Table from Logical to Physical Structure
Version: 4
See also: [Multidimensional ArXe Table]
ArXe Theory proposes a fundamental correspondence between logical structures and the dimensional architecture of physics. At its core, it suggests that each level of logical complexity maps directly to a specific physical dimension — and that dimension, in turn, can be read off directly in the language of Arity Numbers once expressed in its natural (Planck) units.
This article updates the original logical→physical table with the completed dimensional framework: the conversion rule that turns any physical dimension into a pure power of time, the matrix form of that reduction, and the empirical confirmation that dimensional constants — masses, energies, lengths — factor into ArXe-pure arities once expressed in Planck units.
1. The Key Concept
Each number of exentation (n) represents a level in a recursive logical hierarchy. Starting from an initial point (n = 1), each new level is built by systematically applying logical operations to the previous one, generating an infinite ladder of increasing complexity.
2. The Dimensional Connection
Through a precise mathematical formula, each of these logical levels (n) is transformed into a dimensional exponent (k). This exponent defines fundamental temporal dimensions of the form T^k, where:
- T⁰ — the dimensionless (the origin point)
- T¹ — Time
- T² — Length (space)
- T³ — Mass
Conversion formula:
$$e(n) = (-1)^{n} cdot lfloor n/2 rfloor, quad n > 1$$
$$e(1) = 0$$
This generates the sequence: 0, 1, −1, 2, −2, 3, −3, 4, −4…
Remarkable feature: positive exponents (1, 2, 3…) correspond to the “direct” fundamental dimensions (time, length, mass), while negative exponents (−1, −2, −3…) generate their “variations” (frequency, curvature, density).
Deeper implication: the dimensional structure of physics is not arbitrary but emerges naturally from the architecture of logical recursion.
3. The ArXe Dimensional Rule (V4 formalization)
The V4 framework makes the T = T¹, L = T², M = T³ assignment fully explicit and gives it a general conversion rule.
If a magnitude $X$ has conventional physical dimension:
$$[X] = M^{alpha} L^{beta} T^{gamma}$$
then, under the ArXe hierarchy:
$$[X]_{text{ArXe}} = T^{,3alpha + 2beta + gamma}$$
Derivation, step by step:
- Basic hierarchical substitution. Each physical dimension is defined as an exponentiation of the temporal one: $L = T^2$, $M = T^3$.
- Complete expansion. For $X$ with dimension $M^{alpha} L^{beta} T^{gamma}$:
$$[X] = (T^3)^{alpha}(T^2)^{beta}T^{gamma}$$ - Simplification. Adding exponents of $T$:
$$[X] = T^{3alpha + 2beta + gamma}$$ - Result. Every physical magnitude reduces to a single power of hierarchical time, where $n = 3alpha + 2beta + gamma$ is its ArXe exentation level.
As a linear projection (transformation matrix)
$$n = begin{bmatrix} 3 & 2 & 1end{bmatrix}begin{bmatrix}alpha beta gammaend{bmatrix}$$
This matrix is a dimensional collapser: it takes any combination $(M,L,T)$ and returns a single hierarchical exponent $T^n$.
