P12
the action Lie algebroid of the symmetry-reducible sector, and the problem of time's ``wrong sign'' as the substrate's coset signature
General relativity's hypersurface-deformation algebra carries structure functions: the coefficient in the normal–normal bracket is the inverse spatial metric, a field rather than a constant, and that is what makes the canonical generator of normal deformation a constraint to be solved rather than a Hamiltonian that evolves. The algebra is therefore a Lie algebroid and not a Lie algebra—a fact standardly recognized as the obstruction at the heart of the problem of time.
An action Lie algebroid is a Lie algebra acting on a base manifold, with an anchor sending algebra elements to vector fields on the base and structure functions varying over it. General relativity's constraint algebra has the structure functions but has never been given the base they vary over, nor a section selecting a definite flow. This paper supplies both, from the geometry of Cosmological Relativity: the base is the space of cuts of a de Sitter substrate, the acting algebra is the substrate's isometry algebra so (5,1), the anchor is the slicing operator's cut-to-stress-energy map, and the section is the cosmic clock that turns the constraint into a true Hamiltonian. The claim is a recognition rather than an addition.
The bracket closes, and the shape that makes the Dirac algebra puzzling is the shape of a symmetric-space coset. At the symmetric cut so (5,1)=fh⊕fm with fh= so (4,1) the cut-fixing isotropy, and the symmetric-space relation [fm,fm]⊂fh holds—two normal deformations bracketing into a tangential one is two coset directions bracketing into the isotropy. They are the same grading, term for term. The grading is also the lapse–shift split the cosmology computes on, so that reading is not a separate posit.
The structure function is identified with the coset metric, and what makes that more than dimensional bookkeeping is the signature: the indefinite sign is supplied by the substrate's own geometry rather than inserted.
As the cut moves off the symmetric vacuum the isotropy drops, and its strata are the range paper's Petrov classes. Read as a count of first integrals, the stratification says where the construction's hidden symmetry is needed and where it is not. The discrete skeleton at those strata is the substrate's own Weyl group: both factors of Aut(A2)=S3×Z2≅D6 are realized geometrically, the S3 permuting the three horizon roots and the Z2 the central inversion that swaps the substrate's two null rulings—so the double-ruling swap, the orientation parity, the diagram automorphism and the graviton's chirality are one and the same Z2.
The verification is on the symmetry-reducible sector throughout, and we are explicit about it: so (5,1) is finite-dimensional, the full Dirac algebra is not, and what is established is the finite reduction rather than a claim about the entire infinite-dimensional algebra. The algebroid hands off at the wall to the propagating sector, which is the companion dynamics paper's subject.
In the canonical formulation of general relativity, deformations of a spatial hypersurface generate an algebra [Dirac1964, ADM1962, Teitelboim1973]—an algebra whose content, and not only whose form, bears on what follows, since Teitelboim's reading of it is that the brackets are the embeddability condition for the hypersurface in a spacetime, and Hojman, Kucha\v{r} and Teitelboim show that requiring a canonical representation of them on (gij,πij) alone recovers the Einstein Hamiltonian [HKT1976]. That forcing is dimension-dependent: the same algebra closes for the Lovelock theories [TeitelboimZanelli1987], which coincide with general relativity only in four dimensions. So the leaf being four-dimensional is doing work here rather than merely being the case, and it is a different point from the one the dimension result guards—that the cut's dimension says nothing about the substrate's. We record the connection and claim no more than it: this paper's algebroid is a recognition of the structure the brackets already have, not a derivation of the field equations [JanzenRange]. Written out, with ⊥ the normal (Hamiltonian) constraint and a the tangential (momentum) constraint, smeared with lapse and shift, the brackets read:
An action Lie algebroid is a Lie algebra acting on a base manifold, with an anchor mapping algebra elements to vector fields on the base and structure functions varying over it. General relativity's constraint algebra has the structure-functions but has never been given the base they vary over, nor a section of the bundle that would select a definite flow. This paper supplies both, from the geometry of Cosmological Relativity. The base is the space of cuts of a de Sitter substrate; the acting algebra is the substrate's isometry so (5,1); the anchor is the slicing operator's cut-to-stress-energy map [JanzenOperator]; and the section is the cosmic clock that turns the constraint into a true Hamiltonian [JanzenCanonicalTime]. The claim is a recognition, not an addition: the algebroid was already there, in GR's own constraint algebra, waiting for the base.
