Construct the object
Build the six marked points and selected arc from the stated parameters.
Example: a = 1, one-arcsecond seed.
Six labeled marks and the kernel export.
Marked geometry and its calculations
From a marked shape to reconstructible calculations
Maps of Vectors included. OMEGA NAUGHT starts with a precisely specified geometric construction. Explore how its measurements enter arithmetic, how recorded remainders recover an integer trajectory, and how selected samples reconstruct averages and fields. Every tool links to its assumptions, mathematical sources, and verification records.
Canonical one-arcsecond object. The inset magnifies the seed; it is not the drawing scale.
One result
Rounding merges possible inputs. The stored remainders identify the original trajectory. Starting from 399, the integer orbit is 399, 135, 45, 15, 5, 1, 0. The first remainder is 639958766566422. This is the integer part of the full walkthrough.
Four tools
Recover a record
Reconstruct an integer trajectory from its terminal state and ordered remainders.
Example: seed 399, first remainder 639958766566422.
Orbit 399, 135, 45, 15, 5, 1, 0.
Average over a sphere
Apply a specified sample rule to polynomial data and inspect its degree limit.
Example: x²y²z².
Continuous mean 1/105; twelve-node sample 0.
Recover an interior field
Reconstruct the declared field family from boundary measurements.
Example: u = 2 + 3x + x² − y², probe J = −11/300.
Twelve samples miss β; the extra sample recovers it.
Evidence
Current source works: 7 preprints and 1 report. Not peer reviewed. Independent third-party reproduction: none recorded. A passing checker is not a global project certificate.
Further work
Six-mark sampling
Cross-ratio, recovery, and a degree-four limit.
Geometric transport
The family that scales N by c² and preserves J.
Drift experiment
A declared numerical experiment, not a theorem.
Research papers
Eight source works, with their own dates and DOIs.