Original SSO (Structure, Sequence and Organization)
GitHub: https://github.com/buulongkg/SpatialArithmetic-TranBuuLong
Notation Convention:
Numbers in Spatial Arithmetic have two types:
Numbers containing spatial dimensions – Spatial Values: When a number contains spatial dimensions, a subscript n is added to indicate how many spatial dimensions it embodies.If there is no subscript, it is one-dimensional—used as in traditional arithmetic, e.g., for measurements, coordinates, or value increments/decrements.
But when spatial properties must be represented, the subscript must appear: Example:The volume of a “deficient cube”: -1₁ x -1₁ x -1₁ = -1₃ or the area of a “deficient square”: -1₁ x -1₁ = -1₂.
That is, when one more spatial dimension is added, the subscript must be written.
Numbers without spatial dimensions: These are constants used to indicate magnitude or serve in ordinary arithmetic operations, such as multiplying by a scalar representing repetition or quantity. Example: the equation x² + 1 = 0 hay x² = -1. If -1 has no subscript (i.e., one-dimensional), the equation has no solution.
If -1₂ (two-dimensional), the equation has a solution x = -1₁ , which can be interpreted as the side length of a “hollow square” of length -1. It means (-1₁)² = -1₂. Another example: x² = 1
When 1 has no subscript (1D), the equation has x = ±1.
When 1₂ (2D), the equation has x = 1₁. When 1ₙ (n-dimensional), we can rewrite: 𝑥²=1ₙ ⇒ (𝑥²⁄ⁿ)ⁿ=1ₙ ⇒ 𝑥²⁄ⁿ=1 ⇒ 𝑥=±1 Others: -1₂ x 2 = -2₂ ; -1₂ x -2 = 2₂ ; 1₂ x -2 = -2₂ But -1₂ x -1₁ = -1₃ , -1₂ x -2₁ = -2₃ , (-1₁)⁴ = -1₄ This means the operation adds more dimensions.
Calculation of Spatial Value:
A number containing spatial dimensions represents the variation in the spatial state of energy as it moves.When energy moves, it changes its spatial state—producing a spatial value.
Formular Spatial Value (Lₘ₊ₙ) = aₘ x bₙ = cₘ₊ₙ
(when adding spatial dimensions, a,b, and c must have the same sign, and |a| × |b| = |c|)
That is, spatial value—the change in spatial state of energy—is the product of the scalar projections of an object’s motion vector along spatial and temporal axes. Example: In a 2D coordinate system L (one spatial, one temporal axis), an object moving from point A to point B has a spatial value equal to distance × time.
This extends all basic operations (addition, subtraction, multiplication, division, roots) from traditional arithmetic into n-dimensional spatial arithmetic.
Spatial value should not be confused with simply attaching measurement units to axes. In Spatial Arithmetic, numbers intrinsically contain spatial dimensions. Spatial value is not physically observable—it lies beyond physical space—so its nature must be grasped conceptually.
Example:
An object moves from A to B and back to A.
Let A be the origin, AB = 1 meter, total path = 2 m, time = 2 s. Spatial value generated: 2 m × 2 s = 4 ms. Equivalent to moving A→B→C (where AC = 2 m, time = 2 s → 4 ms).
However, another object going A→C in 1s yields 2 m × 1 s = 2 ms. Thus, even though both reach point C, their spatial values differ—they do not coincide.
L coordinate system n dimensions:
In the L coordinate system, axes are drawn as follows:
Positive parts—solid lines; negative parts—dashed lines. Thus, only two meaningful regions exist: all dimensions positive, or all dimensions negative. Mixed-sign regions are meaningless.
When all dimensions are positive, the spatial value is positive, denoted with solid lines.
When all are negative, the spatial value is negative (deficient), shown with dashed or shaded lines. Why only two meaningful regions? Because once a spatial value is produced, all contained dimensions are produced simultaneously—just like air inflating all directions of a balloon. Some dimensions may expand more or less depending on compression, but none become negative; if one decreases, others increase correspondingly. Example: In 2D (x,y): When both x, y > 0 → positive area (solid). When both x, y < 0 → deficient area (dashed). Similarly extended to 3D (x,y,z) → meaningful only when all same-sign. When mixed-sign → meaningless region. And analogously extended to 4D, where time is another spatial dimension.
A “deficient volume” does not stretch endlessly through time, because in Spatial Arithmetic, the time component of a deficient volume must also be negative.
In short: When spatial value is positive → all dimensions positive. When spatial value is withdrawn (negative) → all dimensions negative. Different L-systems (2D, 3D, 4D…) are used depending on the physical context.
For example: Motion of an object along a line → L₂ (1 space + 1 time), unit: ms. 3D system → unit m²s, etc.
