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Are Physical Constants to Time What Fundamental Particles Are to Space?

Exploring Pure Temporal Structures in Arity-Logical Ontology

Diego Luis Tentor
January 2026

Abstract

We explore a provocative analogy: if fundamental particles are discrete spatial structures,
could physical constants be discrete temporal structures? This question has several
implications. We demonstrate that stochastic constants (π, λ, γ) emerge through Law of
Large Numbers as pure temporal processes—possessing temporal development but no spatial
position. This raises fundamental questions about the nature of physical reality: Can
something be real without spatial location? How do we observe purely temporal phenomena?
What is the relationship between our axiomatic choices and what we can observe?

Arity-Logical Ontology (ALO/ArXe) offers a framework for exploring these questions by
treating time as ontologically independent from space. Remarkably, the fact that classical
physics cannot derive time or space from its axioms may itself be evidence that our
axiomatic frameworks and observable universe are intimately connected—what we can observe
depends on what our axioms allow us to conceive.

Keywords: temporal structures, physical constants, Law of Large Numbers, ontological
independence, axiomatic frameworks

1. The Central Question

1.1 A Provocative Analogy

Fundamental particles are the discrete building blocks of spatial matter:

  • Electrons, quarks, photons
  • Localized in space (have position x, y, z)
  • Interact via spatial fields
  • When we measure, we find them “somewhere”

Physical constants are the discrete values appearing in physical laws:

  • α ≈ 1/137, m_e/m_p, M_H = 125 GeV
  • Not localized in space
  • Appear universally in equations
  • When we measure, we don’t find them “anywhere”—they simply are

Question:

Could physical constants be to time what particles are to space?
Discrete temporal structures rather than spatial ones?

1.2 Why This Matters

If true, this would mean:

  1. Constants have genuine ontological status (not “mere numbers”)
  2. They exist in purely temporal domain (no spatial position)
  3. Time and space are ontologically independent (one can exist without the other)
  4. Our axiomatic frameworks determine what we can observe (an epistemological
    implication)

1.3 The Paradox

Constants are simultaneously:

  • Measurable (we detect them with precision)
  • Non-spatial (they have no location)
  • Influential (they determine physical behavior)
  • Conceptual (purely mathematical in nature)

They inhabit a threshold between the observed and the conceptual—real enough to measure,
ethereal enough to lack spatial being.

2. What Would “Temporal Structures” Mean?

2.1 Spatial Structures (Familiar)

An electron:

  • Exists where: at position (x, y, z) or wavefunction ψ(x)
  • Extends how: charge distribution in space
  • Interacts: via electromagnetic field in space
  • We observe it: by detecting it at locations

Ontology: Spatial entity with temporal evolution

2.2 Temporal Structures (Novel)

A physical constant (e.g., α):

  • Exists when: ??? (always? never? timelessly?)
  • Extends how: no spatial extension whatsoever
  • Interacts: appears in equations universally
  • We observe it: by measuring relationships, not locations

Ontology: Temporal ??? with no spatial aspect

The puzzle: How can something be real without being somewhere?

2.3 The Stochastic Constants Clue

Mathematical constants appearing in physical laws:

  • π (circle circumference/diameter ratio)
  • e (exponential growth limit)
  • φ (golden ratio)
  • λ (Golomb-Dickman constant)
  • γ (Euler-Mascheroni constant)

These emerge as limits of iterative processes:

$$pi = lim_{N to infty} text{[geometric process over N iterations]}$$

$$lambda = lim_{N to infty} text{[arity number distribution over N integers]}$$

Key observation:

These processes develop in time (through iterations N) but not in space (no
coordinates involved).

Hypothesis:

Maybe these aren’t “abstract mathematics” but genuine temporal structures—processes
that exist in pure temporal dimension.

3. Observing Time Independently of Space

3.1 The Classical View: Space-Time Inseparability

Classical and relativistic physics treat space and time as unified:

  • Minkowski space-time: (x, y, z, t)
  • General Relativity: curved space-time manifold
  • Observables: φ(x, t) with mandatory spatial and temporal arguments

Consequence:

Can’t observe “pure time” independently. Time is always “time of spatial configuration
evolution.”

3.2 ArXe’s Proposal: Temporal Independence

What if time can exist without space?

