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General relativity does not single out a physical foliation. Nothing in the theory says which slicing of spacetime is the present, and the standard reading takes that as a statement about the world: no objective present, no fact about simultaneity, only the block.
Cosmological Relativity begins from the observation that this is a statement about general relativity, not about the world — and that the world, in one specific circumstance, does single one out. Under gravitational collapse the horizon's own geometry fixes a limiting causal direction, and it is generically non-orthogonal to any spacelike slice. That direction is not a convention. It is forced, and it is what gets reassigned as cosmic time.
Followed through, the picture inverts. Our universe is read as the interior of a gravitational collapse — not an explosion from a point but the far side of one, on the conjugate branch: an antimatter progenitor, decelerating to a turnaround, sweeping through the coordinate origin where the species reverses, re-expanding. What we call the beginning is that passage. The origin is not where matter came from; it is a horizon the slicing is anchored to, and the reading that made it a source was inherited from a choice of slicing rather than found in the geometry.
The progenitor is not a free parameter. The collapse selects the one configuration whose limiting direction grazes its own horizon, and that fixes its mass by the cosmological constant alone — M = c²/3√Λ G, about 4×10⁵² kg. Everything on the geometry scales from that single length.
We identify a geometric structure we argue standard general relativity has lacked among its standard definitions: the metric singularity—a null hypersurface along whose generators the spatial extent has, in addition, contracted to zero, so that events on a generator are topologically distinct and causally ordered yet carry no metric separation. Using only standard Lorentzian structure, we prove a Metric Singularity Theorem: two events that are both null-separated and spatially coincident have vanishing temporal separation as well; two of the three separations vanishing forces the third. We distinguish this sharply from the generic case: an ordinary null hypersurface, such as an observer's past light cone, carries frame-dependent but nonzero temporal separations and is not a metric singularity. Specializing to the Schwarzschild geometry, we show that the future event horizon H+ satisfies the theorem's hypotheses—horizon-crossing events on a common generator occur at the same areal radius rh, hence at zero invariant spatial separation—and is therefore a metric singularity. The same forcing extends the conclusion to any Killing horizon, the Kerr family included. Independently of this metric identification, the horizon's defining causal property—that it is the null future boundary of the exterior, H+=∂J-(I+)—forces any temporal slicing adapted to an exterior observer to become asymptotically tangent to a single horizon generator and to meet H+ only in the limit of infinite exterior time: within the causal domain of the external universe, the event horizon occurs solely as a null future boundary approached asymptotically by every admissible “now” slicing. This clarifies the behavior of horizon neighborhoods in Eddington–Finkelstein diagrams, constrains the interpretation of horizon-proximal processes in gravitational collapse, and implies that the densities associated with the central singularity never form at any finite exterior time. Because the globally completed horizon on which the standard black-hole problems rest is, for this reason and in a universe of ongoing mergers and accretion, never physically realised, three of those problems dissolve on causal grounds alone and are drawn out here: no closed trapped surface is realised, so Penrose's singularity theorem, though mathematically correct, has physical preconditions the astrophysical domain never meets, and cosmic censorship becomes unnecessary; the horizon-induced Bogoliubov mode-splitting that yields Hawking radiation has no realised background to be computed on, leaving horizon-induced radiation absent while local, non-horizon particle production is untouched; and the information-loss paradox does not arise, the realised spacetime remaining globally connected, with a global Cauchy surface and unobstructed unitary evolution. Each of these follows from the same standard causal structure and no modification of general relativity.
