Genesis

Cl(3,1)⊗Cl(1,3)

Created by Kristin Tynski Based on Null Vector work with Garret Sobczyk

one null Gram · two signatures · one inversion · Grace φ

scalar vector bivector trivector pseudoscalar

SO(3) lock —

dimension—

witness—

curvature—

boost—

causal depth—

spectral gap—

grade radius

scalar0.00

vector0.00

bivector0.00

trivector0.00

pseudoscalar0.00

0.000time

1.0×

L·S SHELL GAPS

boosts cancelled — angular momentum well-defined

What this is

Created by Kristin Tynski Based on Null Vector work with Garret Sobczyk Grace operator (φ grade-scaling) is this construction’s, not a CGA/STA kit

Press M. You just changed the metric signature of spacetime. Press G. You just turned off the thing that decides which signature survives.

Every course hands you a signature as a choice — Cl(3,1) or Cl(1,3), pick one, argue in the footnotes. This says it is not a choice. It is an outcome.

Run a Lindblad-form kinematic sieve S(M) = Σ_k c_k M c_k on a Clifford multivector. It acts with eigenvalue +2 on compact generators and −2 on non-compact ones: rotations sustained, boosts damped. In Cl(3,1) the late state settles with rotations outliving boosts 4.04 to 1 — and the 4 is exact, the ratio of the sieve floors 2S : 0.5S, nothing fitted. In Cl(1,3), where the flip makes every spacetime bivector compact, the same operator shows no preference at all.

So the world looks rotational and Euclidean at late times not because anyone postulated it, but because that is what a dissipative sieve leaves behind. Signature selection by decoherence, running on your GPU at 60 fps.

The stage is the same claim. The gasket is not drawn — it is the Gram matrix of four null vectors, evaluated per pixel through the sandwich SXS you already write. Conformal geometry is not a symmetry imposed on a stage; it is the packing that made the stage. Six vertices, no mesh, 96 steps, every pixel solving it from scratch.

And φ is not decoration. A grade ladder closed under the algebra’s own composition requires x⁻¹ + x⁻² = 1 — one positive root. Every constant follows: GRACE_k = φ^−k, κ = 2φ⁻³, W_gen = φ⁻². Turn the clock off with G and the state freezes at the undecayed seed, ‖bivector‖ = √3 exactly.

The same operator’s spin–orbit term produces 2, 8, 20, 28, 50, 82 — six of the seven nuclear magic numbers, including the three a harmonic oscillator cannot reach. It gives 114 where nature gives 126. That is the open edge, and it is stated here rather than buried.

Whatever you compute with, you already split a situation into a here and a rest: a product and an inverse, a quadratic form and the form with the opposite sign. This picture is that split, running.

Not two houses

You were taught two five-generator stories as if they were different subjects. Conformal geometric algebra: take Euclidean 3-space, add origin and infinity, get spheres and inversions. Spacetime algebra: take Minkowski space, get rotors and boosts. Those are supposed to be different houses. This is one house.

The operations have names in the nineteenth century. Inversion, sandwich, Descartes tangency, Cl(3,1), Cl(1,3). Naming them is not the theory. Naming them and stopping is the dodge — treating the work as a collage of Hestenes and Sobczyk so you never have to take the identity.

One null Gram. Two signatures. One inversion. The gasket is that Gram. Grace is the φ-scaling that lets the grades settle.

If you leave thinking “they ran CGA on an STA field,” you left with the dodge. If you leave thinking G is a fade, you missed the operator.

One null Gram

Start with generators that square to nothing: c_i² = 0, w_j² = 0. Their inner products are the whole metric. Cohesive Gram G^c, repulsive G^w = −G^c. Signature is not a convention you pick for the sign of time. It is read off those nulls.

A point, in this reading, has no self-measure. “Here” is empty until it is related. Spacetime is a product, not a stage you put fields on.

Two algebras, one inversion

That Gram has two signs, so it has two Clifford algebras, Cl(3,1) and Cl(1,3). They are not two modes sitting next to a fold. They share Node 0 — one inversion, the throat of a lemniscate. Walk the figure-eight and you have not reflected an object from outside. You have gone through the inversion they share and arrived carrying a sign.

M cycles what you are looking at: Cl(3,1), Cl(1,3), their braid. In one signature the boost bivectors die and three-space locks — that is the world that has when and where. In the other those blades stay rotations; nothing is killed the same way. Physics usually throws that child away or calls it a Wick rotate. Both are on.

Their braid is the third thing that exists because the first two do.

The gasket is the Gram

The cauliflower is not a conformal skin glued onto a spacetime field. Descartes’ tangency condition — the one that generates the Apollonian gasket — is D = I − 2G^c. Tangency is the null of the cohesive form. Nested balls are an orbit of the same isometry group that is the Lorentz group of that form. The foam and the field are one object in two grades of drawing.

Lie-sphere geometry already says a sphere is a null vector and tangency is null separation. That is not an analogy for this picture. It is this picture’s conformal sector.

The Grace operator

The fold is the sandwich you already write. What this picture has that a CGA packing or an STA plot does not is the Grace operator: φ-scaling of the grades. Two characteristic polynomials sit on this Gram. Idempotents: x(x−1)=0 — spectrum locked to {0,1}. Golden: x² − x − 1 = 0 — root φ. Grace is that second root, applied: weight grade k by φ^−k, send it at c·φ^−k/2. One clock — gasket recurrence, writhe bound, spectral gap γ = φ⁻². Without it you have two algebras on one inversion and nothing to contract. The late core — scalar plus a φ⁻¹ remnant of volume — does not exist.

