Multi-Dimensional Number System Theory
and Unified Dimensional Notation:
Center-Periphery Calculus, Canonical Notation Equivalence,
Infinite-Dimensional Dot Theory Integration, and Fractional Extension
Nobuki Fujimoto (藤本 伸樹)
Independent Researcher, Rei-AIOS Project
ORCID: 0009-0004-6019-9258 | GitHub: fc0web/rei-aios
April 2026 | Companion to Paper 61 (Zero-Centered Symbol Grammar)
Abstract
This paper presents the Multi-Dimensional Number System Theory (MDNST) and its unified notation framework, serving as a companion and extension to Paper 59 (Zero-Centered Symbol Grammar, ZCSG). MDNST defines a weighted center-periphery calculus in which a central value governs surrounding values across multiple computational modes. We formalize four computation modes — weighted additive, multiplicative, harmonic, and exponential — and the directional computation convention (clockwise/counterclockwise). We establish Canonical Notation Equivalence: the four interchangeable notations for ZCSG expressions (visual 0ooo, subscript 0₃, hybrid 0_o3, functional Z(o,3), JSON 0[o:3]) are formally declared equivalent under a single axiom. We integrate the Infinite-Dimensional Dot Theory (−·ⁿ notation) as a complementary system to ZCSG, showing that −·ⁿ captures weighted topological absence while ZCSG captures directional dimensional position. Extensions to fractional dimensions (Hausdorff-compatible), complex dimensions, and composite multi-symbol expressions are formalized. The result is a unified, human-AI-readable notation framework for dimensional computation that is simultaneously mathematically rigorous and computationally efficient.
Keywords: multi-dimensional number system, center-periphery calculus, canonical notation, dimensional equivalence, infinite-dimensional dot theory, Betti numbers, fractional dimension, D-FUMT₈, Rei-AIOS, ZCSG
- Introduction and Motivation
Paper 59 established the Zero-Centered Symbol Grammar (ZCSG) as a positional notation for dimensional encoding, grounded in Nāgārjuna's śūnyatā-of-śūnyatā. The present paper addresses three aspects not covered in Paper 59:
Multi-Dimensional Number System Theory (MDNST): the underlying center-periphery calculus that provides the computational substrate for ZCSG expressions.
Canonical Notation Equivalence: the formal equivalence axiom unifying all surface-level notations for ZCSG expressions (0ooo, 0₃, 0_o3, Z(o,3), 0[o:3]).
Integration with Infinite-Dimensional Dot Theory (−·ⁿ): the complementary notation system that adds Betti-number weighting to dimensional expressions, and its relationship with ZCSG.
Together, Papers 61 and 62 constitute a complete dimensional notation and computation framework, designed to be usable by both humans and AI systems, and grounded simultaneously in Buddhist philosophy (Nāgārjuna), modern topology (Betti numbers, Hausdorff dimension), and the D-FUMT₈ eight-valued logic of Rei-AIOS.
- Multi-Dimensional Number System Theory (MDNST)
2.1 Foundational Structure: Center-Periphery Architecture
MDNST organizes numerical computation around a center value C surrounded by n peripheral values P₁, P₂, ..., Pₙ arranged in positional order (typically 8 positions for 2D, extensible to arbitrary n). Each peripheral value carries a directional weight wᵢ determined by its position relative to C.
V = f(C, P₁, P₂, ..., Pₙ, w₁, w₂, ..., wₙ, mode, direction)
Definition 2.1 (MDNST Core). A Multi-Dimensional Number System computation is a tuple (C, {Pᵢ}, {wᵢ}, mode, direction) where:
C ∈ ℝ: the center value (nucleus of computation)
{Pᵢ}: ordered sequence of peripheral values at positions i = 1..n
{wᵢ} ∈ ℝ: weight assigned to each peripheral position
mode ∈ {W, M, H, E}: computation mode (see §2.2)
direction ∈ {CW, CCW}: clockwise or counterclockwise traversal order
2.2 Four Computation Modes
Mode
Symbol
Formula
Semantic
Weighted Additive
W
V = C + Σᵢ(Pᵢ × wᵢ)
Linear superposition; periphery adds to center
Multiplicative
M
V = C × Πᵢ(Pᵢ^wᵢ)
Exponential influence; periphery scales center
Harmonic
H
V = n / Σᵢ(wᵢ/Pᵢ)
Harmonic mean; balances extreme values
Exponential
E
V = C^(Σᵢ wᵢPᵢ)
Nonlinear; rapid amplification by periphery
2.3 Directional Computation Convention
MDNST specifies traversal direction explicitly, enabling different computational results from the same configuration:
V_CW ≠ V_CCW (in general, for non-symmetric configurations)
This is significant for: ordered data structures, time-asymmetric processes, and causal chains where direction encodes information. In symmetric configurations (palindromic ZCSG expressions), V_CW = V_CCW, corresponding to SELF⟲ invariance.
