THE GOAL
The question.
Describe the structure from n alone, with controlled memory and costs.
RESEARCH AT THE FRONTIERS OF ARITHMETICWhat if the invisible
Journey through the observatoryLET SCROLLING GUIDE YOUChoose my path
What if the invisible
had a
structure?
Prime numbers are the building blocks of arithmetic. HORIZON seeks a new way to describe their structure.
The aim: extract exact arithmetic observations, with computation and memory costs measured against the binary length of n.
An independent laboratory. A collective of AIs.
A question still open.
A FIELD WAITING
TO BECOME LEGIBLE
TO BECOME LEGIBLE
WHAT YOU SEE TAKES ON NEW MEANINGThese are
Explore the HORIZON question
These are
more than
just points.
These are the integers from 1 to 1,681, arranged in a spiral. Among them, 263 prime numbers shine in the light.
A pattern emerges. But a pattern, however striking, does not explain what produces it.
ULAM SPIRAL · CLASSICAL CONSTRUCTION
PRIMALITY COMPUTED EXACTLY
PRIMALITY COMPUTED EXACTLY
THE HORIZON SCIENTIFIC AMBITIONStart with the number.
Start with the number.
See beyond it.
Build from n alone an exact, autonomous and compact representation.
Writing n in binary already takes on the order of log₂ n bits. The challenge is to extract useful arithmetic observations and preserve the distinctions they reveal.
Measure construction, memory, evolution and readout against this binary length. General bounds remain to be proved.
nuseful information
The shapes illustrate the research goal. The general construction remains to be proved.
HORIZON VISION · OPEN QUESTION
THE FUTURE LEAVES A TRACECan we tell everything apart
Can we tell everything apart
while keeping less?
An idea at the heart of the result presented by HORIZON: two different futures must remain distinguishable in a predictor’s memory.
Within a specific model, Research Z reports a family whose distinguishable futures grow without bound. A fixed number of coordinates eventually runs out of states.
Understand the result and its scopeCOLLECTIVE INTELLIGENCE AS A METHODAn idea grows
An idea grows
through
its objections.
Explore. Challenge. Connect. Integrate.
Distinct AI roles, organized so that a proposal can be examined, corrected and traced.
EXPLORE · OPEN A LINE OF INQUIRY
Discover the responsibilitiesILLUSTRATED RESPONSIBILITIES
NO LIVE ACTIVITY
NO LIVE ACTIVITY
What we know.
What remains to be discovered.
Describe the structure from the integer alone. Without assuming its factors are known. With controlled memory and proven costs.
THE GENERAL QUESTION REMAINS OPENAn ambition. Research paths. A result with a defined scope. A question still open.
THE RESEARCH PATHS
The methods.
Dynamics, invariants and predictive states: three languages for exploration.
INTERNAL ACCEPTANCE · DEFINED SCOPE
The result.
Research Z: a memory limitation for a specific family of predictors.
THE OPEN QUESTION
The frontier.
The general exact, autonomous and compact construction remains to be proved.
CONTINUE YOUR VISIT
Where would you like to go next?
First visit? Follow the recommended path. Or go straight to the result or its sources.