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P11

Why the cut bends

the dynamics of the de Sitter cut, the confined gravitational wave, the generative boundary of the slicing operator, and chirality as the turning of the polarization plane

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Introduction: from the static operator to the bend

The slicing operator of Cosmological Relativity generates the static, spherically symmetric sector of general relativity from one de Sitter substrate [JanzenOperator]. The gap this paper closes was named at the outset of the programme: the essay that first developed this cosmology recorded that “details involving the emergence of dynamical matter in the SdS cosmology have not been worked out” [JanzenFQXi2012]. This paper works them out for the inhomogeneous, time-dependent case. A slicing curve on the substrate is the radial profile of a spatial leaf; the straight cut is vacuum (Schwarzschild–de Sitter, the kernel), and a bend in the curve is matter, with density ρ=m'/4πr² fixed by the bend. The companion range paper extends this from a classifier to a map of reach [JanzenRange]: a geometry is generated as a cut exactly when its isometry group contains a sweep-subgroup of the substrate's, so the range is the symmetry-reducible sector, laid out along a null-degeneracy axis from the Nariai seam to a wall at which continuous symmetry is lost altogether. At the wall, inhomogeneity, free gravitational radiation, and the failure of any global sweep orientation are one and the same boundary; “why the cut bends” is named there as the open dynamical question.

The symmetric sector already has its answer in closed form, and it is worth stating before the inhomogeneous case departs from it. On any comoving worldline of a symmetric cut, at any energy,

with K_G the slicing surface's Gaussian curvature [JanzenSlicing]^P3R12. So the rate at which the symmetric cut's bend changes in time is the bend itself, up to the areal factor: the cut straightens, is momentarily flat, and bends the other way as the geometry passes its one sign change. That is the whole of the symmetric sector's dynamics, and what follows is the first case in which the bend is not symmetric.

This paper answers it, in the regime where the answer can be made concrete and exact. The static operator says what a bend is—matter, the anchor's image. The dynamical question is how a bend evolves: a leaf that is not maximally symmetric must carry its departure from the round S³ forward in time, and the governing law of that advance is what we exhibit. Three ingredients, each already established in the corpus, combine:

1. the anchor—the map from a cut to its stress-energy—whose energy sector is the Hamiltonian constraint and whose momentum sector is the bend in the shift [JanzenOperator, JanzenRange];
2. the cosmic clock—the substrate's preferred foliation, empirically forced [JanzenModernParallax] and structurally distinguished, on which the canonical constraint deparametrizes to a true Hamiltonian [JanzenCanonicalTime, JanzenCRframework];
3. the stratification—the symmetry classes of the range read as the isotropy strata of the construction, from the maximally symmetric vacuum down to the wall [JanzenRange].

We assemble them on the simplest model that carries a genuine propagating degree of freedom—a linearly polarized Gowdy–de Sitter spacetime [Gowdy1974, EinsteinRosen1937]—and then read the result back into the stratification and out to the wall. The register is that of the rest of the programme [Janzen2025, JanzenCRframework]: the field equations are Einstein's, unchanged; the existing object is the evolving three-dimensional layer, and “the cut bends” is a statement about that layer's advance, not about a four-dimensional block. In the layered reading the cosmology runs on, this is the dynamics of the leaf and its rate: the bend is the leaf's content (matter, its departure from the round S³), and the advance that carries it forward is generated by the same lapse whose deparametrized true Hamiltonian is the foliation's rate of advance [JanzenOperator, JanzenCanonicalTime]—so where the static operator gives the leaf and its bend, this paper supplies how that leaf evolves. The wall at the far end of the stratification is the boundary of that layered generation: up to it the leaf is generated by a symmetric sweep of the substrate, and past it, where the last continuous symmetry is gone, it is carried instead by ordinary evolution—the same seam-to-wall span along which the observable cosmology is laid out, the seam its inner (Nariai) end [JanzenRange].

