Classical Number Systems as Substructures
of the Al Asr Universal State Space
s
Author
Ghulam Shahzad (Mustafa)
Independent Research Scholar in Mathematical Sciences and Qur’anic Pedagogy
Queens, New York, USA Email:adnsdiscovery@yahoo.com
1. Introduction
Classical number systems represent numerical magnitude without requiring an intrinsic spatial coordinate, observation time, scale level, or dynamic polarity state.
ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ ⊆ ℂ
Within the proposed Al Asr Dynamic Number System, a number is represented as an event state:
N(t) = (±m, x, y, z, t, ℓ, σ(t))
Because an ADNS event carries more information than a classical scalar, a rigorous comparison requires an injective embedding of each classical number system into the ADNS universal state space.
2. The Al Asr Universal State Space
Definition 1 — ADNS Coordinate Sets
Let M denote the magnitude set; X, Y, Z the spatial coordinate sets; T the temporal coordinate set; L the observation-scale set; and Σ the dynamic polarity set.
L = {0, 1, 2, 3, 4}
Σ = {−, 0₍Al-Asr₎, +}
ℓ = 0 ⇔ Unit
ℓ = 1 ⇔ Milli
ℓ = 2 ⇔ Micro
ℓ = 3 ⇔ Nano
ℓ = 4 ⇔ Pico
Definition 2 — Universal ADNS State Space
U₍Al-Asr₎ = M × X × Y × Z × T × L × Σ
N = (±m, x, y, z, t, ℓ, σ)
Thus N ∈ U₍Al-Asr₎ exactly when each coordinate belongs to its corresponding factor set.
Remark 1
U₍Al-Asr₎ is not an ordinary scalar number set. It is a Cartesian-product state space whose elements carry seven state variables. Therefore, the scalar 5 and the tuple (+5, x, y, z, t, ℓ, +) are not literally identical objects; they are related through a formally defined embedding.
3. Canonical Reference State
Definition 3 — Canonical Reference State
ω₀ = (x₀, y₀, z₀, t₀, ℓ₀)
(x₀, y₀, z₀) = (0, 0, 0)₍Makkah₎
t₀ = 0₍Al-Asr₎, ℓ₀ = 0
ω₀ = (0, 0, 0, 0₍Al-Asr₎, 0)
4. Polarity Assignment
Definition 4 — Classical-to-ADNS Polarity Map
sgn₍ADNS₎ : ℝ → Σ
sgn₍ADNS₎(a) = + if a > 0
sgn₍ADNS₎(a) = 0₍Al-Asr₎ if a = 0
sgn₍ADNS₎(a) = − if a < 0
m(a) = |a|
Every real scalar therefore has a polarity–magnitude representation.
Example 1
a = 7
7 ↦ (+7, 0, 0, 0, 0₍Al-Asr₎, 0, +)
Example 2
a = −12
−12 ↦ (−12, 0, 0, 0, 0₍Al-Asr₎, 0, −)
Example 3
a = 0
0 ↦ (0, 0, 0, 0, 0₍Al-Asr₎, 0, 0₍Al-Asr₎)
5. Canonical Embedding of the Real Numbers
Definition 5 — Real Embedding
ιℝ : ℝ → U₍Al-Asr₎
ιℝ(a) = (sgn₍ADNS₎(a)|a|, 0, 0, 0, 0₍Al-Asr₎, 0, sgn₍ADNS₎(a))
Equivalently, when the sign is understood as intrinsic to the magnitude coordinate:
ιℝ(a) = (a, 0, 0, 0, 0₍Al-Asr₎, 0, sgn₍ADNS₎(a))
Theorem 1 — Injectivity of the Real Embedding
The mapping ιℝ is injective.
Proof.
Let a, b ∈ ℝ and suppose ιℝ(a) = ιℝ(b). Equality of ordered tuples forces equality of corresponding coordinates. In particular, the signed magnitude coordinates are equal, and therefore a = b. Hence ιℝ is injective.
