transverse crossing, so nothing there selects against orthogonality, while the forced member's discriminant vanishes on the merged root at the front seam. The claim is bounded accordingly: what r=0 is on the collapse face is the branch point's, reached on the conjugate leg, and nothing here disturbs the other links of that chain.

Two clarifications follow, both established by the foregoing but obscured by the projection coordinate. First, the cosmological beginning is reached at finite proper time. The “t=-∞” above is the synchronous coordinate τ in which η(X,B)=e^τ/α, not the proper cosmic time; the proper time elapsed from a→0—the branch point r=0, which lies on the throat circle opposite the hinge and is not a turning point of the slicing curve—to any later epoch is the ordinary, convergent flat-ΛCDM age. The cosmological beginning is finite-curvature and a finite proper time in the past; only the conformal/synchronous chart sends it to -∞, exactly as the areal chart reads the finite throat as a divergence at r=0—the coordinate degenerating, not the geometry. Taking the chart's -∞ for the physical age is the time-face of the same doubled category error whose radius-face is taking r=0 for the source. Second, the branch point and the collapse's horizon are of the same finite-curvature species, and they are not the same locus. On the closed cosmogenetic bead the framework proves [JanzenCRframework, JanzenBHcausality] they are distinct loci of one worldline's lap: the lap is entered at a seam, runs the conjugate arc through turnaround and lift, reaches the branch point r=0, and leaves at the same seam point onto the expanding horn. The horizon is the seam at which the lap is entered; the big bang is the branch point the lap reaches after it, and cosmic time elapses between them. The single E=1 geodesic is dust collapse read inward and dust cosmology read outward, and it is one curve carrying both—but the two readings are joined along the lap and not at a point, so the collapse's event-of-events at the close of the antecedent cosmic time and the big bang at the opening of this one are the two ends of that passage rather than one event under two descriptions. The distinction is worth holding because the minimal S³ at the throat carries the whole lap of every worldline in the congruence: on it one worldline's branch point sits where another's seam does, so the sphere is not the beginning—the beginning is the r=0 locus alone, one point per worldline, and it is the labelling that distinguishes it.

The three constant-curvature slicings

The cosmological sector gave the flat leaf—the E=1 congruence of Section 7—and the synchronous section gave the closed cosh leaf as its orthogonal alternative (Section 8). Both are maximally symmetric constant-curvature spatial sections of the one substrate, and they are two of three. The third is the open, negative-curvature leaf; with it the cosmological sector closes over the full Friedmann–Lemaître–Robertson–Walker family of spatial curvatures.

The leaves are the constant-coordinate sections of the embedding hyperboloid -X₀²+X₁²+…+X₄²=α², sorted by the character of the held direction.

The open leaf stacks into a cosmology exactly as the other two do. Carrying the H³ sections through a timelike parameter τ, the embedding

lies on the hyperboloid and induces

the open (k=-1) slicing of de Sitter—the sinh counterpart of the closed leaf's cosh. The constant-X₁ section of Proposition 4 is its leaf at the instant cosh (τ/α)=b/α, with H³ radius √b²-α²=α sinh (τ/α).

The three slicings are one congruence at three energies.^P8R5 The radial timelike geodesics of (5) obey (dr/dτ)²=E²-f=(E²-1)+2M/r+r²/α² (Proposition 3); read as the Friedmann equation a²=8π/3ρa²+Λ/3a²-k with the dust term 2M/r and Λ=3/α², the curvature constant is