The confined bend: a polarized Gowdy–de Sitter wave

Consider the linearly polarized Gowdy–de Sitter metric

with ψ,γ,R functions of (t,z) only. The field ψ is the single transverse-traceless polarization^P11R1—the leaf's shear, the propagating graviton; is the area element of the (x,y) two-torus, the local clock/area variable; γ is the conformal factor on the (t,z) plane. With cosmological constant Λ, the Einstein equations G_ab+Λg_ab=0 reduce to a wave equation, an area equation, two constraints, and a quadrature for γ:

These are verified directly from (2) (clean residuals). Equations (5)–(6) are not evolution equations: they are precisely the ADM Hamiltonian and momentum constraints [ADM1962] for the foliation by constant- leaves, and they say that the wave's energy and momentum are carried entirely by the shear of the spatial leaf—ψ_t²+ψ_z² and ψ_tψ_z. This is the anchor at work in the dynamical setting: matter and radiation are the bend, and here the bend is the leaf's transverse-traceless shear.

A true Hamiltonian, not a frozen constraint. The canonical Hamiltonian density built from (2) generates (3), (4), and the γ-quadrature through Hamilton's equations: the time development of the leaf is reproduced exactly. The lapse N=e^γ-ψ is not a free multiplier to be gauged away but the metrical rate at which the existing layer advances, dτ=N dt, exactly as in the cosmological and canonical-time papers [JanzenCRframework, JanzenCanonicalTime]. The constraint, read on the substrate's foliation rather than an arbitrary one, deparametrizes:

with the reduced Hamiltonian for the propagating mode. This is the same content as the minisuperspace deparametrization of [JanzenCanonicalTime], now carrying a local propagating degree of freedom: the same absolute foliation that makes flat ΛCDM a true-Hamiltonian flow makes the graviton's confined polarization a true-Hamiltonian flow.

The two regimes. At Λ=0 the model degenerates to the classical Gowdy/Einstein–Rosen system [Gowdy1974, EinsteinRosen1937]: the area may be used as an internal clock and the wave equation (3) becomes the cylindrical (Bessel) wave equation, ψ_TT+R^-1ψ_T-ψ_zz=0 in the -time gauge. At Λgt;0 this internal-clock construction fails— no longer separates the dynamics cleanly—and the substrate's cosmic time is the natural clock instead. This is the dynamical analogue of the point made in the canonical-time paper: the absolute foliation is not a convenience but the structure on which the constraint becomes a true Hamiltonian, and it earns that necessity precisely where an internal clock breaks down.

The unpolarized cut: the wave map, and a conserved handedness

The cut above is linearly polarized, and by the criterion of §6 that makes it achiral: its single polarization is pinned to a fixed axis by the residual T². The first chiral member of the reachable sector is therefore not this one but its unpolarized generalization, which we now build—restoring the second polarization as the off-diagonal ω of the torus block,

with ψ,γ,ω,R functions of (t,z) and ω≡0 returning (2). The torus block still has determinant R², so remains the area element, and G_ab+Λg_ab=0 still has exactly six nonvanishing components: the conformal–constraint sector and the torus block in which the second polarization lives.

The two polarizations are a wave map into the hyperbolic plane. In the variables

the torus block's equations are, identically,

which is the harmonic-map system for a map from the (t,z) plane, with density , into the target dP²+e^2PdQ²—a metric of Gaussian curvature -1, the hyperbolic plane^P11R7. The two polarizations are therefore not two decoupled fields but one point moving on a negatively curved surface, the polarized case being the geodesic Q= const and the turning of the polarization plane being motion off it.

And Λ is absent from that sector. Both of (10)–(11) hold with no cosmological term: Λ enters only the area and conformal equations. This is worth stating because it explains the source in (3) above. With Q≡0 the identity (R P_t)_t-(R P_z)_z = 2[(Rψ_t)_t-(Rψ_z)_z]-(R_tt-R_zz) is exact, so the vanishing of (10) is precisely the polarized WAVE equation minus the AREA equation, 2ΛRe^2(γ-ψ)-2ΛRe^2(γ-ψ)=0. In the wave-map variable the graviton propagates freely and Λ drives only the area; the source in (3) is the area equation, changed variables, and it is the second polarization that forces the variable in which this is visible.

The handedness is then the disconnected component of the target's isometry group, and it carries a conserved charge. The helicity-flipping parity of §6—the transverse reflection x↦-x—sends g_xy↦-g_xy and fixes every other component of (8), which in the variables (9) is exactly Q↦-Q with fixed. That is an isometry of dP²+e^2PdQ², and its differential has determinant -1: it reverses orientation, so it lies in the component of Isom(H²) that the identity component does not reach. No connected target isometry identifies the two handednesses. On the homogeneous reduction—-independent, hence a three-dimensional abelian isometry with spacelike orbits, among the classes the range paper lists as reachable [JanzenRange]—equation (11) integrates once,

and the parity acts on this twist as c↦-c. So on the reachable sector the graviton's handedness is the sign of a conserved charge, and the substrate's orientation parity is exactly its sign flip, with c=0 the polarized, achiral cut of §2. Solutions with c≠0 exist and their polarization turns, the first integral (12) reproduced along them^P11R7.

