Xzenoverse Industries
Intelligence you
can stake a
mission on.
We did not teach it the laws. We built it out of them, so it does not stop where our measurements do.
For problems where an approximate answer is not an answer.
01The law drifts
One patch,
two updates.
The same states, the same flow, the same step. Liouville lets the patch take any shape and forbids its area to change. Only one update knows that.
- Area, encoded
- 1.000000
- Area, fitted
- 1.000
- Energy drift, encoded
- 0
- Energy drift, fitted
- 0.000
Step 0 of 652 · Hairer, Lubich & Wanner, Springer (2006)
A guarantee that depends on training going well is not a guarantee.
02The difference
Four ways to
hold a law.
- BaselineConventionalno guarantee
- Level 1Physics-encodedexact
- Level 2Categoricalcomposes
- Level 3Sheaf-theoreticor refuses
- Level 4Topos-theoretictrue where
Conventional no guarantee
The law enters as a penalty. Inside the training window the fit is excellent. Beyond it nothing holds the law, and the same model leaves it.
- Error inside the window
- 0
- Error where you are
- 0
ℒ = ℒdata + λℒphysics · Raissi, Perdikaris & Karniadakis (2019)
Physics-encoded exact
Move the body, then predict. Or predict, then move the answer. The encoded map lands both routes on one body. The learned map lands them on two.
- Residual, encoded
- 0
- Residual, learned
- 0
Rotation · Φ(πgx) = π′gΦ(x) · Cohen & Welling, ICML (2016)
Categorical composes
Most systems hold a property in each stage and lose it in the assembly. Three maps in turn: where each keeps area exactly, so does the whole.
- Area after three, exact stages
- 1.000000
- After three, each off by 0.5%
- 1.015
F(g ∘ f) = F(g) ∘ F(f) · Mac Lane, Categories for the Working Mathematician (1971)
Sheaf-theoretic or refuses
Four sources each see part of a ring. A constant disagreement is a bias and is absorbed. One that varies across its overlap cannot be, and that overlap is named.
- Class round the ring
- 0
- Verdict
- glued
Ȟ¹(𝒰, ℱ) = ker δ¹ / im δ⁰ · Leray (1945) · Bredon, Sheaf Theory (1997)
Topos-theoretic true where
A truth value becomes an extent: the region on which a statement holds. Remove one point from it, negate twice, and the point comes back.
- ⟦¬¬P⟧ ∖ ⟦P⟧
- ∅
⟦P⟧ ∈ Ω, ¬¬P ≠ P · Mac Lane & Moerdijk, Sheaves in Geometry and Logic (1994)
03Invariants
Glue the edges
and the
space
starts counting.
- 01Flat patch
- 02Möbius strip
- 03Torus
- 04Sphere
- 05Cross-cap, ℝP²
- 06Knotted torus
- 07Klein bottle
- 08Three-torus, T³
Flat patch
A Euclidean patch, and the space a conventional network represents in. Every loop drawn on it shrinks to a point, so there is no cycle to preserve and nothing here that a construction could carry exactly.
Möbius strip
One side and one edge. Carry a frame once around and it returns reversed, so no consistent global orientation exists. Follow the wiring across the seam and it comes back mirrored.
Torus
Genus one, and the first surface here with cycles that cannot be shrunk to a point. Two of them, which is what a first Betti number of two counts, and what a cohomology of a cover is built to find.
Sphere
The simplest closed orientable surface, and the one every conservation argument quietly assumes when it says closed system. Closed, but still nothing a loop can catch on.
Cross-cap, ℝP²
An immersion of the real projective plane. Odd Euler characteristic, which no closed orientable surface can have, and a first Betti number of zero even so. The two invariants are independent.
Knotted torus
The same surface as the torus, tied in a trefoil. Every number in this panel is unchanged, because these invariants belong to the space and not to the way it happens to sit in the room.
Klein bottle
Closed and non-orientable. It cannot be embedded in three dimensions at all, so what is drawn is an immersion, and the crossing you can see is an artefact of the room rather than of the surface.
Three-torus, T³
One dimension up. A cube with each face identified with the one opposite, so the network wraps in three directions rather than two and nothing on it has an edge. Leave through any face and you arrive through the other side.
- Euler characteristic
- χ = 1
- First Betti number
- b₁ = 0
- Orientable
- yes
Counted from the gluing: 1519 − 2958 + 1440 = 1
04Architectures
Two of them.
Both validated.
Laboratory validated, TRL 4
05Deployment
Portable by default.
Scalable by demand.
The same construction runs across the whole range of hardware. What must not be handed away is the guarantee, not the compute.
07What we stand for
How the work
is done.
- 01Correctness as architectureA guarantee that depends on training going well is not a guarantee.
- 02Rigor that tries to failEvery claim earns its place by surviving an adversary built to break it.
- 03Quiet executionUnderstood by five people who can check the work, rather than admired by five thousand who cannot.
- 04The long horizonIntegrity held by structure, not by any one person.
We are not building a model that is usually right. We are building one that cannot be wrong in the way that matters, and being honest about everything else it still cannot do.
Agnesh Pandey · Founder and Chief Executive Officer
08Contact
While you read,
it held.
0 steps since you arrived. The encoded system has kept its energy to 0 the whole way. The fitted one is still gaining energy on its way to the rim of its well.The fitted one has left the well 0 times, the first at step 0.