This paper exhibits one homogeneous circle whose two poles are the event horizon and the curvature singularity at r=0, run through by a single analytic continuation, and read entirely from within the Schwarzschild geometry. The Schwarzschild interior, written in Lemaître–Tolman cycloid coordinates, has areal radius r(z) = M(1+ cos z) for z ∈[0,π], taking the horizon value r=2M at z=0 and the standard “singularity” value r=0 at z=π. This same cycloid is, term for term, the scale factor of a closed Friedmann cosmology: the black-hole interior and a closed universe are one curve—the classical seed, present already in bare Schwarzschild, of the collapse–cosmology identity this sequence makes exact. And the curve only ever decelerates: the radial acceleration along the infalling worldline is -M/r2, Newtonian free fall, negative at every radius, so the cycloid has no turning point of its rate. The turn belongs to a positive cosmological constant, not to the collapse, and appears only when this circle is carried to Λ>0. We observe that the two endpoints are analytic critical points of r(z) of identical character: dr/dz = 0 at both, d2r/dz2 = ±M alternately, and r(z) is everywhere C∞ smooth as a function of z. The maximal analytic extension of the cycloid arc is a single analytic curve—hyperbola, circle, hyperbola—in which each critical point opens, by continuation of z into the complex plane, into a two-branch hyperbolic arm: z ↦±iρ at z=0 produces two isometric asymptotically flat exteriors at r>2M, and z ↦π±iρ' at z=π produces two asymptotically flat regions at r<0. The four regions at r ≥0 (the two exteriors and the two interior arcs) are precisely the maximal Kruskal–Szekeres extension, which terminates at r=0; the two regions at r<0 are the continuation through z=π that Kruskal never reaches, because it treats r=0 as a boundary rather than a critical point one continues through. We show that the Kretschmann scalar's divergence at r=0 is generated by the chain rule applied to the composition 48M2/r(z)6 at a non-degenerate critical point of r(z), and is therefore the curvature scalar of the Schwarzschild perspectival metric—the metric as a tensor field over the parameter r, real as that metric's—rather than an invariant of the underlying smooth manifold parametrised by z, which carries none that diverges. The standard asymmetric classification of the horizon as a “removable coordinate singularity” and r=0 as a “true curvature singularity” is not forced by the analytic structure of r(z): the two critical points are the same kind of object, and the standard classification commits a single category error twice. We identify both critical points as metric singularities of one genus—the horizon the finite-curvature species the companion paper establishes for the event horizon, r=0 the conjugate infinite-curvature species—distinguished only by curvature, which is to say only at second order: the two are identical through first order and parted only there, being the two r-poles of one homogeneous circle, so the asymmetry lies entirely in which value the chart's origin assigns each (2M versus 0), not in the curve. The standard reading draws an inference: because the curvature diverges at r=0, no continuation through it exists, so r=0 is an inextendible boundary. The cycloid is a counterexample to that inference: the same analytic continuation that the standard treatment accepts as removing the horizon at z=0 passes smoothly through z=π into the r<0 arm, so the cycloid curve passes through r=0 by the same step that carries through the horizon. The inference standardly drawn from the curvature divergence—that r=0 is an inextendible boundary at which the construction must stop—therefore fails; what is left standing is the rigorous, curvature-independent C0-inextendibility of r=0 established by Sbierski, which the present continuation of the curve leaves untouched, and what the continued curve passes into is set out in Section sec:r0-tee. The discussion places the construction in a wider setting: the single horizon exhibited here is the Λ→0 degeneration of a root triple, so that horizon multiplicity is a reading of the cosmological constant, and the one circle carries both a Schwarzschild and a de Sitter reading. What the continued curve passes into is set out in Section sec:r0-tee: the point the standard reading treats as the terminus of the description is the one the companion papers' subject matter is organised around.
In a companion paper the maximally extended Schwarzschild geometry was shown to be the analytic completion of a single cycloid arc, r(z)=M(1+ cos z), whose two endpoints—the horizon at r=2M and the curvature singularity at r=0—are non-degenerate critical points of identical analytic character on a smooth underlying manifold. This paper carries the construction to the Schwarzschild–de Sitter family, and the vacuum construction it arrives at has a single moving part: one slicing plane swings about one fixed line, and the whole family of cuts is the single arc of that swing.