We verify the construction's defining identities on the symmetry-reducible sector—the cuts a substrate isometry can reach, which is exactly the range of the slicing construction [JanzenRange]—and we are explicit, throughout and in Section 8, that this is where the verification lives: so (5,1) is finite-dimensional, the full Dirac algebra is not, and the closure is the finite, symmetry-reducible reduction, not a claim about the entire infinite-dimensional algebra.
The substrate is a single five-dimensional de Sitter manifold dS5, a hyperboloid in the flat ambient M6, with isometry so (5,1); the four-dimensional geometries of the symmetric sector are generated as cuts of it [JanzenOperator, JanzenSlicing]. (The dimension is bounded below here and pinned elsewhere, and the two should not be run together. The construction generates many distinct four-geometries from one substrate, and slicing a four-dimensional de Sitter space only re-coordinatizes it—which excludes dS4 and yields D≥5, a lower bound and not an equality. No upper bound is established anywhere in this framework. The companion's ontological commitments fix only the existent—a one-parameter family of three-dimensional layers, the sole dimensional statement in any of its axioms—and require of the representation only that it admit them as spacelike hypersurfaces, so that dim M≥4 with the projection explicitly non-unique [JanzenCRframework]; the empirical forcing is of the foliation, likewise a floor [JanzenModernParallax]. dS5 is accordingly the minimal substrate sufficient for the symmetry-reducible sector built here—a modelling economy, not a derived maximum—and the polar structure does not close the gap either: the polar of a spacelike substrate point is dSD-1 in every dimension [JanzenGeometricCore], which relates the rungs without capping them.P12R9
Nor can least-arbitrariness close it: it prefers the structure requiring no choice of how to break a symmetry, and dSD is maximally symmetric and moduli-free for every , so it selects the manifold at fixed dimension and is silent on the dimension itself [JanzenShadowExistence]. Reading dS5 as a ceiling would set a feature by hand, which Rule 3 counts as maximally arbitrary rather than least. And the absence of an upper bound has a positive reason, which is stronger than the absence and which the criterion supplies rather than fails to supply.R Suppose the descent to the cut ran in more than one step. Each intermediate rung is itself the substrate for the steps below it, so Rule 2 applies to it in its own right: a rung that were not maximally symmetric would carry a choice of how to break its symmetry, which is a modulus in the excluding sense.
So every rung above the last must be maximally symmetric—and on this substrate a section carries mass exactly when it is not a plane section [JanzenGeometricCore], so every step above the last is a plane section. A plane section {X:η(n,X)=c} of a dSD of radius α returns dSD-1 of radius √α2-c2 for every admissible , the normal being gauge because the isometry group is transitive on unit spacelike normals. So a step above the last changes the scale and nothing else: the entire tower enters the four-geometry through the single combination that is its α, and α is the one input this framework does not derive. The consequence is neither of the two the question anticipated. Rule 2 does not forbid a second step and does not force one: it empties it. Such a step introduces no modulus in the excluding sense—its normal is gauge, its offset re-enters an input already present, and where the descent starts is a dimension, on which the criterion is silent by the argument just given—and it introduces no content either, the cut fixing α from Λ and r0 from the mass and being blind to everything above. So the dimension is unbounded above because nothing below can see the difference, which turns the missing ceiling from a gap in the argument into a property of the descent, and makes the discipline of arguing from the cut to the dynamics and never from the cut to the substrate a consequence rather than a rule of conduct. One thing is settled, and it is a different object: the cut's dimension. The matter sector shows that the horizon relation collapses to a single multiple-angle in the sky angle only at four and five spacetime dimensions, and that the mass-parity grading chirality exists only in even dimension, so a four-dimensional cut is the only one carrying both a generation count and a handedness [JanzenSlicing, JanzenMatter]. That settles the geometry the slicing operator delivers, not the substrate it is delivered from; the grading below is accordingly the last step of any descent ending at a four-geometry, and constrains the rung immediately above spacetime rather than the top of the ladder.) Crucially, dS5 is a symmetric space [Helgason1978],
and this is the structural fact the whole construction turns on. At a maximally symmetric (vacuum) cut, the subalgebra of so (5,1) fixing the leaf is fh= so (4,1) (ten-dimensional), and the cut-deforming directions are the coset fm (five-dimensional, the normal carrying one leaf to the next): so (5,1)=fh⊕fm.