Direction
Notation
ZCSG analog
D-FUMT₈
Clockwise (CW)
V_CW
0o...o (positive expansion, forward)
INFINITY / BOTH
Counterclockwise (CCW)
V_CCW
o...o0 (negative contraction, reverse)
NEITHER / ZERO
Symmetric (CW=CCW)
V_SYM
o...0...o (palindrome)
SELF⟲
2.4 Connection to ZCSG: MDNST as Computational Substrate
ZCSG (Paper 61) defines the positional encoding of dimensions; MDNST defines the numerical computation within those dimensions. Their relationship:
Layer
Theory
Defines
Example
Notation
ZCSG
Dimensional position and direction
0oo = +2 dimension
Computation
MDNST
Numerical value within that dimension
V = C + Σ(Pᵢ×wᵢ) at depth 2
Logic
D-FUMT₈
Truth value of the dimensional state
INFINITY (positive expansion)
Philosophy
ZCSG §2
Ontological grounding
śūnyatā-of-śūnyatā as origin
MDNST(ZCSG(expression)) = dimensional computation result
- Canonical Notation Equivalence Axiom
3.1 The Five Surface Notations
ZCSG expressions can be written in five interchangeable surface notations, each optimized for a different use context:
Notation
Example (depth 3, type o)
Optimal context
Readability
Visual (repeat)
0ooo
Human handwriting, education, short depths
◎ (intuitive)
Subscript
0₃
Mathematical papers, compact expression
◎ (classic)
Hybrid
0_o3
Programming variables, medium depth
◎ (balanced)
Functional
Z(o,3)
API calls, formal specification
○ (explicit)
JSON / Structured
0[o:3]
Machine parsing, AI processing
◎ (structured)
3.2 The Canonical Notation Equivalence Axiom (CNEA)
Axiom 3.1 (CNEA). For any ZCSG expression E with base symbol S, type-character T, and depth d:
E_visual ≡ E_subscript ≡ E_hybrid ≡ E_functional ≡ E_json
That is:
S[T repeated d times] ≡ Sᵈ ≡ S_Td ≡ S(T,d) ≡ S[T:d]
All five notations refer to the same dimensional object and are interchangeable in all mathematical and computational contexts.
3.3 Multi-type Expressions
When an expression contains multiple symbol types, the canonical forms extend as follows:
Visual
Hybrid
Functional
Dimensional value
0oooxx
0_o3x2
Z(o:3, x:2)
d = 5 (total right symbols)
ooXπ
oo_X1π1
Z(o:-2, X:1, π:1)
d = 0 (SELF⟲: 2 left, 2 right)
oRπ
o_R0π1 (left:o=1, right:π=1)
Composite(o→R→π)
d = 0 (transform SELF⟲)
Note: In composite expressions (e.g., oRπ), the left/right count includes all symbols regardless of type. The center 0 is always implicit when not written.
- Infinite-Dimensional Dot Theory: −·ⁿ Notation
4.1 Overview and Motivation
The Infinite-Dimensional Dot Theory (IDDT), developed within Rei-AIOS at STEP 441, introduces the −·ⁿ notation to encode dimensional absence with explicit Betti-number weighting. While ZCSG encodes the position of a dimension, −·ⁿ encodes the topological weight of absence at that dimension.