That reduction is homogeneous; the member the criterion is really about is the propagating one, and it too is explicit. Restoring the -dependence, the pair (10)–(11) is an unconstrained wave map: the two remaining Einstein equations fix γ by a first-order quadrature whose integrability ∂_tγ_z=∂_zγ_t holds identically on shell, so every (P,Q) Cauchy datum integrates to a genuine vacuum member and the wave carries no hidden constraint. An explicit inhomogeneous single-helicity travelling datum—two propagating modes coupled through the H² metric, not the homogeneous reduction—is then evolved to the roundoff floor with fourth-order convergence, and on it the polarization plane genuinely turns. Written in the orthonormal H²-frame strain-rates (h₊,h_×)=(P_u,e^PQ_u)—the plane wave's two polarizations in the wall limit—the criterion of §6 is the winding χ= d (h₊+ih_×), and it is nonzero with a definite sign on the turning member, identically zero on the polarized cut Q≡0 where the axis is fixed, and reversed by the parity Q↦-Q: the two helicities are exhibited as parity images on a propagating wave, the Cauchy problem the homogeneous reduction had left open^P11R8.

What this adds to the handedness argument of §6 is a third instrument, and the one that lies inside the construction's own reach.} The index theorem on the compact face and the polarization computation on the type-N plane wave both establish that a chirality can live only on the component no connected action reaches; but the plane wave is a geometry the operator provably cannot generate, its full isometry algebra being a five-dimensional Heisenberg algebra where so's unipotent radical is abelian [JanzenRange]. Here the same conclusion is a property of a wave map's target, on a cut the operator does generate—and it upgrades the statement from a feature of a geometry to a conserved charge of a field.

And the sector that criterion was always about—a fermion on this member—is built here rather than deferred to the matter paper's bound modes. Put a massless Dirac field on (8) and separate on the two Killing directions. The twist is then not merely analogous to a spinor's chirality: it is a component of the connection the spinor moves in. The orthonormal coframe e⁰=e^Adt, e¹=e^Adz, e²=e^ψ(dx+ωdy), e³=R e^-ψdy with A=γ-ψ carries ω_{t 23}=- 12 c e^-2ψ and ω_{z 23}=- 12 c_z e^-2ψ, with the conserved twist (12) itself. Rescaling by Ω=e^-A/2R^-1/2—which is exactly the leaf's own volume element, Ω²=1/√h—removes every background-derivative term and leaves four:

where m_x=e^A-ψk_x and m_y=e^A+ψ(k_y-k_xω)/R are the two Killing momenta seen as masses on the reduction and (b_t,b_z)=(c_z,c) e^-2ψ/4 is the twist, carried on γ⁵ alone. Two things follow, and the first is what the frontier item asks for. The principal part is e^-A(γ⁰∂_t+γ¹∂_z), independent of the twist and of the transverse momenta, so the characteristic matrix γ⁰γ¹ is Hermitian with eigenvalues exactly ±1: the characteristics are the light cone, on the inhomogeneous member as much as the homogeneous one, and nothing is trapped. The sector propagates—against the matter sector's wall modes, which bind precisely because a superpotential changes sign, and which have no counterpart here because there is no wall in [JanzenMatter]. The two norms that paper found to disagree agree here, being the same integral: the factor that flattens the reduced operator is the leaf's volume element, and this member carries no horizon in the (t,z) sector at which the two could part. Second, the chirality is the twist's sign. At vanishing Killing momenta (13) splits on γ⁵ into E=±(k-b) for one chirality and E=±(k+b) for the other—opposite momentum shifts—so the orientation parity is a symmetry only as the joint operation c↦-c with γ⁵↦-γ⁵, and neither half alone is. Evolved on a vacuum-Λ member the two chiralities accumulate a relative phase 2 ∫b dt, identically zero on the polarized cut c=0 and reversing with the sign of . So the graviton's handedness and the fermion's are one datum on this background: what §6 argues from two sectors returning the same parity is here a single field's own dispersion relation. The locking is exact in the Killing zero mode—which the torus always carries—and broken by the transverse momenta, which are the terms that break chirality in any case^P11R9L3.