The base is the space of cuts. A cut is fixed by an orientation datum—how the slicing curve is aimed into the geometry (the reticle orientation r0, with the mass a determined function of it, not a free parameter) [JanzenSlicing]—together with a causal-vantage datum. Two distinct operations move between geometries, split on what they do to dimension: slicing, codimension-one and dimension-reducing, which is the operation that forces the substrate up to five dimensions and which the continuous so (5,1) structure realizes; and reassignment, dimension-preserving and discrete, which holds one background geometry fixed and involutes the causal roles of the generating null ruling (the Null–Boundary Correspondence, the f gt;0 hole / f lt;0 cosmos) [JanzenCRframework]. The continuous algebroid is the slicing structure; reassignment is a distinct discrete operation (Section 6).
The anchor maps an infinitesimal cut-deformation to the stress-energy it produces—the slicing operator made infinitesimal. Its four sectors are the ADM data of the cut, read as functions of the cut [JanzenOperator, JanzenRange]:
The four close as one functional of the cut on the spherical class; the general-cut functionals and the dimensional restriction are part of the open scope (Section 8).
The defining test of the action algebroid is whether the cut-deformation bracket closes and the anchor is a homomorphism into the Dirac algebra. On the symmetric space it does, by the symmetric-space property.
The grading has a name in the literature, and naming it costs nothing and buys a great deal. For a homogeneous space the action algebroid of on G/H is the Atiyah algebroid of the principal bundle G→G/H, so with dS5=SO(5,1)/SO(4,1) the sequence
is exact, with fh the kernel of the anchor—the adjoint bundle, ten-dimensional—and fm its image, the five dimensions of the base, closing at 10+5=15= dim so (5,1). All three of the direct sum, the dimension count and [fm,fm]⊂fh are verified on explicit matrices, and with them the exactness of the sequence at every termP12R18L13. And a splitting of that sequence is what a connection is—so the section supplied below to select a definite flow is a connection in the standard sense rather than an object peculiar to this construction.
This grading is also the algebraic form of the split the cosmology computes on. The normal generator ⊥↔fm is smeared by the lapse—the metrical rate of the layer's advance, the existent's own foliation stacking rate—and the tangential a↔fh by the shift, the synchronization convention through which that advance is coordinated and observed [JanzenModernParallax, JanzenCanonicalTime]. So the cosmological reading in which the observable expansion rides the foliation rate (set by the geometry, the density its content) while a chosen synchronization projects it to distance and redshift is not a separate posit: it is this fm/fh grading, read on the preferred foliation, with the leaf's bend the matter the operator reads as the cut's departure from vacuum [JanzenOperator, JanzenCosmogenesis]. The “wrong sign” that is the problem of time and the geometric stacking rate that dissolves the Hubble tension are thus two faces of one object—the substrate's coset metric read, respectively, across the base and along the preferred foliation.
