Laegna Numbers – Frequentials and Octavians

1. Hologram vs. fractal numbers

In the usual decimal system, numbers behave like a hologram: each extra digit can radically change the scale, and the same digit can mean very different things depending on where it appears. The length of the number and the size of the value are only loosely related. This is why decimal expansions naturally support fractals: self‑similar patterns across scales, but with no strict synchronization between digit length and magnitude.

Laegna numbers are designed to be non‑fractal in length: the number of digits \(R\) and the total real range \(T\) are tightly coupled. Length and size are not independent; they are synchronized. This makes Laegna numbers more like a structured hologram with fixed layers: each layer (digit) has a precise, predictable role in the global range.

2. Frequential Laegna numbers (base‑4)

2.1. Discrete log–exp synchronization

In the Frequential system, each digit is one of \(I, O, A, E\), giving \(4^R\) real values for a given length \(R\). We denote:

\[ T = 4^R \]

Here, \(R\) is both:

• Digit length (how many Laegna digits), and
• Discrete logarithm of the range: \(\log_4 T = R\).

This gives a clean, integer‑only synchronization between range and length:

\[ T = 4^R,\quad R = \log_4 T \]

No irrational numbers are needed to move between “how long is the number” and “how wide is the real range”. The system behaves like a discrete log–exp pair, with \(R\) and \(T\) as dual coordinates.

2.2. Fourier‑like positioning and simple encodings

Because Frequential numbers are base‑4 and tightly synchronized with range, they map naturally to Fourier‑like encodings. A simple way to see this is to treat each Laegna digit as a small alphabet and encode it into phases or amplitudes.

For example, a simple encoding could map:

\[ I \mapsto 0,\quad O \mapsto 1,\quad A \mapsto 2,\quad E \mapsto 3 \]

Then a length‑\(R\) Frequential number corresponds to a vector in \(\mathbb{Z}_4^R\). If we interpret each position as a phase or frequency component, we can assign a Fourier wavelength or amplitude to each digit block of size 4. Because the range \(T\) and length \(R\) are synchronized, the mapping from “digit position” to “frequency band” is clean and regular.

In contrast, decimal digits are not synchronized with any simple power of 2 or 4. Their fractal behavior makes them excellent for representing arbitrary real numbers, but less ideal for clean, discrete Fourier‑like positioning. Laegna’s Frequential system is built to align with such structures from the start.

3. Octavian Laegna numbers (base‑2)

3.1. Hilbert spaces and dimensional loss

Octavian numbers are base‑2 Laegna numbers built from digits \(O\) and \(A\), with the same special symbols \(W, U, V, \cap\). For a given length \(R\), there are \(2^R\) real values:

\[ T = 2^R \]

In functional analysis and quantum theory, Hilbert spaces (after David Hilbert) describe infinite‑dimensional spaces where projections from higher to lower dimensions inevitably lose information. A lower‑dimensional space cannot fully represent a higher‑dimensional one.

Octavian Laegna numbers can be seen as a digitwise projection of Frequential numbers: base‑4 structure compressed into base‑2. When operations are performed digitwise between Frequential and Octavian representations, the loss of information (or simplification) can be expressed and tracked via the correlation of \(R\) and \(T\).

3.2. Logical structure in Octavian digits

Octavian digits also carry logical meaning. For \(R = 1\):

\[ O \mapsto \text{False} \quad (\text{“Bad takes place”}), \qquad A \mapsto \text{True} \quad (\text{“Good takes place”}) \]

For \(R \ge 2\), the first two digits encode a 2‑bit logical state:

\[ \begin{aligned} OO &\mapsto \text{False–False} &\quad& \text{“Disillusion Bad (I)”} \\ OA &\mapsto \text{False–True} && \text{“Illusion Bad (O)”} \\ AO &\mapsto \text{True–False} && \text{“Real Good (A)”} \\ AA &\mapsto \text{True–True} && \text{“Illusion Good Hope (E)”} \end{aligned} \]

This gives Octavian numbers a natural interpretation in terms of logical Hilbert‑like states, where each number is both a position in a discrete range and a structured logical configuration.

4. Central roles of 0, 1, 2, 4, 16 and infinities

Laegna numbers emphasize a small set of central relations:

• 0 is a quantum point, not an interval.
• ±1 are the first non‑zero intervals from 0.
• 2 is the base of Octavian growth: \(T = 2^R\).
• 4 is the base of Frequential growth: \(T = 4^R\).
• 16 appears as the Laegna “hex” complex table: \(4^2\).
• ±∞ are explicit boundary symbols \(W\) (−∞) and \(\cap\) (+∞), with structured zero variants \(V\) (log, −0) and \(U\) (exp, +0).

These relations make Laegna numbers particularly suitable for:

• Discrete logarithmic reasoning without irrationals.
• Fourier‑like encodings where digit positions map cleanly to frequencies.
• Hilbert‑space‑inspired projections between base‑4 and base‑2 structures.
• Logical interpretations of number states via Octavian digits.

5. Frequential numbers and zero as quantum element

In Frequential Laegna numbers, zero is not “just another number”. It is a zero‑width quantum element between negative and positive intervals. The numbers \(-1\) and \(+1\) are the first intervals from zero, and the special symbols \(V\) and \(U\) represent logarithmic and exponential zero variants (“log” and “exp” in the unsigned scale).

Digits after the comma are interpreted as percentages on the last digit, not as an infinite fractal tail. This keeps the system coherent for advanced mathematics where intuition about coherence and tautology matters more than arbitrary real‑number precision.