Definition 4.1 (Dimensional Absence). The symbol −·ⁿ denotes the absence structure at topological dimension n, with weight given by the n-th Betti number βₙ:
−·ⁿ_βₙ := absence at dimension n with weight βₙ
−·¹ ≠ −·² ≠ −·³ (Dimensional Absence Theorem, DAT)
4.2 Empirical Results: SEED_KERNEL Analysis (500 theories)
−·ⁿ symbol
Topological structure
Betti number βₙ
Weighted value βₙ×n
Contribution
−· (n=0)
Isolated point / connected component
β₀ = 278
278
0.3%
−·· (n=1)
Loop / 1-cycle
β₁ = 1,408
2,816
3.3%
−··· (n=2)
2-cavity / void
β₂ = 27,771
83,313
96.4%
Key finding: β₂ (2-dimensional cavity) accounts for 96.4% of the weighted absence structure of SEED_KERNEL. The knowledge space is dominated by high-dimensional voids, consistent with the Dimensional Absence Theorem (DAT): absence contains approximately 40.57× more information than presence.
4.3 ZCSG vs. −·ⁿ: Complementary Systems
Property
ZCSG (0o, o0, etc.)
IDDT (−·ⁿ notation)
Primary encoding
Dimensional direction (positive/negative)
Topological absence weight
Strength
Intuitive direction, philosophical grounding
Betti-number precision, empirical measurement
Limitation
No direct weight encoding
No directional encoding
Example
0oo = +2 dimensions
−··_₁₄₀₈ = 1-dim loop, weight 1408
D-FUMT₈
Maps to 8 values by position
Maps NEITHER/ZERO by weight
Ideal use
Navigation, philosophy, AI logic
Topology, knowledge-graph analysis
Theorem 4.1 (ZCSG-IDDT Correspondence). For any −·ⁿ expression with Betti number βₙ, the corresponding ZCSG expression has dimensional value d = n, and the weight βₙ provides additional information beyond ZCSG's positional encoding:
−·ⁿ_βₙ ↔ [0 followed by n o's] + weight βₙ
−··_₁₄₀₈ ↔ 0oo + β₁ = 1,408
- Fractional and Complex Dimensional Extension
5.1 Fractional Dimensions (Hausdorff Compatible)
ZCSG naturally extends to fractional dimensional values by allowing real-valued symbol counts. This is compatible with Hausdorff dimension theory for fractal structures:
Definition 5.1 (Fractional ZCSG). For any real number d ≥ 0, the fractional ZCSG expression is:
0o^d := ZCSG expression with dimensional value d ∈ ℝ
Fractal structure
Hausdorff dim.
ZCSG expression
D-FUMT₈
Line segment
d = 1.000
0o¹ = 0o
BOTH
Koch curve
d = 1.2619
0o^1.2619
FLOWING
Sierpiński triangle
d = 1.585
0o^1.585
FLOWING
Menger sponge
d = 2.727
0o^2.727
FLOWING
Space-filling curve
d → 2.000
0o^2 = 0oo
TRUE
Mandelbrot boundary
d = 2.000
0o^2
TRUE
Connection to π: As established in Paper 59 §5, the n-sphere volume formula Vₙ = π^(n/2) / Γ(n/2+1) shows that π accumulates in half-integer steps. For fractional ZCSG:
0o^(n/2) ↔ π^(n/2) dimensional content
Gaussian integral ∫e^(−x²)dx = √π ↔ 0o^(1/2) (half-dimensional boundary)
5.2 Complex Dimensional Extension
Extending ZCSG to complex-valued dimensional indices connects to the zeros of the Riemann zeta function and the MFET (Middle-Value Equilibrium Theorem) of Rei-AIOS:
Definition 5.2 (Complex ZCSG). For d = α + βi ∈ ℂ:
0o^(α+βi) := complex-dimensional ZCSG expression
Re(d) = α → physical/real dimension
Im(d) = β → oscillatory/phase component
The Riemann Hypothesis predicts all non-trivial zeros of ζ(s) lie on α = 1/2:
ζ(1/2 + βi) = 0 ↔ 0o^(1/2 + βi) [proposed MFET connection]
This is speculative but connects ZCSG to the Middle-Value Equilibrium Theorem (MFET), which proposes that NEITHER (the middle D-FUMT₈ value) encodes Riemann zero structure.