The standard cosmological model installs a global cosmic time, a preferred rest frame, and a spatially uniform expansion as kinematical assumptions, prior to and independent of the dynamical fit to data. We show that these assumptions are not conventions the data merely tolerate but structures the data force. The cosmological redshift is the path integral of the expansion rate, ln (1+z)=∫H dt, so the observed anisotropy of the cosmic microwave background (CMB) separates exactly into a source term, fixed at last scattering, and a cumulative term, set by the integrated expansion along each line of sight. Granted even perfect homogeneity at last scattering, the observed isotropy of the monopole is then a direct measurement of uniform expansion—so the common supposition that the isotropy is supplied by homogeneity at decoupling is a category error. We make the alternative quantitative: were the expansion rate to track the inhomogeneous matter distribution region by region, with no single global scale factor, the accumulated redshift would scatter by ∼10-3 across the sky, against an observed ≲3×10-6—an exclusion by three orders of magnitude. The effect is distinct from the Sachs–Wolfe anisotropies and not a re-derivation of them: when lumpiness clusters upon a single background, a photon's gravitational redshift descending into a potential well is undone climbing out, so the contributions telescope along the path and leave no √N accumulation, whereas a genuine differential expansion accumulates because there is nothing to cancel against—so the observed isotropy selects the telescoping picture over the accumulating one. Every choice in the estimate biases it downward, so the number is a floor. A differential programme can evade it only by requiring the local rates to average to a common value in every direction, which is itself a global uniform expansion in a cosmic time—the very structure such programmes set out to dispense with. What is excluded is an inhomogeneous expansion rate; the matter lumpiness is untouched, the density being the bend of the spatial cut, so uniform expansion and lumpy matter are consistent. The remaining, finely tuned, observer-centred escape is closed by the Copernican principle together with the independently measured isotropy of the expansion history. What the exclusion selects is a global, time-ordered expansion—the cosmic foliation made dynamical—which is logically prior to the notions of space, isotropy, and homogeneity the model otherwise assumes. The same datum thus establishes, from the bottom up, the cosmic time, the uniformity of its advance, and—within the observable region, under the Copernican principle—the maximal symmetry of its slices. Read ontologically, this measured foliation is the lapse of the 3+1 split—an objective rate of advance, the cosmic present—while the relativity of synchrony is the shift; the century-old reading of synchrony's relativity as the absence of any objective present is thereby a modal fallacy, which the measured isotropy falsifies outright. We prove that fixing the physical foliation and reading it so is both necessary and sufficient for a coherent formal description of an existing, evolving world—an augmentation of general relativity that changes none of its equations, its necessary half here measured. We also set down the history the measurement closes. The structure this datum forces was posited by Einstein in February 1917—explicitly “against the spirit of relativity”, and on the empirical ground that stellar proper motions are small compared with c—countered a month later by de Sitter, who placed the choice among candidate universes outside physical argument altogether and pressed as his sharpest objection that in Einstein's solution “time has a separate position”; and defended in 1920 by Eddington on geometric grounds, in the declared absence of any experimental knowledge on cosmical scales. Einstein then went nearly silent on cosmology for the rest of his life, and never addressed how the cosmic time his own assumption distinguished stands to the relativity of simultaneity. The assumption was correct, and the objection was correct as a description of the structure though not as a reason to reject it. This is the empirical counterpart of the conceptual results of the companion papers: as stellar parallax converted the Earth's motion from an interpretive option into a measured fact, the isotropy of the cosmological redshift converts the cosmic foliation from a modelling convenience into a measured feature of the world.
A companion paper exhibited the Schwarzschild–de Sitter family as a single rigid geometry charted from several observer vantages, with the admissible vantages organised into a groupoid whose one invariant is the geometry. That paper established the rigidity and the discrete generation of the morphisms, and left open the relational content of the generated structure. The present paper takes the groupoid as established and develops that content.
We give the groupoid as a category, name its generators in coordinate-independent form, and derive the relations they satisfy: the root-exchange involution σ of order two and the sky-angle periodicity τ of order three, with (στ)2= id, generating D3≅S3—and shown complete, which closes the generation question the slicing paper left open.