The structure function is the substrate's metric. The coefficient hab in (1)—the structure function that makes the algebra an algebroid and is the canonical root of the problem of time—is, on the symmetry-reducible reduction, identified with the coset metric of the symmetric space. This is not a naive tensor equality: hab is the Riemannian inverse spatial 3-metric, whereas the coset metric of SO(5,1)/SO(4,1) is the Lorentzian 5-dimensional form (the Killing form on the coset is ∝diag(+,-,-,-,-), signature (1,4)P12R2); the identification is of the reduced structure function on the symmetric-cut pattern with that coset form. What makes it more than dimensional bookkeeping is the signature: the indefinite, Lorentzian sign carried by the coset metric—supplied by the substrate's geometry, not inserted by hand—is the same indefiniteness standardly recognized as the “wrong sign” at the heart of the problem of time. At the symmetric cut the structure function is constant, so so (5,1) is there a genuine Lie algebra; as the cut moves over it varies, so the object is a genuine Lie algebroid. The problem-of-time obstruction is thus identified with a single geometric fact: the base-dependence of the substrate's symmetric-space metric, whose Lorentzian signature is the obstruction's own. Read on the substrate's preferred foliation, that same content deparametrizes to a true Hamiltonian [JanzenCanonicalTime]; the obstruction and its resolution are one object under the static and dynamic vantages.
As the cut moves off the symmetric vacuum, the cut-fixing isotropy drops, and its strata are the range/Petrov classes of the construction [JanzenRange]: Type O (de Sitter, isotropy so (4,1), dimension ten), Type D (Schwarzschild–de Sitter Rt×SO(3), dimension four; Kerr–de Sitter, dimension two), Type I (Bianchi, dimension three; Zipoy–Voorhees, dimension two), and the wall (Type N, isotropy zero). Read as a count of first integrals the stratification says where the construction's hidden symmetry is needed and where it is not. A Killing vector contributes one linear integral of the geodesic flow and the isometry algebra's Casimir a quadratic one built from those same vectors, so against the four integrals in involution that a four-dimensional geodesic flow requires: Type D Schwarzschild–de Sitter, isotropy four, supplies the norm together with , L2 and Lz and is integrable on its Killing vectors alone; Kerr–de Sitter, isotropy two, supplies only the norm, and Lz and is short by one. That deficit is exactly what the range paper's Killing tensor makes up [JanzenRange]—so the drop in isotropy dimension across this stratum is the precise place at which a hidden symmetry stops being a redundancy and becomes a requirementL14. Proposition 1 pins the base-variation only at the symmetric cut; computing it across these strata sharpens the picture. The so (5,1)-action on is non-transitive: on the Schwarzschild–de Sitter family the diffeomorphism invariant RabcdRabcd=48M2/r6+24/α4 separates different-mass cuts while the scalar curvature 3R=2Λ does notP12R10, so cuts of different mass are non-isometric and no substrate isometry connects them—the mass is a modulus transverse to the orbits (folding at the Nariai seam, where the offset-to-mass map of Section 2 is stationary). The qualifier is connected: no connected substrate isometry connects different-mass cuts, but an orientation-reversing one does.
The reticle reflection r0↦-r0—the reversal of the slicing aim on the observer's celestial sphere—is a substrate isometry lying in O(5,1)∖SO0(5,1), and since 2M=r0-r03 is odd it sends 2M↦-2M, carrying the +M cut to the -M cut across exactly the SO0-orbits the connected action cannot cross. It realizes the mass-reflection Z2—the A2 diagram automorphism that adjoins the Weyl group [JanzenGroupoid]—as the substrate's own orientation parity O(5,1)/ SO0(5,1).
The non-isometry of the +M (black-hole) and -M (naked) Lorentzian charts is then a property of the representational record, not of the existent: the substrate is symmetric under the reflection, the evolving layer it carries is what exists, and the charts are its shadow—granting them the standing of distinct existents is the category error the cosmological ontology dissolves—the “events exist” horn the foundational augmentation closes [JanzenModernParallax, JanzenCanonicalTime, JanzenBHcausality]. This single orientation Z2 recurs beyond the symmetric sector: in the radiating sector as the graviton's two helicities, and past the wall as the un-undoable chirality of the turning polarization plane [JanzenRange, JanzenDynamics]—one parity threading the symmetric and radiative faces of the construction.