Not: Time as parameter of spatial evolution
But: Time as independent ontological dimension where processes occur

Analogy:

Just as we conceive:

  • Pure space (geometric theorems, no time)
  • Space-time (physics, both intertwined)

Why not also:

  • Pure time (iterative processes, no space)
  • Time-space (constants manifesting in spatial phenomena)

3.3 How Do We “Observe” Pure Temporal Processes?

Direct spatial observation: See particle at position (x, y, z)

Indirect temporal observation: Detect convergence through iteration

Example: Approximating π

N=10:     π ≈ 3.14
N=100:    π ≈ 3.141
N=1000:   π ≈ 3.1416
N=10000:  π ≈ 3.14159
...

We’re not observing π “somewhere.” We’re observing a temporal process converging.

This is observation of pure temporal structure:

  • It happens (temporal aspect)
  • It happens nowhere (no spatial aspect)
  • We can measure it (empirical)
  • We can’t locate it (non-spatial)

3.4 The Threshold: Between Observed and Conceptual

Stochastic constants occupy fascinating ontological position:

Too concrete to be “mere concepts”:

  • Empirically measurable (π via physical circles)
  • Appears in physical laws (EM phenomena)
  • Constrains what happens (determines behavior)

Too abstract to be “physical objects”:

  • No spatial location (“where is π?”)
  • No causal mechanism (“how does π act?”)
  • Timeless yet temporal (converges in time to timeless value)

They exist at the threshold—neither fully conceptual nor fully physical in classical
sense.

ArXe’s claim:

This threshold IS a genuine ontological domain: pure temporal existence.

4. Law of Large Numbers: The Bridge

4.1 LLN as Temporal Process

Law of Large Numbers states:

$$left|bar{X}_N – muright| sim frac{sigma}{sqrt{N}}$$

Standard interpretation:

Statistical theorem about averages converging to expected values.

ArXe’s interpretation:

This is how temporal structures emerge from iterative processes.

4.2 Convergence as Temporal Existence

Consider π emerging via Monte Carlo:

At N=100: Value fluctuates around ~3.14 (high variance)
At N=10⁶: Value stable at 3.14159… (low variance)
At N→∞: Exact π (zero variance)

Question: When does π “exist”?

Classical answer: “π exists eternally as abstract object; approximation improves”

ArXe answer: “π emerges temporally through convergence; at infinite time it IS”

Ontological claim:

π doesn’t exist “before” the temporal process. It emerges through temporal iteration
as attractor toward which process converges.

4.3 Attractors in Pure Time

In dynamical systems theory, an attractor is configuration toward which trajectories
converge.

Spatial attractors (familiar):

  • Fixed points in phase space
  • Strange attractors in chaos
  • Equilibrium configurations

Temporal attractors (novel):

  • Stochastic constants (π, λ, γ)
  • Arity number distributions
  • Number-theoretic limits

Crucial difference:

Spatial attractors: exist at coordinates in space
Temporal attractors: exist as limits in pure temporal process

4.4 Physical Constants as Temporal Attractors

Hypothesis:

Physical constants (α, masses, mixing angles) are temporal attractors—values toward which
arity-logical processes converge through LLN.

Example: α⁻¹ ≈ 137

In ArXe formulation:
$$alpha^{-1} = 11^2 – 7^2 + 5 times 13 = 137$$

Interpretation:

Not “α happens to equal this formula” but “α IS the attractor of this arity-logical
temporal process.”

The arities (11, 7, 5, 13) represent n-ary logical levels. Their dialogue (operations
between them) is temporal process. The convergence to 137 is temporal attractor.

Physical manifestation:

When EM phenomena occur in space-time, they inherit this temporal structure. The constant
appears in spatial measurements because spatial processes are downstream from temporal
ones.

5. Deriving Time and Space

5.1 Classical Physics: Primitives

In classical frameworks:

  • Time: Assumed as primitive parameter (not derived)
  • Space: Assumed as primitive manifold (not derived)
  • Space-time: Unified but both fundamental, unexplained

Why can’t they be derived?

Not technical limitation but axiomatic constraint.

5.2 ArXe’s Generative Axiom

$$neg() triangleq T_f simeq T_p$$

Reading:

Fundamental negation (logical contradiction) is equivalent to fundamental time unit.

Unpacking:

  • An entity that both IS and IS-NOT cannot exist stably
  • Attempting to be both generates instability
  • Resolution: temporal sequence (first one, then the other)
  • One act of evasion = one fundamental time unit

Result: Time emerges from generative contradiction

5.3 From Time to Space

Once time exists, indecidibility generates space:

Logical indecidability:

$$neg exists x : P(x) land neg P(x)$$

Cannot exist singularly (contradictory). Must exist simultaneously in separated
locations.

