Interdiscrete Arithmetic. Model 1

Division by Zero as a Generative Principle

Physicists and mathematicians treat division by zero as a forbidden operation — the single absolute boundary in mathematics.

You may add infinities, take imaginary roots, integrate divergences, quantize vacuum fluctuations —

but you may not divide by zero.

In classical mathematics, the rule makes sense:

zero-distance and zero-duration do not exist.

So operations involving them should not exist either.

But this changes the moment you introduce interdiscreteness.

Interdiscreteness is the structure that exists between the quanta of space and time —

the non-space and non-time without which discreteness makes no sense at all.

If temporal quanta can be separated by regions of zero temporal metric,

and spatial quanta can be separated by regions of zero spatial metric,

then division by zero becomes not a paradox —

but a necessary operator of transitions across the interdiscrete zone.

This is the foundation of Interdiscrete Arithmetic, Model 1.

1. Why division by zero must be legal here

In interdiscrete physics, the fundamental quantities we divide by —

distance and time —

can each take a 0-metric.

distance may collapse to zero
time may collapse to zero
both may collapse simultaneously

And yet physical processes still occur and still produce outcomes.

A formula such as

velocity = distance / time

ceases to be pathological when both numerator and denominator may be 0, or ∞, or oscillating between them.

If the world contains genuine 0-zones, then physics must be able to reason within them.

Thus Model 1 begins with the simplest possible postulate:

0-metric regions require new algebraic objects that arise from division by zero.

Not to “fix” the anomaly — but to formalize it.

2. Introducing the Interdiscrete Quotients

In classical mathematics:

X / 0

is undefined.

In Interdiscrete Arithmetic, Model 1, it becomes the generator of a new entity.

We denote these entities as Z-types — each corresponding to a particular ratio of a measurable quantity to a zero-metric interval.

Examples:

1 / 0 = Z₁
d / 0 = Z_d
E / 0 = Z_E
0 / 0 = Z₀

These are not numbers.

These are states of interdiscrete transition, each representing a different way of entering the 0-metric region.

You do not “evaluate” them into a scalar.

You treat them as operators or structural markers, just as quantum mechanics treats |ψ⟩ not as a number but as a state vector.

Interpretation example

Z₁ is “unit-state entry” into the interdiscrete zone
Z_E is “energy-ranked entry”
Z₀ (from 0/0) is the interdiscrete singular state

Just as imaginary numbers were once “impossible,”

Z-types are the next layer of mathematical necessity.

3. Why Z-types are unavoidable

Because interdiscrete physics requires us to evaluate:

0 distance with finite energy
0 time with finite momentum
0 spacetime separation in entanglement phenomena
0-duration transition in quantum tunneling
0-metric regions at singularities
0-metric micro-zones between cosmic or quantum events

Classical mathematics has no way to encode these states.

Interdiscrete Arithmetic does.

The Z-types are the minimal objects capable of representing:

entering a region with no time
entering a region with no space
entering a region where both collapse
entering a region where metric is phase-dependent

They are not exotic. They are necessary.

4. The Core Axiom of Model 1

Division by zero produces a structured interdiscrete element, not an undefined contradiction.

Formally:

X / 0 → Z_X
0 / 0 → Z₀
∞ / 0 → Z_∞ (maximal expansion state)
0 / ∞ → 0 (collapse)
∞ / ∞ → Z_cycle (oscillatory or phase-ambiguous state)

Each Z-type may have its own internal rules,

just as imaginary numbers have their own multiplication table.

This creates a hierarchy of interdiscrete quotients.

5. What Model 1 does not attempt

It does not create new addition or multiplication rules.
It does not define commutators or operators.
It does not involve infinite operator families.
It does not invert the role of numbers and operations.

Those belong to Model 2.

Model 1 is simpler and narrower:

Its sole purpose is to legalize and formalize division by zero

as the generator of new interdiscrete objects.

It is the arithmetic of entry into the interdiscrete zone.

6. Why Model 1 matters

Because division by zero is not an anomaly of mathematics —

it is an anomaly of classical ontology.

Once you allow time and space to collapse into 0-metric regions,

dividing by zero is simply a transformation rule:

from discrete spacetime into interdiscrete structure,
from measurable continuity into non-metric substrate,
from ontic states into non-ontic transitions.

Interdiscrete Arithmetic, Model 1, is the first attempt to give these transitions:

algebraic form,
structural encoding,
and conceptual legitimacy.

Without this foundation, nothing in interdiscrete physics can be formalized.