Testing the symmetric-space grading [fm,fm]⊂fh stratum by stratum, it survives at exactly two: Type O (fh= so (4,1), the symmetric space SO(5,1)/SO(4,1) itself) and Nariai (SO(2,1)×SO(3), the block-diagonal Grassmannian pair). The stratification has invariants of its own, and two are worth recording. First, orbit dimension: since dim so (5,1)=15, the orbit through a cut has dimension 15- dim fh, so the two symmetric strata carry orbits of dimension five and nine against a generic eleven and a principal fifteen where the isotropy is trivialP12R6. The most symmetric cut has the smallest orbit, which places pure de Sitter at the degenerate end of the filtration rather than in its interior, and the ordering 10 gt;6 gt;4 gt;3 gt;2 gt;0 runs from there to the wall—the same axis the range paper traverses from the Nariai seam outward, ordered there by algebraic type and here by isotropy, the two orderings agreeing at the ends that paper names [JanzenRange]. Second, and stated as a question rather than a result: the admissible symmetric-pair dimensions are {6,7,10}, and the construction occupies six and ten. The dimension seven, realised by so (4)⊕ so (1,1) and by so (3,1)⊕ so (2), is not occupied, and the reason is the range paper's own bound rather than an omission of the survey. That bound is sharper than membership in so (5,1): a swept geometry inherits a subgroup of so (4,1) [JanzenRange], and neither dimension-seven pair embeds there. so (4)⊕ so (1,1) cannot, because so (4) already rotates all four spacelike directions and a boost on any of them fails to commute with those rotations; so (3,1)⊕ so (2) cannot, because so (3,1) leaves one spacelike direction free and a rotation needs twoP12R7. So the count {6,7,10 is a fact about the algebra and the two occupied strata are a fact about the reachable class, and the third dimension is excluded by the sweep rather than left untested.} At every generic stratum—Schwarzschild–de Sitter Rt×SO(3), Kerr–de Sitter, the Type-I classes, the wall—the pair is non-symmetric and [fm,fm] acquires a component in fm: that leak is the algebroid connection, the genuine base-variation of the structure function, vanishing exactly at the two symmetric strata. The isotropy dimensions are the Killing-vector counts the construction establishes [JanzenRange], and the triple itself is a fact about so (5,1) rather than about any stratum: its symmetric decompositions are so (5) and so (4,1) at ten, so (1,1)⊕ so (4) at seven, and so (2,1)⊕ so (3) at six, and the verdict is independent of the choice of generator realization: the symmetric-pair isotropy dimensions of so (5,1) are {6,7,10}, so the two symmetric strata sit at the only admissible dimensions and are the geometry-fixed pairs (SO(4,1) at ten, the SO(2,1)×SO(3) Grassmannian at six), while every generic stratum (dimensions four, three, two, zero) is non-symmetric by dimension alone.
The strata boundaries—where the isotropy jumps—meet the corpus's metric-singular seams at the inner end and part from them at the outer end. At the inner end, the Schwarzschild–de Sitter horizon cubic r3-α2 r+2Mα2 has discriminant -4α4(27M2-α2), which vanishes at ΛM2=1/9: the double root rN=1/√Λ, the Nariai seam, where the geometry is dS2×S2 (both factors constant-curvature) and the isotropy jumps from four to six. The isotropy-jump locus is the metric-degenerate locus there. This Nariai double root is the member the cosmogenesis selects [JanzenCosmogenesis]—not its branch point, which is at r=0, a lap away on that same member. It is the stratum boundary at which collapse re-founds as cosmology, and across which—the reassignment fixing the Λ-set rate on the leaf-carried density—the light-element abundances are inherited and produced. At the outer end they diverge: the wall is isotropy zero, Type N—and not a metric singularity. A metric singularity in the 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 null-separated, spatially coincident events have vanishing proper interval (a Killing horizon being the worked sufficient case, not the definition). The wall, a Type-N plane wave, has a non-degenerate metric (no such measure-collapse, so not the finite-curvature species) and vanishing curvature invariants (so not the infinite-curvature species); it is neither. The “isotropy zero” that marks it is the cut-fixing substrate isotropy—which coincides with a geometry's own isometry on the symmetry-reducible sector but diverges here, the Type-N geometry retaining its own large isometry—so the wall is no metric singularity by the measure criterion, not for want of a Killing field. It is a regular radiative boundary, the construction's generative boundary, where generation-by-symmetry hands off to ordinary evolution [JanzenRange]. So “isotropy boundary” and “metric singularity” coincide at the inner seam (Nariai) but are two species of degeneracy at the wall.
