Register 04 · Geometry spaces.

Twelve marks · the ambient mathematics · same construction · 12 marks · one of the 17 registers on the Backstaff.

This register is one face of the Backstaff evidence system — the held instrument that holds the measurement engine, maps compliance controls, and engraves every evidence record into a register of record. The full reference library lives at /corpus; the alphabetical view at /corpus/glossary; the operator view at /corpus/switchboard.

API path

Try the evidence path in the sandbox.

The Console shows the sealed evidence and lets anyone verify a record; the Docs carry the endpoints behind it.

Open the Console Read the Docs Read the capability briefing

01 ·

Marks.

12 source marks in this register.

Euclidean Plane#01 · geometry

ℝ² · flat geometry

Reference Paper

Hyperbolic Disk#02 · geometry

Poincaré model · negative curvature

Reference Paper

Spherical Space#03 · geometry

S² · positive curvature

Reference Paper

Manifold#04 · geometry

smooth chart · saddle curvature

Reference Paper

Vector Space#05 · geometry

ℝⁿ · linear combinations

Reference Paper

Hilbert Space#06 · geometry

ℓ² · infinite-dim inner product

Reference Paper

Torus#07 · geometry

T² · genus-1 topology

Reference Paper

Fractal#08 · geometry

Sierpinski · self-similarity

Reference Paper

Riemann Surface#09 · geometry

multivalued · branched sheets

Reference Paper

Klein Bottle#10 · geometry

non-orientable · 4D immersion

Reference Paper

Lattice#11 · geometry

ℤⁿ · discrete subgroup

Reference Paper

Tangent Bundle#12 · geometry

TM · fibered velocity space

Reference Paper

API ·

When a regulator asks for proof, hand over records — not assurances.

record the duty · seal the receipt · verify offline

Planisphere records each duty event — an output marked, a disclosure shown, a review made — and seals it into a receipt that verifies offline against our published keys. You send hashes, never content. A record is evidence a third party can check; it is not a certification and not a legal determination.

Open the Console Read the Docs Read the capability briefing

See how a record is checked