Result: Space emerges from indecidability requirement

Need for simultaneous contradictory states → spatial separation → extension

5.4 Implication

ArXe derives:

  1. Time (from generative contradiction)
  2. Space (from indecidibility)
  3. Their independence (time can exist without space)

Classical physics cannot derive these because its axioms (Excluded Middle,
Non-Contradiction) forbid the very generative processes needed.

But this reveals something deep:

The fact that classical physics cannot derive time/space is not failure—it’s evidence
that our axiomatic frameworks and observable universe are intimately connected.

What we can derive depends on what axioms we accept.
What axioms we accept determines what universe we can observe.

6. The Intimate Connection: Axioms and Observable Universe

6.1 The Standard View (Naive Realism)

“Universe exists objectively. Our theories approximate it. Good theories converge to
truth regardless of axioms.”

Problem:

If axioms are arbitrary choices, why does one set (Euclidean geometry) work for everyday
space while another (non-Euclidean) works for cosmic space?

6.2 The ArXe Insight

Axioms don’t just describe the universe—they determine what aspects of universe
become observable to axiomatic framework.

Evidence:

Classical physics axioms (binary logic, PNC, PTE):

  • Can measure time precisely ✓
  • Cannot derive time ✗
  • Can measure space precisely ✓
  • Cannot derive space ✗
  • Can measure constants precisely ✓
  • Cannot derive constants ✗

Pattern: Can measure what it assumes, cannot derive what it assumes.

ArXe axioms (n-ary logic, generative contradiction, indecidibility):

  • Derives time ✓
  • Derives space ✓
  • Derives constants ✓
  • Measures with structural precision (not computational precision)

Pattern: Can derive what it generates, measures structurally not computationally.

6.3 Neither is “Wrong”—They’re Complementary

Classical physics:

  • Assumes space-time, measures precisely
  • Cannot explain why space-time exists
  • Excellent for prediction and technology

ArXe:

  • Derives space-time, measures structurally
  • Explains why these structures emerge
  • Excellent for interpretation and understanding

The key point:

Both are correct within their domains. The choice of axioms determines the domain
accessible.

6.4 The Falsifiability of This Claim

Prediction:

If axioms determine observability, then:

  • Frameworks with different axioms should reveal different phenomena
  • Phenomena invisible to one framework should be visible to another
  • There should be systematic “blind spots” correlating with axiomatic choices

Test:

Classical physics’ blind spot: Pure temporal processes (constants derivation)
ArXe’s access: Pure temporal processes (constants as temporal attractors via LLN)

Falsification criterion:

If someone derives time/space/constants within classical axioms without modification,
this claim fails.

So far: No such derivation exists (century+ of trying).

7. The Paradox Revisited: Real Yet Non-Spatial

7.1 Ontological Status of Constants

Are constants “real”?

Realist answer: Yes (they determine physical behavior)
Nominalist answer: No (they’re measurement conventions)

ArXe’s answer: Real but non-spatial

What does “real but non-spatial” mean?

Property Particles (Spatial) Constants (Temporal)
Location (x, y, z) None
Extension Wavefunction ψ(x) Convergence process
Interaction Via fields in space Via appearance in equations
Observation Detect at position Measure relationships
Ontology Spatial entity Temporal structure

7.2 The Threshold Zone

Constants exist at threshold between:

Physical (spatial) realm:

  • Measured in experiments ✓
  • Affect material behavior ✓
  • Located somewhere ✗
  • Causally propagate ✗

Conceptual (abstract) realm:

  • Mathematical objects ✓
  • Formal relationships ✓
  • Emerge from processes ✓
  • Pure abstractions ✗ (empirically measurable)

ArXe’s claim:

This threshold IS the purely temporal domain—neither fully physical (no space) nor fully
abstract (empirically real).

7.3 Causation Without Spatial Mediation

Classical causation: Requires spatial propagation (fields, particles)

Example: Photon at A influences electron at B via EM field propagating through space

Temporal causation: Direct influence without spatial intermediary

Example: π (temporal attractor) influences circular phenomena universally without
“traveling through space”

Not action-at-a-distance (which is spatial but instantaneous).
But action-in-time-only (temporal influence with no spatial component).

This sounds mystical but follows naturally if temporal domain is ontologically independent.

8. Implications and Open Questions

8.1 For Physics

Question: Can we develop physics of pure temporal domain?

