Mathematical foundations and protected technologies for the reliability of critical systems.

Three answers to one question: how do critical systems keep running?

On 21 September 1997, a single division by zero in a shipboard program brought down the propulsion network of the USS Yorktown for more than a day.

Nearly three decades later, unhandled arithmetic edge cases still crash flight systems, control loops, and financial software. FDNLTECH addresses this class of failure at its root: mathematically, then industrially.

An exact algebraic answer to division by zero.

In classical fields, division by zero is undefined. In IEEE-754 floating-point arithmetic, it produces ±∞ or NaN — values that silently propagate through computations and corrupt results. Existing algebraic alternatives (Wheels, Meadows) resolve the issue by introducing new elements or altering the underlying structure.

CDS takes a different, minimal path. It is defined by two axioms:

• CDS-Z∀a  ·  a ⊘ 0 = a
• CDS-R∀a, ∀b ≠ 0  ·  (a ⊘ b) · b = a

Division by a non-zero divisor behaves classically. Division by zero returns the numerator — exactly, deterministically, and without introducing new elements.

Uniqueness

Any structure satisfying CDS-Z and CDS-R forces a ⊘ 0 = a. The definition is not a convention among others; it is the unique solution under these axioms.

Consistency

CDS introduces no contradiction with the ring axioms. The classical contradiction relies on applying the inverse identity for all b; CDS restricts it to b ≠ 0.

Minimality

Unlike Wheels or Meadows, CDS adds no new elements and modifies no existing operation other than the single undefined case. It does not replace existing structures — it fills a precise gap.

CDS modifies only the exact operation of division by zero. Classical analysis — limits, convergence, continuity, differentiation, integration — remains entirely intact.

CDS is the mathematical foundation of FDNL-ZE, our patent-pending technology for securing program execution.

Deterministic continuity for critical software.

A division by zero in running software typically produces one of three outcomes: a program crash, an unhandled exception, or the silent propagation of NaN and infinity values through subsequent computations. In critical environments, none of these outcomes is acceptable. Execution continuity must be guaranteed.

FDNL-ZE secures program execution through a controlled substitution mechanism: when a zero divisor is detected and the security mechanism is active, the division returns the numerator value, allowing the program to continue deterministically. When the mechanism is disabled, standard arithmetic behavior is preserved. The mechanism is activatable and deactivatable — via control register, configuration variable, dedicated instruction, or compilation option — so engineers retain full control over where and when it applies.

Software library

A secured division function integrated into a software library, called automatically when a division is executed — including by interception of standard division symbols at load time, with no modification of the program’s executable code.

Compiler

Detection of every division at compile time, transformed into an equivalent guarded sequence.

Hardware

A computing device with a zero-divisor comparator, a control register with an enable bit, and an output selector — with optional logging of every zero-divisor occurrence for audit and diagnostics.

FDNL-ZE implements, at the engineering level, the Conservative Division Structure — a proven algebraic foundation. Technical evaluation is available to qualified industrial partners and laboratories under NDA.