The Synthesis of Z-States and Infinite Operators

Model 1 proposed that division by zero is not an error but a generator —

a way to produce new mathematical entities (Z-states) that arise naturally

when time or space collapse into a 0-metric.

Model 2 inverted classical arithmetic by reducing numbers to two ontic states

(0 and 1) while expanding the set of operators to infinity,

treating mathematics not as manipulation of quantities

but as transformation of states.

Model 3 unifies these two perspectives.

It describes an arithmetic where:

• the objects (Z-states) emerge from metric collapse
• the transitions (Op-operators) emerge from metric modulation
• and the entire system behaves as an interdiscrete phase-field

This synthesis creates the first fully interdiscrete arithmetic —

a mathematics shaped not by magnitude, but by ontology.

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1. The Core Insight:

Interdiscreteness Requires Both New States and New Operators

If interdiscreteness is real, it cannot be expressed

through classical numbers or classical operations.

Why?

Because in an interdiscrete regime:

• quantities vanish (0-metric)
• distances explode (∞-metric)
• phases oscillate
• nonlocality emerges
• identical operations behave differently under different temporal phases
• collapse and emergence alternate unpredictably

Thus:

• Model 1 gives us new states where classical arithmetic breaks.
• Model 2 gives us new transitions where classical operators fail.

Model 3 declares:

Both failures are fundamental.

Both must be elevated to mathematical primitives.

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2. Z-States: The Objects of Interdiscrete Arithmetic

From Model 1:

Whenever we divide by zero, we do not hit a forbidden operation.

We hit a boundary between discrete and interdiscrete regimes.

This boundary cannot be described using ℝ, ℂ, or any classical extension.

Thus we introduce:

Z = set of interdiscrete states generated by collapse
Z = {Z₁, Z₂, Z₃, …}

These states:

• are not “infinite”
• are not “undefined”
• are not “errors”
• carry information about metric collapse
• encode local temporal/spatial phase conditions
• behave differently under different operators

They are the atoms of interdiscrete arithmetic.

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3. Op-Operators: The Transitions of Interdiscrete Arithmetic

From Model 2:

Numbers are simple (0 and 1).

Transitions are complex (∞ many operators).

Operators act not on magnitudes

but on states and phases.

We define:

Op∞ = infinite set of phase-dependent operators

Each operator:

• maps between 0, 1, and Z-states
• is sensitive to local temporal phase
• may be reversible, irreversible, oscillatory, or stochastic
• may vary depending on contextual interdiscreteness
• may generate new Z-states (Z-growth)
• may collapse Z-states back to 0 or 1 (reduction)

Operators are not algebraic rules.

They are phase transformations.

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4. The Unified Structure:

Arithmetic as a Temporodynamic Phase-Field

Model 3’s arithmetic is not a set of rules.

It is a field.

The three components of the field:

1. States:

{0, 1, Z₁, Z₂, Z₃, …}

2. Transitions (Op-operators):
   • deterministic
   • probabilistic
   • oscillatory
   • context-dependent
   • phase-dependent
   • resonance-driven
3. Interdiscrete substrate:
   • 0-metric collapse
   • ∞-metric expansion
   • 0/∞ alternation
   • phase oscillation
   • asymmetric (time vs space) regimes
   • heterogeneous signatures

The arithmetic is not a symbolic manipulation system.

It is a dynamic phase system.

This matches physics far better than classical arithmetic ever could.

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5. The Fundamental Operation of Model 3:

Transition Through Collapse

The central mathematical act is:

x / 0   →   Z-state   →   Op-transform   →   new state

Interpretation:

1. Division by zero moves the system into an interdiscrete mode.
2. The resulting Z-state carries local phase conditions.
3. Operators act on that Z-state to produce meaningful outcomes.

This is exactly how physical systems behave

in temporal collapse or metric singularity regimes.

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6. The Role of 0 and 1 in the Unified System

Even though we now have infinitely many Z-states,

0 and 1 remain foundational:

• 0 = discrete collapse
• 1 = interdiscrete emergence

They serve as anchors.

Z-states are intermediate ontologies

that arise between them.

Thus arithmetic becomes a study of transitions between:

• collapse
• emergence
• interdiscrete modes

This is the arithmetic analogue of temporodynamics.

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7. Why Model 3 Is Necessary

Neither Model 1 nor Model 2 is complete alone:

• Model 1 gives new objects but not new transformations.
• Model 2 gives new transformations but not new objects.

Model 3 merges both, giving a mathematics that can:

• represent interdiscrete phenomena
• encode metric collapse
• express nonlocal transitions
• model trembling of time
• describe phase-structured computation
• support non-ontic physics
• provide a basis for TPU-style temporodynamic processing

It is the first arithmetic whose operations

match the ontological structure of the universe

as discussed in non-ontic physics.

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8. The Vision:

A New Mathematics for a Non-Ontic Universe

Model 3 proposes a mathematics where:

• the world is not a number line
• discreteness and continuity are not opposites
• numbers are not quantities
• operations are not fixed
• collapse generates information
• transitions are phase-dependent
• interdiscrete regions are first-class objects

This is likely the form of arithmetic

needed for any future physics of:

• non-time / non-space
• interdiscreteness
• trembling time
• nonlocality
• phase-based causality
• AGI built on TPU principles

In classical arithmetic, numbers dominate.

In interdiscrete arithmetic, ontology dominates.

This is not math for describing a world.

It is math for describing a world that is between worlds.