The bound congruence E lt;1 is the closed leaf (k=+1), the marginal E=1 the flat leaf (k=0) of Section 7, and the unbound E gt;1 the open leaf (k=-1). The mass and the cosmological constant are common to all three; only the energy of the comoving congruence—equivalently, the spatial curvature it slices—differs. The flat ΛCDM of Proposition 3 is the marginal member of the family, not a separate construction. Read inward rather than outward, the bound closed member is a collapse, as the marginal member's inward reading is the τ^2/3 dust collapse of Section 7: run inward, the closed member is the cycloid r=M(1+ cos η)—the recollapse curve of a closed dust cosmology, and the curve the companion reads as the Schwarzschild black-hole interior [JanzenCircle]. That identification is exact in the dust limit and a small- reading otherwise: the cycloid solves (dr/dτ)²=2M/r-1 identically, whereas the closed member of this family carries the further term r²/α², whose weight against the dust term is r³/2Mα² and so is negligible deep in the collapse and not at the turnaround. The distinction is not cosmetic: with Λ present the closed and open members have no elementary closed form at all—the substitution r=u² carries their quadrature to a sextic under the root—so the marginal member is the only non-degenerate one that integrates, and the cycloid and τ^2/3 are the two forms the family takes when Λ is switched off^L14. The operator thus carries the classical Oppenheimer–Snyder seed—that homogeneous dust collapse traces the interior of a closed Friedmann universe [Oppenheimer1939b]—as the closed member of its own family: the black-hole interior is the inward reading of the closed cosmology, exactly as the τ^2/3 collapse is the inward reading of the flat.

The dust is carried in each case as the bend, with no new mechanism. For a general spatial leaf the bend-density identity of Section 4 is the Hamiltonian constraint 16πρ=³R+K²-K_ijK^ij-2Λ; for an FLRW leaf K_ij=-(a/a)h_ij and ³R=6k/a², and it reads

the Friedmann equation whose curvature term is the leaf's own intrinsic curvature ³R=6k/a². The open leaf bends to carry pressureless dust exactly as the flat leaf does in (15); the open ΛCDM cosmology is the matter-filled E gt;1 slicing, not merely the empty open chart of de Sitter.

Two scope notes fix what this is and is not. First, the observed cosmology is the flat member, and this is not a free choice: E=1 is fixed by the field, at the radius where the potential vanishes (Proposition 3), and E=1 is the flat leaf k=0 (Proposition 4); the framework paper's reassigned ruling carries the sinh^2/3 law of that leaf and not the closed-slicing law, and its identification of Nariai as the unique non-pivoting cosmological cut [JanzenCRframework] fixes the mass. So the flat leaf is the construction's output rather than its input, and the two selections the reader might expect to be independent—the curvature and the mass—are each fixed, by the field and by the tangency respectively; the flat member's expansion history is developed in full in the cosmology paper [JanzenCRcosmology]. The open and closed leaves are the substrate's other admissible cosmological slicings—shown to carry matter through the same bend, not merely the empty sinh and cosh charts of de Sitter—and not the physical universe. Second, the open leaf is the spatial hyperbolic 3-space H³, the Riemannian maximally symmetric negative-curvature section, and not Lorentzian anti–de Sitter spacetime. AdS₄ carries two timelike directions and embeds in M^2,3; the one-time substrate M^1,4 admits no such section. The trichotomy is over the spatial curvature of the FLRW leaf, never over the signature of the spacetime.

Scope and open problems

The results above are established within the static, spherically symmetric sector and its spherical-cosmological reassignment: the vacuum kernel (Theorem 1), the bend-density identity (Proposition 1), the lapse split (Proposition 2), the geometric dictionary (Section 6), and the cosmological geodesic with its collapse limit and synchronous-space identification (Sections 7–8). Within that sector the operator is complete: the cut generates the geometry, the bend generates the density, the lapse generates the pressure, the vantage generates the signature, and the substrate is invariant under all of it. We close by naming what lies outside that sector, and where each of those now stands.