Inside the continuous so (5,1) sits a finite discrete skeleton: the S3≅D3 permuting the three roots of the horizon cubic of Section 5.
That cubic r3-α2 r+2Mα2 has no quadratic term, so its three roots sum to zero—the A2 root configuration [Humphreys1972]; written in the gauge α=1 with the mass carried by the reticle orientation (2M=r0-r03, the determined mass of Section 2) it reads r3-r-(r03-r0)=0 and factors as (r-r0)(r2+r r0+r02-1), the quadratic factor the fundamental ellipse r2+r r0+r02=1 [JanzenGroupoid, JanzenSlicing].
The discrete operations are distinct (they are different involutions, not one), and each is anchored at a stratum, organized by the cubic's root structure: the Nariai seam is the fixed point of the root-permutation transposition (two roots collide, the discriminant vanishes); the cosmogenesis horizon is the locus of the null↔timelike reassignment [JanzenCRframework]; the Riemannian↔Lorentzian seam is the locus of the signature flip. The wall carries no such marker—no colliding cubic roots, no degenerating Killing field, no measure-collapse—consistent with its being the purely continuous boundary. The continuous face is thus the flow through the symmetry-reducible sector—whose slicing curve, run through the seam and backward through the r=0 branch point with the signed areal radius, closes into the single cosmogenetic bead the framework proves one object [JanzenCRframework]; the discrete face is the set of distinct operations marking its special strata. (We make no claim that these unify into a single discrete action, nor that the A2 configuration realizes a continuous su(3) isometry. The root structure is the established skeleton; the continuous realization is a separate question, and on the established five-dimensional substrate it is settled negatively. The three roots summing to zero furnish a Cartan element of su(3) and the S3 its Weyl group—the Cartan–Weyl skeleton, a necessary ingredient—but a continuous su(3) isometry would have to embed in the substrate's rotation group, and SU(3)⊄ SO(5) (its smallest faithful real representation is six-dimensional, on C3= R6), so no such isometry exists on dS5: there the A2 structure is the discrete Weyl(A2)=S3 shadow only. Read on a fermion sector, this discrete shadow is realized as matter—the three zero-sum weights as a threeness of fermion content, the orientation Z2 as its chirality, so that D6=S3×Z2 reads as a threeness times two chiralities: three chiral families forced within CR, built in the matter-sector paper [JanzenMatter]. The two threes are distinct, and the matter paper proves them affinely inequivalent: these weights carry the within-state index, while the generations are carried by the turnaround's deck Z3 (ibid.). A continuous, geometric su(3) would require a six-dimensional substrate (SU(3)⊂SO(6)), which the construction permits but does not force—and a structure a framework merely permits, as against one it requires, is on the programme's criterion of necessity no explanation but a description [JanzenShadowExistence]; the resonance is necessary, not sufficient, and silent on the dimension. We assert neither the continuous su(3) nor the dimensional rise.)
Geometric realization of Aut(A2)=S3×Z2. Both factors of the root system's full automorphism group are realized on the substrate in the pivot-and-ruling frame.
The Weyl factor S3 permutes the three horizon roots, which—the cubic having no quadratic term—are the three zero-sum weights and sit 120∘ apart on the throat-image circle of radius 2/√3: in the sky angle of [JanzenSlicing] they are rk=2/√3 cos (-2π/3k), with the reflection w↔π/3-w about the Nariai crest the order-two generator.
The diagram-automorphism factor Z2 is the central inversion r0↦-r0 of Section 5—the parity that carries the weight triangle to its negative, completing the A2 hexad—and it has a clean geometric form.
The one-sheeted dS5 is doubly ruled by null generators [JanzenOperator]—the comoving and synchronous congruences of the cosmology—and the central inversion exchanges the two families: restricted to the ruled section, an orientation-reversing isometry swaps the two rulings while an orientation-preserving one fixes each. Explicitly it is R= diag(1,1,-1,1,1,1)∈O(5,1)∖SO0, the reflection of the cut's displacement transverse to the symmetry axis (the offset r0=2/√3 sin w): one verifies directly that it is an isometry, has determinant -1 both globally and on the ruled three-block, swaps the two rulings, and sends r0↦-r0 hence 2M↦-2M.