The nonlinear Λgt;0 regime: the de Sitter attractor and classical admissibility

The Λgt;0 regime is where the dynamics is most distinctively CR's, and it can be carried through. The model has an exact isotropic de Sitter background, and that background fixes the foliation on which the nonlinear evolution runs. Isotropy is the equality of the three spatial scale factors—γ=2ψ and R=e^2ψ—and under it the homogeneous field equations (3)–(6) reduce consistently (clean residuals) to a single constraint and a single dynamical law,

solved by ψ_t=√Λ/3 e^ψ. The lapse is N=e^γ-ψ=e^ψ, so the scale factor a=e^ψ and cosmic time dτ=N dt=a dt give a(τ)=a₀ e^Hτ with H²=Λ/3—exact de Sitter. The internal area-clock R=t that organizes the Λ=0 Gowdy system is inconsistent here (AREA then forces 0=2Λt e^2(γ-ψ)≠0); the substrate's de Sitter cosmic time is the clock that replaces it, established positively and not merely by the failure of the alternative. The graviton is the anisotropic departure from this background, and the question of whether its nonlinear back-reaction forces a new (quantum) structure or only admits one—as the clean Λ=0 case does—is the question of that departure's evolution on the cosmic-time foliation.

In the homogeneous sector the answer is exact. Writing the shear shape s=ψ- 12 ln R (so a_x/a_y=e^2s, with s=0 on the background), the combination AREA-2 WAVE cancels the Λ source identically and leaves a conserved quantity,

the canonical shear charge^P11R4 (C₀=- 12(p_γ+p_ψ), verified on shell). Since 2R s_t=-C₀, in cosmic time—with spatial volume V=R e^γ-ψ—the shear rate obeys ds/dτ=-C₀/2V: as the background expands, V→∞ and the anisotropy rate decays as the inverse volume. This is cosmic no-hair [Wald1983]—the isotropic de Sitter background is an attractor, the shape freezing as the expansion-rate anisotropy redshifts away. The homogeneous nonlinear back-reaction thus admits clean classical dynamics: a conserved charge and a monotone decay, with nothing forcing a new structure.

The propagating mode admits as well, at linear order. Linearizing the wave equation (3) for δψ(t,z) on the de Sitter background and passing to cosmic time gives

a de Sitter wave equation: Hubble friction 3H, a gradient term k²/a² that redshifts away, and an effective mass m²=2Λ=6H² (principal series) in this fixed-background (transverse-traceless) truncation. This is a massive scalar on de Sitter and admits a clean, unitary Bunch–Davies quantization [BunchDavies1978]. Restoring the constraint back-reaction—the metric perturbations δγ,δR the truncation drops—and reducing to the gauge-invariant combination yields, in conformal time, the massless minimally-coupled de Sitter Mukhanov equation [MukhanovFeldmanBrandenberger1992]

with no a²m² term. The physical effective mass is therefore exactly zero^P11R5—the apparent shift 6H² →4H² is a gauge/truncation artifact, the ∝a solution being pure residual gauge—and the gauge-invariant perturbation is bounded—its two super-horizon branches going as a^-3 and as a constant, the latter the standard frozen mode, so that the a^-2 decay belongs to the rescaled variable W=a δψ in which the equation is written rather than to itself^P11R2L10. The propagating mode is thus a healthy massless de Sitter scalar, with no ghost, tachyon, or runaway: the admissibility is established at the gauge-invariant level by computation, not merely asserted robust to an ambiguous shift.

