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.
The reading order is in README.md, which follows the corpus's own causal spine rather than the numbering.
The forcing. P1 shows the event horizon is a metric singularity — a missing definition in general relativity, argued from causal structure alone. P4 shows the redshift-isotropy floor forces the cosmic foliation empirically. The two are a complementary pair: one proves it, the other measures it.
The substrate. P2 gives the Schwarzschild and de Sitter circle — one homogeneous ring, the intrinsic
⛔ THE DIMENSIONS, because the corpus is subtle here and a reader who guesses gets it wrong. The SUBSTRATE is dS₅ = SO(5,1)/SO(4,1) — a hyperboloid in flat ambient M⁶, the thing that is sliced. The BACKGROUND is dS₄ — the maximally symmetric four-geometry the substrate's leaves carry, itself a cut, the thing that is reassigned. Below those sit the exact solutions (Schwarzschild, Nariai, the Friedmann geometries), and below those the layer 𝒮ₜ — the three-dimensional existent, with the four-manifold M its representation and not an existent. The fifth dimension is forced: slicing a four-dimensional de Sitter space only re-coordinatizes it, so a dS₄ substrate would generate nothing. "The substrate is four-dimensional" is a retracted error. geometry. P3 gives the substrate and its slicing curve, horizons as turning points. P5 gives the description groupoid: generators, relations, and Schwarzschild as one member. P17 is the geometric core — the maximally symmetric substrate, reached through the imaginary and real by construction.
The framework. P7 — collapsed matter must become a universe: the necessary and sufficient augmentation of general relativity. P8 gives the slicing operator and the covariance of geometries over the substrate; P9 its range, surjective onto the symmetry-reducible sector; P11 why the cut bends.
Time and the constraint algebra. P10 treats the canonical problem of time as a category error, with an empirically forced cosmic time. P12 identifies GR's constraint algebra as the symmetric-space structure of the substrate — and the problem of time's "wrong sign" as the substrate's coset signature.
Matter. P13 maps four converging routes to a geometric boundary on what the substrate's isometry does and does not force. P14 gives the fermion sector: three chiral generations from the maximally symmetric substrate.
Cosmology. P15 gives the expansion history from causal reassignment. P16 gives cosmogenesis — the Big Bang as a synthesis of the framework's deductively forced consequences.
P6 stands apart and holds the rest: the shadow of existence — scientific theory-choice as an empirically grounded discipline. It supplies the epistemic altitude the whole corpus is held at.
And P18 closes the set by doing what none of the others can. What the construction delivers, what it declines, and where its edge is — the synthesis, written last and from the outside. It collects what the corpus derives, separates that from what it dissolves and what it declines to claim, and carries the frontier: the open questions stated with what would discharge each. It is the paper to read if you want to know what this is worth rather than how it works, and it is the only one whose subject is the corpus itself.
The geometric core is P17 throughout. An older tag, p0, appears in retired notes and is deprecated.
Two documents are corpus in ledger form rather than paper form: the geometry ledger (58 proved statements across 25 receipts) and the combinatorics ledger (the corpus's numbers are two families, the boundary between them the Standard Model's arrangement).
Every computed claim cites a script that can be re-run.
Loading the matrix…
(The live table is BOOK_INTRO_cosmiCave/assets/dependency_matrix.html, regenerated from the corpus by scripts/depmatrix.py; the same data is P7's tab:dependency-matrix.)
Read down the columns and one thing stands out: exactly two papers are coupled to the whole corpus — a full row and a full column each. P7, the framework, and P17, the geometric core. (Verified from the source .tex rather than the rendered table: each cites all sixteen others and is cited by all sixteen. P3, the waypoint, is cited by fifteen and has the largest column sum, 147 — recomputed r1612 at 146 and again r1632 after P3 gained a citation; the figure had read 134, stale since the r1600–r1608 placements added citations without re-running scripts/depmatrix.py.) Everything feeds P7 and P7 feeds everything; P17 is built from every paper and points back to every paper. The other fifteen cluster and gap; these two touch all.
That is why the corpus resisted being put in an order for so long, and why it has no first page. A one-hub corpus is a tree with a root, and a root sits at the front. A two-hub corpus has no linear slot — it has a between: fifteen papers stretched between two synthesis poles. P17 kept wanting to be "the border" for exactly this reason.
Two things follow, and they answer different questions. The doors are per-reader: one arrives with an interest and enters where it is answered. The guided arc after them is one route for a room — ordered by structural depth rather than by readership, and deliberately not opening with P1 and P4, which carry the load and also raise the guard before there is anything to weigh them against.