G is the experiment, not a style switch. It gates the whole φ clock: weights, decay, speeds, couplings. The fold stays. What you get is not a rawer field. The grades do not settle. The GPU will not explode. You get mud. That is how you see that Grace is doing work.

What the machine is doing

The fold you watch is the inversion: four spheres sandwich a point, mix the unflipped location with the inverted rest, and remain a 1-vector. That is the sandwich you already write, X ↦ V X Ṽ, staying in 1-vectors. The demo still does it in the CGA embedding you know. That is a picture of the inversion, not a proof that origin and infinity were derived on the GPU. The claim underneath is the stronger one: those two nulls are not extras glued onto spacetime. They are supposed to come out of the same primitives that made the Gram. Grace is the other half of what the machine is doing: the φ clock on those grades. The sandwich without it is not this picture.

If you came from CGA

You add n₀ and n_∞ to Euclidean 3-space so you can talk about spheres, flats, and inversions in one algebra. You are not wrong about the sandwich. You are being asked to stop treating origin and infinity as a modeling kit you attach after 3-space is already there. If the nulls are primitive, CGA is not “Euclidean 3-space plus two extra directions.” It is the conformal sector of a Gram that also has Minkowski children. Your packing, your CAD spheres, your inversive interpolations — those are the gasket of G^c, not a separate chapter from spacetime.

If you came from STA

You picked a signature, Cl(1,3) or Cl(3,1), and you have arguments about which is physics. Hestenes made a house out of one of them. This picture refuses the pick. The two signatures are opposite children of one Gram. Boost cancellation in Cl(3,1) is how three-space becomes speakable; Cl(1,3) is not a Wick rotate and not a mostly-minus convention. It is the other lobe, on the same inversion. If your Dirac operator needed a complex structure or a Wick turn to breathe, that was a request for the missing child.

If you came from PGA, robotics, kinematics

You already treat points as derived and infinity as a distinguished plane. You sandwich motors. You are closer than the “add two nulls” story, because you refused to start with points. The next refusal: infinity is not a Euclidean plane you promoted. It is a null of the same Gram that gives you the Euclidean motions as a subalgebra. Inverse kinematics already inverts a product. This fold is that invert, kept, mixed with the thing you inverted from.

If you came from packing, Soddy, inversive geometry

Descartes, kissing circles, inversive geometry — you were told this is recreational nineteenth century. The identity D = I − 2G^c says the kissing condition is the null cone of the cohesive Gram. The Apollonian group sits inside O(3,1). If you already compute Lorentz isometries, you already own the gasket’s bookkeeping group. Nested spheres in the demo are not texture. They are that orbit.

If you came from rotors and nothing else

You trust one formula: sandwich. Rotations, boosts, translations, inversions — different versors, same pattern. Watch the fold. That is your formula, applied to “here” with “the rest” inverted. Everything else in this note is what that formula is a formula of, once the metric is a Gram of nulls with two signs.

If you came from spinors, Dirac, quantum

A spinor is a module of a Clifford algebra. Two algebras on one inversion means two modules sharing a fold, not a tensor factor you add by hand to make “the environment.” Mixing the unflipped point with its inverted rest is the holographic move without a bulk you glued on. You do not have to believe this is physics. You do have to notice that the operation is already in your sandwich, and the second signature is already in your Wick rotation.

How to watch

7 the gasket. 4 the core. ↑/↓ cycle vantages. M the two algebras and the braid. G the Grace operator. Hit G. The sandwich remains. The grades do not settle. That is the point of the toggle. The late core is what does not die: scalar plus oriented volume. The claim, if it is physics, is that spacetime is a product of nulls, Cl(1,3) is not discarded, and conformal geometry is not a symmetry of a stage — it is the packing of the Gram that made the stage.

The GPU does not prove a cosmos. It runs the inversion with Grace on. The identity is the thing to take or refuse: gasket = Gram, two signatures, one inversion, Grace φ.

Technical notes

There is no mesh of the gasket and no film. The stage is a WebGL 2 fullscreen quad. Every pixel runs a fragment shader: intersect the camera ray with the ball the field occupies, then sphere-trace a signed field. Hit colour is grade energy, not a texture lookup.

At a sample the shader does the sandwich you already write. Four tetrahedral spheres in the usual conformal embedding — origin and infinity included as the CGA pair you know — invert a point, mix the unflipped location with the inverted rest, and stay in 1-vectors. Then it evaluates Cl(3,1), Cl(1,3), or both (the braid). Time, pose, and mode are uniforms. So is Grace: uGrace gates φ^−k weights, decay, speeds, and couplings. The fold is not gated. The HUD clocks are CPU readouts of those same weights, not a second solver.

That is a live picture of the inversion, not a proof that those two nulls were derived on the GPU. Backing-store resolution is capped on retina displays so the march stays interactive. A small volumetric overlay used for chamber glow is skipped on phones; the fold and the two algebras are not.

H note · ↑/↓ cycle vantages · 1–9 jumps · M mode · G Grace · TeX in reader.tex