These computations, with the canonical structure of §2 already in hand, settle the classical question. The full nonlinear Gowdy–de Sitter system is a genuine true-Hamiltonian system—Hamilton's equations from the canonical H reproduce the full evolution—and it is exact vacuum-Λ general relativity, so its energy and momentum equations (5)–(6) are the first-class Hamiltonian and momentum constraints of GR. A first-class constrained system evolves consistently to all orders by the contracted Bianchi identity: there is no classical dynamical obstruction at any order. Two exact structures sharpen this beyond the Bianchi count. First, the homogeneous shear charge C₀=R_t-2Rψ_t extends to the full propagating system: AREA-2 WAVE is identically a continuity equation, ∂_t(R_t-2Rψ_t)=∂_z(R_z-2Rψ_z) with the Λ source cancelling, so ∫(R_t-2Rψ_t) dz is conserved in the full inhomogeneous model, not only the homogeneous sector. Second, the graviton's reduced energy e_grav=2R(ψ_t²+ψ_z²)≥0 is positive-definite, so the propagating sector is ghost-free to all orders. These close the ghost and zero-mode runaways exactly. The one structure they do not by themselves close is a propagating nonlinear parametric resonance, and here the natural monotone instrument fails for a precise reason: the total field energy grows as 2Λa⁴, carrying the background's own expanding energy, so no total energy is monotone. The right object is the gauge-invariant perturbation energy, which decays as a^-2 with the monotone (never periodic) coefficient of (17), excluding single-mode resonance at linear order and isolating the residual to nonlinear mode–mode transfer against the de Sitter detuning. Every tractable sector—the homogeneous nonlinear back-reaction and the linearized propagating mode—returns admissibility. The verdict, recorded at the scope it is earned:

The residual just isolated—whether the propagating nonlinear evolution no-hairs away rather than resonating into runaway—is exactly the future stability of de Sitter, and at the level of the physics it is a settled theorem rather than an open frontier. Friedrich proved the nonlinear stability of de Sitter in vacuum with Λgt;0: data near de Sitter have geodesically-complete maximal developments asymptotic to it, a proof of cosmic no-hair in the vacuum case [Friedrich1986]. His method—the conformal field equations, trading the global-in-time problem for a local one at J⁺—is also why no exact energy in this model closes the question: energy is the wrong instrument, and the failure of any monotone energy above is the expected shape of that fact, not a sign of openness. Friedrich is a small-data result, exactly the perturbative regime of the propagating graviton, so it settles the in-regime all-orders stability directly; Andréasson and Ringström extend cosmic no-hair and future stability to all data in the T³-Gowdy class with Λgt;0 [AndreassonRingstrom2016]—with Vlasov matter, a corroboration rather than a vacuum theorem—and the one non-generic exception, the Nariai branch, unstable within the Gowdy class [Beyer2009], is precisely the locus the range construction already isolates as the no-pivot seam [JanzenRange]. The classical propagating no-hair therefore holds for generic data, externally grounded, with the non-generic Nariai branch the genuine boundary. This is convergence across results—vacuum small-data, all-data Gowdy with matter, vacuum Nariai-genericity—rather than a single theorem covering the exact vacuum polarized Gowdy–Λ case, and the in-model computations above are the right in-model rigor the theorems ground rather than replace.

Two cautions fix the scope. This is not a constructed full non-perturbative quantization—that object is neither built here nor needed for the verdict; and it is not the claim that no structure could ever be forced, only that no classical dynamical obstruction exists, so that any forcing must live elsewhere than in the continuous evolution. The one remaining candidate is the discrete root structure of the construction—the horizon cubic's S₃ permutation symmetry and its A₂ root system (§7)—a separate, discrete object rather than a feature of the continuous dynamics, and developed in the algebroid construction [JanzenAlgebroid] rather than here. Within the dynamics proper the verdict stands: the cut bends by a true-Hamiltonian flow that admits quantization throughout the symmetry-reducible sector.

Where the bend lives: the Type-I edge of the stratification

The range paper classifies the reachable geometries by the symmetry a cut retains—read infinitesimally, the subalgebra of the substrate isometry that fixes the cut, the isotropy [JanzenRange]. This isotropy is the cut-fixing substrate isotropy; on the symmetry-reducible sector it coincides with the generated geometry's own isometry (the residual symmetry there is inherited from the substrate isometry that sweeps the cut), and it is this that falls, through the Type-D and Type-I classes, to zero at the wall (Type N). At the wall the two notions part company—a point we will need: the substrate isotropy is zero, while the Type-N geometry retains its own (large) plane-wave isometry that is not substrate-inherited. The isotropy is maximal at the vacuum cut (the de Sitter / FLRW stratum). The confined wave (2) sits at a definite stratum, and we can name it exactly.

This places the dynamics where the range paper said the confined radiation lives: “intermediate symmetry confines radiation (the Type-I Einstein–Rosen and Gowdy waves, on the edge)” [JanzenRange]. The wave is the worked first bend at the last confined stratum: a single residual symmetry (the T²) still pins the polarization to a fixed global orientation, so the wave is self-consistent only while it propagates transverse to that orientation. The dynamics is then the motion of the cut along the strata: as one descends from the symmetric vacuum, the bend turns on, generated by the true Hamiltonian, and the confined Gowdy–de Sitter wave is the explicit instance at the edge.