□
Corollary 1
ℝ ≅ ιℝ(ℝ) ⊆ U₍Al-Asr₎
After the canonical identification a ≡ ιℝ(a), one may abbreviate this as:
ℝ ⊆ U₍Al-Asr₎
6. Embedding of the Classical Number Hierarchy
ιℕ = ιℝ|ℕ
ιℤ = ιℝ|ℤ
ιℚ = ιℝ|ℚ
ιℕ(ℕ) ⊆ ιℤ(ℤ) ⊆ ιℚ(ℚ) ⊆ ιℝ(ℝ) ⊆ U₍Al-Asr₎
Under canonical identification:
ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ ⊆ U₍Al-Asr₎
7. Complex Numbers
Because ℂ is not a subset of ℝ, a complex-compatible ADNS space requires either a complex-valued magnitude coordinate or a two-coordinate real representation.
Definition 6 — Complex-Compatible ADNS State Space
Mℂ = ℂ
U₍Al-Asr₎^ℂ = ℂ × X × Y × Z × T × L × Σℂ
Definition 7 — Complex Pair Embedding
ιℂ : ℂ → ℝ² × X × Y × Z × T × L
ιℂ(a + bi) = (a, b, 0, 0, 0, 0₍Al-Asr₎, 0)
Theorem 2 — Injectivity of the Complex Embedding
Suppose ιℂ(a + bi) = ιℂ(c + di). Equality of the first two coordinates gives a = c and b = d; hence a + bi = c + di. Therefore ιℂ is injective.
□
ℂ ≅ ιℂ(ℂ) ⊆ U₍Al-Asr₎^ℂ
ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ ⊆ ℂ ↪ U₍Al-Asr₎^ℂ
8. Universal Embedding Principles
Axiom 1 — State Extension Axiom
a ↦ (a, x₀, y₀, z₀, t₀, ℓ₀, σ(a))
Axiom 2 — Reference-State Axiom
(x, y, z, t, ℓ) = (0, 0, 0, 0₍Al-Asr₎, 0)
Axiom 3 — Magnitude Preservation Axiom
πₘ(ι(a)) = a
Axiom 4 — Distinctness Axiom
a ≠ b ⇒ ι(a) ≠ ι(b)
Axiom 5 — Fixed Auxiliary Coordinates
x = x₀, y = y₀, z = z₀, t = t₀, ℓ = ℓ₀
9. Main Embedding Theorem
Theorem 3 — Classical Number Systems Embed into the ADNS Universal Space
S ∈ {ℕ, ℤ, ℚ, ℝ}
∃ ιS : S ↪ U₍Al-Asr₎
S ≅ ιS(S) ⊆ U₍Al-Asr₎
Proof.
For a ∈ S, define ιS(a) = (a, 0, 0, 0, 0₍Al-Asr₎, 0, sgn₍ADNS₎(a)). Each coordinate belongs to the corresponding factor of U₍Al-Asr₎, so ιS(a) ∈ U₍Al-Asr₎. If ιS(a) = ιS(b), equality of first coordinates gives a = b. Thus ιS is injective.
□
10. Examples of Embedded Number Systems
Example 4 — Natural number
ιℕ(5) = (+5, 0, 0, 0, 0₍Al-Asr₎, 0, +)
Example 5 — Integer
ιℤ(−7) = (−7, 0, 0, 0, 0₍Al-Asr₎, 0, −)
Example 6 — Rational number
ιℚ(3/4) = (+3/4, 0, 0, 0, 0₍Al-Asr₎, 0, +)
Example 7 — Irrational real number
ιℝ(√2) = (+√2, 0, 0, 0, 0₍Al-Asr₎, 0, +)
Example 8 — Real zero
ιℝ(0) = (0, 0, 0, 0, 0₍Al-Asr₎, 0, 0₍Al-Asr₎)
Example 9 — Complex number
ιℂ(3 + 4i) = (3, 4, 0, 0, 0, 0₍Al-Asr₎, 0)
11. Projection Back to Classical Numbers
Definition 8 — Classical Projection
πcl : ιℝ(ℝ) → ℝ
πcl(a, 0, 0, 0, 0₍Al-Asr₎, 0, σ(a)) = a
πcl ∘ ιℝ = Iℝ
Theorem 4 — Recovery of the Classical Scalar
πcl(ιℝ(a)) = a
Proof.