This Z2 is distinct from the Weyl reflections that generate S3: those fix 2M and permute the three horizon roots within a mass level—their realisation on the real roots regime-dependent, all of S3 under-critically and only the order-two complex conjugation of the conjugate pair over-critically [JanzenGroupoid]—while the central inversion sends the whole root triple to its negative and swaps the two rulings. The two factors thus act on independent structures—the three roots and the two rulings—and the inversion is central, which is why Aut(A2) is the direct product S3×Z2≅D6 the groupoid paper obtains algebraically [JanzenGroupoid].
The double-ruling swap, the orientation parity O(5,1)/ SO0, the A2 diagram automorphism, and the graviton chirality of Section 5 are therefore one and the same Z2.
That one Z2 carries a further, fermionic face on the substrate's spin structure: dS5≃R×S4 is spin with a unique spin structure [LawsonMichelsohn1989], inherited by the four-dimensional cut, and —reflecting the transverse cut-normal (r0) direction and so fixing all four spacetime legs—acts on the cut's natural spinor as the chirality operator γ5 itself, so the orientation parity is the chirality grading, measuring left against right as it carries the graviton's helicities [JanzenGeometricCore]. (Grounded: the parity measures chirality, the exchange reading excluded by explicit computation. The same Z2 grades mass in both faces—the geometric mass-reflection 2M↦-2M above and the -odd fermion mass term on a cut spinor—so mass, geometric or fermionic, is its one -odd datum; the chiral content this parity would grade is not built here.)
The two factors moreover descend onto any such spinor sector by different kind: the Z2, being a substrate isometry, acts on the cut spinor as the gauged grading γ5; the Weyl S3 is no substrate isometry—its continuous parent excluded by su(3)⊄ so(5,1) above, and a single Dirac spinor carrying no three-fold to realise it—so a family symmetry it would grade is global, not gauged.
The same partition falls out of the substrate's null structure, and by a route that mentions neither spinors nor su(3): the sky-angle periodicity τ is two steps along a null ruling of the substrate—so it is a closed path of light, and τ3= id is that path closing—whereas the Weyl reflection σ is a loop in the complex 2M-plane about the Nariai point [JanzenGroupoid], and 2M labels the family rather than pointing along the manifold.
The rulings walk exactly the isometries and nothing else, which is this paragraph's different kind read off the light cone [JanzenSlicing, JanzenGeometricCore]. And the light cone and the throat are one reading, not two: the substrate's null rulings are the tangent lines to its waist—“doubly ruled by straight null lines” and “every tangent to the throat is null” are one statement, the power of a point with respect to that circle being the square of its height [JanzenGeometricCore]. So “two steps along a null ruling” is two steps along a tangent to the throat, and the two structures the factors act on are the two relations a figure can bear to that one circle: the three roots are the three special points on it—the offset line meets it at and , with r=0 fixed at the back [JanzenSlicing]—and the two rulings are the lines tangent to it. The automorphism group factorises because on and tangent are independent relations; and the factors differ in kind for the same reason, a tangent being a line of the substrate while a root labels a different cut. The pivot-and-ruling geometry this section reads the skeleton off is, in that reading, one circle read two ways. Where those two factors become matter—the inversion as a bound state's chirality eigenvalue, the Weyl S3 as a count of three—the reading is stated on the built sector [JanzenMatter]. Gauged chirality with global flavour is the Standard Model's own arrangement, here not assumed but following from which of the two factors is an isometry and which the Weyl/monodromy symmetry of the solution space [JanzenGeometricCore]. With the pivot supplying S3 and the ruling swap supplying Z2, the order-twelve Aut(A2)=S3×Z2—the symmetry group of the A2 hexad—is realized in full on the substrate, the discrete skeleton read off its pivot-and-ruling geometry. The slicing paper draws this realisation as a figure [JanzenSlicing]: the pivot's three vantage-hinges—each designating one root—as an equilateral whose incircle is the throat, and the ruling structure as a skew hexagon of null generators laced across the substrate's two horns; whether that six-hinge figure is the same six-fold as the six-Nariai root hexad is established in the slicing paper as a resonance, not an identity—the horn-swap (T:X0↦-X0) and the offset parity (R:r0↦-r0) distinct reflections that share the fundamental triple but complete it distinctly [JanzenSlicing].