The wall: a generative boundary, not a metric singularity

The loss of the last Killing vector—isotropy from two to zero—is the wall: the Type-N, freely radiating boundary where no global sweep orientation survives and the polarization must reorient from place to place [JanzenRange]. Two questions stand at the wall. Does the dynamics simply continue across it? And is the wall a metric singularity—a place where, as at a black-hole or cosmological horizon, a clock can be re-founded by the Null-Boundary Correspondence [JanzenBHcausality, JanzenCRframework]? We settle both.

A metric singularity, in the precise sense of [JanzenBHcausality], is a collapse of the metric's measure: a null hypersurface along whose generators the spatial extent contracts to zero, so that two events that are null-separated (Δs²=0) and spatially coincident (Δx=0) are assigned vanishing temporal separation (Δτ=0)—distinct, causally ordered points the metric measures as one place at one instant. A Killing horizon is the worked sufficient case, not the definition: for Schwarzschild–de Sitter the static Killing field ζ=∂_t has |ζ|²=g_tt=-f, which vanishes at each horizon f=0, so its orbits become the null generators of zero invariant spatial extent and the measure-collapse hypotheses are met, while the Kretschmann scalar K=48M²/r⁶+24/α⁴ stays finite there—the finite-curvature species. Worldlines pass through, and the freed Killing direction is reassigned as the new timelike clock—this is cosmogenesis, the re-founding of a flat-ΛCDM clock on the reassigned horizon null direction [JanzenCRframework].

The consequence sharpens the whole picture rather than complicating it. The Null-Boundary Correspondence re-founds a clock by reassigning the freed null direction of a degenerating Killing field; the wall has no degenerating Killing field—its null Killing vector is null throughout, never crossing from non-null to null—so the cosmogenesis mechanism has nothing to act on there. Cosmogenesis lives at the finite-curvature Killing-horizon strata—the cosmological/black-hole horizon and the Nariai seam—and not at the wall. That cosmogenesis, the re-founding of the clock at the Nariai seam, is carried to a quantitative edge in the cosmogenesis paper [JanzenCosmogenesis]: the matter re-expanding from the seam produces the primordial light-element abundances, a fossil of the collapse that made it. The wall is instead the slicing operator's generative boundary: the locus where generation-by-symmetry, the sweep of a substrate isometry, runs out and hands off to ordinary inhomogeneous evolution. “Why the cut bends past the wall” is therefore a genuinely different question from cosmogenesis—it is the question of free, fully inhomogeneous gravitational dynamics, with no residual symmetry to confine it—and the dynamics of this paper, Hamiltonian throughout the symmetry-reducible sector, is exactly the dynamics up to that boundary.

The handedness: chirality as the turning of the polarization plane

At the wall the two transverse-traceless polarizations are free, so whether the radiation carries a handedness—a chirality no isometry can undo—is settled by direct computation rather than asserted. Resolve the harmonic profile of §5 into the two polarizations,

the Ricci tensor vanishing for any h₊(u),h_×(u)^P11R6—the two polarizations independent and unconstrained (the trace part of would be the matter bend, absent in vacuum). Two transverse operations act on the pair h₊+ih_×. The reflection x↦-x sends h_×↦-h_×, i.e. (h₊+ih_×)↦(h₊-ih_×): it is the helicity-flipping parity, exchanging the two circular polarizations, and is a symmetry of the wave iff h_×=0. A transverse rotation by θ sends (h₊+ih_×)↦e^-2iθ(h₊+ih_×), the spin-2 transformation, with polarization angle 12 (h₊+ih_×).

The handedness criterion follows. If the polarization is fixed—h_×=0, or any constant ratio h₊:h_×, a linear polarization along a fixed (possibly tilted) axis—that axis furnishes a fixed reflection symmetry, parity identifies the two helicities, and there is no genuine chirality; this is the polarized Gowdy wave of §2, its single polarization pinned to a global orientation by the residual T² isotropy of §4. If instead the polarization turns—12 (h₊+ih_×) varying with , the plane reorienting along the wave—no fixed axis is a reflection symmetry for all , no isometry identifies the helicities, and the radiation is genuinely chiral. The slogan of §5, that past the wall the polarization must reorient from place to place, is thus the exact criterion: chirality is the turning of the polarization plane, the handedness being the Z₂ sign of d/du (h₊+ih_×)—left- versus right-circular, helicity ±2.