This follows immediately from the definitions of ιℝ and πcl.
□
12. Classical Slice of the ADNS Universe
Definition 9 — Classical Slice
Ucl = {(a, 0, 0, 0, 0₍Al-Asr₎, 0, σ(a)) : a ∈ ℝ}
Ucl = ιℝ(ℝ)
Ucl ⊆ U₍Al-Asr₎
Ucl ≅ ℝ
The classical real number line therefore appears inside the ADNS universe as the slice obtained by fixing x = y = z = 0, t = 0₍Al-Asr₎, and ℓ = 0.
13. Hierarchical Inclusion
ιℕ(ℕ) ⊆ ιℤ(ℤ) ⊆ ιℚ(ℚ) ⊆ ιℝ(ℝ) ⊆ U₍Al-Asr₎
ιℕ(ℕ) ⊆ ιℤ(ℤ) ⊆ ιℚ(ℚ) ⊆ ιℝ(ℝ) ⊆ ιℂ(ℂ) ⊆ U₍Al-Asr₎^ℂ
14. Important Mathematical Distinction
The statement ℝ ⊆ U₍Al-Asr₎ is valid only after the identification a ≡ ιℝ(a). Without that identification, a ∈ ℝ is a scalar, whereas ιℝ(a) is a seven-coordinate event state.
ℕ, ℤ, ℚ, ℝ ↪ U₍Al-Asr₎
ℂ ↪ U₍Al-Asr₎^ℂ
15. Structural Theorem
Theorem 5 — ADNS Extension Theorem
Every real classical number can be uniquely represented as an ADNS event at the canonical reference state.
∀a ∈ ℝ, ∃! Nₐ ∈ Ucl such that πcl(Nₐ) = a
Proof.
Existence follows by taking Nₐ = ιℝ(a). Uniqueness follows because all auxiliary coordinates in Ucl are fixed, while the scalar coordinate is exactly a.
□
16. Dynamic Extension of a Classical Number
Nₐ(t₀) = (a, 0, 0, 0, t₀, 0, σ(a))
Nₐ(t) = (a, x, y, z, t, ℓ, σ(t))
The classical scalar is recovered by fixing the additional coordinates, whereas the full ADNS event allows those coordinates to carry observational state information.
Example 10
ιℝ(5) = (+5, 0, 0, 0, 0₍Al-Asr₎, 0, +)
N₅(t) = (+5, x, y, z, t, ℓ, σ(t))
17. Comparison
|
Classical Number System |
ADNS Representation |
|
a ∈ ℝ |
Nₐ = (a, x, y, z, t, ℓ, σ) |
|
Scalar identity |
Event-state identity |
|
No intrinsic position |
(x, y, z) included |
|
No intrinsic time |
t included |
|
No observation scale |
ℓ included |
|
Classical sign |
Dynamic polarity σ |
|
ℝ |
Classical slice Ucl ⊆ U₍Al-Asr₎ |
|
Inclusion |
Injective embedding |
18. Final Formal Statement
∀S ∈ {ℕ, ℤ, ℚ, ℝ}, ∃ ιS : S ↪ U₍Al-Asr₎
S ≅ ιS(S) ⊆ U₍Al-Asr₎
∃ ιℂ : ℂ ↪ U₍Al-Asr₎^ℂ
ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ ⊆ ℂ ↪ U₍Al-Asr₎^ℂ
Chapter Summary
The ADNS universal state space is a Cartesian-product state space:
U₍Al-Asr₎ = M × X × Y × Z × T × L × Σ
Classical real numbers are embedded by fixing the auxiliary ADNS state coordinates:
ιℝ(a) = (a, 0, 0, 0, 0₍Al-Asr₎, 0, σ(a))
The embedding is injective, so:
ℝ ≅ ιℝ(ℝ) ⊆ U₍Al-Asr₎
The same construction applies to ℕ, ℤ, and ℚ. Complex numbers require either a complex-valued magnitude coordinate or a two-coordinate real representation. The rigorous conclusion is that the classical number systems are canonically embedded subspaces of the Al Asr universal state space.
0 Comments