The range. Everything proven is spherically symmetric. The vacuum sector is exactly the planar cuts; the single curve reaches the p_r=-ρ equation of state; the lapse carries general radial pressure and the cosmology. One question the spherical results place but do not themselves answer is the range of the operator over the rest of general relativity. The sharp question is whether every solution—rotating (Kerr), generally anisotropic, the radiation-filled early universe—arises as some slicing of the de Sitter substrate, or whether only a privileged class closes up coherently. The mechanism is proven here; the range is settled in the companion range paper, and its answer is a theorem with physical content. If the operator is surjective, Cosmological Relativity is a generating theory for the geometries of general relativity, with the slicing curve the universal gauge object, and the structural reading of Section 1 is earned in full. If it reaches only a subclass, the boundary of what is reachable—those solutions the substrate “permits”—is itself physical content, a constraint on admissible matter rather than a defect of the construction. The operator is stated precisely enough there that the spherically symmetric sector is a rigorous instance and the Einstein tensor of a general (non-spherical) slicing is computable as a functional of the cut [JanzenRange]: the range is the symmetry-reducible sector of general relativity—a geometry is a cut of the de Sitter substrate exactly when its isometry group contains a sweep-subgroup of the substrate's—and within each such class the operator is surjective across all algebraic types, the vacuum members being the substrate's own family in the class and matter the bend. The vacuum kernels run from the one-parameter SdS family in the spherical case to the rotating Kerr–NUT–(A)dS family and the functional Weyl class; the boundary of the range is the loss of continuous symmetry.

The emergence of the bend. The bend-density identity is exact—(7) in the spherical case, (8) in general—but it states what the bend is, not why a cut bends as it does. The slicing operator is kinematic: it generates the stress-energy from the curve, not the curve's own dynamics.

What is open is narrower than that framing suggests, and the boundary is sharp. The confined case is exhibited: on a wave that one residual isometry still pins, the leaf's transverse-traceless mode evolves by a wave equation, with the wave's energy carried by the shear of the leaf [JanzenDynamics]. The boundary at which free gravitational radiation begins is the wall of [JanzenRange], identified as the rigidity of a single global sweep. Beyond it the framework leaves the dynamics of general relativity unchanged, so the general inhomogeneous evolution is ordinary dynamical evolution of the leaf rather than a new generative law, and its canonical formulation—the cut's advance generated by a true Hamiltonian—is the companion canonical-time paper's [JanzenCanonicalTime]. The bend crossing the cosmogenesis branch point is well posed there—the substrate's curvature finite at r=0 however the chart-borne geometry's invariants behave, and the tortoise measure convergent so the crossing carries no scale—and structurally governed [JanzenCRframework], with the worldline dynamics taken up for a concrete matter model in [JanzenCosmogenesis].

What closes it is a constitutive relation, and the construction supplies everything else. The contracted Bianchi identity is not imposed here but entailed by the cut's own geometry—it is what fixes the angular component from the radial pair (§2), so the two free functions of (9) carry exactly two independent stress components. Add an equation of state and the system closes into ordinary differential equations on the cut itself: in the spherical class, m'=4πr²ρ together with

What the construction does not supply is the equation of state—and general relativity does not supply it either. There the contracted Bianchi identity likewise gives conservation and not the constitutive law, and a fluid needs its own closure exactly as here. The difference between the two readings is therefore not that this one supplies less: it is that the content is read leftward off the cut rather than fed into it, which is the whole of Proposition 1. Where a generative law for the content itself would have to come from is the matter sector, whose gauge representations and mass spectrum are the boundary the companion papers draw [JanzenBoundary, JanzenMatter].