The matrix is not only a self-portrait; it says as data where a reader should enter. There is no single first page, so here are the doors — and whichever you take, the road runs through P3, the largest column sum in the matrix and where the substrate's geometry is actually built.
If you want what is forced — the sceptic's door: P1, then P4. The two forcing keystones, and the matrix marks them a complementary pair: P1 proves the foliation from causal structure alone, needing none of the cosmology; P4 measures it, from the redshift-isotropy floor. Nothing downstream is load-bearing until these hold.
If you are a cosmologist: P1, P4, then P7, P15, P16. The forcing, then the framework, then the expansion history from causal reassignment, then cosmogenesis — the Big Bang as a synthesis of forced consequences, with the light-element abundances reproduced and the Hubble tension resolved across the low-redshift distance ladder.
If you want to know what it is worth before you spend time on it: P18. The synthesis, and it is a door rather than a step. It collects what the construction derives, separates that from what it dissolves and what it declines to claim, and carries the frontier — the open questions stated with what would discharge each. It will not teach you the construction, and read first it will look like a set of verdicts on arguments you have not met. But it is the only paper that answers is this worth my time, and where does it fail, and a reader who wants that answered first is entitled to it.
If you come from philosophy, epistemology, or the history of science: P6. It is the door to the foundational drivers, and it argues something prior to the physics — that "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." A physical theory is chosen, ahead of any decisive measurement, by coherence, by requiring rather than permitting the phenomena, by consolidation and resistance to patchwork. P6 also supplies the altitude the whole corpus is held at, so it repays reading early even by those who came for the geometry.
If you work in canonical general relativity — the esoteric door: P12, and P10 beside it. P12 is entered directly by anyone who knows the hypersurface-deformation algebra: it establishes that the Dirac algebra is not a Lie algebra but a Lie algebroid, that the symmetric-space grading is that algebra term for term, and that the problem of time's "wrong sign" is the substrate's coset signature. P10 treats the canonical problem of time as a category error, with an empirically forced cosmic time.
If you come for the Standard Model: P13, then P14. P13 is a synthesis as much as a boundary — four converging routes to one wall, with three operations co-located at the branch point pried apart. P14 gives three chiral generations on the discrete residue.
If you want the whole: either pole — P17 or P7. From either, the rest is one step away. These two are what make the corpus a dipole, and each is a full row and column of the matrix.
(A different order exists and serves a different purpose: README.md steps 3–6 read the corpus along its causal spine, the order in which the results are forced. That is for someone who must be able to stand on each result before the next leans on it. The doors above are interest-ordered; that order is dependency-ordered. Neither is the reading order — they are answers to different questions.)
The doors above are per-reader, each entering where their own question is answered. This is the other thing: one route, for a room of them. It is ordered by what is most structurally arresting to someone who already holds the machinery — and it deliberately does not open with P1 and P4. Those two carry the load, and they are also the two that put a reader's guard up before there is anything to weigh them against. They come at the end, when the structure they force has already been seen.
① P12 — the constraint algebra is a Lie algebroid, and the "wrong sign" is a coset signature. Start here because it is checkable in an afternoon by anyone who knows hypersurface deformations, and because it costs nothing to accept. The Dirac algebra's structure function makes it not a Lie algebra; read as a Lie algebroid the anchor is the ADM data as functions of the cut — energy IS the Hamiltonian constraint, the bend of the leaf. And the symmetric-space grading so(5,1)= h⊕m is that algebra term for term, with the notorious ±qab sign the substrate's Lorentzian coset metric. A recognition, not an addition — that is the paper's own weight-marking.
② P10, immediately beside it — what the algebroid licenses about time. The problem of time as a category error: the bare formalism singles out no foliation, and CR's selection is measured rather than posited. Deparametrized, the constraint is solved for a true Hamiltonian that generates the advance, and quantum evolution in cosmic time is unitary — the frozen constraint and the true Hamiltonian are the same canonical content under two readings. And the scale factor's lone self-adjoint extension freedom is closed with no free parameter, by the de Sitter horizon's own Hartle–Hawking state at κ=1/α — which is the substrate read on its other real form, the global Wick carrying dS5 to the compact S5, the two being real forms of one SO(6,C).
③ P8 — matter is a bend of the cut, and the offset IS the mass. This is the mechanism everything above was implicitly using, and it is the paper the whole mass reading rests on. The slicing operator carries geometries over the substrate: a geodesic planar section is vacuum; an offset section — a parallel plane not through the centre — is SdS with M≠0, and 2M=α ((r0/α)-(r0/α)3). The offset is the mass. And that single identification is the sharpest thing anyone gets out of this corpus about the asymptotic-mass problem: an asymptotic charge is built to measure a property the spacetime possesses, read off at its boundary — but if the mass IS where the cut sits, there is nothing at infinity to measure. The quantity being sought is a placement of the section, not a content of the geometry. The standing difficulty that no conserved charge is well defined in an asymptotically-de Sitter spacetime is not a gap in the definitions; the constructions differ over how to subtract a de Sitter background, and on this reading there is no background to subtract.
