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ni-number  

High-precision computation of the Ni constant (η_ν) — the quantum energy scattering constant — in Rust, with multiple selectable backends.

η_ν = 1.88937666040491913115597775087642096081019761538215...

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Concept of the Quantum Energy Scattering Constant (Ni Number)

Author of the hypothesis: NiZaMinius

1. Definition and Fundamental Meaning

The quantum energy scattering constant $\eta_{\nu}$ (Ni Number) is a fundamental mathematical quantity that defines the total amplitude of gauge-field attenuation at sub-Planckian distances.

The value of the Ni Number is strictly fixed mathematically and is approximately equal to 1.8893766...

Modern physics postulates the existence of hidden, compactified spatial dimensions (Calabi–Yau manifolds). This concept asserts that when particles approach each other at distances smaller than the size of an atomic nucleus, a fraction of the virtual bosons (interaction carriers) scatters into that hidden volume. The Ni Number is the exact measure of the energy fraction that leaves the three-dimensional brane.

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2. Mathematical Structure of the Constant

The constant is defined by a series normalized over all Kaluza–Klein spectral modes:

$$\eta_{\nu} = \sum_{n=1}^{\infty} \frac{\pi^n}{n! \cdot 2^{n^2}}$$

The index $n$ (1, 2, 3, ...) does not denote the ordinal number of a hidden dimension (since all dimensions exist simultaneously), but rather the quantum mode number (energy level). Just as a guitar string produces a set of harmonics, field quanta resonate inside a single multidimensional manifold at discrete frequencies. The infinite series sums all possible quantum states of energy leakage.

Each element of the formula has a rigorous justification in differential geometry and string theory:

2.1. Geometry of Volume Resonances (numerator $\pi^n$)

Powers of $\pi$ define the geometric factor of multi-dimensionality. The growing exponent shows how the wavefront area expands as energy penetrates deeper into new coordinates at each successive harmonic.

2.2. Topological Symmetry (factorial $n!$)

The appearance of the factorial is a direct consequence of the differential geometry of complex Kähler manifolds (Calabi–Yau spaces).

To compute the physical volume of such a manifold, one must integrate its fundamental volume form, which is built from the Kähler 2-form ($\omega$). The exact volume formula is:

$$V = \int_{M} \frac{\omega^n}{n!}$$

Due to the anticommutativity of the exterior product (which accounts for axis orientations), multiplication produces a vast number of overlapping coordinate permutations. Mathematics requires dividing the integral by the count of these permutations ($n!$). Thus, the factorial in the Ni Number naturally collapses the redundant symmetries of the compactified space, preventing the volume and leaking energy from growing infinitely.

2.3. Quantum Blocking (base $2$ and exponent $n^2$)

In the physics of continuous processes, damping is typically governed by Euler's number $e$. However, at sub-Planckian scales, space is discrete.

In the matrix models of M-theory (e.g., BFSS), interactions between branes are described by $n \times n$ matrices, where $n$ is the mode quantum number. The matrix size $n^2$ determines the number of degrees of freedom of the quantum channel. Since at the fundamental level quantum information is binary (a qubit: connection present or absent, spin up or down), two is the basis of the state.

The total number of microstates is $2^{n^2}$. Using $e$ would mean applying classical statistics to quantum gravity. The base 2 is the signature of the fundamental informational nature of space. The quadratic growth of connections causes an avalanche-like resistance at high frequencies, ensuring superfast convergence of the series.

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3. Modification of Interaction Laws at Ultra-Small Distances

According to this hypothesis, the classical inverse-square laws (Newton and Coulomb) break down at sub-Planckian distances, as predicted by models with extra dimensions (e.g., Randall–Sundrum theory).

The Ni Number ($\eta_{\nu}$) does not merely halve the force in the macroworld — it acts as a dynamic integral coefficient in the field equation itself, serving as the transition amplitude between the brane (the 3D world) and the bulk (the higher-dimensional space).

The modified force law is:

$$F(r) = \frac{G \cdot m_1 m_2}{r^2} \cdot \frac{1}{1 + \eta_{\nu} \cdot \left(\frac{R_{\mathrm{Planck}}}{r}\right)^d}$$

where $R_{\mathrm{Planck}}$ is the Planck length, $r$ is the distance, and $d$ is the effective number of hidden dimensions.