A physical theory is chosen, ahead of any decisive measurement, by criteria—coherence, the requiring rather than the permitting of the phenomena, consolidation, resistance to patchwork—that are usually treated as the province of philosophy, standing outside science and adjudicating it from above. We argue that this is a category placement, and the wrong one. The epistemology of scientific theory-choice is a discipline of the same kind as the sciences it grounds, with the same object and the same method: it reads the structure behind appearances—here, which inference rules reliably track the world—by the science's own procedure, off the science's own record. Its empirical engine is historiography, and its data are episodes in which a structure favoured by these criteria ahead of a decisive, non-local measurement was then vindicated by that measurement. We make the inference rule explicit (ask what world must exist for the appearances to arise), its operable form (four rules of reasoning), its characteristic dual (the modal fallacy: no local discriminator is not the absence of the fact), and its constructive ordering (ontology from evidence, kinematics from ontology, coordinates from kinematics). We add the constraint that an admissible world must explain the perspectival appearances—exhibit the projection under which they arise—rather than discard them or merely reproduce them, and we identify the ontological register of the criterion of necessity: least-arbitrariness, under which a structure carrying an unforced modulus is not a single world but a family, and is inadmissible on that ground, the maximally symmetric structure being the unique one that requires its own configuration. The register's boundary is drawn with it, since a criterion claimed to apply everywhere is as suspect as one that applies only where it was formulated: a modulus fixes how a symmetry is broken, so an unforced choice that leaves the symmetry maximal—the dimension of a maximally symmetric substrate is the programme's first such case—lies outside the register, and where form is silent content may still decide.
General relativity supplies a four-geometry and field equations but does not single out which of the foliations its formalism admits is the physical one. A companion paper shows that observation does: the isotropy of the cosmological redshift forces a global cosmic time and a uniform expansion. Fixing that foliation and reading it ontologically—the lapse the objective rate at which an existing spatial layer advances, the shift the relativity of synchrony—is both necessary and sufficient for a coherent description of an existing, evolving world, and it changes none of general relativity's equations. This paper takes that augmented theory, Cosmological Relativity, and develops it: its axioms, the worked constructions that flesh them out, its central theorem, and what follows for the rest of the programme.
Companion papers exhibited a single slicing curve on a de Sitter substrate whose turning points are the horizons of the Schwarzschild–de Sitter family, and read the family as one geometry charted from several vantages. This paper asks what the construction is as an operator: what it takes as input, what it returns, and what the matter content of a returned geometry is a property of.
Working in the construction gauge ds2=-f dt2+dr2/f+r2 dΩ2 that the single curve enforces, we form the Einstein tensor and read the matter content off the curve. Three results follow as derivations rather than identifications.
The vacuum kernel. The condition Tμν=0 is the first-order linear ordinary differential equation rf'+f-1+Λr2=0, whose entire solution space is f=1-2M/r-Λr2/3—the whole Schwarzschild–de Sitter family, with M the single constant of integration. The vacuum sector is exactly the kernel of the matter functional, derived rather than matched: straight cuts are vacuum.
A companion paper read the de Sitter slicing construction as an operator with three independent data—the leaf, the stacking and the vantage—and showed that its vacuum sector is the kernel of its own matter functional. This paper asks how far the operator reaches: which exact solutions of general relativity are cuts of the one substrate, and which are not.
The bound. A swept geometry inherits the sweep's symmetry, and the sweep is by isometries of the substrate, so every reachable geometry carries an isometry group containing a sweep-subgroup of so. The range is therefore bounded above by the symmetry-reducible sector of general relativity, and the body of the paper shows the bound is attained.
Filling the sector. Within a reachable symmetry class the operator's four data supply exactly the metric functions the class admits, so the cut spans the class, the vacuum members are the substrate's own family in that class, and matter is the bend. The homogeneous class returns the Kantowski–Sachs form with the Schwarzschild–de Sitter interior as its vacuum kernel; and that interior and the flat cosmology are the same geometry read at two slicings, differing by a single rest-energy term—so collapse and cosmology sit in the range as cuts of one substrate rather than as separate catalogue entries.
In the canonical formulation of general relativity the Hamiltonian is a sum of constraints that vanish on the physical phase space: no preferred time survives, and the generator of evolution annihilates physical states rather than evolving them. This is the canonical “problem of time.” We argue that it is not a technical defect awaiting a technical repair, but the canonical symptom of a category error—treating the four-dimensional manifold of occurrences as the thing that exists, the block universe.
In Cosmological Relativity the exact spherically symmetric and stationary geometries of general relativity are generated as symmetry-breaking cuts of a single de Sitter substrate, with vacuum the straight cut and matter the bend, and the range of that construction is the symmetry-reducible sector, bounded by a wall at which continuous symmetry is lost and free gravitational radiation begins. This paper takes the next step: the dynamics—why and how the cut bends in time. The gap it closes was named at the outset of the programme, in the essay that first developed this cosmology and recorded that the details of dynamical matter had not been worked out.