④ P9, its range — the Carter constant as the substrate's own symmetry. For anyone whose instinct is that hidden symmetries are where the truth is. The operator's range runs onto Kerr–NUT–(A)dS, and the Killing–Yano structure that gives Carter his constant is not imported — it is what the substrate carries. ⌗ And the sharpest thing in the paper is a warning about a word: its "acceleration" is the Plebański–Demiański parameter — a bare accelerating mass with an irremovable conical strut — not the cosmic acceleration, and the two are different objects.
⑤ P11 — why and how the cut bends in time. Having seen that matter is a bend, the dynamics is the obvious next question, and P11 is the paper that asks it. In the symmetric sector the answer is closed-form and worth stating for its shape: d2r/ d τ2=rKG — the rate at which the cut's bend changes in time is the bend itself, up to the areal factor, so the cut straightens, is momentarily flat, and bends the other way as the geometry passes its one sign change. Then the inhomogeneous case, which is where the paper actually lives — the Gowdy sector, free gravitational radiation, and the wall at which it begins, distinct from the seam. And the standard puzzle it dissolves is one this readership will recognise: gravitational-wave energy non-localizability, which stops being a defect once energy is the Hamiltonian constraint — a local functional of the leaf's own bend — rather than a charge owed at a boundary.
⑥ P13 → P14 — the boundary, then what survives it. P13 is four converging routes to one wall: su(3)⊄ so(5,1), so colour is not a realised continuous isometry. A negative result of unusual quality — it says exactly what the substrate does and does not force, and it pries apart three operations that sit on top of each other at the seam. Then P14 shows what the wall leaves standing, and it is not small: a Dirac field on the slicing curve with dim ker+=3, dim ker-=0 — three chiral generations, forced by least-arbitrariness rather than posited, protected as a γ5-graded index under any deformation preserving the three-wall structure. A one-hinge truncation is excluded not as disfavoured but as carrying an unfixed modulus. ⌗ AND THE SAME CONSTRUCTION, READ IN A GENERAL DIMENSION, SPEAKS ABOUT THE DIMENSION (r2376+c54.10). The fold the count reads is D-1; the horizon relation collapses to a single multiple-angle only at D=4 and D=5; and the mass-parity that grades chirality exists only at even D. ⇒ four dimensions is the only one carrying both a generation count and a chirality, so three generations and four-dimensional spacetime are one fact in CR read at two ends. At the sector's own altitude — forced within CR, not a proof about the world — and it settles the dimension of the cut, never the substrate's, which stays bounded below only.
⑦ P17 — the one scale, and the Standard Model's own shape read off a circle. This is the paper to sit with. Maximal symmetry worn seven ways, and two of them are the reason to make the trip. The gravitational–cosmological–quantum sector's geometric constants spend no free dimensionless constant: c is the null-ruling slope, G appears only as GM/c2 with the mass fixed to Nariai by Λ — ΛG2M2/c4=1/9 — and ℏ enters only at the branch point, scaled by Λ alone. (The graviton tower's mode sums do spend one, computed at PO-23: a regularisation rather than a gauge, and the audit above does not reach it.) And the waist: the power of a point with respect to it is the square of the point's height, so Euclid III.36 and the null condition are the same equation, the minus sign in the metric doing all the work — which makes the double ruling and the classical power law one statement, set by α and nothing else. Then the line worth the whole section: Aut(A2)=S3×Z2 factorises because its factors act on the two things a figure can be to a circle — the three roots are ON the waist, the two rulings are TANGENT to it. A ruling is a line of the substrate, so exchanging them is an isometry: chirality descends gauged. A root labels a different cut, so permuting them is no motion at all: flavour is global. The Standard Model's arrangement of a gauged chirality against a global flavour, read as on versus tangent.