Smaller remainders. Three further items are noted for completeness, and they stand differently: the first is not owed, the second is settled below, and the third is an empirical discriminant rather than a construction. First, the explicit higher-dimensional embedding of the bent (non-vacuum) and Nariai cuts. The framework paper's analysis dissolves the need for it—the fundamental metric is the five-dimensional closed-de Sitter sphere and the SdS cosmology a derivative metric on the same manifold, not a higher-dimensional object [JanzenCRframework, Janzen2015]—so nothing rests on drawing one. Its value would be pedagogical: an explicit embedding picture would make visible how a matter cut bends off the planar vacuum profile and where the Nariai cut sits, a visualisation of the construction's central image rather than an owed result. Second, where the forced scale 2/√3 sits, and what it is not. In magnitude it is the lone Nariai root—the backward-radial root the closed slicing (the cosmogenetic lap) runs out onto and closes on, which is how the value enters the bead—standing against the merged pair of magnitude 1/√3.^P8R4 The signs belong to a vantage and the enumeration must not privilege them [JanzenCircle]: on the M gt;0 member the cosmology selects, the merged pair is degenerate at the single value +α/√3 and the lone root is -2α/√3, with the backward-radial reflection carrying the whole set to its conjugate reading. And the magnitude is shared by quantities that are not roots at all: the Nariai offset |r₀|=2/√3, which is a slicing parameter, and the horizon-locus ellipse's focal distance—the two foci lying at exactly that distance on the backward-radial major axis, at sky-angle w=π/4, the bisector of the Nariai crest (w=π/6) and the throat-tangent rulings (w=π/3) [JanzenSlicing]. The ellipse foci, the two null generators A,B of Section 8, and the lap-close root are thus all carried at the single forced scale 2/√3=2α/√3=2/√Λ—twice the merged-pair radius α/√3, the one gauge α times the pure number the equilateral cubic forces, no second scale. It is an item of the substrate's structural inventory—roots, rulings, foci, sky-angles—kept on record for the matter sector's mapping (receipt storyboard_receipts/foci_ruling_2sqrt3.py). Third, the empirical discriminant: that the expansion rate is geometric—set by Λ and the cut's offset, with radiation carried as content rather than sourcing the rate [JanzenCRframework]—is the in-principle distinction not merely from ΛCDM's parameter choices but from Einsteinian cosmology as such. The Friedmann–Lemaître–Robertson–Walker construction rests on four kinematic assumptions—a global cosmic time, hypersurface orthogonality of its flow, spatial isotropy, spatial homogeneity—together with one dynamical principle, that the scale factor evolves by the Einstein field equations sourced by the content. That last one is Einstein's 1917 move itself: his static solution fixes the radius from the density, R=1/√λ with λ=κρ/2 [Einstein1917], and Friedmann's equations carry the same direction of determination into the expanding case. Two of those five are not shared, and they fail differently. The dynamical one is not shared: here the rate is a property of the substrate's foliation and the content rides on it, rather than a scale factor evolving by the field equations sourced by the content. And hypersurface orthogonality is excluded rather than merely relaxed—synchrony is exactly the content the relativity of synchrony leaves free, a convention re-choosable without touching the geometry, so imposing it as a first principle fixes an unforced modulus and is maximal arbitrariness rather than least [JanzenCRframework, JanzenShadowExistence]. The remaining three are not assumed here either but derived: the cosmic time and the uniformity of its advance follow from the redshift-isotropy floor, with the maximal symmetry of the slices following under a Copernican premise [JanzenModernParallax]. So the distinction is from the whole family and not from a member of it. It is decided by early-universe data and not by the geometry—the observational content the cosmology paper develops in full [JanzenCRcosmology].

The central constructions are established and verified at the stated scope, which is the static, spherically symmetric and spherical-cosmological sector. The slicing operator generates that sector—vacuum as the straight cut, mass as the offset, matter as the bend, pressure as the unlocked stacking, the cosmos as the timelike vantage—over one fixed substrate whose only scale is the throat radius α=√3/Λ. The covariance of geometries it exhibits is, at the representational level, general relativity's covariance lifted one rung; that rung carries the symmetry-reducible sector of general relativity—its boundary the loss of continuous symmetry—as the companion range paper establishes [JanzenRange], the reach the construction earns. That covariance is the gravitational face of the wider unification the framework paper assembles: one maximally symmetric substrate read three ways—on its cuts, general relativity's solution space (this operator); on its discrete residue, the and charge-conjugation structure; on its two real forms, the gauge algebra and the quantum of action—of which this operator supplies the first [JanzenCRframework].