The mechanism places this on the stratification. The reflection x↦-x is the radiative descendant of the slicing reflection that fixes a comoving worldline in the symmetric sector; there the swept SO(3) completes it to an orientation-preserving rotation and rotates it away, so the two handednesses are identified—a mirror, not a chirality. The criterion therefore bites not only at the wall but from the loss of that swept SO(3) onward: the polarized Gowdy edge (§4, residual T², one fixed axis) is still achiral, the unpolarized turning wave is the first chiral case, and the wall (no residual isometry) is where chirality is generic^P9R9. So handedness is achieved on a cut the operator reaches, and not deferred to a sector it does not generate: what the wall adds is that chirality there is generic, not that it begins there.

The handedness is the orientation parity of the substrate—the reflection lying outside the connected isometry group that sweeps the symmetric sector—surfacing in the radiating sector as the graviton's two helicities; it is not one of the cubic-organized markers of §7 but the radiating-sector face of that single orientation Z₂—the A₂ diagram automorphism (the mass-reflection parity R=γ⁵ of the groupoid paper) identified there with the de Sitter↔Schwarzschild vantage-swap, the outer involution completing the Weyl group to Aut(A₂)=D₆ [JanzenGroupoid]—whose development in the radiative sector belongs to the algebroid construction [JanzenAlgebroid].

That this handedness rides the disconnected orientation parity, outside any connected-group action, is also why it lies beyond the reach of the matter-sector index obstruction that renders a connected-gauge fermion spectrum vector-like [JanzenBoundary]: the gravitational sector is chiral precisely through the component that obstruction cannot touch. The agreement is closer than a shared conclusion: the two are one mechanism read in two sectors. The obstruction bites because a positive-dimensional connected group contains a circle whose action forces the equivariant index to vanish; the criterion above is achiral exactly while the swept SO(3) supplies a continuous rotation that completes the reflection and identifies the two helicities. In each a connected isometry identifies the handednesses, and a chirality survives only on the component no connected action reaches—which is what the boundary paper means by saying where geometric chirality can live at all. The two are established independently and by different instruments: there by an index theorem on the compact face, here by direct computation on (18) in the real radiating sector. So the handedness settled here is not merely consistent with that boundary; it is the same statement computed where the index theorem does not reach, and the stratification supplies what neither instrument alone does—the locus at which the identification is lost, the swept SO(3) of §4. (The helicity–parity relation is itself standard for gravitational waves; what is specific here is the placement—identified by the swept SO(3) in the symmetric sector, released past it—read off the programme's own stratification.) That orientation parity is, in the geometric core [JanzenGeometricCore], the discrete residue the substrate's exhausted continuous symmetry leaves—matter's one geometric opening; the handedness settled here is its face in the radiating sector, the same Z₂ that acts on the cut's fermion as the chirality operator γ⁵ [JanzenBoundary], here read dynamically. This Z₂ is, moreover, the gauged factor of the discrete residue's full Aut(A₂)=S₃×Z₂≅D₆: being a substrate isometry, it grades a chirality that descends gauged—graviton helicity here, fermion γ⁵ there—whereas the Weyl S₃ is the monodromy symmetry of the solution space and no isometry, so a family symmetry it would grade is global. Gauged chirality with global flavour is the Standard Model's own arrangement, following from which of the two factors is an isometry [JanzenAlgebroid, JanzenGeometricCore].

The strata and their discrete markers

Read across the whole sector, the picture is a continuous flow punctuated by discrete boundaries. The substrate isometry so (5,1) acts continuously on the space of cuts, carrying the layer through the symmetry-reducible sector; the true Hamiltonian advances it. The special strata are marked by distinct discrete operations organized by the horizon cubic's root structure: the Nariai seam is the locus where two of the three horizon roots collide (a fixed point of the root-permutation symmetry [JanzenSlicing, JanzenGroupoid]); the cosmogenesis horizon is the locus of the null↔timelike reassignment [JanzenCRframework]; the Riemannian↔Lorentzian seam is the locus of the signature flip [JanzenSlicing]. The wall is the one boundary carrying no such discrete marker—no colliding roots, no degenerating Killing field, no measure-collapse—consistent with its being purely the end of continuous generation. These markers are distinct operations, not one; but the strata they mark are the ordered turning points of a single closed slicing curve—the operations distinct, the object one [JanzenSlicing, JanzenCRframework]. The cosmogenesis marker carries one further structure the boundary paper reads off it: composed with the antilinear reality involution τ↦ τ of that closed bead, the orientation parity R=γ⁵—graded in §6 as the graviton's handedness—is the geometric factor of charge conjugation's kinematic closure, C=(Q↦-Q)_field∘(R∘K)_geometric, so that the same r=0 crossing which re-founds the clock here supplies, read in the matter sector rather than the radiative one, the Feynman–Stückelberg face of [JanzenBoundary]. (The detailed development belongs to the algebroid construction [JanzenAlgebroid] and is summarized here only to place the dynamics.)