⑧ P3, with P5 and P2 beside it — the geometry all of that was read off, and its group. P5 is the algebraic half and it is the one to put beside P17, because it does in group terms what §⑦ did in figure terms: the description groupoid — generators, relations, and Schwarzschild as one member — with the partition that matters, the invariant de Sitter geometry is the group's R-even part and the Schwarzschild mass its R-odd perspectival artefact. That is the shadow-reading in closed form: not a description of a projection but a projection exhibited, as a group action with an even and an odd part. And it supplies the algebraic content of P3's geometric result — the horizon–singularity asymmetry as a sweep-pivot artefact — so the two hold one dissolution at two levels rather than one citing the other. Its own sharpening is worth having: α is the unique chart-invariant quantity in the groupoid, so any fully invariant gravitational mass would have to be built from α — there is no invariant mass because the invariant is not a mass. Then P3 itself: the slicing curve, horizons as its turning points, and mass as a reading — The slicing curve, horizons as its turning points, and mass as a reading: 2M=α sin u cos2u, linear in α with a dimensionless slicing profile as coefficient, so "M is not an intrinsic coefficient of a spacetime; it is the throat radius projected through a turning of the slicing" — and the profile's maximum is the Nariai value, so the largest mass any slicing can carry is set by the throat. P2 then does the economical thing: it undoes the standing verdict that the Schwarzschild curvature singularity is an inextendible boundary not by any claim about which manifold is fundamental, but by exhibiting the continuation that carries the curve through it.
⑨ P7, then P15 and P16 — the framework, and what it predicts. P7 is the necessary-and-sufficient augmentation and the pole everything feeds. Then the cosmology, and here the arc's tone changes from structure to consequence: a geometric expansion rate, the Hubble tension resolved across the low-redshift distance ladder and not by the acoustic angle alone, and light-element abundances reproduced from a collapse excursion — deuterium at D/H≃2.5×10-5, Yp≃0.25.
⑩ And only now, P1 and P4. Held to the end on purpose. P1: no event horizon completes at finite exterior time — argued from causal structure alone, needing none of the cosmology, and it is where the programme's whole weight sits. P4: the redshift-isotropy floor forces the cosmic foliation empirically, ≲3×10-6 after secondaries. Read first, these two are a fight. Read eighth, they are the two load-bearing legs under a structure the reader has already found interesting — and P4's own clause is the one to end on: "because the ontology's necessary half is measured here rather than posited, those dissolutions are not free-standing coherence but the consequences of a distinction whose foundation the redshift isotropy already forces."
(P6 sits outside the arc and can be read at any point — it supplies the altitude the whole thing is held at, and its prin:reclass is the test every dissolution above is graded against: exhibit the projection under which the appearance arises; do not merely reproduce it, and do not discard it.)
The arc above is the theoretical physicist's. A second one runs alongside it and is less developed, and it is worth drawing because the objects it visits are the same objects — met as mathematics rather than as physics, and in almost the reverse order. It can be read in parallel, by someone whose interest is the structure rather than the world, and it opens somewhere the other cannot: with a theorem from Euclid.
Ⓐ Classical plane geometry, and a two-thousand-year-old theorem doing modern work. P17 §sec:power. The power of a point with respect to a circle — Euclid III.36, Steiner's invariant, |X|2-α2 — and the hyperboloid's own equation says that quantity is the square of the point's height. So the tangent from any point runs exactly as far across as the point stands high, and ds2=0: the tangent–secant relation and the null condition are the same equation, and the only thing turning one into the other is the minus sign in the metric. Bounded honestly, and the bound is the interesting part: it is a fact about ONE circle — a power taken with respect to any other is a perfectly good Euclidean quantity and is not a height, because the identity is the hyperboloid's equation and the hyperboloid has one waist. Then the waist keeps paying: it is the incircle of an equilateral triangle and that triangle's nine-point circle, so the hinge distance 2α is an output rather than a stipulation; the triangle's three sides are null rulings; and r=0 sits on the circle.
Ⓑ Trigonometry that turns out to be forced. P3 prop:gnomonic, prop:triple. The mass–offset relation is 2M=α(u-u3), and under the gnomonic chart it becomes the pure triple angle, 2M=2/3√3α sin 3w. The scale 2/√3 is not chosen — it is the unique value for which the relation is a pure multiple of sin 3w with no residual sin w harmonic. So the cubic r0-r03 is the cubic in sin w that the triple-angle identity collapses; and the profile's maximum is the Nariai configuration, which the identity therefore returns of its own accord.
Ⓒ Complex analysis: a continuation where the literature has a boundary. P2. The Schwarzschild curvature singularity is standardly an inextendible boundary. P2 undoes that verdict not by any claim about which manifold is fundamental but by exhibiting the continuation that carries the curve through it — "the reclassification in its most economical form, and a reminder that dissolving a puzzle by identity need not wait on settling an ontology." And the interior cycloid is a closed Friedmann scale factor: one homogeneous circle whose two poles are the horizon and the origin. With the monodromy that goes with it: r3 has period 2πiα/3, so r=(r3)1/3 lives on the three-sheeted cube-root cover and closes only after 2πiα.
