Author: Jack Kowalski <jack@entropment.com>

Overview

This code defines a deterministic numerical transformation producing a scale-dependent, highly stable floating-point invariant ("fraction"), computed as a ratio of two structured sums involving fractional powers.

The construction is intentionally simple, but it exhibits a nontrivial and repeatable behavior when evaluated in finite-precision arithmetic (IEEE-754 floating point), which is qualitatively different from it's behavior in the real numbers \mathbb R.

The goal of this code is not cryptography, but the exploration of numerical structure arising from:

This makes it suitable as: - a numerical stability probe, - a scale-sensitive fingerprint, - a didactic example of projection-induced structure.

Definition

For a given integer key k and a finite number of rounds N:

Left side:

L (k) = k + \sum k^{\frac{1}{p}}

for p in even-indexed primes

Right side:

R (k) = \sqrt{k} + \sum k^{\frac{1}{p}}

for p in odd-indexed primes

The resulting fraction is:

F (k) = \dfrac{L(k)}{R(k)}

The sums are truncated after N terms.

Behavior in \mathbb R (Idealized Analysis)

In exact real arithmetic:

This convergence does not require analytic number theory or zeta-function machinery; any balanced construction with symmetric fractional powers will exhibit similar asymptotic behavior.

In \mathbb R, the limit exists and is trivial.

Behavior in IEEE-754 Floating Point

In finite-precision arithmetic, the behavior changes qualitatively:

  1. The ratio does not converge to 1.
  2. After a sufficient number of rounds (typically ~1024–2048), F(k) stabilizes at a finite, key-dependent value.
  3. Further iterations no longer change the result (numerical convergence).

This stabilized value is a quasi-attractor induced by projection onto the floating-point lattice.

Key empirical properties:

Rounding-Mode Independence

Changing IEEE-754 rounding modes: - round-to-nearest-even, - toward zero, - toward + \infty, - toward - \infty,

affects the final value only at the level of \approx 1\text{e-}5 to \approx 1\text{e-} 3 relative error, even after thousands of iterations.

This indicates that:

Role of Mantissa Size

Two independent mantissas are relevant:

  1. Mantissa of the key representation.
  2. Mantissa of the floating-point arithmetic.

Important regimes:

Thus, collision behavior is not mysterious -- it is a direct consequence of projection from a higher-resolution space into a lower-resolution one.

Why So Many Iterations?

Iterations serve to:

The required iteration count scales with mantissa size and key magnitude. Stopping earlier yields transient values, not the attractor.

Geometric Intuition

Geometrically, the construction can be viewed as:

The attractor is not a fixed point in \mathbb R, but a fixed orbit under projection.

What This Is NOT

This function does NOT claim:

It should not be labeled or marketed as encryption.

What This IS

This is:

The observed behavior arises from the algebra itself, not from implementation bugs or randomness.

Interpretation

In \mathbb R, the limit exists and is trivial. In floating-point arithmetic, the limit is replaced by a stable projection.

This illustrates a general principle:

Deterministic + finite resolution + projection
⇒ apparent irreversibility without randomness.

Intended Use

Final Note

This code treats floating-point arithmetic as a first-class algebraic structure, not as a flawed approximation of \mathbb R.

If one insists on interpreting it through the lens of ideal real analysis, the behavior appears anomalous.

If one accepts the algebra actually being used, the behavior is expected.

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