Scope, and where what is not claimed here is settled

The distinction the list below turns on is the programme's own: a result may ground a reading without entailing it, and the two are not to be run together [JanzenShadowExistence]. Each item is placed accordingly.

We have exhibited the dynamics of the cut in the regime where it is exact and Hamiltonian: the confined Gowdy–de Sitter wave, the leaf's transverse-traceless shear advanced by a true Hamiltonian on the substrate's cosmic foliation, located at the Type-I edge of the stratification, with the wall identified as a regular radiative boundary rather than a metric singularity. Three things are explicitly not claimed here, and they stand differently: the first is registered as an open problem of the programme, the second lies outside the construction by a result proved below, and the third is carried out in the companion canonical-time paper:

1. the full nonlinear Λgt;0 evolution is resolved here on its classical side (§3, Proposition 1): the de Sitter background is an attractor, every tractable sector admits, and the first-class system is consistent to all orders, so the continuous dynamics admits rather than forces a quantum structure. What is not claimed is a closed-form nonlinear solution, nor a full non-perturbative quantization—neither is built here, and the latter is not needed for the admissibility verdict. The non-perturbative quantization is the definition of the interacting tower, which the framework paper carries as an open problem of the programme [JanzenCRframework];
2. the dynamics strictly beyond the wall, in the fully inhomogeneous Type-N regime. This is not owed work but work of a different kind: by Proposition 3 the wall is a regular radiative boundary and what lies past it is ordinary free evolution, not a stratum-crossing this construction generates;
3. the lift of the deparametrized to the closed-S³ model appropriate to the cosmological sector — carried out in the companion canonical-time paper [JanzenCanonicalTime], where the identification of the cosmic-microwave-background frame with the comoving congruence and the null-boundary S³ is made precise and the transverse-traceless graviton tower deparametrizes to a unitary cosmic-time evolution; the planar Gowdy model of the present paper is the companion field-theoretic realization in the symmetry-reducible sector.

None of these unsettles the result obtained: within the symmetry-reducible sector the cut bends by a true-Hamiltonian flow, and the boundary of that sector is a generative boundary, not a singular one. The cut's dynamics at the other boundary is the matter counterpart of the confined bend developed here. That boundary is the cosmogenesis branch point, and it is not this paper's wall (§5): its substrate curvature is finite while the geometry read over the areal chart diverges, where the wall carries neither. There the crossing of matter through the seam is shown well posed and structurally governed—the foliation-preserving reassignment fixing the Λ-set rate while the density crosses as inherited content—in the companion cosmology paper [JanzenCRcosmology], and the worldline side is settled with it: a comoving worldline reaches the branch point at finite proper time with divergent tidal stretch and terminates, what continues being the analytic continuation, so the crossing is lossless for content and fatal for bodies [JanzenCRframework]. The scalar (density) perturbation sector—the counterpart to the transverse-traceless tensor sector developed here, and the primordial source of the cosmic-microwave-background anisotropies—is treated in the companion paper [JanzenCRcosmology].

Read against the whole, this dynamics sets the framework's one synthesis in motion. The maximally symmetric de Sitter substrate is read at once as general relativity's solution space—generated on its cuts by the slicing operator—as the discrete and charge residue, and as the gauge algebra on its conjugate real form [JanzenCRframework]; the present paper advances the first face, the cut's true-Hamiltonian flow through the symmetry-reducible sector, Hamiltonian up to the generative boundary and ordinary general relativity beyond it, and surfaces the second in the radiating sector—the graviton's handedness the discrete orientation parity that grades the cut's fermion as γ⁵ [JanzenGeometricCore]. The continuous dynamics and the discrete residue are two faces of the one substrate, read here in motion and in the radiating sector.