The three levels of the symmetry structure. The discrete skeleton read off here is the substrate's linear face—real reflections acting as isometries on the ruled section. It does not exhaust the symmetry structure the substrate carries. Composed with the antilinear reality involution τ↦ τ of the cosmogenetic bead—antilinear and geometric, living in the substrate's complex-analytic structure rather than in its isometry group—the orientation parity yields charge conjugation's kinematic (Feynman–Stückelberg) face, so that charge conjugation factorises, C=(Q↦-Q)field∘(R∘K)geometric with K: τ↦ τ, the geometry supplying its whole kinematic content and only the electric-charge sign closing from the matter field [JanzenBoundary, JanzenMatter]. The substrate's symmetry structure is thereby three-levelled—the discrete linear skeleton of this section (substrate isometries), the antilinear complex-analytic conjugation face, and the field-level charge—of which the order-twelve Aut(A2)=D6 realized here is the first and load-bearing face, the one the constraint algebroid's base-and-section reads directly; the geometric factor of the conjugation is the same cosmogenetic bead the continuous flow closes into, so the discrete skeleton and the conjugation share the one r=0 crossing.
The corpus reads its discrete skeleton off the horizon cubic, whose three roots realise A2, giving Aut(A2)=S3×Z2=D6 of order twelve. That is a sub-root-system of one the substrate already carries, and the difference is supplied by a holonomy rather than by an assumption.
The residue pairing on functions over the root triple—diagonal, entries 1/f'(ri), signature (2,1)—has a non-trivial holonomy about the Nariai points [AmbroseSinger1953], the Klein four-group V4 of even sign changes, whose origin is the per-root resolution of √Δ [JanzenGroupoid]. Adjoining it to the monodromy S3 closes a group of order twenty-four; adjoining the orientation parity as well closes one of order forty-eightP12R5. Neither is an abstract coincidence of order. Computing the element profile: the six order-four elements are all improper, none a proper rotation, which is the signature of the full tetrahedral group Td and not of the chiral octahedral group; the transpositions appear as reflections, the V4 as the three proper two-fold rotations, the three-cycles as three-fold rotationsP12R4. So the group is W(A3) in its Weyl embedding, not merely a group isomorphic to S4.
And A3 is not a new structure. The substrate's isometry algebra is so (5,1), a real form of so (6,C), whose root system is D3≅A3; and A2 sits inside A3 by deleting one node. So the sector's discrete group is the Weyl group of the substrate's own complexified isometry algebra, and what the corpus had been using is the sub-root-system obtained by reading only the horizon cubic. The enlargement is therefore not an addition to the construction but a recognition of what the construction already contains, reached by transporting a form the substrate itself supplies.
We mark the scope. This concerns the discrete structure alone. It says nothing about a continuous su (3), which the boundary paper's obstruction continues to address [JanzenBoundary]; a finite group, however enlarged, is not a Lie algebra.
The verification above is, throughout, on the symmetry-reducible sector. We state plainly what that means and what it leaves.
Within that scope the result stands as a recognition: general relativity's constraint algebra is the symmetric-space structure of a de Sitter substrate, an action Lie algebroid whose base is the space of cuts, whose structure function is the substrate's own metric, and whose problem-of-time obstruction is that metric's base-dependence. This symmetric-space root is, in the geometric core [JanzenGeometricCore], read as one rung of the substrate's maximal symmetry: the same exhaustion of SO(5,1) that there locks the cosmological constants and walls a continuous colour symmetry is what closes the constraint algebra as the coset grading and dissolves the problem of time. That the rungs are one fact is a reading advanced there, at that reading's weight; here the algebroid rung stands on its own, on the symmetry-reducible sector.