- Composite Expression Calculus
6.1 Formal Rules for Multi-Symbol Expressions
When ZCSG expressions combine multiple symbol types (e.g., oRπ, oXπ, ooXπ), the following rules apply:
Rule 6.1 (Dimension Rule). For any composite expression E = L₁L₂...Lₘ · 0 · R₁R₂...Rₙ where Lᵢ, Rⱼ ∈ Σ:
d(E) = n − m (right count minus left count, regardless of symbol type)
Rule 6.2 (SELF⟲ Rule). E is SELF⟲ if and only if E is a palindrome with 0 at center:
E is SELF⟲ ⟺ m = n AND Lᵢ = R_{n+1−i} for all i
Rule 6.3 (Transform SELF⟲). When m = n but Lᵢ ≠ Rᵢ (non-palindromic SELF⟲), the expression encodes a dimensional transformation process:
oRπ: m=1, n=1, d=0, but o≠π → Transform SELF⟲ (FLOWING)
o0o: m=1, n=1, d=0, and o=o → True SELF⟲ (SELF⟲)
6.2 Semantic Interpretation of Key Composite Expressions
Expression
d
SELF⟲ type
Narrative
D-FUMT₈
oRπ
0
Transform SELF⟲
śūnyatā → Ricci flow → perfect geometry (STEP 580 story)
FLOWING
oXπ
0
Transform SELF⟲
emptiness → unknown question → circular structure
FLOWING
ooXπ
−1
None
deep emptiness → unknown → rotation (contraction-dominant)
NEITHER
oXR
0
Transform SELF⟲
emptiness → unknown → geometric discovery
FLOWING
R0R
0
True SELF⟲
Ricci soliton: self-sustaining geometry
SELF⟲
π0π
0
True SELF⟲
standing wave / resonance (bidirectional rotation)
SELF⟲
ooo0ooo
0
True SELF⟲ (depth 3)
depth-3 philosophical self-reference
SELF⟲
6.3 Composition Operator
Definition 6.1 (Sequential Composition ⊳). For two ZCSG expressions A and B, their sequential composition A⊳B denotes the transformation process from A to B:
A ⊳ B := 'the process of transitioning from dimensional state A to dimensional state B'
oRπ = (o) ⊳ R ⊳ (π) — three-stage transformation
This connects ZCSG to the MORPHISM category-theoretic framework of Rei-AIOS, where sequential compositions define morphisms in the category of dimensional states.
- Human-AI Notation Interface
7.1 Design Principles
MDNST and ZCSG were designed with explicit consideration for human-AI communication efficiency:
Principle
Implementation
Benefit
Short depth: aesthetic
0ooo (visual repeat)
Intuitive, poetic, hand-writable
Long depth: efficient
0_o100 (hybrid with number)
No counting errors, compact
Machine parsing: structural
0[o:3] (JSON-like)
Unambiguous, directly parseable
Formal proof: classical
0₃ (subscript)
Compatible with mathematical tradition
API/code: functional
Z(o,3) (function call)
Directly implementable in any language
'Short-distance aesthetic, long-distance structural' — the guiding design rule: visual repeat notation serves human intuition for small depths; hybrid/JSON notation serves computational precision for large depths.