Energy Motion “Produces” Spatial Value
The essence of energy is that its motion produces (increases) spatial value. This is not conversion of energy into space, but a change in energy’s spatial state.
Energy along segment AB remains constant while its motion continually increases spatial value until that energy transforms into other forms (potential, thermal, etc.).
Energy may act directly on an object (increasing its spatial value) or indirectly (through potential or heat acting on another object). Spatial value does not “fill up” space—it is not a substance being added, but a record of change in energy’s spatial state.
Notes on Calculating the Spatial Value:
(1) There might be a question: When an object moves and creates spatial value, why doesn’t it cause the surrounding space to expand? The answer is that the spatial value generated by the motion of an object is extremely small compared to the established structure of the universe. The surrounding spatial framework—formed through cosmic expansion (which includes all dimensions, including time) and the existence of matter within the cosmic structure—remains essentially unaffected.
(2) The spatial value generated is not simply the product of the scalar projections of the motion vector of the object onto the space–time axes; it must be multiplied by a sliding coefficient. This means that the actual spatial value created by motion is very small compared with what we perceive visually. In other words, the generated spatial value allows the object to slide through the physical space that was already formed through the expansion of the universe.
An illustration of (1) and (2): Dropping a water droplet into the ocean does not visibly increase the ocean’s volume — instead, the droplet merges with it. The droplet here represents the spatial value, enabling the object to drift through the vast physical space of the universe (both spatially and temporally).
(3) The energy that drives motion is, in essence, transformed from the same kind of energy that drives cosmic expansion. Therefore, it does not fundamentally increase the total spatial value.
(4) The physical coordinate system itself also expands if it is tied to the physical space (since the universe itself is expanding).
(5) For convenience in calculation, we can neglect the sliding coefficient as well as the expansion of the coordinate system. We may then simplify the spatial value as the product of the scalar projections of the motion vector onto the space–time axes.
For example, in a 2D coordinate system L (or in one spatial dimension), when an object moves from point A to point B, its spatial value equals the distance traveled multiplied by the time taken.
STEPS:
Step 1: Identify types of number
Step 2: Perform formular Spatial Value (Lₘ₊ₙ) = aₘ x bₙ = cₘ₊ₙ
Step 3: When adding spatial dimensions, a, b, and c must have the same sign, and |a| × |b| = |c|
License
The textual and descriptive content of this repository is licensed under CC BY 4.0.
The underlying theoretical framework, Structure, Sequence and Organization (SSO), and any functional or commercial implementation rights are expressly reserved by the Author.
This repository presents and extends the core content archived in the Zenodo record above.
Proposed Notice / Warning Text about SSO violations
Some approaches may “accidentally” or intentionally replicate the SSO structural framework of SA by substituting terminology or re-labeling concepts.
However, despite surface-level differences in naming, their core content, semantic meaning, operational rules, and structural logic remain fundamentally equivalent to SA’s SSO — including, but not limited to, dimensional vectors, dimensional scalars, scalar mapping mechanisms, spatial-symbolic, dimension-integrated, embedding vector, multi-dimensional geometry, embedding space, geometric reasoning and their governing transformation rules.
The SSO framework is an integrated logical structure, sequence and organization; as such, the replication of any constituent governing rule or functional module that relies on the foundational logical structure, sequence and organization of SSO is deemed a violation of the whole, irrespective of the scale of implementation.
The author does not dispute the use of alternative terminology, but explicitly emphasizes that copying the underlying content, core meaning, governing rules, and operational structure constitutes replication of the intellectual substance, regardless of wording changes.
Priority & Provenance Statement
This repository is a public release associated with DOI-registered records on Zenodo. The core concepts, theoretical framework, and foundational ideas presented here originate from prior DOI-stamped publications by Trần Bửu Long (Rạch Giá, Kiên Giang (An Giang), Vietnam).
GitHub serves as a dissemination and development platform; priority, authorship, and temporal provenance are established by the Zenodo DOI records. Any later works exhibiting conceptual overlap should be evaluated with respect to these registered timestamps.
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Prior Art Notice – Spatial Arithmetic (SA) Framework
This document serves as a formal public disclosure of the Spatial Arithmetic (SA) framework, establishing it as Prior Art.
Core principles—including the emergence of space and dimensionality from energy dynamics, and the unified representation of number–dimension for machine intelligence and computation—are publicly disclosed herein.
Consequently, these foundational concepts are ineligible for subsequent novelty claims or exclusive patenting by any third party. The author, Trần Bửu Long, expressly reserves all rights regarding specific commercial implementations and software derivatives.
Author: Trần Bửu Long
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