4. Comparative Dimensional Table
| Magnitude | Physical Dimension | Exponents $(M,L,T)$ | ArXe Dimension |
|---|---|---|---|
| $c$ | $LT^{-1}$ | $(0,1,-1)$ | $T^{1}$ |
| $t_P$ (Planck time) | $T$ | $(0,0,1)$ | $T^{1}$ |
| $l_P$ (Planck length) | $L$ | $(0,1,0)$ | $T^{2}$ |
| $hbar$ | $ML^2T^{-1}$ | $(1,2,-1)$ | $T^{6}$ |
| $G$ | $M^{-1}L^3T^{-2}$ | $(-1,3,-2)$ | $T^{1}$ |
| $m_P$ (Planck mass) | $M$ | $(1,0,0)$ | $T^{3}$ |
| $E_P$ (Planck energy) | $ML^2T^{-2}$ | $(1,2,-2)$ | $T^{5}$ |
| Frequency | $T^{-1}$ | $(0,0,-1)$ | $T^{-1}$ |
| Pressure | $ML^{-1}T^{-2}$ | $(1,-1,-2)$ | $T^{-1}$ |
| Force | $MLT^{-2}$ | $(1,1,-2)$ | $T^{3}$ |
| Momentum | $MLT^{-1}$ | $(1,1,-1)$ | $T^{4}$ |
| Power | $ML^2T^{-3}$ | $(1,2,-3)$ | $T^{4}$ |
| Acceleration | $LT^{-2}$ | $(0,1,-2)$ | $T^{0}$ |
Consistency checks:
- $l_P = c,t_P ;Rightarrow; T^2 = T^1cdot T^1$ ✓
- $t_P = sqrt{hbar G/c^5} ;Rightarrow; T^1 = sqrt{T^6cdot T^1/T^5} = T^1$ ✓
- $m_P = sqrt{hbar c/G} Rightarrow T^3$, and $E_P = m_P c^2 Rightarrow T^5$ ✓
5. Three Structural Equivalences (why the table collapses categories)
The dimensional collapse is not just bookkeeping — it forces conventionally distinct physical quantities into the same ArXe level, and each coincidence has an ontological reading:
Equivalence 1 — $c = G = $ velocity $rightarrow T^1$.
The speed of light and the gravitational constant share the same ArXe dimension. Both are “conversion factors” between the temporal scale ($T^1$) and the spatial scale ($T^2$). Setting $c=G=1$ in Planck units eliminates two redundancies at the same level.
Equivalence 2 — Force $=$ Mass $rightarrow T^3$.
$F = ma$, and acceleration has dimension $T^0$ — dimensionless in ArXe. Force is therefore not an independent ontological entity: it is mass acting along a path with no dimension of its own.
Equivalence 3 — Momentum $=$ Power $rightarrow T^4$.
Two quantities that look very different in conventional physics ($ML/T$ vs. $ML^2/T^3$) sit at the same ArXe level. The computational/informational level $T^4$ unifies them.
6. From Table to Data: Dimensional Constants Read in Planck Units
The dimensional table is not only a formal reduction — it is a prediction: physical constants expressed in their natural (Planck) units should factor into the same small arities that already organize the dimensionless constants (the ALO grammar). This was tested directly.
The rule applied to constants:
Step 1: Compute the ArXe exponent n = 3α + 2β + γ for constant C
Step 2: Express C in Planck units: C_Planck = C / (m_P^α × l_P^β × t_P^γ)
Step 3: C_Planck is a pure number — factor it into arity numbers
Step 4: Read the arity number grammar in the context of level T^n6.1 Masses (level T³)
| Particle | m/m_P | P | Factorization | Class |
|---|---|---|---|---|
| Electron | 4.186×10⁻²³ | 42 | 2×3×7 | ArXe ✓ |
| Muon | 8.654×10⁻²¹ | 865 | 5×173 | human arity |
| Tau | 1.455×10⁻¹⁹ | 146 | 2×73 | ArXe ✓ |
| Up quark | 1.769×10⁻²² | 177 | 3×59 | ArXe ✓ |
| Down quark | 3.825×10⁻²² | 383 | 383 | human arity |
| Strange quark | 7.650×10⁻²¹ | 765 | 3²×5×17 | ArXe ✓ |
| Charm quark | 1.040×10⁻¹⁹ | 104 | 2³×13 | ArXe ✓ |
| Bottom quark | 3.424×10⁻¹⁹ | 342 | 2×3²×19 | ArXe ✓ |
| Top quark | 1.415×10⁻¹⁷ | 142 | 2×71 | ArXe ✓ |
| W boson | 6.583×10⁻¹⁸ | 66 | 2×3×11 | ArXe ✓ |
| Z boson | 7.469×10⁻¹⁸ | 75 | 3×5² | ArXe ✓ |
| Higgs boson | 1.026×10⁻¹⁷ | 103 | 103 | human arity |
| Proton | 7.685×10⁻²⁰ | 77 | 7×11 | ArXe ✓ |
| Neutron | 7.696×10⁻²⁰ | 77 | 7×11 | ArXe ✓ |
12 of 14 masses (86%) have pure ArXe factorizations — confirming the prediction that Planck units are the natural language for dimensional constants.