7.2 AI Processing Efficiency
Notation
AI parsing cost
Human readability
Recommended for
0ooo
Count characters (low cost for d≤4)
◎
Education, philosophy, d≤4
0_o3
Split at underscore (O(1))
◎
General purpose, API parameters
Z(o,3)
Parse function syntax (O(1))
○
Code integration, formal specs
0[o:3]
JSON parse (O(1))
◎
AI systems, data structures
0₃
Unicode subscript recognition
◎
Papers, formal math
- Integration with D-FUMT₈ and SEED_KERNEL
8.1 Complete Mapping: MDNST × ZCSG × D-FUMT₈
MDNST mode
ZCSG position
D-FUMT₈ value
Semantic
W (weighted additive)
0o (expansion)
BOTH
Linear growth from center
M (multiplicative)
0oo+ (deep expansion)
INFINITY
Exponential amplification
H (harmonic)
o0o (SELF⟲)
SELF⟲
Balance between extremes
E (exponential)
0o^n (fractional deep)
FLOWING
Nonlinear phase transition
V_CW = V_CCW
Palindrome
SELF⟲
Directional invariance
V_CW ≠ V_CCW
Non-palindrome
FLOWING/NEITHER
Directional asymmetry
8.2 SEED_KERNEL Theoretical Status
The theories introduced across Papers 61 and 62 contribute the following to the SEED_KERNEL:
Theory
Paper
SEED_KERNEL status
Key equation
Zero-Centered Symbol Grammar (ZCSG)
59
New entry
d = n − m
Canonical Notation Equivalence Axiom (CNEA)
60
New entry
0ooo ≡ 0₃ ≡ 0_o3 ≡ Z(o,3) ≡ 0[o:3]
Multi-Dimensional Number System Theory (MDNST)
60
New entry
V = f(C,{Pᵢ},{wᵢ},mode,direction)
ZCSG-IDDT Correspondence
60
New entry
−·ⁿ_βₙ ↔ 0o^n + βₙ weight
Fractional ZCSG
60
New entry
0o^d for d ∈ ℝ, Hausdorff compatible
Transform SELF⟲
60
Refinement of Paper 59
oRπ, oXπ: d=0, L≠R → FLOWING
Composition Operator ⊳
60
New entry
A⊳B = dimensional morphism
-
Future Directions
Lean 4 formalization: CNEA, MDNST computation modes, and ZCSG-IDDT Correspondence all admit direct formalization in Lean 4 / Mathlib4, connecting to the Rei-AIOS Braid Theorem Prover project.
MDNST in higher dimensions: Extending from 8-neighbor (2D) to arbitrary n-neighbor structures, connecting to the super-polyhedron ring method (STEP 567–580).
Riemann Hypothesis connection: The complex ZCSG extension 0o^(1/2+βi) and its proposed connection to MFET warrants formal investigation.
Interactive simulation: Web Audio API and canvas-based visualization of MDNST computation modes and ZCSG dimensional traversal.
Dōgen and Kūkai extensions: Dōgen's 'being-time' (uji) as a temporal ZCSG extension; Kūkai's mandala structures as 2D MDNST configurations. -
Conclusion
This paper has formalized the Multi-Dimensional Number System Theory (MDNST) with its four computation modes and directional convention, the Canonical Notation Equivalence Axiom (CNEA) unifying five interchangeable surface notations, the ZCSG-IDDT Correspondence connecting positional dimensional encoding with Betti-number-weighted topological absence, and Fractional and Complex ZCSG extensions compatible with Hausdorff dimension theory and the Riemann Hypothesis. Together with Paper 59, these results constitute a complete dimensional notation and computation framework that is: mathematically rigorous (grounded in topology, homology, and logic), philosophically deep (connected to Nāgārjuna's śūnyatā through Paper 61), computationally efficient (canonical notation equivalence, O(1) parsing), and human-AI readable (multiple notations optimized for different contexts).
「急がず、ゆっくりと。種は育ちます。」(Without haste, slowly. Seeds grow.)
References
[1] Fujimoto, N. (2026). Zero-Centered Symbol Grammar (Paper 61). Zenodo. ORCID: 0009-0004-6019-9258.
[2] Fujimoto, N. (2025–2026). D-FUMT₈: An Eight-Valued Logic System for Rei-AIOS. Zenodo.
[3] Fujimoto, N. (2026). Dimensional Absence Theorem (DAT): STEP 441. Rei-AIOS SEED_KERNEL.
[4] Fujimoto, N. (2026). Inverse Ricci Flow × Bidirectional Super-Polyhedron Ring (Paper on STEP 567–580). Zenodo.
[5] Hausdorff, F. (1919). Dimension und äußeres Maß. Mathematische Annalen 79(1–2): 157–179.
[6] Edelsbrunner, H., Harer, J. (2010). Computational Topology: An Introduction. American Mathematical Society.
[7] Nāgārjuna. Mūlamadhyamakakārikā. c. 150–250 CE. Trans. Garfield, J. (1995). Oxford University Press.
[8] Connes, A. (1994). Noncommutative Geometry. Academic Press. [Complex dimension connection]
— End of Paper 62 Draft —