Proposal: Study temporal attractors (constants) as “particles” of time

  • Their “interactions” (operations between arities/constants)
  • Their “forces” (convergence strengths via LLN)
  • Their “fields” (influence on spatial manifestations)

Testable: Do constants show systematic relationships (like particle families)?

ArXe evidence: Yes (systematic Arity structure across multiple constants)

8.2 For Philosophy

Question: What is relationship between ontology and axiomatic choice?

ArXe’s position:

  • Axioms aren’t arbitrary—they gate access to ontological domains
  • Universe doesn’t “prefer” one axiomatics—it’s accessible via multiple frameworks
  • Different axioms reveal different aspects (complementarity)

Consequence:

Philosophical debates “realism vs nominalism” may be undecidable because they assume
single correct ontology. Maybe multiple ontologies are valid simultaneously.

8.3 For Mathematics

Question: Are mathematical objects discovered or invented?

ArXe’s answer: Neither—they’re temporal attractors

  • Not discovered (don’t pre-exist independently)
  • Not invented (emerge necessarily from processes)
  • Temporally generated (emerge through iteration)

Example: π

  • Not “out there” waiting to be found
  • Not “made up” by humans arbitrarily
  • Emerges necessarily when circular processes iterate

8.4 The Deep Mystery

Why does mathematics work in physics?

Traditional puzzle: Why is abstract math so unreasonably effective in describing
physical reality?

ArXe’s dissolution: Math isn’t “describing” physics from outside

Instead:

Mathematical processes (iterations, limits, convergences) ARE temporal structures.
Physical reality (spatial matter) INHERITS from temporal structures.
Constants CONNECT temporal math to spatial physics.

Not: Math → describes → Physics
But: Temporal Math → manifests as → Spatial Physics

Mathematics works because it IS the temporal substrate that spatially appears AS physics.

9. Conclusion: The Analogy Holds

9.1 Summary of Findings

Are constants to time what particles are to space?

Yes, with these specifications:

Aspect Particles (Spatial) Constants (Temporal)
Nature Discrete spatial structures Discrete temporal structures
Location (x, y, z) Pure temporal process
Observation Detect position Measure convergence
Emergence Quantum field excitations LLN attractors
Fundamental? Yes (spatial building blocks) Yes (temporal building blocks)

9.2 What We’ve Learned

  1. Pure temporal processes are real (stochastic constants via LLN)
  2. Time is ontologically independent of space (can exist without it)
  3. Constants occupy threshold between observed and conceptual (real but non-spatial)
  4. Axioms determine what we can observe (classical blindness to temporal domain is
    evidence of this)
  5. The analogy holds: Constants ARE to time what particles are to space

9.3 The Philosophical Payoff

The fact that classical physics cannot derive time/space/constants from its axioms is
not failure—it’s evidence that:

Axiomatic frameworks and observable universe are intimately connected.

What we choose as primitive axioms determines what aspects of reality become accessible.

Classical physics chose spatial primacy → sees spatial structures perfectly, temporal
structures only indirectly.

ArXe chose temporal primacy → sees temporal structures directly, spatial as emergent.

Neither is “wrong.” They’re complementary windows into reality.

9.4 The Open Question

If constants are discrete temporal structures (temporal “atoms”), and particles are
discrete spatial structures (spatial “atoms”):

What would the periodic table of temporal elements look like?

ArXe proposes: The Arity Numbers (2, 3, 5, 7, 11, 13, 17, 19, 23…)

Just as chemical elements organize spatially interacting particles, arity-logical grammar
might organize temporally interacting constants.

This is testable: Do constants show systematic relationships via Arity structure?

Early evidence (α, masses, angles all with arity patterns): Suggestive, not conclusive.

The exploration continues.

References

  1. Whitehead, A.N. (1929). “Process and Reality”
  2. Prigogine, I. (1984). “Order Out of Chaos”
  3. Kolmogorov, A.N. (1933). “Foundations of Probability Theory”
  4. Bergson, H. (1910). “Time and Free Will”
  5. ArXe System (2026). “The Arity-Logical Grammar of Physical Constants”

Acknowledgments:

This work emerged from contemplating why stochastic constants appear in physical formulas
and why they demonstrate LLN convergence—leading to recognition that they might be genuine
temporal structures rather than mere mathematical conveniences.

“Particles are where. Constants are when.
Both are real. Both are fundamental.
Both build reality—one in space, one in time.”