We work the first inhomogeneous, time-dependent bend explicitly. In a polarized Gowdy–de Sitter model the spatial leaf carries a single transverse-traceless shear—the propagating graviton—advanced by a true Hamiltonian on the substrate's cosmic foliation, with the cosmological constant driving the area and the wave carrying its own energy and momentum in the leaf's shear.
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 slicing operator of Cosmological Relativity (CR) generates the spherically symmetric, stationary sector of general relativity as symmetry-breaking cuts of a single de Sitter substrate. It is natural to ask whether the same substrate geometry yields the Standard Model—colour SU (3), the full gauge group, and chiral matter—as a continuous isometry, the way it yields the gravitational solutions as cuts. This paper maps a precise negative boundary and grounds it in the established literature. The substrate's continuous symmetry is exhausted by SO (5,1), and su (3)⊄ so (5,1) structurally; the gauge structure can live only on the compact (Wick) face, reached by a change of signature and not by any real-substrate operation—the candidate real involution is a Weyl reflection, demonstrably not the Wick rotation.
The companion papers establish that the black-hole/de Sitter substrate of Cosmological Relativity (CR) is a maximally symmetric manifold sliced perspectivally, and that the Standard Model gauge group does not arise as a continuous isometry of it: the geometric route to su (3) is excluded, leaving the substrate's discrete orientation structure as the one opening for matter.
This paper puts a Dirac spinor field on the slicing structure and asks what that discrete opening delivers.
Three results follow, each stated at the weight it is earned.
Cosmological Relativity (CR) obtains the observed cosmological expansion by a causal reassignment on a de Sitter background at the cosmogenesis branch point, with the Einstein field equations unchanged. This paper develops that cosmology and completes it with its scalar perturbation sector. The reassignment selects the Nariai member of Schwarzschild–de Sitter, whose proper frame is a homogeneous, observationally isotropic, non-synchronous cosmology with areal radius r(τ)∝ sinh2/3—exactly the flat-ΛCDM scale factor—its rate fixed by the cosmological constant alone, with radiation and matter inherited content read off the cosmic clock rather than terms that source the expansion. The cosmic beginning is the branch point r=0—a genuine curvature singularity of the SdS metric read over the r-chart, and not a singularity of the underlying substrate, whose throat is the defining constant α; the divergence of the r-chart invariants there is the areal coordinate degenerating; the constant-proper-time slices are exactly Euclidean (Ωk=0) while the cosmological layers are a closed S3—spatial curvature and dark energy one Λ read on two slicings. The acoustic scale and the apparent Hubble tension resolve as consequences of a rate fixed by the substrate's geometry: there is no second H0 to reconcile, and the scale is met at the directly measured H0 by a single inherited datum, ρr/ρm≈2—a one-parameter accommodation, the structural analogue of the baryon-to-photon ratio. The light-element composition is produced on the same footing: the infalling matter heats above the deuterium bottleneck and re-expands through it, so the cooling leg is a standard big-bang nucleosynthesis fixing the composition with the inherited baryon-to-photon ratio η—a second boundary datum of the same handover, distinct from the radiation amplitude ρr/ρm that sets the scale. Helium-4 and deuterium sit at their observed values—the freeze-out temperature the data demands being the bottleneck the standard rate delivers, un-tuned; the near-zero metallicity of the oldest systems follows from the handover; and because the run is standard, lithium-7 carries the standard threefold over-prediction, the lithium problem shared with flat ΛCDM rather than dissolved.
In Cosmological Relativity (CR) the cosmic beginning is not a singular origin but a cosmogenesis: the collapse of matter in a previous universe, continued through the branch point r=0 at the close of the lift—singular for the r-chart metric, smooth on the substrate and re-expanding as our own, with the Einstein field equations unchanged. This paper does two things. First, it exhibits the Big Bang as the deductively forced synthesis of consequences the corpus already establishes—the measured foliation forces the augmentation; the augmentation forces that collapse cannot terminate but becomes a universe; that completion is the branch point r=0 at the close of the lift, crossed by a foliation-preserving causal reassignment that re-founds the horizon's null direction as the cosmic time of the expanding phase; the degeneracy the reassignment uses is the forced Nariai member's—the horizon cubic's double root, at which the substrate's isotropy jumps and the geometry is dS2×S2 —a property of the geometry rather than an event on any worldline, and to be kept apart from the crossing itself; the reassignment fixes a geometric flat-ΛCDM rate and carries the leaf's matter across as inherited content; the discrete matter sector and the acoustic peaks follow—each arrow a result proved elsewhere in the corpus, the Big Bang their conjunction rather than a separate posit. Second, it carries that synthesis to a quantitative, falsifiable edge: the primordial abundances of the light elements, read as a fossil of the previous universe's collapse.