Two readings worth pulling out:
- Electron, P = 42 = 2×3×7 (DIFF × CYC × CPX): binary differentiation × minimal cycle × internal complexity — the simplest structure that can carry charge.
- Proton = Neutron, P = 77 = 7×11 (CPX × REG): color confinement × electromagnetic regulation. At this precision the proton and the neutron are the same ontological structure; their 0.14% mass difference requires finer resolution to distinguish.
6.2 Energies (level T⁵)
| Quantity | E/E_P | P | Factorization | Class |
|---|---|---|---|---|
| Electron rest energy | 4.186×10⁻²³ | 42 | 2×3×7 | ArXe ✓ |
| Proton rest energy | 7.685×10⁻²⁰ | 77 | 7×11 | ArXe ✓ |
| Rydberg energy | 1.114×10⁻²⁷ | 111 | 3×37 | ArXe ✓ |
| Λ_QCD | 1.741×10⁻²⁰ | 174 | 2×3×29 | ArXe ✓ |
| Electroweak VEV | 2.017×10⁻¹⁷ | 202 | 2×101 | human arity |
| GUT scale (est.) | 8.191×10⁻⁵ | 82 | 2×41 | ArXe ✓ |
5 of 6 (83%) pure ArXe.
6.3 Lengths (level T²) — the exception that proves the rule
| Quantity | l/l_P | Note |
|---|---|---|
| Bohr radius | 3.274×10²⁴ | enormous, conventional P |
| Compton wavelength | 1.501×10²³ | same |
| Classical electron radius | 1.744×10²⁰ | same |
| Proton charge radius | 5.206×10¹⁹ | same |
Lengths do not produce clean ArXe factorizations. Atomic and nuclear length scales sit enormously far from the Planck length, and that distance in $T$ accumulates convention along the way — exactly what the framework predicts for a magnitude far from its natural scale. The proposed remedy is to compare lengths to each other rather than to $l_P$ directly (e.g. $a_0/lambda_e = alpha^{-1}$, already dimensionless and already in the ArXe-pure corpus).
7. Summary Table: Dimensional Naturality
| Type | Level | Total | Pure ArXe P | % |
|---|---|---|---|---|
| Masses | T³ | 14 | 12 | 86% |
| Energies | T⁵ | 6 | 5 | 83% |
| Lengths | T² | 5 | 0 | 0% |
This defines a hierarchy of “dimensional naturality”:
- Dimensionless constants — the purest language, no units at all.
- Masses and energies in Planck units (T³, T⁵) — predominantly ArXe-pure.
- Lengths in Planck units (T²) — accumulate convention because of the large separation between atomic/nuclear scale and the Planck scale.
A further confirmed prediction: dimensionless constants often need mathematical anchors (π, φ) because they encode relations between levels — mixing angles, ratios, density fractions. Masses and energies in Planck units, by contrast, need no anchors at all: the electron is 2×3×7, the W boson is 2×3×11, the proton is 7×11. Arity numbers alone suffice — dimensional grammar turns out to be simpler than dimensionless grammar, and closer to ArXe’s native language.
8. Hierarchical Interpretation
Under this framework:
- All physical magnitudes reduce to powers of $T$.
- $L = T^2$ and $M = T^3$ imply that space and mass are hierarchical exentations of time.
- $c = T^1$ is the hierarchical equivalence operator between consecutive temporal levels.
- The system is dimensionally closed and self-referential: every magnitude is expressible solely through powers of $T$, and — once expressed in Planck units — that power’s numerical content is legible directly in the arity number grammar shared with the dimensionless constants.
For the full dimensional derivation: arxe_dimensional_correspondence_en.md
For the applied constant-by-constant analysis: plov2_dimensional_framework_s_en.md
For the underlying grammar: Grammar_V4_s_en.md (ALO Pure Grammar)