Physical meaning:

Regime	Physics
$r \gg R_{\mathrm{Planck}}$ (macroworld)	The Planck fraction → 0, denominator → 1, classical law is recovered
$r \to R_{\mathrm{Planck}}$ (microworld)	Multi-dimensional physics activates; η_ν sets the "bandwidth" of the quantum portal; interaction carriers leak into the geometric pores of the manifold, causing a sharp drop in the registered force

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Installation

Add to your Cargo.toml:

# Default — pure Rust, works everywhere, no system dependencies
[dependencies]
ni-number = "0.2"

Backends

ni-number ships three selectable computation backends:

Feature	Backend	System deps	Precision
backend-dashu	pure Rust (default)	none	arbitrary
backend-rug	GNU MPFR	GMP + MPFR	arbitrary
backend-f64	native f64	none	~15 digits

Default (pure Rust, recommended):

[dependencies]
ni-number = "0.2"

Maximum performance via GNU MPFR:

[dependencies]
ni-number = { version = "0.2", default-features = false, features = ["backend-rug"] }

Minimal footprint (embedded / WASM):

[dependencies]
ni-number = { version = "0.2", default-features = false, features = ["backend-f64"] }

System dependencies for backend-rug

Only needed if you explicitly enable features = ["backend-rug"].

Ubuntu / Debian

sudo apt-get install libgmp-dev libmpfr-dev libmpc-dev

macOS (Homebrew)

brew install gmp mpfr libmpc

Windows (MSYS2)

1. Install MSYS2
2. In the MSYS2 MinGW64 terminal:

pacman -S --needed base-devel mingw-w64-x86_64-toolchain m4 make mingw-w64-x86_64-gmp mingw-w64-x86_64-mpfr

3. Add to PATH in Git Bash (~/.bashrc):

export PATH="/c/msys64/mingw64/bin:/c/msys64/usr/bin:$PATH"

4. Switch Rust toolchain:

rustup toolchain install stable-x86_64-pc-windows-gnu
rustup default stable-x86_64-pc-windows-gnu

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Usage

use ni_number::{NI_F64, ni_number_digits, ni_number, bits_for_digits};
use ni_number::backend::NiFloat;

fn main() {
    // Fast f64 constant — no computation, pre-baked (~15 digits)
    println!("η_ν ≈ {:.15}", NI_F64);

    // 100 decimal digits as String — result is cached after first call
    let s = ni_number_digits(100);
    println!("η_ν = {}", s);

    // Arbitrary-precision value for further computation
    let eta = ni_number(bits_for_digits(500));
    println!("{}", eta.to_decimal_string(500));
}

Run the examples

# Basic demo
cargo run --example basic --release

# High-precision benchmark
cargo run --example high_precision --release

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API

Item	Returns	Description
NI_F64	f64	Pre-computed constant at double precision
NI_F32	f32	Pre-computed constant at single precision
NI_50_DIGITS	&str	First 50 decimal digits, static string
ni_number(bits)	backend float	Full arbitrary-precision value, cached
ni_number_digits(n)	String	Decimal string with n digits after the point, cached
ni_series(bits)	NiSeries	Lazy iterator over series terms
bits_for_digits(n)	u32	Bit precision needed for n decimal digits
clear_cache()	()	Free cached values from memory

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Performance

The series converges in fewer than 10 iterations for any practical precision level — viable even on low-end hardware. Results are cached after the first call — subsequent calls at the same precision are instant.

| Time (release build, dashu) |

Digits	dashu (series)	dashu (total)	rug (total)
100	~0.14 ms	~0.14 ms	~0.1 ms
1 000	~0.9 ms	~0.9 ms	~0.6 ms
5 000	~13.5 ms	~13.5 ms	~10.9 ms
10 000	~146 ms	~449 s	~52 ms

*dashu: series only. Formatting 10 000+ digits requires backend-rug for practical use.

Testing

# Default backend (dashu)
cargo test

# With rug backend
cargo test --no-default-features --features backend-rug

# With f64 backend
cargo test --no-default-features --features backend-f64

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License

MIT — see LICENSE.