THE RESEARCH / PRESERVING WHAT DISTINGUISHES

See the structure.
Without knowing it
in advance.

An integer fits into a few digits. Reaching its arithmetic structure can be far harder.

HORIZON seeks an exact, self-contained and compact representation, constructed from n alone.

↓Approach the problemFROM THE NUMBER TO USEFUL INFORMATION

INTEGERS 1—1,681
THE LUMINOUS POINTS ARE PRIMES

THE STARTING POINT / n, NOTHING ELSE

The answer
must not be hidden
in the input.

A method that assumes the factors of n are already known has shifted part of the problem elsewhere.

The ambition is to construct the useful observation from the integer alone. Hidden factorization, an exhaustive table or memory of uncontrolled size do not meet that goal.

nstructureCONSTRUCT · DISTINGUISH · MEASURE

THREE LANGUAGES FOR THE SAME QUESTION

Change perspective.
Keep the rigor.

01 / DYNAMICS & TWISTED FROBENIUS

Let evolution speak.

Transforming an algebraic object can reveal behavior and symmetries linked to its arithmetic structure. The useful observation still needs to be produced from n alone.

See a concrete example · 697

Take 697, without supplying its factors.

A fixed probe produces four successive states, with determinant 94,812,400.

Their greatest common divisor then reveals a factor: 17.

GCD(697, 94,812,400) = 17

Chosen probe: r = 5, x₀ = 1 + 2T + 2T² + 4T³.

Research A · 17.09.2026 · exact calculation

Explore this language +

Dynamics studies successive transformations. The twisted Frobenius constructions explored by the project modify an algebraic action within a specific framework. This overview does not present a completed general construction.

02 / QUOTIENTS & INVARIANTS

Remove redundancy.
Preserve the difference.

Several descriptions can express the same structure. Grouping them reduces stored information, provided it preserves what distinguishes the relevant cases.

Explore this language +

A quotient identifies objects under an equivalence relation. An invariant preserves a property despite certain changes of description. The work is to specify that equivalence and the observations it preserves.

03 / PREDICTIVE STATES

Distinguish by the future.

Two seemingly similar states can produce different observations tomorrow. Distinguishing them constrains a predictor’s minimum memory.

Explore this language +

Two states are distinguishable when a future evolution separates them through the observation defined in the model. Counting these classes of futures gives a bound on the number of states required.

THE FULL COST

Nothing disappears
behind the scenes.

A method does not become compact by hiding its cost elsewhere.

Initialize

Construct the state from n.

Store

Measure the information retained.

Transform

Count the evolution operations.

Read

Extract the conclusion exactly.

THE CENTRAL CHALLENGE REMAINS OPEN

At the research frontier documented on 18 October 2026, self-contained initialization, exact separating readout and complete cost bounds remain to be established within a declared family.

See the source and date of this assessment ↗

CONTINUE EXPLORING / THE RESULT

The future demands
memory.

Continue to the result

An internal result identifies a limitation within a class of predictors.

See the structure.Without knowing itin advance.

The answermust not be hiddenin the input.

Change perspective.Keep the rigor.

Let evolution speak.

Remove redundancy.Preserve the difference.

Distinguish by the future.

Nothing disappearsbehind the scenes.

Initialize

Store

Transform

Read

The future demandsmemory.

See the structure.
Without knowing it
in advance.

The answer
must not be hidden
in the input.

Change perspective.
Keep the rigor.

Remove redundancy.
Preserve the difference.

Nothing disappears
behind the scenes.

The future demands
memory.