Ⓓ Galois theory of a one-parameter family. P5, and the energy family. The turning cubics of the family r3+pr+q, p=(E2-1)α2, q=2Mα2: the roots are equilateral iff p=0, and — since p is M-independent while q is linear in M — the discriminant is a perfect square in M iff p=0 as well. One condition, met at E=1 alone. So at E lt;1 the roots are colinear with no symmetry as a figure but full S3 as monodromy of the cover; at E=1 they form the equilateral triangle carrying S3 as its own figure symmetry, monodromy dropping to Z/3. Neither end lacks the S3; the deformation exchanges the manner in which it is carried — from monodromy of the cover to symmetry of the figure. And lem:twoturnings forbids the shortcut: no affine change of variable identifies the two threefold symmetries.
Ⓔ Root systems, and a factorisation with a geometric reason. P17 §sec:unification. Aut(A2)=S3×Z2, and the factorisation is not formal — its factors act on the two things a figure can be to a circle. The three roots are the special points ON the waist; the two rulings are the lines TANGENT to it; on and tangent are independent, so the residue factorises. And the two kinds differ for the same reason: a ruling is a line of the substrate, so exchanging them is a motion of it — an isometry; a root labels a different cut, so permuting them is no motion at all. Six Nariai geometries form the A2 hexad; the three hinges join into one skew hexagon — resonance with the hexad, not identity.
Ⓕ Symmetric spaces, and two real forms of one complex group. P17, P13. dS5=SO(5,1)/SO(4,1), and the global Wick x0↦ix0 carries it to the compact S5=SO(6)/SO(5) — the two real forms of one SO(6,C), meeting at the horizon where β=2πα. The Lorentzian form carries the real-geometric gauges, the compact form the thermal ones; and su(3)⊂so(6) while su(3)⊄ so(5), which is the whole colour boundary in one containment.
Ⓖ Lie algebroids — where the physicist's arc began. P12. The hypersurface-deformation algebra's structure function makes it not a Lie algebra; read as a Lie algebroid over the space of cuts, the anchor is the ADM data and the symmetric-space grading is that algebra term for term. This is station ① of the other arc, reached here from the opposite end.
Ⓗ An index theorem, as the destination rather than the entrance. P14. A Dirac field on the slicing curve: dim ker+=3, dim ker-=0, a γ5-graded index protected under any deformation preserving the three-wall structure. The count is topological; what makes it three is the substrate's own three-foldness, and a one-hinge truncation is excluded because it carries an unfixed modulus.
⌗ The two arcs meet at P17 and P12 and run in opposite directions — the physicist enters at the algebroid and arrives at Euclid; the mathematician enters at Euclid and arrives at the algebroid. Neither is the reading order (that is README steps 3–6, the causal spine), and neither is the doors. Three orderings, three purposes: what is FORCED, what INTERESTS, and what is STRUCTURALLY DEEP.
The corpus takes any number of approaches — which is a property of a structure read many ways rather than an accident of arrangement, and the matrix is how you find your own way in.
The register. Most of what the corpus establishes is forced within CR — the structures follow from the maximal-symmetry principle that defines the programme, and a framework declining that principle is not obliged to accept them. That scope is what each result is stated at, and the discipline it rests on is P6 rather than a preface.
What is settled. The forcing arguments; the substrate's geometry and its slicing structure; the operator's range; the constraint algebra's identification; three chiral generations on the discrete residue; the expansion history; the cosmogenesis synthesis with the light-element abundances reproduced. On the observational axis the programme has crossed from coherence to empirically favoured: the Hubble tension is resolved across the low-redshift distance ladder, and not by the acoustic angle alone.
What is open — five families, and they are named rather than implied. The inherited datum · the scalar-perturbation sector to a verdict · the propagating fermion sector, which is the largest undertaking and the gate the others wait on · the world-correspondence · and the interacting quantum tower, which is not CR-specific. Two families have left this list since it was written, and a list that shrinks and says why is worth more than one that was always right: the matter branch-point crossing dynamics closed at r2376+c54.113 and the irreducible interior reassignments at c54.118.
What would decide it. Two things are held out to the world rather than argued: no event horizon completes at finite exterior time — structural, resting on causal structure alone, and where the programme's whole weight sits; and the geometric expansion rate, the nearest-term discriminator on expansion-history data.
And one thing that must always remain a conjecture: that the framework the corpus has uncovered is the framework our actual universe rides. No amount of internal coherence converts it. Empirical facts are primary, and the work proceeds within those bounds.