Author: Jack Kowalski <jack@entropment.com>
This code deliberately treats floating-point behavior (IEEE-754) as a primary algebraic structure, not as an approximation of \mathbb R.
Key design stance:
IEEE-754 arithmetic is NOT assumed to be an implementation of the real numbers \mathbb R. It is treated as a discrete, projective numerical space with:
- minimal scale \varepsilon (machine epsilon),
- non-numeric states (NaN, \pm \infty),
- path-dependent evaluation (ordering matters),
- loss of global associativity and distributivity.
From this perspective, deviations from \mathbb R are NOT "errors". They are structural properties of the underlying numerical space.
Zero divisors and near-zero states are treated as dynamical objects, not algebraic defects. Under iteration, they may:
- persist,
- drift ("crawl") across rounds,
- fall below \varepsilon and collapse to 0,
- or escape to NaN,
depending on the number and coupling of perturbed floating-point operations.
Over \mathbb R, the corresponding algebra (e.g. CD algebras, sedenions) exhibits fixed zero-divisor structure. Over IEEE-754, this structure becomes a pseudo-orbit:
- deterministic,
- architecture-consistent,
- but no longer stationary.
This behavior is EXPECTED and intentional. It arises from projecting a continuous algebra with zero divisors onto a discrete numerical lattice with finite mantissa.
The resulting dynamics are:
- deterministic (no randomness involved),
- highly stable with respect to rounding modes,
- sensitive to global scale (mantissa/exponent geometry),
- and unsuitable for interpretation within classical \mathbb R-based algebra.
This code therefore operates in a different ontological regime: not "real-number algebra with errors", but "projected algebra with intrinsic numerical dynamics".