The papers of this programme construct, read, and apply a single object: the maximally symmetric de Sitter substrate. This paper is about the object itself. We make explicit three facts the corpus has carried implicitly and never stated in one place. First, the substrate is everywhere intrinsically real by construction: it is a real manifold with a real coordinate basis—the five-dimensional dS5=SO(5,1)/SO(4,1) of the ladder below, of which the four-dimensional dS4 is the background its leaves carry—whose Lorentzian signature is an intrinsic property of its positive curvature, not a signature imposed on it from outside; the imaginary variables the construction uses to reach it—the fifth embedding coordinate, the conjugacy circle, the equatorial seam's analytic continuation—are instruments of visualisation and continuation over a geometry that is real at every point they land on. Second, that substrate is the universal standard of physics: its curvature radius α=√3/Λ and its locked null cone are the intrinsic length and causal structure against which every material structure is what it is—Eddington's reason the cosmological constant cannot vanish, here corrected (its timelike half as real as its spatial one) and grounded (a real intrinsic ground state, not a property of the operation of measurement); and, read at the last-scattering surface, this same standard dissolves the horizon problem—the microwave background's uniformity is the substrate's maximal symmetry, not the residue of an early causal contact. Third, the maximal symmetry of that substrate is a single fact from which the corpus's separately-established results descend as one: the parameter-free rigidity of the cosmology, the fundamental constants standing as unit gauges over the substrate's single scale—a singleness with a consequence the ledger should state, since neither real form supplies a second invariant and a dimensionless magnitude needs two: the global Wick rotation carries the defining quadric to S5 of the same radius, acting on the coordinate and leaving the right-hand side untouched, and every curvature invariant on either face is a pure power of 1/α2. So the construction cannot force a coupling, and its silence about magnitudes is a property of a one-constant theory rather than a gap awaiting work—the common root of three verdicts reached separately, that the winding quantises without measuring, the flat bundle selects without coupling, and the branch point filters without supplying—, the necessity and sufficiency of the augmentation of general relativity, the universality of physics, and the wall against a continuous geometric matter symmetry are the same property seen at several rungs. Read at that constant rung, the single scale renders the traditional Planck values gauge-combinations rather than physical scales, and dissolves the cosmological-constant and coincidence problems by that one fact—no Planck scale for Λ to be small against, and no bare-Λ-versus-vacuum-energy split for the fine cancellation to act on—with Λ's value the ledger's one input scale, not predicted. We collect the source—a real coordinate basis in which the would-be timelike embedding coordinate is not a coordinate of the manifold at all, and a curvature radius that is the manifold's one scale—and we show that the straight null rulings of the surface, the light cone locked to the equilateral profile that maximal symmetry forces, are the visible peak of exactly this intrinsically real geometry.
This paper states, in one place, what the Cosmological Relativity corpus establishes, at the weight each result is established at, together with what the construction declines to claim and where its open edge lies. Nothing here is new: every claim is drawn from one of seventeen papers or from the ledgers those papers cite, and is stated at the register its source gives it—a proposition as a proposition, a reading as a reading, a conjecture as a conjecture. What is new is the assembly, and the three things an assembly can supply that a set of papers cannot. The joins, where a result in one paper is load-bearing for a result in another, or where two papers reach one structure by arguments sharing no premise. The bounds, carried from each source to sit beside the claim they bound — a distillation that drops them has not shortened the corpus but misreported it, and several of the sharpest statements here are the limits their own papers place on their own results. And the relative size of the frontier against the body of work that surrounds it.
What the papers rest on, and what can be re-run.
This is not a summary written for the page. It is fetched from the programme's own frontier, which is generated from its register of open problems, and it changes when the work does.
Fetching the current frontier…